Research ArticleStability Analysis of a Helicopter with an ExternalSlung Load System

Kary Thanapalan

Faculty of Computing Engineering and Science University of South Wales Pontypridd CF37 1DL UK

Correspondence should be addressed to KaryThanapalan karythanapalansouthwalesacuk

Received 20 November 2015 Revised 18 April 2016 Accepted 4 May 2016

Academic Editor Francisco Gordillo

Copyright copy 2016 KaryThanapalan This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

This paper describes the stability analysis of a helicopter with an underslung external load systemThe Lyapunov second method isconsidered for the stability analysis The system is considered as a cascade connection of uncertain nonlinear system The stabilityanalysis is conducted to ensure the stabilisation of the helicopter system and the positioning of the underslung load at hovercondition Stability analysis and numerical results proved that if desired condition for the stability is met then it is possible tolocate the load at the specified position or its neighbourhood

1 Introduction

Recently research on helicopter carrying external underslungloads has gained great attention in the aerospace researchcommunity for the past few decades due to the reevaluationand extension of the ADS-33 and the inherent stabilityproblems associated with this system [1ndash3] Helicopters havethe ability to carry large and bulky loads externally on a slingThis capability is important in many applications rangingfrom lifting heavy loads to saving life Importantly whenlives are under risk and rapid rescue operations are neededthis operation is vital The stability of the helicopter will bedisturbed by the underslung load which is a huge obstaclefor an accurate pickup or placement of the loads [4] Thus itis necessary to resolve the stability problems associated withthe system to ensure the stabilisation of the helicopter systemand the positioning of the underslung load under variouscomplicated situations

From the review of popular helicopter control methodsit is clear that in the past years considerable attention hasbeen paid to the design of controller to obtain a satisfactoryhelicopter handling quality [5] The control problem hasbeen tackled using different approaches ranging from linearquadratic control [6] eigenstructure assignment [7] classicalSISO techniques [8] to sliding mode control [9] Apartfrom the methods emphasised above there are many other

techniques which are reported for complex modern controlsystem design ranging from quantitative feedback theory tosingular perturbation method [10]

The extensive studies of the reported controller designmethods evidenced that the helicopter control and the controlof a helicopter with an external underslung load are veryactive research areasThe research in this area is mainlymoti-vated by the factor that the current control methods cannotprovide full satisfaction to the desired design requirementson flight handling quality stability robustness and so forth

In this paper stability analysis for the helicopter withan underslung external load system is discussed The keyadvantage of the proposed method is that the analysis takesthe system uncertainty into account The proposed methodcan give a guaranteed stability region for the systems consid-ered The paper begins by presenting a mathematical modelof the system and then describing the stability analysis witha numerical example to illustrate the applicability accuracyand effectiveness of the proposed method

2 System Model

Considering the control of a helicopter with an underslungload the dynamical models of both the helicopter and loadhave some terms which are uncertain The uncertaintiesmay arise from the helicopter to carry an unknown load or

Hindawi Publishing CorporationJournal of Control Science and EngineeringVolume 2016 Article ID 4195491 14 pageshttpdxdoiorg10115520164195491

2 Journal of Control Science and Engineering

the immeasurable parameters in the dynamical models Theuncertainties may also arise from computational errors ofthe dynamical effects such as aerodynamics Therefore fora realistic model uncertainties must be taken into accountduring the controller design

A mathematical model of the helicopter described in [11]and an underslung load model presented in [4] are adoptedin this work Considering the two models a mathematicalmodel for a helicopter carrying an underslung load can beobtained

Firstly the underslung load is considered to be suspendedfrom a single suspension point that is subject to motionand therefore modelled as a driven spherical pendulum Theequations that describe the load dynamics are obtained byfirst considering motion with reference to the longitudinalsuspension angle 120579

119871in the 119883-119885 plane (Figure 1) This is then

repeated for the lateral case involving 120593119871and the 119884-119885 plane

These are then combined to obtain the model for the motionof the load The underslung load system has six inputslongitudinal lateral and vertical velocities together withthe corresponding accelerations of the helicopter whilst theoutputs are the longitudinal and lateral directional suspen-sion angles The load is subject to an isotropic aerodynamicforce (proportional to the square of its airspeed) such aswhat would be experienced by a spherical shaped loadAerodynamic interaction with the helicopter that may occurfor example due to rotor downwash has been ignoredFinally the sling itself is assumed to be rigid and contributezero aerodynamic force of its own With these assumptionsthe equations governing the load motion can be derived asfollows

For the case of the longitudinal motion in the119883-119885 planethe mathematical model is described below

119871= minus

119892

119897119909

sin 120579119871+cos 120579119871

119897119909

0+sin 120579119871

119897119909

0

+119896119863sign (

119871) cos 120579

119871

119872119871119897119909

2

0+119896119863sign (

119871) sin 120579

119871

119872119871119897119909

2

0

minus2119896119863

119872119871

(sign (119871) cos2 120579

1198710

+ sign (119871) sin2 120579

1198710) 119871minus 119896120579119871

+119896119863119897119909

119872119871

[sign (119871) cos3 120579

119871+ sign (

119871) sin3 120579

119871] 2

119871

(1)

Define 119871= [120579119871

119871]119879

= [1205791198711

1205791198712

]119879 then the load model can

be rewritten as follows

1198711

= 1205791198712

(2a)

1198712

=minus119892

119897119909

sin 1205791198711

+119896119863119897119909

119872119871

[sign (119871) cos3 120579

1198711

+ sign (119871) sin3 120579

1198711

] 1205792

1198712

minus 1198961205791205791198712

+ (119896119863sign (

119871) cos 120579

1198711

119872119871119897119909

)2

0

+ (119896119863sign (

119871) sin 120579

1198711

119872119871119897119909

) 2

0+cos 1205791198711

119897119909

0

+sin 1205791198711

119897119909

0minus2119896119863

119872119871

(sign (119871) cos2 120579

1198711

0

+ sign (119871) sin2 120579

1198711

0) 1205791198712

(2b)

The helicopter model is considered as the second subsystemTo simplify the analysis the linear helicopter model [4] isconsideredwhich is expressed in the state space form

119867(119905) =

119860119909119867(119905) + 119861119906(119905)

[[[[[[[[[[[[[[[[[[[[

[

V

119903

]]]]]]]]]]]]]]]]]]]]

]

=

[[[[[[[[[[[[[[[[[[[[

[

0 0 0 0 0 0 1 0 0

0 0 0 0 0 0 0 1 0

0 0 0 0 0 0 0 0 1

0 minus119892 cos 1205791198900 119883119906

119883V 119883119908

119883119901

(119883119902minus 119908119890) (119883119903minus V119890)

119892 0 0 119884119906

119884V 119884119908

(119884119901+ 119908119890) 119884

119902(119884119903minus 119906119890)

0 minus119892 sin 120579119890

0 119885119906

119885V 119885119908

(119885119901minus V119890) (119885

119902+ 119906119890) 119885

119903

0 0 0 119871119906

119871V 119871119908

119871119901

119871119902

119871119903

0 0 0 119872119906

119872V 119872119908

119872119901

119872119902

119872119903

0 0 0 119873119906

119873V 119873119908

119873119901

119873119902

119873119903

]]]]]]]]]]]]]]]]]]]]

]

[[[[[[[[[[[[[[[[[[[

[

120601

120579

120595

119906

V

119908

119901

119902

119903

]]]]]]]]]]]]]]]]]]]

]

+

[[[[[[[[[[[[[[[[[[[

[

0 0 0 0

0 0 0 0

0 0 0 0

1198831205791119888

1198831205791119904

1198831205790

1198831205790119879

1198841205791119888

1198841205791119904

1198841205790

1198841205790119879

1198851205791119888

1198851205791119904

1198851205790

1198851205790119879

1198711205791119888

1198711205791119904

1198711205790

1198711205790119879

1198721205791119888

1198721205791119904

1198721205790

1198721205790119879

1198731205791119888

1198731205791119904

1198731205790

1198731205790119879

]]]]]]]]]]]]]]]]]]]

]

[[[[[

[

1205791119888

1205791119904

1205790

1205790119879

]]]]]

]

(3)

For linearization it is assumed that the external forces 119883119884 and 119885 and moments 119871 119872 and 119873 can be represented as

analytic functions of the disturbedmotion variables and theirderivatives [4] Thus the forces and moments can be written

Journal of Control Science and Engineering 3

X

Z

120579Llx

(XL YL ZL)

(X0 Y0 Z0)

Figure 1 Coordinate system for the longitudinal motion in the119883-119885plane

in Taylorrsquos expansion form Then the linearized equations ofmotion about a general trim condition can be written as inthe state space form = 119860119909 + 119861119906 and the system matrix 119860

and control matrix 119861 are derived from the partial derivativesof the nonlinear function 119865 that is

= 119865 (119909 119906 119905)

119860 = (120597119865

120597119909)

119909=119909119890

119861 = (120597119865

120597119906)

119909=119909119890

(4)

Now the longitudinal rotational motion is described by thepitch angle 120579 and pitch rate 119902 together with the translationmotion components 119906 119908 so the equation of longitudinalmotion can be written as follows

[[[[[

[

]]]]]

]

=

[[[[[[

[

0 1 0 0

0 119872119902

119872119906

119872119908

minus119892 cos 120579119890(119883119902minus 119908119890) 119883119906

119883119908

minus119892 sin 120579119890

(119885119902minus 119906119890) 119885119906

119885119908

]]]]]]

]

[[[[[

[

120579

119902

119906

119908

]]]]]

]

+

[[[[[

[

0 0

1198721205791119904

1198721205790

1198831205791119904

1198831205790

1198851205791119904

1198851205790

]]]]]

]

[1205791119904

1205790

]

(5)

By applying a linear transformation119879 such that119909119867(119905) = 119879119911(119905)

and 119879 is defined by

119879 =

[[[[[

[

1 0 0 0

0 1 11988611

11988612

0 0 1 0

0 0 0 1

]]]]]

]

(6)

where

11988611

= minus1198851205791119904

1198721205790

minus 1198851205790

1198721205791119904

1198831205791119904

1198851205790

minus 1198851205791119904

1198831205790

11988612

= minus1198831205790

1198721205791119904

minus 1198831205791119904

1198721205790

1198831205791119904

1198851205790

minus 1198851205791119904

1198831205790

(7)

and letting 119902 = (119902 minus 11988611119906 minus 11988612119908) then the system equation

(5) is transformed into the following form

[[[[[[

[

]]]]]]

]

=

[[[[[

[

11988311

11988312

11988313

11988314

11988321

11988322

11988323

11988324

11988331

11988332

11988333

11988334

11988341

11988342

11988343

11988344

]]]]]

]

[[[[[

[

120579

119902

119906

119908

]]]]]

]

+

[[[[[

[

0 0

11986121

11986122

1198831205791119904

1198831205790

1198851205791119904

1198851205790

]]]]]

]

[1205791119904

1205790

]

(8)

where

11988311

= 0

11988312

= 1

11988313

= 11988611

11988314

= 11988612

11988321

= (11988611119892 cos 120579

119890+ 11988612119892 sin 120579

119890)

11988322

= (119872119902minus 11988611(119883119902minus 119908119890) minus 11988612(119885119902minus 119906119890))

11988323

= (119872119902minus 11988611(119883119902minus 119908119890) minus 11988612(119885119902+ 119906119890)) 11988611

+ (119872119906minus 11988611119883119906minus 11988612119885119906)

11988324

= (119872119902minus 11988611(119883119902minus 119908119890) minus 11988612(119885119902+ 119906119890)) 11988612

+ (119872119908minus 11988611119883119908minus 11988612119885119908)

11988331

= minus119892 cos 120579119890

11988332

= (119883119902minus 119908119890)

11988333

= (119883119902minus 119908119890) 11988611+ 119883119906

11988334

= (119883119902minus 119908119890) 11988611+ 119883119908

11988341

= (minus119892 sin 120579119890)

11988342

= (119885119902+ 119906119890)

11988343

= (119885119902+ 119906119890) 11988611+ 119885119906

11988344

= (119885119902+ 119906119890) 11988612+ 119885119908

4 Journal of Control Science and Engineering

11986121

= (1198721205791119904

minus 119886111198831205791119904

minus 119886121198851205791119904

)

11986122

= (1198721205790

minus 119886111198831205790

minus 119886111198851205790

)

(9)

Using (8) the system model can be rearranged to include thevariables 120579 and 119902 into the loadmodel For the stability analysispurpose an extra term is introduced into the system modelwhich is zero with the expression

1205811119896119863sign (

119871) cos 120579

1198711

119906

119872119871119897119909

+1205812119896119863sign (

119871) cos 120579

1198711

119908

119872119871119897119909

minus1205811119896119863sign (

119871) cos 120579

1198711

119906

119872119871119897119909

minus1205812119896119863sign (

119871) cos 120579

1198711

119908

119872119871119897119909

(10)

where 120581119894gt 0 (119894 = 1 2) are small positive constants

With this arrangement for the longitudinal motion of thehelicopter with an underslung load combined system modelcan be written as follows

120579119871 (119905)

= 1198911(120579119871 (119905))

+ 1198661(120579119871 (119905)) [119901 (119909

119867 (119905)) + 119902 (120579119871 (119905) 119909119867 (119905))]

+ 119867(119905 120579119871 119909119867 (119905))

(11a)

119867 (119905) = 119891

2(120579119871 (119905) 119909119867 (119905)) + 119866

2 (119905) (11b)

where

120579119871 (119905) = [120579119871

1

1205791198712

120579 119902]119879

119909119867= [119906 119908]

119879

(119905) = [1205791119904 1205790]119879

119901 (119909119867) = [119906

2+ 1205811119906 1199082+ 1205812119908]119879

1198911(120579119871 (119905)) =

[[[[[[[

[

1205791198712

minus119892

119897119909

sin 1205791198711

+119896119863119897119909

119872119871

(sign (119871) cos3 120579

1198711

+ sign (119871) sin3 120579

1198711

) 1205792

1198712

minus 1198961205791205791198712

119902

(11988321120579 + 119883

22119902)

]]]]]]]

]

1198661(120579119871 (119905)) =

[[[[[[[[[

[

0 0

119896119863sign (

119871) cos 120579

1198711

119872119871119897119909

119896119863sign (

119871) sin 120579

1198711

119872119871119897119909

0 0

0 0

]]]]]]]]]

]

119902 (120579119871 (119905) 119909119867 (119905))

=[[

[

0

cos 1205791198711

119897119909

+sin 1205791198711

119897119909

minus2119896119863

119872119871

(sign (119871) cos2 120579

1198711

119906 + sign (119871) sin2 120579

1198711

119908) 1205791198712

minus1205811119896119863sign (

119871) cos 120579

1198711

119906

119872119871119897119909

minus1205812119896119863sign (

119871) cos 120579

1198711

119908

119872119871119897119909

]]

]

119867(119905 120579119871 (119905) 119909119867 (119905)) =

[[[[[

[

0

0

(11988611119906 + 11988612119908)

11988323119906 + 119883

24119908

]]]]]

]

1198912(120579119871 (119905) 119909119867 (119905)) = [

(11988331120579 + 119883

32119902 + 119883

33119906 + 119883

34119908)

(11988341120579 + 119883

42119902 + 119883

43119906 + 119883

44119908)

]

1198662= [

1198831205791119904

1198831205790

1198851205791119904

1198851205790

]

(12)

Journal of Control Science and Engineering 5

Y

Z

ly

(XL YL ZL)

(X0 Y0 Z0)

120593L

Figure 2 Coordinate system for the lateralmotion in the119884-119885 plane

It is assumed that the longitudinal motion is primarilycontrolled by longitudinal cyclic commands (120579

1119904) and main

rotor collective 1205790

For the case of the lateral motion in the 119884-119885 plane withthe coordinate system described in Figure 2 the load modelis

119871119910119911

= minus119892

119897119910

sin120593119871119910119911

+

cos120593119871119910119911

119897119910

0+

sin120593119871119910119911

119897119910

0

+

119896119863sign (

119871) cos120593

119871119910119911

119872119871119897119910

2

0

+

119896119863sign (

119871) sin120593

119871119910119911

119872119871119897119910

2

0

minus2119896119863

119872119871

(sign (119871) cos2 120593

119871119910119911

0

+ sign (119871) sin2 120593

119871119910119911

0) 119871minus 119896120593119871

+119896119863119897119910

119872119871

[sign (119871) cos3 120593

119871119910119911

+ sign (119871) sin3 120593

119871119910119911

]

sdot 2

119871119910119911

(13)

Define 119871= [120593119871

119871]119879

= [1205931198711

1205931198712]119879 then the model can be

written as follows

1198711

= 1205931198712

(14a)

1198712

=minus119892

119897119910

sin1205931198711

+119896119863119897119910

119872119871

[sign (119871) cos3 120593

1198711

+ sign (119871) sin3 120593

1198711

] 1205932

1198712

minus 1198961205931205931198712

+ (119896119863sign (

119871) cos120593

1198711

119872119871119897119910

) 2

0

+ (119896119863sign (

119871) sin120593

1198711

119872119871119897119910

) 2

0+cos1205931198711

119897119910

0

+sin1205931198711

119897119910

0minus2119896119863

119872119871

(sign (119871) cos2 120593

1198711

0

+ sign (119871) sin2 120593

1198711

0) 1205931198712

(14b)

Now for the helicopter model the lateral rotational motion isdescribed by the roll angle 120601 and roll rate 119901 together with thetranslationmotion components V 119908 so the equation of lateralmotion can be written as follows

[[[[[

[

V

]]]]]

]

=

[[[[[[

[

0 1 0 0

0 119871119901

119871V 119871119908

119892 (119884119901minus 119908119890) 119884V 119884

119908

0 (119885119901minus V119890) 119885V 119885

119908

]]]]]]

]

[[[[[

[

120601

119901

V

119908

]]]]]

]

+

[[[[[

[

0 0

1198711205791119888

1198711205790119879

1198841205791119888

1198841205790119879

1198851205791119888

1198851205790119879

]]]]]

]

[1205791119888

1205790119879

]

(15)

now using a linear transformation 1198791such that 119909

119867(119905) =

1198791119911(119905) and 119879

1is defined by

1198791=

[[[[[

[

1 0 0 0

0 1 11988711

11988712

0 0 1 0

0 0 0 1

]]]]]

]

(16)

where

11988711

= minus1198851205791119888

1198711205790119879

minus 1198851205790119879

1198711205791119888

1198841205791119888

1198851205790119879

minus 1198851205791119888

1198841205790119879

11988712

= minus1198841205790119879

1198711205791119888

minus 1198841205791119888

1198711205790119879

1198841205791119888

1198851205790119879

minus 1198851205791119888

1198841205790119879

(17)

6 Journal of Control Science and Engineering

Let 119901 = (119901 minus 11988711V minus 11988712119908) then (15) can be written in the

following form

[[[[[[

[

V

]]]]]]

]

=

[[[[[

[

11988411

11988412

11988413

11988414

11988421

11988422

11988423

11988424

11988431

11988432

11988433

11988434

11988441

11988442

11988443

11988444

]]]]]

]

[[[[[

[

120601

119901

V

119908

]]]]]

]

+

[[[[[

[

0 0

11987321

11987322

1198841205791119888

1198841205790119879

1198851205791119888

1198851205790119879

]]]]]

]

[1205791119888

1205790119879

]

(18)

where

11988411

= 0

11988412

= 1

11988413

= 11988711

11988414

= 11988712

11988421

= minus (11988711119892)

11988422

= (119871119901minus 11988711(119884119901+ 119908119890) minus 11988712(119885119901minus V119890))

11988423

= (119871119901minus 11988711(119884119901+ 119908119890) minus 11988712(119885119901minus V119890)) 11988711

+ (119871119901minus 11988711119884V minus 119887

12119885V)

11988424

= (119871119901minus 11988711(119884119901+ 119908119890) minus 11988712(119885119901minus V119890)) 11988712

+ (119871119901minus 11988711119884119908minus 11988712119885119908)

11988431

= 119892

11988432

= (119884119901+ 119908119890)

11988433

= ((119884119901+ 119908119890) 11988711+ 119884V)

11988434

= ((119884119901+ 119908119890) 11988712+ 119884119908)

11988441

= 0

11988442

= (119885119901minus V119890)

11988443

= ((119885119901minus V119890) 11988711+ 119885V)

11988444

= ((119885119901minus V119890) 11988712+ 119885119908)

11987321

= (1198711205791119888

minus 119887111198841205791119888

minus 119887121198851205791119888

)

11987322

= (1198711205790119879

minus 119887111198841205790119879

minus 119887121198851205790119879

)

(19)

Using (18) the helicopter with underslung load for lateralmotion in the 119884-119885 plane can be described by

119871(119905)

= 1198911(120593119871(119905))

+ 1198661(120593119871(119905)) [119901 (119909

119867 (119905)) + 119902 (120593119871(119905) 119909119867 (119905))]

+ 119867 (119905 120593119871 119909119867 (119905))

(20a)

119867 (119905) = 119891

2(120593119871(119905) 119909119867 (119905)) + 119866

2 (119905) (20b)

where

119871(119905) = [

1198711

1198712

]119879

119901 (119909119867 (119905)) = [V2 119908

2]119879

1198911(120593119871(119905)) =

[[[[[[[

[

1205931198712

minus119892

119897119910

sin1205931198711

+119896119863119897119910

119872119871

(sign (119871) cos3 120593

1198711

+ sin (119871) sin3 120593

1198711

) 1205932

1198712

minus 1198961205931205931198712

119901

(minus11988711119892120601 + (119871

119901minus 11988711(119884119901+ 119908119890) minus 11988712(119885119901minus V119890)) 119901)

]]]]]]]

]

1198661(120593119871(119905)) =

[[[[[[[[

[

0 0

(119896119863sign (

119871) cos120593

1198711

119872119871119897119910

) (119896119863sign (

119871) sin120593

1198711

119872119871119897119910

)

0 0

0 0

]]]]]]]]

]

119902 (120593119871(119905) 119909119867 (119905)) =

[[

[

0

cos1205931198711

119897119910

V +sin1205931198711

119897119910

minus2119896119863

119872119871

(sign (119871) cos2 120593

1198711

V + sign (119871) sin2 120593

1198711

119908)1205931198712

]]

]

Journal of Control Science and Engineering 7

119867(119905 120593119871(119905) 119909119867 (119905)) =

[[[[[

[

0

0

( minus 119901)

(11988423V + 11988424119908)

]]]]]

]

(119905) = [1205791119888 1205790119879]119879

1198912(120593119871(119905) 119909119867 (119905)) = [

(11988431120601 + 11988432119901 + 11988433V + 11988434119908)

(11988442119901 + 11988443V + 11988444119908)

]

1198662= [

1198841205791119888

1198841205790119879

1198851205791119888

1198851205790119879

]

(21)

It is assumed that the lateral motion is primarily controlledby lateral cyclic commands (120579

1119888) and the tail rotor collective

1205790119879

3 Stability Analysis

The goal is to analyse the stability of the combined systemto ensure that it is possible to stabilise the helicopter withunderslung load system modelled by (11a) (11b) (20a) and(20b) in a real environment with uncertainties In this paperfor the stability analysis the Lyapunov second method isapplied for the helicopter with an underslung external loadsystem The analysis for the longitudinal motion is discussedfirst The system equations can be considered to have twomain parts that is known and unknown (or partly known)The known terms formed the nominal part of the systemmodelThe unknown or partly known part can be consideredas the uncertainty to the system The whole system is thenmodelled by a nominal part with the addition of uncertaintyIn fact the known elements in the subsystem (11a) arecharacterised by the prescribed triple (119891

1 1198661 119901) and it is

desired that the nominal part of the system is stableThe basic notations and concepts required for the analysis

are described first The state space is denoted by119883 fl R119899 andthe control space by119906 fl R119898 where 1 le 119898 le 119899TheEuclideaninner product (on 119883 or 119906 as appropriate) and induced normare denoted by ⟨sdot sdot⟩ and sdot respectively Let 119862(R119901R119902) and1198621(R119901R119902) denote the space of all continuous functions and

the space of continuous functions with continuous first-orderpartial derivatives respectively and let 119862infin(R119901R119902) denotethe space of functions whose partial derivatives of any orderexist and are continuous mapping R119901 rarr R119902 For a real-valued continuous scalar function 119909 rarr V(119909) defined onR119899 nabla V rarr nablaV isin R119899 denotes the gradient map The Liederivative of V along a vector field 119891 R119899 rarr R119899 is denoted by119871119891V R119899 rarr R which is defined by

(119871119891V) (119909) = ⟨nablaV (119909) 119891 (119909)⟩ (22)

The Lie bracket of vector fields 119891 119892 isin 119862infin(R119899R119899) is the

vector field [119891 119892] isin 119862infin(R119899R119899) defined by [119891 119892] = (119863119892)119891 minus

(119863119891)119892 where (119863119891)denotes the Jacobianmatrix of119891 and (119863119892)denotes the Jacobian matrix of 119892

In this paper nonlinear systemswith the following formatare considered

(119905) = 119891 (119909 (119905)) + 119866 (119909 (119905)) (119905) (23)

where 119909(119905) isin R119899 isin R119898 In general mathematical modelsof dynamical systems are usually imprecise due to modellingerrors and exogenous disturbances [12] Equation (23) can beconsidered as the nominal part of the system model and theuncertainty can be modelled as an additive perturbation tothe nominal systemmodel more specifically the structure ofthe system has the form

(119905) = 119891 (119909 (119905)) + 119866 (119909 (119905)) (119905) + 120599 (119909 (119905) 119906 (119905)) (24)

where 120599(119909(119905) 119906(119905))models the uncertainty in the systemSystem (24) is globally asymptotically stable to the zero

state if the system exhibits the following properties

(i) Existence and Continuation of Solutions For each 119909 isin R119899there exists a local solution119909 [0 119905

1) rarr R119899 (ie an absolutely

continuous function satisfying (24) almost everywhere (ae)and 119909(0) = 119909

0) and every such solution can be extended intoa solution on [0infin)

(ii) Boundedness of Solutions For each ℏ gt 0 there exists119903(ℏ) gt 0 such that 119909(119905) isin 119903(ℏ)119861

119899 for all 119905 ge 0 on every

solution 119909 [0infin) rarr R119899 with 1199090isin ℏ119861119899 where 119861

119899denote

the open unit ball centred at the origin in R119899

(iii) Stability of the State Origin For each 120575 gt 0 there exists119889(120575) gt 0 such that 119909(119905) isin 120575119861

119899for all 119905 ge 0 on every solution

119909 [0infin) rarr R119899 with 1199090isin 119889(120575)119861

119899

(iv) Global Attractivity of the State Origin For each ℏ gt 0 and120576 gt 0 there exists 119879(ℏ 120576) ge 0 such that 119909(119905) isin 119861

119899for all

119905 ge 119879(ℏ 120576) on every solution 119909 [0infin) rarr R119899 with 1199090isin ℏ119861119899

8 Journal of Control Science and Engineering

Consider a nonlinear system described by the ordinarydifferential equation as follows

(119905) = 119891 (119905 119909 (119905))

119909 (1199050) = 1199090

(25)

where 119891 R times 119883 rarr 119883 and 119891(119905 0) = 0 for all 119905To analyse the stability of (25) Lyapunovrsquos second stabilityanalysis method is applicable The Lyapunov approach is toshow that a candidate ldquoLyapunov functionrdquo is nonincreasingalong all solution to (25) by means that do not requireexplicit knowledge of solutions to (25) From this appropriateconclusion can be drawn regarding stability concepts relatingto solutions of the differential equation (25) An essentialpart of Lyapunovrsquos method is the determination of the timederivative of the candidate ldquoLyapunov functionrdquo along allsolution of the dynamical system

Consider a Lyapunov candidate (119905 119909) rarr V(119905 119909) Rtimes119883 rarr

R which satisfies the condition V isin 1198621(R times 119883) in which case

its time derivative along solutions to (25) is given by

V (119905 119909 (119905)) =120597V (119905 119909 (119905))

120597119905+ ⟨nablaV (119905 119909 (119905)) 119891 (119905 119909 (119905))⟩ (26)

for almost all 119905 isin RLet 119882(119909(119905)) denote a positive definite function If

V(119905 119909(119905)) satisfies

(i) V(119905 0) = 0 for all 119905 ge 0

(ii) 119882(119909(119905)) le V(119905 119909(119905)) for all 119909(119905) isin Φ 0 sub Φ sub 119883 andall 119905 ge 0

(iii) V(119905 119909(119905)) lt 0 in Φ

then V(119905 119909(119905)) is said to be a Lyapunov function in Φ IfV(119905 119909(119905)) in Φ then V(119905 119909(119905)) is said to be a weak Lyapunovfunction

A set Ε is said to be an invariant set with respect to thedynamical system = 119891(119909) if

119909 (0) isin Ε 997891997888rarr 119909 (119905) isin Ε forall119905 isin R+ (27)

In other words Ε is the set of points such that if a solution of = 119891(119909) belongs to Ε at some instant initialized points at119905 = 0 then it belongs to Ε for all future time

Now a setΕ sub 119909 is said to be a local invariantmanifold for(25) if for any 1199090 isin Ε 119909(119905) with 119909(0) = 119909

0 is in Ε for |119905| lt 119879

where 119879 gt 0 If 119879 = infin then Ε is said to be an invariantmanifold

Now considering the longitudinal motion (Figure 1) forthe helicopter with an underslung external load systemchoose a Lyapunov function candidate for the first subsystemV1as

V1(120579119871) =

1

2[12058911205792

1198711

+ 12058921205792

1198712

+ (1205893120579 minus 1205894119902)2+ 12058951199022] (28)

where 120589119894(119894 = 1 2 3 4 5) are design parameters to be

determined Then

V1(120579119871) = ⟨nablaV

1(120579119871) 1198911(120579119871) + 1198661(120579119871 (119905))

sdot [119901 (119909119867 (119905)) + 119902 (120579

119871 (119905) 119909119867 (119905))]

+ 119867(119905 120579119871 119909119867 (119905))⟩ = (119871

1198911

V1) (120579119871)

+ ⟨nablaV1(120579119871) 1198661(120579119871 (119905))

sdot [119901 (119909119867 (119905)) + 119902 (120579

119871 (119905) 119909119867 (119905))]

+ 119867(119905 120579119871 119909119867 (119905))⟩

(29)

From (28) nablaV1(120579119871) can be obtained as

nablaV1(120579119871)

= [12058911205791198711

12058921205791198712

1205893(1205893120579 minus 1205894119902) minus120589

4(1205893120579 minus 1205894119902) + 120589

5119902]119879

(30)

Substituting nablaV1(120579119871) and 119891

1(120579119871) into the derivative of

(1198711198911

V1)(120579119871) we have

(1198711198911

V1) (120579119871) = 12058911205791198711

1205791198712

minus 11989612057912058921205792

1198712

minus 1205892

119892

119897119909

sin 1205791198711

1205791198712

+119896119863119897119909

119872119871

1205892[sign (

119871) cos3 120579

1198711

+ sign (119871) sin3 120579

1198711

]

sdot 1205793

1198712

+ 1205893[1205893120579 minus 1205894119902] 119902 + (minus120589

31205894120579 + (120589

2

4+ 1205895) 119902)

sdot [11988321120579 + 119883

22119902]

(31)

Checking the term of minus sin (1205791198711

)1205791198712

it can be seen thatminus sin (120579

1198711

)1205791198712

le minus(2120587)1205791198711

1205791198712

when 1205791198711

and 1205791198712

are bothpositive or negative The situation of 120579

1198711

and 1205791198712

havingdifferent signs will help with the system stability Then

(1198711198911

V1) (120579119871) le 12058911205791198711

1205791198712

minus 11989612057912058921205792

1198712

minus 1205892

119892

119897119909

2

1205871205791198711

1205791198712

+119896119863119897119909

119872119871

sdot 1205892[sign (

119871) cos3 120579

1198711

+ sign (119871) sin3 120579

1198711

] 1205793

1198712

+ 1205893[1205893120579 minus 1205894119902] 119902 + (minus120589

31205894120579 + (120589

2

4+ 1205895) 119902)

sdot [11988321120579 + 119883

22119902]

(32)

If the design parameter 1205891is chosen as 120589

1= (2119892119897

119909120587)1205892

(1198711198911

V1)(120579119871) satisfies the following

(1198711198911

V1) (120579119871) le minus119896

12057912058921205792

1198712

+119896119863119897119909

119872119871

sdot 1205892[sign (

119871) cos3 120579

1198711

+ sign (119871) sin3 120579

1198711

] 1205793

1198712

+ 1205893[1205893120579 minus 1205894119902] 119902 + (minus120589

31205894120579 + (120589

2

4+ 1205895) 119902)

sdot [11988321120579 + 119883

22119902]

(33)

Journal of Control Science and Engineering 9

Further analysis on (33) will start from examining the firsttwo terms Rewrite these two terms So

minus 11989612057912058921205792

1198712

+119896119863119897119909

119872119871

1205892[sign (

119871) cos3 120579

1198711

+ sign (119871)

sdot sin3 1205791198711

] 1205793

1198712

= minus11989612057912058921205792

1198712

+119896119863119897119909

119872119871

12058921205791198712

[sign (119871)

sdot cos3 1205791198711

+ sign (119871) sin3 120579

1198711

] 1205792

1198712

le minus11989612057912058921205792

1198712

+119896119863119897119909

119872119871

1205892

10038161003816100381610038161003816max (120579

1198712

) (sign (119871) cos3 120579

1198711

+ sign (119871) sin3 120579

1198711

)100381610038161003816100381610038161205792

1198712

= minus12058921205792

1198712

119896120579

minus119896119863119897119909

119872119871

10038161003816100381610038161003816max (120579

1198712

)

sdot (sign (119871) cos3 120579

1198711

+ sign (119871) sin3 120579

1198711

)10038161003816100381610038161003816

(34)

If the hinge friction is big enough to satisfy the following

119896120579gt

119896119863119897119909

119872119871

10038161003816100381610038161003816max (120579

1198712

)

sdot (sign (119871) cos3 120579

1198711

+ sign (119871) sin3 120579

1198711

)10038161003816100381610038161003816

(35)

then

120582119871= 119896120579minus119896119863119897119909

119872119871

10038161003816100381610038161003816max (120579

1198712

)

sdot (sign (119871) cos3 120579

1198711

+ sign (119871) sin3 120579

1198711

)10038161003816100381610038161003816 ge 0

minus 11989612057912058921205792

1198712

+119896119863119897119909

119872119871

1205892[sign (

119871) cos3 120579

1198711

+ sign (119871)

sdot sin3 1205791198711

] 1205793

1198712

le minus12058921205792

1198712

120582119871

(36)

Now the following analysis has been conducted for (33) andthe load movement for the longitudinal motion is describedby

119871= 0minus 1198971199091205791198712

cos 1205791198711

119871= 0minus 1198971199091205791198712

sin 1205791198711

(37)

Around the hover condition1198830= 0 and 119885

0= 0 the signs for

119871and

119871are opposite to the one of 120579

1198712

provided that 1205791198711

ispositive

Considering the term (sign (119871) cos3 120579

1198711

+

sign (119871) sin3 120579

1198711

)1205793

1198712

for all possible combinations ofthe signs of 120579

1198712

and 1205791198711

the analysis follows below

(i) If 1205791198711

gt 0 and 1205791198712

gt 0 then sign (119871) = minus1

sign (119871) = minus1 cos3 120579

1198711

gt 0 and sin3 1205791198711

gt 0 There-fore (sign (

119871) cos3 120579

1198711

+ sign (119871) sin3 120579

1198711

)1205793

1198712

lt 0

(ii) If 1205791198711

lt 0 and 1205791198712

gt 0 then sign (119871) = minus1

sign (119871) = +1 cos3 120579

1198711

gt 0 and sin31205791198711

lt 0 So(sign (

119871) cos3 120579

1198711

+ sign (119871) sin3 120579

1198711

)1205793

1198712

lt 0

(iii) If 1205791198711

gt 0 and 1205791198712

lt 0 then sign (119871) = +1

and sign (119871) = +1 cos3 120579

1198711

gt 0 and sin3 1205791198711

gt

0 So the following is true (sign (119871) cos3 120579

1198711

+

sign (119871) sin3 120579

1198711

)1205793

1198712

lt 0

(iv) If 1205791198711

lt 0 and 1205791198712

lt 0 then sign (119871) = +1 and

sign (119871) = minus1 cos3 120579

1198711

gt 0 and sin3 1205791198711

lt 0 So(sign (

119871) cos3 120579

1198711

+ sign (119871) sin3 120579

1198711

)1205793

1198712

lt 0

Therefore

minus 11989612057912058921205792

1198712

+119896119863119897119909

119872119871

sdot 1205892[sign (

119871) cos3 120579

1198711

+ sign (119871) sin3 120579

1198711

] 1205793

1198712

le minus11989612057912058921205792

1198712

(38)

Now examining the rest of the terms of (1198711198911

V1)(120579119871) then

1205893[1205893120579 minus 1205894119902] 119902

+ (minus12058931205894120579 + (120589

2

4+ 1205895) 119902) [119883

21120579 + 119883

22119902] = 120589

2

3120579119902

minus 120589312058941199022minus 12058931205894119883211205792minus 1205893120589411988322120579119902

+ (1205892

4+ 1205895)11988321120579119902 + (120589

2

4+ 1205895)119883221199022

= minus [12058931205894minus (1205892

4+ 1205895)11988321] 1199022minus 12058931205894119883211205792

+ [1205892

3minus 1205893120589411988322+ (1205892

4+ 1205895)11988321] 120579119902

(39)

For the system 120579 and 1205791198711

always have different signs 120579 and 119902

always have different signs and 119902 and 1205791198711

have the same signsSo

[1205892

3minus 1205893120589411988322+ (1205892

4+ 1205895)11988321] 120579119902 le 0

if [1205892

3minus 1205893120589411988322+ (1205892

4+ 1205895)11988321] ge 0

(40)

Therefore

1205893[1205893120579 minus 1205894119902] 119902

+ (minus12058931205894120579 + (120589

2

4+ 1205895) 119902) [119883

21120579 + 119883

22119902]

le minus [12058931205894minus (1205892

4+ 1205895)11988321] 1199022minus 12058931205894119883211205792

(41)

Choose the design parameters to satisfy

[12058931205894minus (1205892

4+ 1205895)11988321] gt 0 (42)

If11988321

gt 0 then

minus [12058931205894minus (1205892

4+ 1205895)11988321] 1199022minus 12058931205894119883211205792lt 0 (43)

10 Journal of Control Science and Engineering

If11988321

lt 0 then

minus [12058931205894minus (1205892

4+ 1205895)11988321] 1199022minus 12058931205894119883211205792lt 0 (44)

in the region ofminus[12058931205894minus(1205892

4+1205895)11988321]1199022lt 12058931205894119883211205792 Since 120579 is

caused by the loadmotion here the value of 120579 should bemuchsmaller than 119902 The inequality (44) is easy to be satisfied

In summary of the above analysis and by defining

Ψ1(120579119871) = 11989612057912058921205792

1198712

+ [12058931205894minus (1205892

4+ 1205895)11988321] 1199022

+ 12058931205894119883211205792

Ψ2(120579119871) = [120589

31205894minus (1205892

4+ 1205895)11988321] 1199022+ 12058931205894119883211205792

(45)

the following lemma can be derived

Lemma 1 Defining a Lyapunov function (28) and choosingthe design parameters to satisfy 120589

1= (2119892119897

119909120587)1205892 [12058931205894minus (1205892

4+

1205895)11988321] gt 0 and [1205892

3minus1205893120589411988322+(1205892

4+1205895)11988321] ge 0 then within

the region specified by minus[12058931205894minus (1205892

4+ 1205895)11988321]1199022lt 12058931205894119883211205792

(i) V1(0) = 0 and V

1(120579119871) gt 0 forall120579

119871= 0

(ii) V1(120579119871) rarr infin as 120579

119871 rarr infin

(iii) (1198711198911

V1)(120579119871) le minusΨ

1(120579119871) forall120579119871 around the hover condi-

tion or (1198711198911

V1)(120579119871) le minusΨ

2(120579119871) forall120579119871 if (35) holds

Both functions Ψ1and Ψ

2are nonnegative

Recall the control term 119901(119909119867(119905)) in the first subsystem

of (11a) and (11b) it can be seen that [(119863119901)(119909119867)]minus1 exists

for all 119909119867(119905) The unknown vector fields 119902(120579

119871(119905) 119909119867) and

119867(119905 120579119871(119905) 119909119867) model the uncertainties imposed onto the

system Since 119902(120579119871(119905) 119909119867) is directly mapped into the ldquocon-

trolrdquo space of 119909119867(119905) it can be considered as a matched

uncertainty119867(119905 120579119871(119905) 119909119867) is unknown and it does belong to

the control space of 119909119867(119905) so it represents the mismatched

uncertainty in the system [12]Generally the range of the (longitudinal) load suspension

angle is within minus1205872 lt 120579119871lt 1205872 The helicopter velocities

and load suspension angle havemaximumoperational valuestherefore the uncertainties in the system are bounded Withthe Lyapunov function defined in (28)

(1198711198921

V1) (120579119871) = ⟨nablaV

1(120579119871) 1198921⟩ =

119896119863

119872119871119897119909

12058921205791198712

(46)

In 1198661(120579119871) the columns 119892

1and 119892

2are the same therefore

(1198711198922

V1) (120579119871) =

119896119863

119872119871119897119909

12058921205791198712

(47)

By considering the maximum values of cos (sdot) and sin (sdot)functions the bounding values relating to the uncertainty119902(120579119871(119905) 119909119867) can be estimated as follows

10038171003817100381710038171003817119902 (120579119871 119909119867)10038171003817100381710038171003817le

10038161003816100381610038161003816100381610038161003816

1

119897119909

( + )

+1

119872119871

(minus119896119863(2 +

120581

119897119909

) (119906 + 119908) +119896120579

119897119909

)1205791198712

10038161003816100381610038161003816100381610038161003816

le

10038161003816100381610038161003816100381610038161003816

119896120579

119897119909

1205791198712

10038161003816100381610038161003816100381610038161003816+

10038161003816100381610038161003816100381610038161003816

119896119863

119872119871

(2 +120581

119897119909

) (119906 + 119908)

10038161003816100381610038161003816100381610038161003816+

10038161003816100381610038161003816100381610038161003816

1

119897119909

( + )

10038161003816100381610038161003816100381610038161003816

le119896120579

119897119909

100381710038171003817100381710038171205791198712

10038171003817100381710038171003817+

119896119863

119872119871

(2 +120581

119897119909

)1003817100381710038171003817119901 (119909119867)1003817100381710038171003817 + 1205722 (119905)

le 1205831

10038161003816100381610038161003816(1198711198922

V1) (120579119871)10038161003816100381610038161003816+ 1205721

1003817100381710038171003817119901 (119909119867)1003817100381710038171003817 + 1205722 (119905)

(48)

where 1205831

= 119872119871119896120579119896119863 1205721

= (119896119863119872119871)(2 + 120581119897

119909) (120581 =

max [1205811 1205812]) and 120572

2(119905) is defined by 120572

2(119905) = max |(1119897

119909)( +

)| gt 0For the mismatched uncertainty 119867(119905 120579

119871(119905) 119909119867) the fol-

lowing analysis is conducted to obtain its bounding functionThe mismatched uncertainty can be rewritten as

119867(119905 120579119871 (119905) 119909119867) =

[[[[[

[

0

0

(11988611119906 + 11988612119908)

(11988323119906 + 119883

24119908)

]]]]]

]

=

[[[[[

[

0 0

0 0

11988611

11988612

11988323

11988324

]]]]]

]

[119906

119908]

(49)

Let 119860 = [1198861111988612

1198832311988324] then

10038171003817100381710038171003817119867 (119905 120579

119871 119909119867)10038171003817100381710038171003817le1003817100381710038171003817100381711986010038171003817100381710038171003817

1003817100381710038171003817119909119867 (119905)1003817100381710038171003817 (50)

Define a positive function as

1006704120579 (120579119871) = radic(1205792

1198711

+ 12057921198712

+ 1205792 + 1199022+ 120576) (51)

where 120576 is a very small positive constant Choosing1205731

= (1198721198711198971199091198961198631205892)119860 then the mismatched uncertainty

119867(119905 120579119871(119905) 119909119867) is bounded by the following inequality

10038171003817100381710038171003817119867 (119905 120579

119871 119909119867)10038171003817100381710038171003817

le 1006704120579minus1

(120579119871) 1205731

2

sum

119894=1

(10038161003816100381610038161003816(119871119892119894

V1) (120579119871)10038161003816100381610038161003816)1003816100381610038161003816119901119894 (119909119867)

1003816100381610038161003816

(52)

where 119892119894denotes the 119894th column of the matrix function

1198661(120579119871) and 119901

119894is the 119894th component of 119901(119909

119867) respectively As

1205731has a design parameter 120589

2involved it is easy to have 120573

1to

lead (52) to be true For the matched uncertainty we have

10038171003817100381710038171003817119902 (120579119871 119909119867)10038171003817100381710038171003817le

119896120579

119897119909

100381610038161003816100381610038161205791198712

10038161003816100381610038161003816+ 1205721

1003817100381710038171003817119901 (119909119867)1003817100381710038171003817 + 1205722 (119905) (53)

Journal of Control Science and Engineering 11

Following the above analysis the following lemma can bederived

Lemma 2 The uncertainties (52) and (53) are bounded andsatisfy if 120572

1 1205731ge 0 and 120572

1+ 1205731lt 1 for 120572

1= (119896119863119872119871)(2 +

120581119897119909) and 120573

1= (1198721198711198971199091198961198631205892)119860

4 Numerical Results

Using a simulation software the linearized model for a UH-60 helicopter is obtained with the following state and inputmatrices

119860 =

[[[[[[[[[[[[[[[[[[[

[

minus00000 00003 minus00000 minus00000 00000 minus00000 10000 minus00023 00503

minus00003 minus00000 minus00000 minus00000 minus00000 minus00000 minus00000 09989 00463

minus00001 00000 minus00000 minus00000 00000 00000 minus00000 minus00464 10002

minus19386 minus28593 minus00030 minus00195 minus00126 00168 minus12000 57076 02683

328694 minus03640 00027 00126 minus00311 minus00013 minus41348 minus22350 03713

03138 minus14034 00004 00150 00034 minus03340 02866 06307 23410

minus09144 minus04783 00017 00226 minus00246 00007 minus65893 minus21301 minus00598

00631 minus06744 00008 00029 00046 00025 02059 minus14255 minus01636

minus07549 minus00119 minus00002 00009 00028 00006 minus00749 minus01032 minus02214

]]]]]]]]]]]]]]]]]]]

]

119861 =

[[[[[[[[[[[[[[[[[[[

[

minus00000 00000 minus00000 00000

00000 00000 minus00000 minus00000

minus00000 00000 minus00000 minus00000

minus22190 05734 03170 minus00019

35381 minus00453 minus00963 02007

minus15629 00237 minus60740 minus01014

19943 minus00714 minus00541 00731

00446 minus01097 00553 minus00281

minus08197 minus00082 01573 minus00635

]]]]]]]]]]]]]]]]]]]

]

(54)

The desired condition 1205721+ 1205731lt 1 can be verified by consid-

ering a typical case that the UH-60 helicopter carries a loadweight 1000 lb by a 15 ft slung length and let 119896

119863= 75 lbft

For the UH-60 helicopter model with reference to thegeneral mathematical model presented in Section 2 we have1198831205791119904

= 05734 1198831205790

= 03170 1198851205791119904

= 00237 1198851205790

= minus607401198721205791119904

= minus01097 and 1198721205790

= 00553 So the followingparameters can be calculated to have the values 119886

11= minus02

and 11988612

= minus002 For 119883119906= minus00195 119885

119906= 00150 119872

119906=

00029 119872119902= minus14255 (119883

119902minus 119908119890) = 57076 and (119885

119902+ 119906119890) =

06307 we have 11988323

= 006 And also for 119883119908

= 00168119885119908

= minus03340 and 119872119908

= 00025 we have 11988324

= 001Based on all the above parameters 120590max(119860) = 00041 Inthis case let 120581 = 001 120572

1= (119896119863119872119871)(2 + 120581119897

119909) asymp 015

and 1205731= (119872

119871119897119909119896119863)120590max(119860) = 082 Therefore 120572

1+ 1205731=

015 + 082 = 097 which satisfies 1205721+ 1205731

lt 1 Anotherexample considers that the UH-60 helicopter carries a loadweight 500 lb by a 15 ft slung length and let 119896

119863= 50 lbft then

1205721= 02 and 120573

1= 062 therefore (120572

1+ 1205731) = 082 Therefore

this assumption 1205721+ 1205731

lt 1 is realistic for the systemdiscussed in here The numerical values vary with respectto the slung load configuration It may represent the system

dynamic variations To investigate the stability characteristicsof dynamic variations two series of simulation studies werecarried out

Firstly the slung length was kept constant at 15 ft andthe load varied between the limits 500ndash2000 lb with 119896

119863=

75 lbft producing the system poles listed in Table 1 Thecorresponding root loci are sketched in Figures 3 and 4Figure 4 clearly shows that the positions of the system polescrossed the imaginary axis as the load weight increased

Secondly load weight was kept constant and the slunglength varied between the limit of 10ndash20 ft Table 2 sum-marises the results of variation of the system poles and thecorresponding root loci are illustrated in Figures 5 and 6

The pole locus indicates that the system poles have thevariation trends of moving further towards right direction onthe 119904-plane with increase of load weight and slung lengthsFrom the simulation results it is clearly shown that thesystem becomes less stable when the load andor the slunglength increases The simulation results are compared withthe stability analysis presented above

Now consider the case of fixed slung length of 15 ft withdifferent load weights In the cases of a UH-60 helicopter that

12 Journal of Control Science and Engineering

Table 1 System poles locations for a fixed slung length (119897 = 15ft) but with different load weights

Helicopter 500 lb 1000 lb 1500 lb 2000 lbminus10919 minus11159 minus07848 minus06221 minus09369minus63938 minus64301 minus92203 minus79848 minus75613minus00032 minus00048 minus00034 minus00052 minus00052minus00977 minus00818 minus00888 minus00972 minus00694minus03045 minus03036 minus03394 minus03318 minus03333minus03159 plusmn 04363119894 minus03199 plusmn 04628119894 minus02295 plusmn 05050119894 minus02366 plusmn 05974119894 minus01834 plusmn 04968119894minus00489 plusmn 03898119894 minus00211 plusmn 03868119894 00074 plusmn 03445119894 00119 plusmn 04406119894 00264 plusmn 03199119894mdash minus00316 plusmn 15043119894 minus01020 plusmn 17380119894 minus00368 plusmn 18753119894 minus01160 plusmn 20003119894mdash minus00736 plusmn 15019119894 minus07662 plusmn 11680119894 minus07349 plusmn 19064119894 minus12367 plusmn 08995119894

Table 2 System poles for a fixed load (1000 lb) but with different slung lengths

Helicopter 10 ft 15 ft 20 ftminus10919 minus10560 minus07848 minus08687minus63938 minus99252 minus92203 minus85565minus00032 minus00061 minus00034 minus00114minus00977 minus02012 minus00888 minus00953minus03045 minus02606 minus03394 minus03479minus03159 plusmn 04363119894 minus01809 plusmn 06219119894 minus02295 plusmn 05050119894 minus01636 plusmn 06092119894minus00489 plusmn 03898119894 00643 plusmn 04577119894 00074 plusmn 03445119894 00799 plusmn 04594119894mdash minus01055 plusmn 19883119894 minus01020 plusmn 17380119894 minus01247 plusmn 14470119894mdash minus08847 plusmn 12464119894 minus07662 plusmn 11680119894 minus06352 plusmn 10384119894

Re(poles)

Im(poles)

minus10 minus5 0 5minus08minus06minus04minus02

002040608

Figure 3 System root-locus diagram showing pole locus forconstant slung length (15 ft) as load weight is increased

minus04 minus035 minus03 minus025 minus02 minus015 minus01 minus005 0 005 01

Re(poles)

Im(poles)

minus08minus06minus04minus02

002040608

Figure 4 Enlarged view of Figure 3 showing poles locus near toRe(poles) = 0 axis

minus10 minus8 minus6 minus4 minus2 0 2

Re(poles)

Im(poles)

minus08minus06minus04minus02

002040608

Figure 5 System root-locus showing pole locus for constant loadweight (1000 lb) as slung length is increased

minus04 minus035 minus03 minus025 minus02 minus015 minus01 minus005 0 005 01

Re(poles)

Im(poles)

minus08minus06minus04minus02

002040608

Figure 6 Enlarged view of Figure 5 showing poles locus near toRe(poles) = 0 axis

Journal of Control Science and Engineering 13

carries a loadweight 500 lb and 1000 lb then1205721= 03 and 041

with 1205731= 041 and 082 respectively therefore 120572

1+ 1205731lt 1

thus the system is stable But when the load weight is 1500 lband 2000 lb then 120572

1= 01 and 0075 and also 120573

1= 123 and

164 respectively therefore 1205721+ 1205731gt 1 thus the system is

unstable For the case of the constant load weight (1000 lb) asslung length is increased then for a 10 ft and 15 ft slung length1205721= 015 and 120573

1= 055 and 082 respectively therefore

1205721+ 1205731lt 1 thus the system is stable But when the slung

length increased to 20 ft then 1205721

= 015 and 1205731

= 109therefore (120572

1+ 1205731) = 124 that is 120572

1+ 1205731gt 1 thus the

system is unstable However the system can be marginallystable if the UH-60 helicopter carries a load weight 725 lb bya 20 ft slung length then 120572

1= 02 and 120573

1= 08 therefore

(1205721+ 1205731) = 1 For this set of analysis 119896

119863is assumed to

be 119896119863

= 75 lbft However 119896119863can vary depending on the

weather condition and an example of 119896119863= 50 lbft is shown

above for illustrative purpose This analysis suggests that themaximum value of 20 ft slung length may be used for slungload operation However the load weight limit can vary withrespect to the capacity of the load carrying hooks helicopterweight environmental conditions and so forth

The examples presented here show that the simulationresults are correlated with the stability analysis Thus desiredcondition for the stability is met and it can locate the load atthe specified position or its neighbourhood

5 Conclusions

In this paper stability analysis of a helicopter with an under-slung load system is investigatedThe investigation identifiedthe conditions for the stabilisation of the system and thepositioning of the underslung load at hover condition Sta-bility analysis and numerical results proved that the desiredcondition for the stability is met then it is possible to locatethe load at the specified position or its neighbourhood Anillustration example is also given in the paper The methodresults in a guaranteed load positioning accuracy whichdepends on the design parameters and the method can beapplicable to any individual helicopter with external loadsystem

Nomenclature

119871119872119873 Overall helicopter rolling pitching andyawing moments

119901 119902 119903 Helicopter roll pitch and yaw rates aboutbody reference axes

119883119884 119885 Overall helicopter force components120601 120579 120595 Roll pitch and yaw angles119906 V 119908 Helicopter velocity components at centre

of gravity1205790 Main rotor collective

1205791119904 Longitudinal cyclic commands

1205791119888 Lateral cyclic commands

1205790119879 Tail rotor collective

1198830 1198840 1198850 Location of the suspension point withrespect to earth referenced 119909 119910 and 119911

directions

0 0 0 The helicopter velocity in the 119909 119910 and 119911

directions0 0 0 The helicopter acceleration in the 119909 119910 and119911 directions

120579119871 Load suspension angle in the119883-119885 plane

with respect to 119885-axis120601119871 Load suspension angle in the 119884-119885 plane

with respect to 119885-axis119872119871 Mass of the suspended load

119892 Acceleration due to gravity119897 Slung length119896119863 Aerodynamic drag force constant

(119896119863= (12)120588119878119862

119863)

120588 Air density119878 The load area presented to the airflow119862119863 Drag coefficient for the load

Competing Interests

The author declares that there are no competing interestsregarding the publication of this paper

References

[1] K Thanapalan and F Zhang ldquoModelling and simulation studyof a helicopter with an external slung load systemrdquo I-ManagersJournal on Instrumentation amp Control Engineering vol 1 no 1pp 9ndash17 2013

[2] M Bisgaard J D Bendtsen and A La Cour-Harbo ldquoModelingof generic slung load systemrdquo Journal of Guidance Control andDynamics vol 32 no 2 pp 573ndash585 2009

[3] Handling Qualities Requirements for Military Rotorcraft ADS-33-D US Army Aviation and Troop Command St Louis MissUSA 1996

[4] K Thanapalan and T M Wong ldquoModeing of a helicopter withan under-slung load systemrdquo in Proceedings of the 29th ChineseControl Conference (CCC rsquo10) pp 1451ndash1456 Beijing China July2010

[5] B Feng ldquoRobust control for lateral and longitudinal channels ofsmall-scale unmanned helicoptersrdquo Journal of Control Scienceand Engineering vol 2015 Article ID 483096 8 pages 2015

[6] J J Gribble ldquoLinear quadratic Gaussianloop transfer recoverydesign for a helicopter in low-speed flightrdquo Journal of GuidanceControl and Dynamics vol 16 no 4 pp 754ndash761 1993

[7] M A Manness J J Gribble and D J M Smith ldquoMultivariablemethods for helicopter flight control law designrdquo in Proceedingsof the 16th Annual Society European Rotorcraft Forum PaperNoIII52 Glasgow UK September 1990

[8] W L Garrard E Low and S Prouty ldquoDesign of attitude andrate command systems for helicopters using eigenstructureassignmentrdquo Journal of Guidance Control and Dynamics vol12 no 6 pp 783ndash791 1989

[9] Y B Shtessel and I A Shkolnikov ldquoAeronautical and spacevehicle control in dynamic sliding manifoldsrdquo InternationalJournal of Control vol 76 no 9-10 pp 1000ndash1017 2003

[10] D FThompson J S Pruyn andA Shukla ldquoFeedback design forrobust tracking and robust stiffness in flight control actuatorsusing a modified QFT techniquerdquo International Journal ofControl vol 72 no 16 pp 1480ndash1497 1999

14 Journal of Control Science and Engineering

[11] K Thanapalan ldquoModelling of a helicopter systemrdquo in Proceed-ings of the 1st Virtual Control Conference (VCC rsquo10) pp 45ndash52Aalborg University 2010

[12] D P Goodall and J Wang ldquoStabilization of a class of uncertainnonlinear affine systems subject to control constraintsrdquo Interna-tional Journal of Robust and Nonlinear Control vol 11 no 9 pp797ndash818 2001

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

2 Journal of Control Science and Engineering

the immeasurable parameters in the dynamical models Theuncertainties may also arise from computational errors ofthe dynamical effects such as aerodynamics Therefore fora realistic model uncertainties must be taken into accountduring the controller design

A mathematical model of the helicopter described in [11]and an underslung load model presented in [4] are adoptedin this work Considering the two models a mathematicalmodel for a helicopter carrying an underslung load can beobtained

Firstly the underslung load is considered to be suspendedfrom a single suspension point that is subject to motionand therefore modelled as a driven spherical pendulum Theequations that describe the load dynamics are obtained byfirst considering motion with reference to the longitudinalsuspension angle 120579

119871in the 119883-119885 plane (Figure 1) This is then

repeated for the lateral case involving 120593119871and the 119884-119885 plane

These are then combined to obtain the model for the motionof the load The underslung load system has six inputslongitudinal lateral and vertical velocities together withthe corresponding accelerations of the helicopter whilst theoutputs are the longitudinal and lateral directional suspen-sion angles The load is subject to an isotropic aerodynamicforce (proportional to the square of its airspeed) such aswhat would be experienced by a spherical shaped loadAerodynamic interaction with the helicopter that may occurfor example due to rotor downwash has been ignoredFinally the sling itself is assumed to be rigid and contributezero aerodynamic force of its own With these assumptionsthe equations governing the load motion can be derived asfollows

For the case of the longitudinal motion in the119883-119885 planethe mathematical model is described below

119871= minus

119892

119897119909

sin 120579119871+cos 120579119871

119897119909

0+sin 120579119871

119897119909

0

+119896119863sign (

119871) cos 120579

119871

119872119871119897119909

2

0+119896119863sign (

119871) sin 120579

119871

119872119871119897119909

2

0

minus2119896119863

119872119871

(sign (119871) cos2 120579

1198710

+ sign (119871) sin2 120579

1198710) 119871minus 119896120579119871

+119896119863119897119909

119872119871

[sign (119871) cos3 120579

119871+ sign (

119871) sin3 120579

119871] 2

119871

(1)

Define 119871= [120579119871

119871]119879

= [1205791198711

1205791198712

]119879 then the load model can

be rewritten as follows

1198711

= 1205791198712

(2a)

1198712

=minus119892

119897119909

sin 1205791198711

+119896119863119897119909

119872119871

[sign (119871) cos3 120579

1198711

+ sign (119871) sin3 120579

1198711

] 1205792

1198712

minus 1198961205791205791198712

+ (119896119863sign (

119871) cos 120579

1198711

119872119871119897119909

)2

0

+ (119896119863sign (

119871) sin 120579

1198711

119872119871119897119909

) 2

0+cos 1205791198711

119897119909

0

+sin 1205791198711

119897119909

0minus2119896119863

119872119871

(sign (119871) cos2 120579

1198711

0

+ sign (119871) sin2 120579

1198711

0) 1205791198712

(2b)

The helicopter model is considered as the second subsystemTo simplify the analysis the linear helicopter model [4] isconsideredwhich is expressed in the state space form

119867(119905) =

119860119909119867(119905) + 119861119906(119905)

[[[[[[[[[[[[[[[[[[[[

[

V

119903

]]]]]]]]]]]]]]]]]]]]

]

=

[[[[[[[[[[[[[[[[[[[[

[

0 0 0 0 0 0 1 0 0

0 0 0 0 0 0 0 1 0

0 0 0 0 0 0 0 0 1

0 minus119892 cos 1205791198900 119883119906

119883V 119883119908

119883119901

(119883119902minus 119908119890) (119883119903minus V119890)

119892 0 0 119884119906

119884V 119884119908

(119884119901+ 119908119890) 119884

119902(119884119903minus 119906119890)

0 minus119892 sin 120579119890

0 119885119906

119885V 119885119908

(119885119901minus V119890) (119885

119902+ 119906119890) 119885

119903

0 0 0 119871119906

119871V 119871119908

119871119901

119871119902

119871119903

0 0 0 119872119906

119872V 119872119908

119872119901

119872119902

119872119903

0 0 0 119873119906

119873V 119873119908

119873119901

119873119902

119873119903

]]]]]]]]]]]]]]]]]]]]

]

[[[[[[[[[[[[[[[[[[[

[

120601

120579

120595

119906

V

119908

119901

119902

119903

]]]]]]]]]]]]]]]]]]]

]

+

[[[[[[[[[[[[[[[[[[[

[

0 0 0 0

0 0 0 0

0 0 0 0

1198831205791119888

1198831205791119904

1198831205790

1198831205790119879

1198841205791119888

1198841205791119904

1198841205790

1198841205790119879

1198851205791119888

1198851205791119904

1198851205790

1198851205790119879

1198711205791119888

1198711205791119904

1198711205790

1198711205790119879

1198721205791119888

1198721205791119904

1198721205790

1198721205790119879

1198731205791119888

1198731205791119904

1198731205790

1198731205790119879

]]]]]]]]]]]]]]]]]]]

]

[[[[[

[

1205791119888

1205791119904

1205790

1205790119879

]]]]]

]

(3)

For linearization it is assumed that the external forces 119883119884 and 119885 and moments 119871 119872 and 119873 can be represented as

analytic functions of the disturbedmotion variables and theirderivatives [4] Thus the forces and moments can be written

Journal of Control Science and Engineering 3

X

Z

120579Llx

(XL YL ZL)

(X0 Y0 Z0)

Figure 1 Coordinate system for the longitudinal motion in the119883-119885plane

in Taylorrsquos expansion form Then the linearized equations ofmotion about a general trim condition can be written as inthe state space form = 119860119909 + 119861119906 and the system matrix 119860

and control matrix 119861 are derived from the partial derivativesof the nonlinear function 119865 that is

= 119865 (119909 119906 119905)

119860 = (120597119865

120597119909)

119909=119909119890

119861 = (120597119865

120597119906)

119909=119909119890

(4)

Now the longitudinal rotational motion is described by thepitch angle 120579 and pitch rate 119902 together with the translationmotion components 119906 119908 so the equation of longitudinalmotion can be written as follows

[[[[[

[

]]]]]

]

=

[[[[[[

[

0 1 0 0

0 119872119902

119872119906

119872119908

minus119892 cos 120579119890(119883119902minus 119908119890) 119883119906

119883119908

minus119892 sin 120579119890

(119885119902minus 119906119890) 119885119906

119885119908

]]]]]]

]

[[[[[

[

120579

119902

119906

119908

]]]]]

]

+

[[[[[

[

0 0

1198721205791119904

1198721205790

1198831205791119904

1198831205790

1198851205791119904

1198851205790

]]]]]

]

[1205791119904

1205790

]

(5)

By applying a linear transformation119879 such that119909119867(119905) = 119879119911(119905)

and 119879 is defined by

119879 =

[[[[[

[

1 0 0 0

0 1 11988611

11988612

0 0 1 0

0 0 0 1

]]]]]

]

(6)

where

11988611

= minus1198851205791119904

1198721205790

minus 1198851205790

1198721205791119904

1198831205791119904

1198851205790

minus 1198851205791119904

1198831205790

11988612

= minus1198831205790

1198721205791119904

minus 1198831205791119904

1198721205790

1198831205791119904

1198851205790

minus 1198851205791119904

1198831205790

(7)

and letting 119902 = (119902 minus 11988611119906 minus 11988612119908) then the system equation

(5) is transformed into the following form

[[[[[[

[

]]]]]]

]

=

[[[[[

[

11988311

11988312

11988313

11988314

11988321

11988322

11988323

11988324

11988331

11988332

11988333

11988334

11988341

11988342

11988343

11988344

]]]]]

]

[[[[[

[

120579

119902

119906

119908

]]]]]

]

+

[[[[[

[

0 0

11986121

11986122

1198831205791119904

1198831205790

1198851205791119904

1198851205790

]]]]]

]

[1205791119904

1205790

]

(8)

where

11988311

= 0

11988312

= 1

11988313

= 11988611

11988314

= 11988612

11988321

= (11988611119892 cos 120579

119890+ 11988612119892 sin 120579

119890)

11988322

= (119872119902minus 11988611(119883119902minus 119908119890) minus 11988612(119885119902minus 119906119890))

11988323

= (119872119902minus 11988611(119883119902minus 119908119890) minus 11988612(119885119902+ 119906119890)) 11988611

+ (119872119906minus 11988611119883119906minus 11988612119885119906)

11988324

= (119872119902minus 11988611(119883119902minus 119908119890) minus 11988612(119885119902+ 119906119890)) 11988612

+ (119872119908minus 11988611119883119908minus 11988612119885119908)

11988331

= minus119892 cos 120579119890

11988332

= (119883119902minus 119908119890)

11988333

= (119883119902minus 119908119890) 11988611+ 119883119906

11988334

= (119883119902minus 119908119890) 11988611+ 119883119908

11988341

= (minus119892 sin 120579119890)

11988342

= (119885119902+ 119906119890)

11988343

= (119885119902+ 119906119890) 11988611+ 119885119906

11988344

= (119885119902+ 119906119890) 11988612+ 119885119908

4 Journal of Control Science and Engineering

11986121

= (1198721205791119904

minus 119886111198831205791119904

minus 119886121198851205791119904

)

11986122

= (1198721205790

minus 119886111198831205790

minus 119886111198851205790

)

(9)

Using (8) the system model can be rearranged to include thevariables 120579 and 119902 into the loadmodel For the stability analysispurpose an extra term is introduced into the system modelwhich is zero with the expression

1205811119896119863sign (

119871) cos 120579

1198711

119906

119872119871119897119909

+1205812119896119863sign (

119871) cos 120579

1198711

119908

119872119871119897119909

minus1205811119896119863sign (

119871) cos 120579

1198711

119906

119872119871119897119909

minus1205812119896119863sign (

119871) cos 120579

1198711

119908

119872119871119897119909

(10)

where 120581119894gt 0 (119894 = 1 2) are small positive constants

With this arrangement for the longitudinal motion of thehelicopter with an underslung load combined system modelcan be written as follows

120579119871 (119905)

= 1198911(120579119871 (119905))

+ 1198661(120579119871 (119905)) [119901 (119909

119867 (119905)) + 119902 (120579119871 (119905) 119909119867 (119905))]

+ 119867(119905 120579119871 119909119867 (119905))

(11a)

119867 (119905) = 119891

2(120579119871 (119905) 119909119867 (119905)) + 119866

2 (119905) (11b)

where

120579119871 (119905) = [120579119871

1

1205791198712

120579 119902]119879

119909119867= [119906 119908]

119879

(119905) = [1205791119904 1205790]119879

119901 (119909119867) = [119906

2+ 1205811119906 1199082+ 1205812119908]119879

1198911(120579119871 (119905)) =

[[[[[[[

[

1205791198712

minus119892

119897119909

sin 1205791198711

+119896119863119897119909

119872119871

(sign (119871) cos3 120579

1198711

+ sign (119871) sin3 120579

1198711

) 1205792

1198712

minus 1198961205791205791198712

119902

(11988321120579 + 119883

22119902)

]]]]]]]

]

1198661(120579119871 (119905)) =

[[[[[[[[[

[

0 0

119896119863sign (

119871) cos 120579

1198711

119872119871119897119909

119896119863sign (

119871) sin 120579

1198711

119872119871119897119909

0 0

0 0

]]]]]]]]]

]

119902 (120579119871 (119905) 119909119867 (119905))

=[[

[

0

cos 1205791198711

119897119909

+sin 1205791198711

119897119909

minus2119896119863

119872119871

(sign (119871) cos2 120579

1198711

119906 + sign (119871) sin2 120579

1198711

119908) 1205791198712

minus1205811119896119863sign (

119871) cos 120579

1198711

119906

119872119871119897119909

minus1205812119896119863sign (

119871) cos 120579

1198711

119908

119872119871119897119909

]]

]

119867(119905 120579119871 (119905) 119909119867 (119905)) =

[[[[[

[

0

0

(11988611119906 + 11988612119908)

11988323119906 + 119883

24119908

]]]]]

]

1198912(120579119871 (119905) 119909119867 (119905)) = [

(11988331120579 + 119883

32119902 + 119883

33119906 + 119883

34119908)

(11988341120579 + 119883

42119902 + 119883

43119906 + 119883

44119908)

]

1198662= [

1198831205791119904

1198831205790

1198851205791119904

1198851205790

]

(12)

Journal of Control Science and Engineering 5

Y

Z

ly

(XL YL ZL)

(X0 Y0 Z0)

120593L

Figure 2 Coordinate system for the lateralmotion in the119884-119885 plane

It is assumed that the longitudinal motion is primarilycontrolled by longitudinal cyclic commands (120579

1119904) and main

rotor collective 1205790

For the case of the lateral motion in the 119884-119885 plane withthe coordinate system described in Figure 2 the load modelis

119871119910119911

= minus119892

119897119910

sin120593119871119910119911

+

cos120593119871119910119911

119897119910

0+

sin120593119871119910119911

119897119910

0

+

119896119863sign (

119871) cos120593

119871119910119911

119872119871119897119910

2

0

+

119896119863sign (

119871) sin120593

119871119910119911

119872119871119897119910

2

0

minus2119896119863

119872119871

(sign (119871) cos2 120593

119871119910119911

0

+ sign (119871) sin2 120593

119871119910119911

0) 119871minus 119896120593119871

+119896119863119897119910

119872119871

[sign (119871) cos3 120593

119871119910119911

+ sign (119871) sin3 120593

119871119910119911

]

sdot 2

119871119910119911

(13)

Define 119871= [120593119871

119871]119879

= [1205931198711

1205931198712]119879 then the model can be

written as follows

1198711

= 1205931198712

(14a)

1198712

=minus119892

119897119910

sin1205931198711

+119896119863119897119910

119872119871

[sign (119871) cos3 120593

1198711

+ sign (119871) sin3 120593

1198711

] 1205932

1198712

minus 1198961205931205931198712

+ (119896119863sign (

119871) cos120593

1198711

119872119871119897119910

) 2

0

+ (119896119863sign (

119871) sin120593

1198711

119872119871119897119910

) 2

0+cos1205931198711

119897119910

0

+sin1205931198711

119897119910

0minus2119896119863

119872119871

(sign (119871) cos2 120593

1198711

0

+ sign (119871) sin2 120593

1198711

0) 1205931198712

(14b)

Now for the helicopter model the lateral rotational motion isdescribed by the roll angle 120601 and roll rate 119901 together with thetranslationmotion components V 119908 so the equation of lateralmotion can be written as follows

[[[[[

[

V

]]]]]

]

=

[[[[[[

[

0 1 0 0

0 119871119901

119871V 119871119908

119892 (119884119901minus 119908119890) 119884V 119884

119908

0 (119885119901minus V119890) 119885V 119885

119908

]]]]]]

]

[[[[[

[

120601

119901

V

119908

]]]]]

]

+

[[[[[

[

0 0

1198711205791119888

1198711205790119879

1198841205791119888

1198841205790119879

1198851205791119888

1198851205790119879

]]]]]

]

[1205791119888

1205790119879

]

(15)

now using a linear transformation 1198791such that 119909

119867(119905) =

1198791119911(119905) and 119879

1is defined by

1198791=

[[[[[

[

1 0 0 0

0 1 11988711

11988712

0 0 1 0

0 0 0 1

]]]]]

]

(16)

where

11988711

= minus1198851205791119888

1198711205790119879

minus 1198851205790119879

1198711205791119888

1198841205791119888

1198851205790119879

minus 1198851205791119888

1198841205790119879

11988712

= minus1198841205790119879

1198711205791119888

minus 1198841205791119888

1198711205790119879

1198841205791119888

1198851205790119879

minus 1198851205791119888

1198841205790119879

(17)

6 Journal of Control Science and Engineering

Let 119901 = (119901 minus 11988711V minus 11988712119908) then (15) can be written in the

following form

[[[[[[

[

V

]]]]]]

]

=

[[[[[

[

11988411

11988412

11988413

11988414

11988421

11988422

11988423

11988424

11988431

11988432

11988433

11988434

11988441

11988442

11988443

11988444

]]]]]

]

[[[[[

[

120601

119901

V

119908

]]]]]

]

+

[[[[[

[

0 0

11987321

11987322

1198841205791119888

1198841205790119879

1198851205791119888

1198851205790119879

]]]]]

]

[1205791119888

1205790119879

]

(18)

where

11988411

= 0

11988412

= 1

11988413

= 11988711

11988414

= 11988712

11988421

= minus (11988711119892)

11988422

= (119871119901minus 11988711(119884119901+ 119908119890) minus 11988712(119885119901minus V119890))

11988423

= (119871119901minus 11988711(119884119901+ 119908119890) minus 11988712(119885119901minus V119890)) 11988711

+ (119871119901minus 11988711119884V minus 119887

12119885V)

11988424

= (119871119901minus 11988711(119884119901+ 119908119890) minus 11988712(119885119901minus V119890)) 11988712

+ (119871119901minus 11988711119884119908minus 11988712119885119908)

11988431

= 119892

11988432

= (119884119901+ 119908119890)

11988433

= ((119884119901+ 119908119890) 11988711+ 119884V)

11988434

= ((119884119901+ 119908119890) 11988712+ 119884119908)

11988441

= 0

11988442

= (119885119901minus V119890)

11988443

= ((119885119901minus V119890) 11988711+ 119885V)

11988444

= ((119885119901minus V119890) 11988712+ 119885119908)

11987321

= (1198711205791119888

minus 119887111198841205791119888

minus 119887121198851205791119888

)

11987322

= (1198711205790119879

minus 119887111198841205790119879

minus 119887121198851205790119879

)

(19)

Using (18) the helicopter with underslung load for lateralmotion in the 119884-119885 plane can be described by

119871(119905)

= 1198911(120593119871(119905))

+ 1198661(120593119871(119905)) [119901 (119909

119867 (119905)) + 119902 (120593119871(119905) 119909119867 (119905))]

+ 119867 (119905 120593119871 119909119867 (119905))

(20a)

119867 (119905) = 119891

2(120593119871(119905) 119909119867 (119905)) + 119866

2 (119905) (20b)

where

119871(119905) = [

1198711

1198712

]119879

119901 (119909119867 (119905)) = [V2 119908

2]119879

1198911(120593119871(119905)) =

[[[[[[[

[

1205931198712

minus119892

119897119910

sin1205931198711

+119896119863119897119910

119872119871

(sign (119871) cos3 120593

1198711

+ sin (119871) sin3 120593

1198711

) 1205932

1198712

minus 1198961205931205931198712

119901

(minus11988711119892120601 + (119871

119901minus 11988711(119884119901+ 119908119890) minus 11988712(119885119901minus V119890)) 119901)

]]]]]]]

]

1198661(120593119871(119905)) =

[[[[[[[[

[

0 0

(119896119863sign (

119871) cos120593

1198711

119872119871119897119910

) (119896119863sign (

119871) sin120593

1198711

119872119871119897119910

)

0 0

0 0

]]]]]]]]

]

119902 (120593119871(119905) 119909119867 (119905)) =

[[

[

0

cos1205931198711

119897119910

V +sin1205931198711

119897119910

minus2119896119863

119872119871

(sign (119871) cos2 120593

1198711

V + sign (119871) sin2 120593

1198711

119908)1205931198712

]]

]

Journal of Control Science and Engineering 7

119867(119905 120593119871(119905) 119909119867 (119905)) =

[[[[[

[

0

0

( minus 119901)

(11988423V + 11988424119908)

]]]]]

]

(119905) = [1205791119888 1205790119879]119879

1198912(120593119871(119905) 119909119867 (119905)) = [

(11988431120601 + 11988432119901 + 11988433V + 11988434119908)

(11988442119901 + 11988443V + 11988444119908)

]

1198662= [

1198841205791119888

1198841205790119879

1198851205791119888

1198851205790119879

]

(21)

It is assumed that the lateral motion is primarily controlledby lateral cyclic commands (120579

1119888) and the tail rotor collective

1205790119879

3 Stability Analysis

The goal is to analyse the stability of the combined systemto ensure that it is possible to stabilise the helicopter withunderslung load system modelled by (11a) (11b) (20a) and(20b) in a real environment with uncertainties In this paperfor the stability analysis the Lyapunov second method isapplied for the helicopter with an underslung external loadsystem The analysis for the longitudinal motion is discussedfirst The system equations can be considered to have twomain parts that is known and unknown (or partly known)The known terms formed the nominal part of the systemmodelThe unknown or partly known part can be consideredas the uncertainty to the system The whole system is thenmodelled by a nominal part with the addition of uncertaintyIn fact the known elements in the subsystem (11a) arecharacterised by the prescribed triple (119891

1 1198661 119901) and it is

desired that the nominal part of the system is stableThe basic notations and concepts required for the analysis

are described first The state space is denoted by119883 fl R119899 andthe control space by119906 fl R119898 where 1 le 119898 le 119899TheEuclideaninner product (on 119883 or 119906 as appropriate) and induced normare denoted by ⟨sdot sdot⟩ and sdot respectively Let 119862(R119901R119902) and1198621(R119901R119902) denote the space of all continuous functions and

the space of continuous functions with continuous first-orderpartial derivatives respectively and let 119862infin(R119901R119902) denotethe space of functions whose partial derivatives of any orderexist and are continuous mapping R119901 rarr R119902 For a real-valued continuous scalar function 119909 rarr V(119909) defined onR119899 nabla V rarr nablaV isin R119899 denotes the gradient map The Liederivative of V along a vector field 119891 R119899 rarr R119899 is denoted by119871119891V R119899 rarr R which is defined by

(119871119891V) (119909) = ⟨nablaV (119909) 119891 (119909)⟩ (22)

The Lie bracket of vector fields 119891 119892 isin 119862infin(R119899R119899) is the

vector field [119891 119892] isin 119862infin(R119899R119899) defined by [119891 119892] = (119863119892)119891 minus

(119863119891)119892 where (119863119891)denotes the Jacobianmatrix of119891 and (119863119892)denotes the Jacobian matrix of 119892

In this paper nonlinear systemswith the following formatare considered

(119905) = 119891 (119909 (119905)) + 119866 (119909 (119905)) (119905) (23)

where 119909(119905) isin R119899 isin R119898 In general mathematical modelsof dynamical systems are usually imprecise due to modellingerrors and exogenous disturbances [12] Equation (23) can beconsidered as the nominal part of the system model and theuncertainty can be modelled as an additive perturbation tothe nominal systemmodel more specifically the structure ofthe system has the form

(119905) = 119891 (119909 (119905)) + 119866 (119909 (119905)) (119905) + 120599 (119909 (119905) 119906 (119905)) (24)

where 120599(119909(119905) 119906(119905))models the uncertainty in the systemSystem (24) is globally asymptotically stable to the zero

state if the system exhibits the following properties

(i) Existence and Continuation of Solutions For each 119909 isin R119899there exists a local solution119909 [0 119905

1) rarr R119899 (ie an absolutely

continuous function satisfying (24) almost everywhere (ae)and 119909(0) = 119909

0) and every such solution can be extended intoa solution on [0infin)

(ii) Boundedness of Solutions For each ℏ gt 0 there exists119903(ℏ) gt 0 such that 119909(119905) isin 119903(ℏ)119861

119899 for all 119905 ge 0 on every

solution 119909 [0infin) rarr R119899 with 1199090isin ℏ119861119899 where 119861

119899denote

the open unit ball centred at the origin in R119899

(iii) Stability of the State Origin For each 120575 gt 0 there exists119889(120575) gt 0 such that 119909(119905) isin 120575119861

119899for all 119905 ge 0 on every solution

119909 [0infin) rarr R119899 with 1199090isin 119889(120575)119861

119899

(iv) Global Attractivity of the State Origin For each ℏ gt 0 and120576 gt 0 there exists 119879(ℏ 120576) ge 0 such that 119909(119905) isin 119861

119899for all

119905 ge 119879(ℏ 120576) on every solution 119909 [0infin) rarr R119899 with 1199090isin ℏ119861119899

8 Journal of Control Science and Engineering

Consider a nonlinear system described by the ordinarydifferential equation as follows

(119905) = 119891 (119905 119909 (119905))

119909 (1199050) = 1199090

(25)

where 119891 R times 119883 rarr 119883 and 119891(119905 0) = 0 for all 119905To analyse the stability of (25) Lyapunovrsquos second stabilityanalysis method is applicable The Lyapunov approach is toshow that a candidate ldquoLyapunov functionrdquo is nonincreasingalong all solution to (25) by means that do not requireexplicit knowledge of solutions to (25) From this appropriateconclusion can be drawn regarding stability concepts relatingto solutions of the differential equation (25) An essentialpart of Lyapunovrsquos method is the determination of the timederivative of the candidate ldquoLyapunov functionrdquo along allsolution of the dynamical system

Consider a Lyapunov candidate (119905 119909) rarr V(119905 119909) Rtimes119883 rarr

R which satisfies the condition V isin 1198621(R times 119883) in which case

its time derivative along solutions to (25) is given by

V (119905 119909 (119905)) =120597V (119905 119909 (119905))

120597119905+ ⟨nablaV (119905 119909 (119905)) 119891 (119905 119909 (119905))⟩ (26)

for almost all 119905 isin RLet 119882(119909(119905)) denote a positive definite function If

V(119905 119909(119905)) satisfies

(i) V(119905 0) = 0 for all 119905 ge 0

(ii) 119882(119909(119905)) le V(119905 119909(119905)) for all 119909(119905) isin Φ 0 sub Φ sub 119883 andall 119905 ge 0

(iii) V(119905 119909(119905)) lt 0 in Φ

then V(119905 119909(119905)) is said to be a Lyapunov function in Φ IfV(119905 119909(119905)) in Φ then V(119905 119909(119905)) is said to be a weak Lyapunovfunction

A set Ε is said to be an invariant set with respect to thedynamical system = 119891(119909) if

119909 (0) isin Ε 997891997888rarr 119909 (119905) isin Ε forall119905 isin R+ (27)

In other words Ε is the set of points such that if a solution of = 119891(119909) belongs to Ε at some instant initialized points at119905 = 0 then it belongs to Ε for all future time

Now a setΕ sub 119909 is said to be a local invariantmanifold for(25) if for any 1199090 isin Ε 119909(119905) with 119909(0) = 119909

0 is in Ε for |119905| lt 119879

where 119879 gt 0 If 119879 = infin then Ε is said to be an invariantmanifold

Now considering the longitudinal motion (Figure 1) forthe helicopter with an underslung external load systemchoose a Lyapunov function candidate for the first subsystemV1as

V1(120579119871) =

1

2[12058911205792

1198711

+ 12058921205792

1198712

+ (1205893120579 minus 1205894119902)2+ 12058951199022] (28)

where 120589119894(119894 = 1 2 3 4 5) are design parameters to be

determined Then

V1(120579119871) = ⟨nablaV

1(120579119871) 1198911(120579119871) + 1198661(120579119871 (119905))

sdot [119901 (119909119867 (119905)) + 119902 (120579

119871 (119905) 119909119867 (119905))]

+ 119867(119905 120579119871 119909119867 (119905))⟩ = (119871

1198911

V1) (120579119871)

+ ⟨nablaV1(120579119871) 1198661(120579119871 (119905))

sdot [119901 (119909119867 (119905)) + 119902 (120579

119871 (119905) 119909119867 (119905))]

+ 119867(119905 120579119871 119909119867 (119905))⟩

(29)

From (28) nablaV1(120579119871) can be obtained as

nablaV1(120579119871)

= [12058911205791198711

12058921205791198712

1205893(1205893120579 minus 1205894119902) minus120589

4(1205893120579 minus 1205894119902) + 120589

5119902]119879

(30)

Substituting nablaV1(120579119871) and 119891

1(120579119871) into the derivative of

(1198711198911

V1)(120579119871) we have

(1198711198911

V1) (120579119871) = 12058911205791198711

1205791198712

minus 11989612057912058921205792

1198712

minus 1205892

119892

119897119909

sin 1205791198711

1205791198712

+119896119863119897119909

119872119871

1205892[sign (

119871) cos3 120579

1198711

+ sign (119871) sin3 120579

1198711

]

sdot 1205793

1198712

+ 1205893[1205893120579 minus 1205894119902] 119902 + (minus120589

31205894120579 + (120589

2

4+ 1205895) 119902)

sdot [11988321120579 + 119883

22119902]

(31)

Checking the term of minus sin (1205791198711

)1205791198712

it can be seen thatminus sin (120579

1198711

)1205791198712

le minus(2120587)1205791198711

1205791198712

when 1205791198711

and 1205791198712

are bothpositive or negative The situation of 120579

1198711

and 1205791198712

havingdifferent signs will help with the system stability Then

(1198711198911

V1) (120579119871) le 12058911205791198711

1205791198712

minus 11989612057912058921205792

1198712

minus 1205892

119892

119897119909

2

1205871205791198711

1205791198712

+119896119863119897119909

119872119871

sdot 1205892[sign (

119871) cos3 120579

1198711

+ sign (119871) sin3 120579

1198711

] 1205793

1198712

+ 1205893[1205893120579 minus 1205894119902] 119902 + (minus120589

31205894120579 + (120589

2

4+ 1205895) 119902)

sdot [11988321120579 + 119883

22119902]

(32)

If the design parameter 1205891is chosen as 120589

1= (2119892119897

119909120587)1205892

(1198711198911

V1)(120579119871) satisfies the following

(1198711198911

V1) (120579119871) le minus119896

12057912058921205792

1198712

+119896119863119897119909

119872119871

sdot 1205892[sign (

119871) cos3 120579

1198711

+ sign (119871) sin3 120579

1198711

] 1205793

1198712

+ 1205893[1205893120579 minus 1205894119902] 119902 + (minus120589

31205894120579 + (120589

2

4+ 1205895) 119902)

sdot [11988321120579 + 119883

22119902]

(33)

Journal of Control Science and Engineering 9

Further analysis on (33) will start from examining the firsttwo terms Rewrite these two terms So

minus 11989612057912058921205792

1198712

+119896119863119897119909

119872119871

1205892[sign (

119871) cos3 120579

1198711

+ sign (119871)

sdot sin3 1205791198711

] 1205793

1198712

= minus11989612057912058921205792

1198712

+119896119863119897119909

119872119871

12058921205791198712

[sign (119871)

sdot cos3 1205791198711

+ sign (119871) sin3 120579

1198711

] 1205792

1198712

le minus11989612057912058921205792

1198712

+119896119863119897119909

119872119871

1205892

10038161003816100381610038161003816max (120579

1198712

) (sign (119871) cos3 120579

1198711

+ sign (119871) sin3 120579

1198711

)100381610038161003816100381610038161205792

1198712

= minus12058921205792

1198712

119896120579

minus119896119863119897119909

119872119871

10038161003816100381610038161003816max (120579

1198712

)

sdot (sign (119871) cos3 120579

1198711

+ sign (119871) sin3 120579

1198711

)10038161003816100381610038161003816

(34)

If the hinge friction is big enough to satisfy the following

119896120579gt

119896119863119897119909

119872119871

10038161003816100381610038161003816max (120579

1198712

)

sdot (sign (119871) cos3 120579

1198711

+ sign (119871) sin3 120579

1198711

)10038161003816100381610038161003816

(35)

then

120582119871= 119896120579minus119896119863119897119909

119872119871

10038161003816100381610038161003816max (120579

1198712

)

sdot (sign (119871) cos3 120579

1198711

+ sign (119871) sin3 120579

1198711

)10038161003816100381610038161003816 ge 0

minus 11989612057912058921205792

1198712

+119896119863119897119909

119872119871

1205892[sign (

119871) cos3 120579

1198711

+ sign (119871)

sdot sin3 1205791198711

] 1205793

1198712

le minus12058921205792

1198712

120582119871

(36)

Now the following analysis has been conducted for (33) andthe load movement for the longitudinal motion is describedby

119871= 0minus 1198971199091205791198712

cos 1205791198711

119871= 0minus 1198971199091205791198712

sin 1205791198711

(37)

Around the hover condition1198830= 0 and 119885

0= 0 the signs for

119871and

119871are opposite to the one of 120579

1198712

provided that 1205791198711

ispositive

Considering the term (sign (119871) cos3 120579

1198711

+

sign (119871) sin3 120579

1198711

)1205793

1198712

for all possible combinations ofthe signs of 120579

1198712

and 1205791198711

the analysis follows below

(i) If 1205791198711

gt 0 and 1205791198712

gt 0 then sign (119871) = minus1

sign (119871) = minus1 cos3 120579

1198711

gt 0 and sin3 1205791198711

gt 0 There-fore (sign (

119871) cos3 120579

1198711

+ sign (119871) sin3 120579

1198711

)1205793

1198712

lt 0

(ii) If 1205791198711

lt 0 and 1205791198712

gt 0 then sign (119871) = minus1

sign (119871) = +1 cos3 120579

1198711

gt 0 and sin31205791198711

lt 0 So(sign (

119871) cos3 120579

1198711

+ sign (119871) sin3 120579

1198711

)1205793

1198712

lt 0

(iii) If 1205791198711

gt 0 and 1205791198712

lt 0 then sign (119871) = +1

and sign (119871) = +1 cos3 120579

1198711

gt 0 and sin3 1205791198711

gt

0 So the following is true (sign (119871) cos3 120579

1198711

+

sign (119871) sin3 120579

1198711

)1205793

1198712

lt 0

(iv) If 1205791198711

lt 0 and 1205791198712

lt 0 then sign (119871) = +1 and

sign (119871) = minus1 cos3 120579

1198711

gt 0 and sin3 1205791198711

lt 0 So(sign (

119871) cos3 120579

1198711

+ sign (119871) sin3 120579

1198711

)1205793

1198712

lt 0

Therefore

minus 11989612057912058921205792

1198712

+119896119863119897119909

119872119871

sdot 1205892[sign (

119871) cos3 120579

1198711

+ sign (119871) sin3 120579

1198711

] 1205793

1198712

le minus11989612057912058921205792

1198712

(38)

Now examining the rest of the terms of (1198711198911

V1)(120579119871) then

1205893[1205893120579 minus 1205894119902] 119902

+ (minus12058931205894120579 + (120589

2

4+ 1205895) 119902) [119883

21120579 + 119883

22119902] = 120589

2

3120579119902

minus 120589312058941199022minus 12058931205894119883211205792minus 1205893120589411988322120579119902

+ (1205892

4+ 1205895)11988321120579119902 + (120589

2

4+ 1205895)119883221199022

= minus [12058931205894minus (1205892

4+ 1205895)11988321] 1199022minus 12058931205894119883211205792

+ [1205892

3minus 1205893120589411988322+ (1205892

4+ 1205895)11988321] 120579119902

(39)

For the system 120579 and 1205791198711

always have different signs 120579 and 119902

always have different signs and 119902 and 1205791198711

have the same signsSo

[1205892

3minus 1205893120589411988322+ (1205892

4+ 1205895)11988321] 120579119902 le 0

if [1205892

3minus 1205893120589411988322+ (1205892

4+ 1205895)11988321] ge 0

(40)

Therefore

1205893[1205893120579 minus 1205894119902] 119902

+ (minus12058931205894120579 + (120589

2

4+ 1205895) 119902) [119883

21120579 + 119883

22119902]

le minus [12058931205894minus (1205892

4+ 1205895)11988321] 1199022minus 12058931205894119883211205792

(41)

Choose the design parameters to satisfy

[12058931205894minus (1205892

4+ 1205895)11988321] gt 0 (42)

If11988321

gt 0 then

minus [12058931205894minus (1205892

4+ 1205895)11988321] 1199022minus 12058931205894119883211205792lt 0 (43)

10 Journal of Control Science and Engineering

If11988321

lt 0 then

minus [12058931205894minus (1205892

4+ 1205895)11988321] 1199022minus 12058931205894119883211205792lt 0 (44)

in the region ofminus[12058931205894minus(1205892

4+1205895)11988321]1199022lt 12058931205894119883211205792 Since 120579 is

caused by the loadmotion here the value of 120579 should bemuchsmaller than 119902 The inequality (44) is easy to be satisfied

In summary of the above analysis and by defining

Ψ1(120579119871) = 11989612057912058921205792

1198712

+ [12058931205894minus (1205892

4+ 1205895)11988321] 1199022

+ 12058931205894119883211205792

Ψ2(120579119871) = [120589

31205894minus (1205892

4+ 1205895)11988321] 1199022+ 12058931205894119883211205792

(45)

the following lemma can be derived

Lemma 1 Defining a Lyapunov function (28) and choosingthe design parameters to satisfy 120589

1= (2119892119897

119909120587)1205892 [12058931205894minus (1205892

4+

1205895)11988321] gt 0 and [1205892

3minus1205893120589411988322+(1205892

4+1205895)11988321] ge 0 then within

the region specified by minus[12058931205894minus (1205892

4+ 1205895)11988321]1199022lt 12058931205894119883211205792

(i) V1(0) = 0 and V

1(120579119871) gt 0 forall120579

119871= 0

(ii) V1(120579119871) rarr infin as 120579

119871 rarr infin

(iii) (1198711198911

V1)(120579119871) le minusΨ

1(120579119871) forall120579119871 around the hover condi-

tion or (1198711198911

V1)(120579119871) le minusΨ

2(120579119871) forall120579119871 if (35) holds

Both functions Ψ1and Ψ

2are nonnegative

Recall the control term 119901(119909119867(119905)) in the first subsystem

of (11a) and (11b) it can be seen that [(119863119901)(119909119867)]minus1 exists

for all 119909119867(119905) The unknown vector fields 119902(120579

119871(119905) 119909119867) and

119867(119905 120579119871(119905) 119909119867) model the uncertainties imposed onto the

system Since 119902(120579119871(119905) 119909119867) is directly mapped into the ldquocon-

trolrdquo space of 119909119867(119905) it can be considered as a matched

uncertainty119867(119905 120579119871(119905) 119909119867) is unknown and it does belong to

the control space of 119909119867(119905) so it represents the mismatched

uncertainty in the system [12]Generally the range of the (longitudinal) load suspension

angle is within minus1205872 lt 120579119871lt 1205872 The helicopter velocities

and load suspension angle havemaximumoperational valuestherefore the uncertainties in the system are bounded Withthe Lyapunov function defined in (28)

(1198711198921

V1) (120579119871) = ⟨nablaV

1(120579119871) 1198921⟩ =

119896119863

119872119871119897119909

12058921205791198712

(46)

In 1198661(120579119871) the columns 119892

1and 119892

2are the same therefore

(1198711198922

V1) (120579119871) =

119896119863

119872119871119897119909

12058921205791198712

(47)

By considering the maximum values of cos (sdot) and sin (sdot)functions the bounding values relating to the uncertainty119902(120579119871(119905) 119909119867) can be estimated as follows

10038171003817100381710038171003817119902 (120579119871 119909119867)10038171003817100381710038171003817le

10038161003816100381610038161003816100381610038161003816

1

119897119909

( + )

+1

119872119871

(minus119896119863(2 +

120581

119897119909

) (119906 + 119908) +119896120579

119897119909

)1205791198712

10038161003816100381610038161003816100381610038161003816

le

10038161003816100381610038161003816100381610038161003816

119896120579

119897119909

1205791198712

10038161003816100381610038161003816100381610038161003816+

10038161003816100381610038161003816100381610038161003816

119896119863

119872119871

(2 +120581

119897119909

) (119906 + 119908)

10038161003816100381610038161003816100381610038161003816+

10038161003816100381610038161003816100381610038161003816

1

119897119909

( + )

10038161003816100381610038161003816100381610038161003816

le119896120579

119897119909

100381710038171003817100381710038171205791198712

10038171003817100381710038171003817+

119896119863

119872119871

(2 +120581

119897119909

)1003817100381710038171003817119901 (119909119867)1003817100381710038171003817 + 1205722 (119905)

le 1205831

10038161003816100381610038161003816(1198711198922

V1) (120579119871)10038161003816100381610038161003816+ 1205721

1003817100381710038171003817119901 (119909119867)1003817100381710038171003817 + 1205722 (119905)

(48)

where 1205831

= 119872119871119896120579119896119863 1205721

= (119896119863119872119871)(2 + 120581119897

119909) (120581 =

max [1205811 1205812]) and 120572

2(119905) is defined by 120572

2(119905) = max |(1119897

119909)( +

)| gt 0For the mismatched uncertainty 119867(119905 120579

119871(119905) 119909119867) the fol-

lowing analysis is conducted to obtain its bounding functionThe mismatched uncertainty can be rewritten as

119867(119905 120579119871 (119905) 119909119867) =

[[[[[

[

0

0

(11988611119906 + 11988612119908)

(11988323119906 + 119883

24119908)

]]]]]

]

=

[[[[[

[

0 0

0 0

11988611

11988612

11988323

11988324

]]]]]

]

[119906

119908]

(49)

Let 119860 = [1198861111988612

1198832311988324] then

10038171003817100381710038171003817119867 (119905 120579

119871 119909119867)10038171003817100381710038171003817le1003817100381710038171003817100381711986010038171003817100381710038171003817

1003817100381710038171003817119909119867 (119905)1003817100381710038171003817 (50)

Define a positive function as

1006704120579 (120579119871) = radic(1205792

1198711

+ 12057921198712

+ 1205792 + 1199022+ 120576) (51)

where 120576 is a very small positive constant Choosing1205731

= (1198721198711198971199091198961198631205892)119860 then the mismatched uncertainty

119867(119905 120579119871(119905) 119909119867) is bounded by the following inequality

10038171003817100381710038171003817119867 (119905 120579

119871 119909119867)10038171003817100381710038171003817

le 1006704120579minus1

(120579119871) 1205731

2

sum

119894=1

(10038161003816100381610038161003816(119871119892119894

V1) (120579119871)10038161003816100381610038161003816)1003816100381610038161003816119901119894 (119909119867)

1003816100381610038161003816

(52)

where 119892119894denotes the 119894th column of the matrix function

1198661(120579119871) and 119901

119894is the 119894th component of 119901(119909

119867) respectively As

1205731has a design parameter 120589

2involved it is easy to have 120573

1to

lead (52) to be true For the matched uncertainty we have

10038171003817100381710038171003817119902 (120579119871 119909119867)10038171003817100381710038171003817le

119896120579

119897119909

100381610038161003816100381610038161205791198712

10038161003816100381610038161003816+ 1205721

1003817100381710038171003817119901 (119909119867)1003817100381710038171003817 + 1205722 (119905) (53)

Journal of Control Science and Engineering 11

Following the above analysis the following lemma can bederived

Lemma 2 The uncertainties (52) and (53) are bounded andsatisfy if 120572

1 1205731ge 0 and 120572

1+ 1205731lt 1 for 120572

1= (119896119863119872119871)(2 +

120581119897119909) and 120573

1= (1198721198711198971199091198961198631205892)119860

4 Numerical Results

Using a simulation software the linearized model for a UH-60 helicopter is obtained with the following state and inputmatrices

119860 =

[[[[[[[[[[[[[[[[[[[

[

minus00000 00003 minus00000 minus00000 00000 minus00000 10000 minus00023 00503

minus00003 minus00000 minus00000 minus00000 minus00000 minus00000 minus00000 09989 00463

minus00001 00000 minus00000 minus00000 00000 00000 minus00000 minus00464 10002

minus19386 minus28593 minus00030 minus00195 minus00126 00168 minus12000 57076 02683

328694 minus03640 00027 00126 minus00311 minus00013 minus41348 minus22350 03713

03138 minus14034 00004 00150 00034 minus03340 02866 06307 23410

minus09144 minus04783 00017 00226 minus00246 00007 minus65893 minus21301 minus00598

00631 minus06744 00008 00029 00046 00025 02059 minus14255 minus01636

minus07549 minus00119 minus00002 00009 00028 00006 minus00749 minus01032 minus02214

]]]]]]]]]]]]]]]]]]]

]

119861 =

[[[[[[[[[[[[[[[[[[[

[

minus00000 00000 minus00000 00000

00000 00000 minus00000 minus00000

minus00000 00000 minus00000 minus00000

minus22190 05734 03170 minus00019

35381 minus00453 minus00963 02007

minus15629 00237 minus60740 minus01014

19943 minus00714 minus00541 00731

00446 minus01097 00553 minus00281

minus08197 minus00082 01573 minus00635

]]]]]]]]]]]]]]]]]]]

]

(54)

The desired condition 1205721+ 1205731lt 1 can be verified by consid-

ering a typical case that the UH-60 helicopter carries a loadweight 1000 lb by a 15 ft slung length and let 119896

119863= 75 lbft

For the UH-60 helicopter model with reference to thegeneral mathematical model presented in Section 2 we have1198831205791119904

= 05734 1198831205790

= 03170 1198851205791119904

= 00237 1198851205790

= minus607401198721205791119904

= minus01097 and 1198721205790

= 00553 So the followingparameters can be calculated to have the values 119886

11= minus02

and 11988612

= minus002 For 119883119906= minus00195 119885

119906= 00150 119872

119906=

00029 119872119902= minus14255 (119883

119902minus 119908119890) = 57076 and (119885

119902+ 119906119890) =

06307 we have 11988323

= 006 And also for 119883119908

= 00168119885119908

= minus03340 and 119872119908

= 00025 we have 11988324

= 001Based on all the above parameters 120590max(119860) = 00041 Inthis case let 120581 = 001 120572

1= (119896119863119872119871)(2 + 120581119897

119909) asymp 015

and 1205731= (119872

119871119897119909119896119863)120590max(119860) = 082 Therefore 120572

1+ 1205731=

015 + 082 = 097 which satisfies 1205721+ 1205731

lt 1 Anotherexample considers that the UH-60 helicopter carries a loadweight 500 lb by a 15 ft slung length and let 119896

119863= 50 lbft then

1205721= 02 and 120573

1= 062 therefore (120572

1+ 1205731) = 082 Therefore

this assumption 1205721+ 1205731

lt 1 is realistic for the systemdiscussed in here The numerical values vary with respectto the slung load configuration It may represent the system

dynamic variations To investigate the stability characteristicsof dynamic variations two series of simulation studies werecarried out

Firstly the slung length was kept constant at 15 ft andthe load varied between the limits 500ndash2000 lb with 119896

119863=

75 lbft producing the system poles listed in Table 1 Thecorresponding root loci are sketched in Figures 3 and 4Figure 4 clearly shows that the positions of the system polescrossed the imaginary axis as the load weight increased

Secondly load weight was kept constant and the slunglength varied between the limit of 10ndash20 ft Table 2 sum-marises the results of variation of the system poles and thecorresponding root loci are illustrated in Figures 5 and 6

The pole locus indicates that the system poles have thevariation trends of moving further towards right direction onthe 119904-plane with increase of load weight and slung lengthsFrom the simulation results it is clearly shown that thesystem becomes less stable when the load andor the slunglength increases The simulation results are compared withthe stability analysis presented above

Now consider the case of fixed slung length of 15 ft withdifferent load weights In the cases of a UH-60 helicopter that

12 Journal of Control Science and Engineering

Table 1 System poles locations for a fixed slung length (119897 = 15ft) but with different load weights

Helicopter 500 lb 1000 lb 1500 lb 2000 lbminus10919 minus11159 minus07848 minus06221 minus09369minus63938 minus64301 minus92203 minus79848 minus75613minus00032 minus00048 minus00034 minus00052 minus00052minus00977 minus00818 minus00888 minus00972 minus00694minus03045 minus03036 minus03394 minus03318 minus03333minus03159 plusmn 04363119894 minus03199 plusmn 04628119894 minus02295 plusmn 05050119894 minus02366 plusmn 05974119894 minus01834 plusmn 04968119894minus00489 plusmn 03898119894 minus00211 plusmn 03868119894 00074 plusmn 03445119894 00119 plusmn 04406119894 00264 plusmn 03199119894mdash minus00316 plusmn 15043119894 minus01020 plusmn 17380119894 minus00368 plusmn 18753119894 minus01160 plusmn 20003119894mdash minus00736 plusmn 15019119894 minus07662 plusmn 11680119894 minus07349 plusmn 19064119894 minus12367 plusmn 08995119894

Table 2 System poles for a fixed load (1000 lb) but with different slung lengths

Helicopter 10 ft 15 ft 20 ftminus10919 minus10560 minus07848 minus08687minus63938 minus99252 minus92203 minus85565minus00032 minus00061 minus00034 minus00114minus00977 minus02012 minus00888 minus00953minus03045 minus02606 minus03394 minus03479minus03159 plusmn 04363119894 minus01809 plusmn 06219119894 minus02295 plusmn 05050119894 minus01636 plusmn 06092119894minus00489 plusmn 03898119894 00643 plusmn 04577119894 00074 plusmn 03445119894 00799 plusmn 04594119894mdash minus01055 plusmn 19883119894 minus01020 plusmn 17380119894 minus01247 plusmn 14470119894mdash minus08847 plusmn 12464119894 minus07662 plusmn 11680119894 minus06352 plusmn 10384119894

Re(poles)

Im(poles)

minus10 minus5 0 5minus08minus06minus04minus02

002040608

Figure 3 System root-locus diagram showing pole locus forconstant slung length (15 ft) as load weight is increased

minus04 minus035 minus03 minus025 minus02 minus015 minus01 minus005 0 005 01

Re(poles)

Im(poles)

minus08minus06minus04minus02

002040608

Figure 4 Enlarged view of Figure 3 showing poles locus near toRe(poles) = 0 axis

minus10 minus8 minus6 minus4 minus2 0 2

Re(poles)

Im(poles)

minus08minus06minus04minus02

002040608

Figure 5 System root-locus showing pole locus for constant loadweight (1000 lb) as slung length is increased

minus04 minus035 minus03 minus025 minus02 minus015 minus01 minus005 0 005 01

Re(poles)

Im(poles)

minus08minus06minus04minus02

002040608

Figure 6 Enlarged view of Figure 5 showing poles locus near toRe(poles) = 0 axis

Journal of Control Science and Engineering 13

carries a loadweight 500 lb and 1000 lb then1205721= 03 and 041

with 1205731= 041 and 082 respectively therefore 120572

1+ 1205731lt 1

thus the system is stable But when the load weight is 1500 lband 2000 lb then 120572

1= 01 and 0075 and also 120573

1= 123 and

164 respectively therefore 1205721+ 1205731gt 1 thus the system is

unstable For the case of the constant load weight (1000 lb) asslung length is increased then for a 10 ft and 15 ft slung length1205721= 015 and 120573

1= 055 and 082 respectively therefore

1205721+ 1205731lt 1 thus the system is stable But when the slung

length increased to 20 ft then 1205721

= 015 and 1205731

= 109therefore (120572

1+ 1205731) = 124 that is 120572

1+ 1205731gt 1 thus the

system is unstable However the system can be marginallystable if the UH-60 helicopter carries a load weight 725 lb bya 20 ft slung length then 120572

1= 02 and 120573

1= 08 therefore

(1205721+ 1205731) = 1 For this set of analysis 119896

119863is assumed to

be 119896119863

= 75 lbft However 119896119863can vary depending on the

weather condition and an example of 119896119863= 50 lbft is shown

above for illustrative purpose This analysis suggests that themaximum value of 20 ft slung length may be used for slungload operation However the load weight limit can vary withrespect to the capacity of the load carrying hooks helicopterweight environmental conditions and so forth

The examples presented here show that the simulationresults are correlated with the stability analysis Thus desiredcondition for the stability is met and it can locate the load atthe specified position or its neighbourhood

5 Conclusions

In this paper stability analysis of a helicopter with an under-slung load system is investigatedThe investigation identifiedthe conditions for the stabilisation of the system and thepositioning of the underslung load at hover condition Sta-bility analysis and numerical results proved that the desiredcondition for the stability is met then it is possible to locatethe load at the specified position or its neighbourhood Anillustration example is also given in the paper The methodresults in a guaranteed load positioning accuracy whichdepends on the design parameters and the method can beapplicable to any individual helicopter with external loadsystem

Nomenclature

119871119872119873 Overall helicopter rolling pitching andyawing moments

119901 119902 119903 Helicopter roll pitch and yaw rates aboutbody reference axes

119883119884 119885 Overall helicopter force components120601 120579 120595 Roll pitch and yaw angles119906 V 119908 Helicopter velocity components at centre

of gravity1205790 Main rotor collective

1205791119904 Longitudinal cyclic commands

1205791119888 Lateral cyclic commands

1205790119879 Tail rotor collective

1198830 1198840 1198850 Location of the suspension point withrespect to earth referenced 119909 119910 and 119911

directions

0 0 0 The helicopter velocity in the 119909 119910 and 119911

directions0 0 0 The helicopter acceleration in the 119909 119910 and119911 directions

120579119871 Load suspension angle in the119883-119885 plane

with respect to 119885-axis120601119871 Load suspension angle in the 119884-119885 plane

with respect to 119885-axis119872119871 Mass of the suspended load

119892 Acceleration due to gravity119897 Slung length119896119863 Aerodynamic drag force constant

(119896119863= (12)120588119878119862

119863)

120588 Air density119878 The load area presented to the airflow119862119863 Drag coefficient for the load

Competing Interests

The author declares that there are no competing interestsregarding the publication of this paper

References

[1] K Thanapalan and F Zhang ldquoModelling and simulation studyof a helicopter with an external slung load systemrdquo I-ManagersJournal on Instrumentation amp Control Engineering vol 1 no 1pp 9ndash17 2013

[2] M Bisgaard J D Bendtsen and A La Cour-Harbo ldquoModelingof generic slung load systemrdquo Journal of Guidance Control andDynamics vol 32 no 2 pp 573ndash585 2009

[3] Handling Qualities Requirements for Military Rotorcraft ADS-33-D US Army Aviation and Troop Command St Louis MissUSA 1996

[4] K Thanapalan and T M Wong ldquoModeing of a helicopter withan under-slung load systemrdquo in Proceedings of the 29th ChineseControl Conference (CCC rsquo10) pp 1451ndash1456 Beijing China July2010

[5] B Feng ldquoRobust control for lateral and longitudinal channels ofsmall-scale unmanned helicoptersrdquo Journal of Control Scienceand Engineering vol 2015 Article ID 483096 8 pages 2015

[6] J J Gribble ldquoLinear quadratic Gaussianloop transfer recoverydesign for a helicopter in low-speed flightrdquo Journal of GuidanceControl and Dynamics vol 16 no 4 pp 754ndash761 1993

[7] M A Manness J J Gribble and D J M Smith ldquoMultivariablemethods for helicopter flight control law designrdquo in Proceedingsof the 16th Annual Society European Rotorcraft Forum PaperNoIII52 Glasgow UK September 1990

[8] W L Garrard E Low and S Prouty ldquoDesign of attitude andrate command systems for helicopters using eigenstructureassignmentrdquo Journal of Guidance Control and Dynamics vol12 no 6 pp 783ndash791 1989

[9] Y B Shtessel and I A Shkolnikov ldquoAeronautical and spacevehicle control in dynamic sliding manifoldsrdquo InternationalJournal of Control vol 76 no 9-10 pp 1000ndash1017 2003

[10] D FThompson J S Pruyn andA Shukla ldquoFeedback design forrobust tracking and robust stiffness in flight control actuatorsusing a modified QFT techniquerdquo International Journal ofControl vol 72 no 16 pp 1480ndash1497 1999

14 Journal of Control Science and Engineering

[11] K Thanapalan ldquoModelling of a helicopter systemrdquo in Proceed-ings of the 1st Virtual Control Conference (VCC rsquo10) pp 45ndash52Aalborg University 2010

[12] D P Goodall and J Wang ldquoStabilization of a class of uncertainnonlinear affine systems subject to control constraintsrdquo Interna-tional Journal of Robust and Nonlinear Control vol 11 no 9 pp797ndash818 2001

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Journal of Control Science and Engineering 3

X

Z

120579Llx

(XL YL ZL)

(X0 Y0 Z0)

Figure 1 Coordinate system for the longitudinal motion in the119883-119885plane

in Taylorrsquos expansion form Then the linearized equations ofmotion about a general trim condition can be written as inthe state space form = 119860119909 + 119861119906 and the system matrix 119860

and control matrix 119861 are derived from the partial derivativesof the nonlinear function 119865 that is

= 119865 (119909 119906 119905)

119860 = (120597119865

120597119909)

119909=119909119890

119861 = (120597119865

120597119906)

119909=119909119890

(4)

Now the longitudinal rotational motion is described by thepitch angle 120579 and pitch rate 119902 together with the translationmotion components 119906 119908 so the equation of longitudinalmotion can be written as follows

[[[[[

[

]]]]]

]

=

[[[[[[

[

0 1 0 0

0 119872119902

119872119906

119872119908

minus119892 cos 120579119890(119883119902minus 119908119890) 119883119906

119883119908

minus119892 sin 120579119890

(119885119902minus 119906119890) 119885119906

119885119908

]]]]]]

]

[[[[[

[

120579

119902

119906

119908

]]]]]

]

+

[[[[[

[

0 0

1198721205791119904

1198721205790

1198831205791119904

1198831205790

1198851205791119904

1198851205790

]]]]]

]

[1205791119904

1205790

]

(5)

By applying a linear transformation119879 such that119909119867(119905) = 119879119911(119905)

and 119879 is defined by

119879 =

[[[[[

[

1 0 0 0

0 1 11988611

11988612

0 0 1 0

0 0 0 1

]]]]]

]

(6)

where

11988611

= minus1198851205791119904

1198721205790

minus 1198851205790

1198721205791119904

1198831205791119904

1198851205790

minus 1198851205791119904

1198831205790

11988612

= minus1198831205790

1198721205791119904

minus 1198831205791119904

1198721205790

1198831205791119904

1198851205790

minus 1198851205791119904

1198831205790

(7)

and letting 119902 = (119902 minus 11988611119906 minus 11988612119908) then the system equation

(5) is transformed into the following form

[[[[[[

[

]]]]]]

]

=

[[[[[

[

11988311

11988312

11988313

11988314

11988321

11988322

11988323

11988324

11988331

11988332

11988333

11988334

11988341

11988342

11988343

11988344

]]]]]

]

[[[[[

[

120579

119902

119906

119908

]]]]]

]

+

[[[[[

[

0 0

11986121

11986122

1198831205791119904

1198831205790

1198851205791119904

1198851205790

]]]]]

]

[1205791119904

1205790

]

(8)

where

11988311

= 0

11988312

= 1

11988313

= 11988611

11988314

= 11988612

11988321

= (11988611119892 cos 120579

119890+ 11988612119892 sin 120579

119890)

11988322

= (119872119902minus 11988611(119883119902minus 119908119890) minus 11988612(119885119902minus 119906119890))

11988323

= (119872119902minus 11988611(119883119902minus 119908119890) minus 11988612(119885119902+ 119906119890)) 11988611

+ (119872119906minus 11988611119883119906minus 11988612119885119906)

11988324

= (119872119902minus 11988611(119883119902minus 119908119890) minus 11988612(119885119902+ 119906119890)) 11988612

+ (119872119908minus 11988611119883119908minus 11988612119885119908)

11988331

= minus119892 cos 120579119890

11988332

= (119883119902minus 119908119890)

11988333

= (119883119902minus 119908119890) 11988611+ 119883119906

11988334

= (119883119902minus 119908119890) 11988611+ 119883119908

11988341

= (minus119892 sin 120579119890)

11988342

= (119885119902+ 119906119890)

11988343

= (119885119902+ 119906119890) 11988611+ 119885119906

11988344

= (119885119902+ 119906119890) 11988612+ 119885119908

4 Journal of Control Science and Engineering

11986121

= (1198721205791119904

minus 119886111198831205791119904

minus 119886121198851205791119904

)

11986122

= (1198721205790

minus 119886111198831205790

minus 119886111198851205790

)

(9)

Using (8) the system model can be rearranged to include thevariables 120579 and 119902 into the loadmodel For the stability analysispurpose an extra term is introduced into the system modelwhich is zero with the expression

1205811119896119863sign (

119871) cos 120579

1198711

119906

119872119871119897119909

+1205812119896119863sign (

119871) cos 120579

1198711

119908

119872119871119897119909

minus1205811119896119863sign (

119871) cos 120579

1198711

119906

119872119871119897119909

minus1205812119896119863sign (

119871) cos 120579

1198711

119908

119872119871119897119909

(10)

where 120581119894gt 0 (119894 = 1 2) are small positive constants

With this arrangement for the longitudinal motion of thehelicopter with an underslung load combined system modelcan be written as follows

120579119871 (119905)

= 1198911(120579119871 (119905))

+ 1198661(120579119871 (119905)) [119901 (119909

119867 (119905)) + 119902 (120579119871 (119905) 119909119867 (119905))]

+ 119867(119905 120579119871 119909119867 (119905))

(11a)

119867 (119905) = 119891

2(120579119871 (119905) 119909119867 (119905)) + 119866

2 (119905) (11b)

where

120579119871 (119905) = [120579119871

1

1205791198712

120579 119902]119879

119909119867= [119906 119908]

119879

(119905) = [1205791119904 1205790]119879

119901 (119909119867) = [119906

2+ 1205811119906 1199082+ 1205812119908]119879

1198911(120579119871 (119905)) =

[[[[[[[

[

1205791198712

minus119892

119897119909

sin 1205791198711

+119896119863119897119909

119872119871

(sign (119871) cos3 120579

1198711

+ sign (119871) sin3 120579

1198711

) 1205792

1198712

minus 1198961205791205791198712

119902

(11988321120579 + 119883

22119902)

]]]]]]]

]

1198661(120579119871 (119905)) =

[[[[[[[[[

[

0 0

119896119863sign (

119871) cos 120579

1198711

119872119871119897119909

119896119863sign (

119871) sin 120579

1198711

119872119871119897119909

0 0

0 0

]]]]]]]]]

]

119902 (120579119871 (119905) 119909119867 (119905))

=[[

[

0

cos 1205791198711

119897119909

+sin 1205791198711

119897119909

minus2119896119863

119872119871

(sign (119871) cos2 120579

1198711

119906 + sign (119871) sin2 120579

1198711

119908) 1205791198712

minus1205811119896119863sign (

119871) cos 120579

1198711

119906

119872119871119897119909

minus1205812119896119863sign (

119871) cos 120579

1198711

119908

119872119871119897119909

]]

]

119867(119905 120579119871 (119905) 119909119867 (119905)) =

[[[[[

[

0

0

(11988611119906 + 11988612119908)

11988323119906 + 119883

24119908

]]]]]

]

1198912(120579119871 (119905) 119909119867 (119905)) = [

(11988331120579 + 119883

32119902 + 119883

33119906 + 119883

34119908)

(11988341120579 + 119883

42119902 + 119883

43119906 + 119883

44119908)

]

1198662= [

1198831205791119904

1198831205790

1198851205791119904

1198851205790

]

(12)

Journal of Control Science and Engineering 5

Y

Z

ly

(XL YL ZL)

(X0 Y0 Z0)

120593L

Figure 2 Coordinate system for the lateralmotion in the119884-119885 plane

It is assumed that the longitudinal motion is primarilycontrolled by longitudinal cyclic commands (120579

1119904) and main

rotor collective 1205790

For the case of the lateral motion in the 119884-119885 plane withthe coordinate system described in Figure 2 the load modelis

119871119910119911

= minus119892

119897119910

sin120593119871119910119911

+

cos120593119871119910119911

119897119910

0+

sin120593119871119910119911

119897119910

0

+

119896119863sign (

119871) cos120593

119871119910119911

119872119871119897119910

2

0

+

119896119863sign (

119871) sin120593

119871119910119911

119872119871119897119910

2

0

minus2119896119863

119872119871

(sign (119871) cos2 120593

119871119910119911

0

+ sign (119871) sin2 120593

119871119910119911

0) 119871minus 119896120593119871

+119896119863119897119910

119872119871

[sign (119871) cos3 120593

119871119910119911

+ sign (119871) sin3 120593

119871119910119911

]

sdot 2

119871119910119911

(13)

Define 119871= [120593119871

119871]119879

= [1205931198711

1205931198712]119879 then the model can be

written as follows

1198711

= 1205931198712

(14a)

1198712

=minus119892

119897119910

sin1205931198711

+119896119863119897119910

119872119871

[sign (119871) cos3 120593

1198711

+ sign (119871) sin3 120593

1198711

] 1205932

1198712

minus 1198961205931205931198712

+ (119896119863sign (

119871) cos120593

1198711

119872119871119897119910

) 2

0

+ (119896119863sign (

119871) sin120593

1198711

119872119871119897119910

) 2

0+cos1205931198711

119897119910

0

+sin1205931198711

119897119910

0minus2119896119863

119872119871

(sign (119871) cos2 120593

1198711

0

+ sign (119871) sin2 120593

1198711

0) 1205931198712

(14b)

Now for the helicopter model the lateral rotational motion isdescribed by the roll angle 120601 and roll rate 119901 together with thetranslationmotion components V 119908 so the equation of lateralmotion can be written as follows

[[[[[

[

V

]]]]]

]

=

[[[[[[

[

0 1 0 0

0 119871119901

119871V 119871119908

119892 (119884119901minus 119908119890) 119884V 119884

119908

0 (119885119901minus V119890) 119885V 119885

119908

]]]]]]

]

[[[[[

[

120601

119901

V

119908

]]]]]

]

+

[[[[[

[

0 0

1198711205791119888

1198711205790119879

1198841205791119888

1198841205790119879

1198851205791119888

1198851205790119879

]]]]]

]

[1205791119888

1205790119879

]

(15)

now using a linear transformation 1198791such that 119909

119867(119905) =

1198791119911(119905) and 119879

1is defined by

1198791=

[[[[[

[

1 0 0 0

0 1 11988711

11988712

0 0 1 0

0 0 0 1

]]]]]

]

(16)

where

11988711

= minus1198851205791119888

1198711205790119879

minus 1198851205790119879

1198711205791119888

1198841205791119888

1198851205790119879

minus 1198851205791119888

1198841205790119879

11988712

= minus1198841205790119879

1198711205791119888

minus 1198841205791119888

1198711205790119879

1198841205791119888

1198851205790119879

minus 1198851205791119888

1198841205790119879

(17)

6 Journal of Control Science and Engineering

Let 119901 = (119901 minus 11988711V minus 11988712119908) then (15) can be written in the

following form

[[[[[[

[

V

]]]]]]

]

=

[[[[[

[

11988411

11988412

11988413

11988414

11988421

11988422

11988423

11988424

11988431

11988432

11988433

11988434

11988441

11988442

11988443

11988444

]]]]]

]

[[[[[

[

120601

119901

V

119908

]]]]]

]

+

[[[[[

[

0 0

11987321

11987322

1198841205791119888

1198841205790119879

1198851205791119888

1198851205790119879

]]]]]

]

[1205791119888

1205790119879

]

(18)

where

11988411

= 0

11988412

= 1

11988413

= 11988711

11988414

= 11988712

11988421

= minus (11988711119892)

11988422

= (119871119901minus 11988711(119884119901+ 119908119890) minus 11988712(119885119901minus V119890))

11988423

= (119871119901minus 11988711(119884119901+ 119908119890) minus 11988712(119885119901minus V119890)) 11988711

+ (119871119901minus 11988711119884V minus 119887

12119885V)

11988424

= (119871119901minus 11988711(119884119901+ 119908119890) minus 11988712(119885119901minus V119890)) 11988712

+ (119871119901minus 11988711119884119908minus 11988712119885119908)

11988431

= 119892

11988432

= (119884119901+ 119908119890)

11988433

= ((119884119901+ 119908119890) 11988711+ 119884V)

11988434

= ((119884119901+ 119908119890) 11988712+ 119884119908)

11988441

= 0

11988442

= (119885119901minus V119890)

11988443

= ((119885119901minus V119890) 11988711+ 119885V)

11988444

= ((119885119901minus V119890) 11988712+ 119885119908)

11987321

= (1198711205791119888

minus 119887111198841205791119888

minus 119887121198851205791119888

)

11987322

= (1198711205790119879

minus 119887111198841205790119879

minus 119887121198851205790119879

)

(19)

Using (18) the helicopter with underslung load for lateralmotion in the 119884-119885 plane can be described by

119871(119905)

= 1198911(120593119871(119905))

+ 1198661(120593119871(119905)) [119901 (119909

119867 (119905)) + 119902 (120593119871(119905) 119909119867 (119905))]

+ 119867 (119905 120593119871 119909119867 (119905))

(20a)

119867 (119905) = 119891

2(120593119871(119905) 119909119867 (119905)) + 119866

2 (119905) (20b)

where

119871(119905) = [

1198711

1198712

]119879

119901 (119909119867 (119905)) = [V2 119908

2]119879

1198911(120593119871(119905)) =

[[[[[[[

[

1205931198712

minus119892

119897119910

sin1205931198711

+119896119863119897119910

119872119871

(sign (119871) cos3 120593

1198711

+ sin (119871) sin3 120593

1198711

) 1205932

1198712

minus 1198961205931205931198712

119901

(minus11988711119892120601 + (119871

119901minus 11988711(119884119901+ 119908119890) minus 11988712(119885119901minus V119890)) 119901)

]]]]]]]

]

1198661(120593119871(119905)) =

[[[[[[[[

[

0 0

(119896119863sign (

119871) cos120593

1198711

119872119871119897119910

) (119896119863sign (

119871) sin120593

1198711

119872119871119897119910

)

0 0

0 0

]]]]]]]]

]

119902 (120593119871(119905) 119909119867 (119905)) =

[[

[

0

cos1205931198711

119897119910

V +sin1205931198711

119897119910

minus2119896119863

119872119871

(sign (119871) cos2 120593

1198711

V + sign (119871) sin2 120593

1198711

119908)1205931198712

]]

]

Journal of Control Science and Engineering 7

119867(119905 120593119871(119905) 119909119867 (119905)) =

[[[[[

[

0

0

( minus 119901)

(11988423V + 11988424119908)

]]]]]

]

(119905) = [1205791119888 1205790119879]119879

1198912(120593119871(119905) 119909119867 (119905)) = [

(11988431120601 + 11988432119901 + 11988433V + 11988434119908)

(11988442119901 + 11988443V + 11988444119908)

]

1198662= [

1198841205791119888

1198841205790119879

1198851205791119888

1198851205790119879

]

(21)

It is assumed that the lateral motion is primarily controlledby lateral cyclic commands (120579

1119888) and the tail rotor collective

1205790119879

3 Stability Analysis

The goal is to analyse the stability of the combined systemto ensure that it is possible to stabilise the helicopter withunderslung load system modelled by (11a) (11b) (20a) and(20b) in a real environment with uncertainties In this paperfor the stability analysis the Lyapunov second method isapplied for the helicopter with an underslung external loadsystem The analysis for the longitudinal motion is discussedfirst The system equations can be considered to have twomain parts that is known and unknown (or partly known)The known terms formed the nominal part of the systemmodelThe unknown or partly known part can be consideredas the uncertainty to the system The whole system is thenmodelled by a nominal part with the addition of uncertaintyIn fact the known elements in the subsystem (11a) arecharacterised by the prescribed triple (119891

1 1198661 119901) and it is

desired that the nominal part of the system is stableThe basic notations and concepts required for the analysis

are described first The state space is denoted by119883 fl R119899 andthe control space by119906 fl R119898 where 1 le 119898 le 119899TheEuclideaninner product (on 119883 or 119906 as appropriate) and induced normare denoted by ⟨sdot sdot⟩ and sdot respectively Let 119862(R119901R119902) and1198621(R119901R119902) denote the space of all continuous functions and

the space of continuous functions with continuous first-orderpartial derivatives respectively and let 119862infin(R119901R119902) denotethe space of functions whose partial derivatives of any orderexist and are continuous mapping R119901 rarr R119902 For a real-valued continuous scalar function 119909 rarr V(119909) defined onR119899 nabla V rarr nablaV isin R119899 denotes the gradient map The Liederivative of V along a vector field 119891 R119899 rarr R119899 is denoted by119871119891V R119899 rarr R which is defined by

(119871119891V) (119909) = ⟨nablaV (119909) 119891 (119909)⟩ (22)

The Lie bracket of vector fields 119891 119892 isin 119862infin(R119899R119899) is the

vector field [119891 119892] isin 119862infin(R119899R119899) defined by [119891 119892] = (119863119892)119891 minus

(119863119891)119892 where (119863119891)denotes the Jacobianmatrix of119891 and (119863119892)denotes the Jacobian matrix of 119892

In this paper nonlinear systemswith the following formatare considered

(119905) = 119891 (119909 (119905)) + 119866 (119909 (119905)) (119905) (23)

where 119909(119905) isin R119899 isin R119898 In general mathematical modelsof dynamical systems are usually imprecise due to modellingerrors and exogenous disturbances [12] Equation (23) can beconsidered as the nominal part of the system model and theuncertainty can be modelled as an additive perturbation tothe nominal systemmodel more specifically the structure ofthe system has the form

(119905) = 119891 (119909 (119905)) + 119866 (119909 (119905)) (119905) + 120599 (119909 (119905) 119906 (119905)) (24)

where 120599(119909(119905) 119906(119905))models the uncertainty in the systemSystem (24) is globally asymptotically stable to the zero

state if the system exhibits the following properties

(i) Existence and Continuation of Solutions For each 119909 isin R119899there exists a local solution119909 [0 119905

1) rarr R119899 (ie an absolutely

continuous function satisfying (24) almost everywhere (ae)and 119909(0) = 119909

0) and every such solution can be extended intoa solution on [0infin)

(ii) Boundedness of Solutions For each ℏ gt 0 there exists119903(ℏ) gt 0 such that 119909(119905) isin 119903(ℏ)119861

119899 for all 119905 ge 0 on every

solution 119909 [0infin) rarr R119899 with 1199090isin ℏ119861119899 where 119861

119899denote

the open unit ball centred at the origin in R119899

(iii) Stability of the State Origin For each 120575 gt 0 there exists119889(120575) gt 0 such that 119909(119905) isin 120575119861

119899for all 119905 ge 0 on every solution

119909 [0infin) rarr R119899 with 1199090isin 119889(120575)119861

119899

(iv) Global Attractivity of the State Origin For each ℏ gt 0 and120576 gt 0 there exists 119879(ℏ 120576) ge 0 such that 119909(119905) isin 119861

119899for all

119905 ge 119879(ℏ 120576) on every solution 119909 [0infin) rarr R119899 with 1199090isin ℏ119861119899

8 Journal of Control Science and Engineering

Consider a nonlinear system described by the ordinarydifferential equation as follows

(119905) = 119891 (119905 119909 (119905))

119909 (1199050) = 1199090

(25)

where 119891 R times 119883 rarr 119883 and 119891(119905 0) = 0 for all 119905To analyse the stability of (25) Lyapunovrsquos second stabilityanalysis method is applicable The Lyapunov approach is toshow that a candidate ldquoLyapunov functionrdquo is nonincreasingalong all solution to (25) by means that do not requireexplicit knowledge of solutions to (25) From this appropriateconclusion can be drawn regarding stability concepts relatingto solutions of the differential equation (25) An essentialpart of Lyapunovrsquos method is the determination of the timederivative of the candidate ldquoLyapunov functionrdquo along allsolution of the dynamical system

Consider a Lyapunov candidate (119905 119909) rarr V(119905 119909) Rtimes119883 rarr

R which satisfies the condition V isin 1198621(R times 119883) in which case

its time derivative along solutions to (25) is given by

V (119905 119909 (119905)) =120597V (119905 119909 (119905))

120597119905+ ⟨nablaV (119905 119909 (119905)) 119891 (119905 119909 (119905))⟩ (26)

for almost all 119905 isin RLet 119882(119909(119905)) denote a positive definite function If

V(119905 119909(119905)) satisfies

(i) V(119905 0) = 0 for all 119905 ge 0

(ii) 119882(119909(119905)) le V(119905 119909(119905)) for all 119909(119905) isin Φ 0 sub Φ sub 119883 andall 119905 ge 0

(iii) V(119905 119909(119905)) lt 0 in Φ

then V(119905 119909(119905)) is said to be a Lyapunov function in Φ IfV(119905 119909(119905)) in Φ then V(119905 119909(119905)) is said to be a weak Lyapunovfunction

A set Ε is said to be an invariant set with respect to thedynamical system = 119891(119909) if

119909 (0) isin Ε 997891997888rarr 119909 (119905) isin Ε forall119905 isin R+ (27)

In other words Ε is the set of points such that if a solution of = 119891(119909) belongs to Ε at some instant initialized points at119905 = 0 then it belongs to Ε for all future time

Now a setΕ sub 119909 is said to be a local invariantmanifold for(25) if for any 1199090 isin Ε 119909(119905) with 119909(0) = 119909

0 is in Ε for |119905| lt 119879

where 119879 gt 0 If 119879 = infin then Ε is said to be an invariantmanifold

Now considering the longitudinal motion (Figure 1) forthe helicopter with an underslung external load systemchoose a Lyapunov function candidate for the first subsystemV1as

V1(120579119871) =

1

2[12058911205792

1198711

+ 12058921205792

1198712

+ (1205893120579 minus 1205894119902)2+ 12058951199022] (28)

where 120589119894(119894 = 1 2 3 4 5) are design parameters to be

determined Then

V1(120579119871) = ⟨nablaV

1(120579119871) 1198911(120579119871) + 1198661(120579119871 (119905))

sdot [119901 (119909119867 (119905)) + 119902 (120579

119871 (119905) 119909119867 (119905))]

+ 119867(119905 120579119871 119909119867 (119905))⟩ = (119871

1198911

V1) (120579119871)

+ ⟨nablaV1(120579119871) 1198661(120579119871 (119905))

sdot [119901 (119909119867 (119905)) + 119902 (120579

119871 (119905) 119909119867 (119905))]

+ 119867(119905 120579119871 119909119867 (119905))⟩

(29)

From (28) nablaV1(120579119871) can be obtained as

nablaV1(120579119871)

= [12058911205791198711

12058921205791198712

1205893(1205893120579 minus 1205894119902) minus120589

4(1205893120579 minus 1205894119902) + 120589

5119902]119879

(30)

Substituting nablaV1(120579119871) and 119891

1(120579119871) into the derivative of

(1198711198911

V1)(120579119871) we have

(1198711198911

V1) (120579119871) = 12058911205791198711

1205791198712

minus 11989612057912058921205792

1198712

minus 1205892

119892

119897119909

sin 1205791198711

1205791198712

+119896119863119897119909

119872119871

1205892[sign (

119871) cos3 120579

1198711

+ sign (119871) sin3 120579

1198711

]

sdot 1205793

1198712

+ 1205893[1205893120579 minus 1205894119902] 119902 + (minus120589

31205894120579 + (120589

2

4+ 1205895) 119902)

sdot [11988321120579 + 119883

22119902]

(31)

Checking the term of minus sin (1205791198711

)1205791198712

it can be seen thatminus sin (120579

1198711

)1205791198712

le minus(2120587)1205791198711

1205791198712

when 1205791198711

and 1205791198712

are bothpositive or negative The situation of 120579

1198711

and 1205791198712

havingdifferent signs will help with the system stability Then

(1198711198911

V1) (120579119871) le 12058911205791198711

1205791198712

minus 11989612057912058921205792

1198712

minus 1205892

119892

119897119909

2

1205871205791198711

1205791198712

+119896119863119897119909

119872119871

sdot 1205892[sign (

119871) cos3 120579

1198711

+ sign (119871) sin3 120579

1198711

] 1205793

1198712

+ 1205893[1205893120579 minus 1205894119902] 119902 + (minus120589

31205894120579 + (120589

2

4+ 1205895) 119902)

sdot [11988321120579 + 119883

22119902]

(32)

If the design parameter 1205891is chosen as 120589

1= (2119892119897

119909120587)1205892

(1198711198911

V1)(120579119871) satisfies the following

(1198711198911

V1) (120579119871) le minus119896

12057912058921205792

1198712

+119896119863119897119909

119872119871

sdot 1205892[sign (

119871) cos3 120579

1198711

+ sign (119871) sin3 120579

1198711

] 1205793

1198712

+ 1205893[1205893120579 minus 1205894119902] 119902 + (minus120589

31205894120579 + (120589

2

4+ 1205895) 119902)

sdot [11988321120579 + 119883

22119902]

(33)

Journal of Control Science and Engineering 9

Further analysis on (33) will start from examining the firsttwo terms Rewrite these two terms So

minus 11989612057912058921205792

1198712

+119896119863119897119909

119872119871

1205892[sign (

119871) cos3 120579

1198711

+ sign (119871)

sdot sin3 1205791198711

] 1205793

1198712

= minus11989612057912058921205792

1198712

+119896119863119897119909

119872119871

12058921205791198712

[sign (119871)

sdot cos3 1205791198711

+ sign (119871) sin3 120579

1198711

] 1205792

1198712

le minus11989612057912058921205792

1198712

+119896119863119897119909

119872119871

1205892

10038161003816100381610038161003816max (120579

1198712

) (sign (119871) cos3 120579

1198711

+ sign (119871) sin3 120579

1198711

)100381610038161003816100381610038161205792

1198712

= minus12058921205792

1198712

119896120579

minus119896119863119897119909

119872119871

10038161003816100381610038161003816max (120579

1198712

)

sdot (sign (119871) cos3 120579

1198711

+ sign (119871) sin3 120579

1198711

)10038161003816100381610038161003816

(34)

If the hinge friction is big enough to satisfy the following

119896120579gt

119896119863119897119909

119872119871

10038161003816100381610038161003816max (120579

1198712

)

sdot (sign (119871) cos3 120579

1198711

+ sign (119871) sin3 120579

1198711

)10038161003816100381610038161003816

(35)

then

120582119871= 119896120579minus119896119863119897119909

119872119871

10038161003816100381610038161003816max (120579

1198712

)

sdot (sign (119871) cos3 120579

1198711

+ sign (119871) sin3 120579

1198711

)10038161003816100381610038161003816 ge 0

minus 11989612057912058921205792

1198712

+119896119863119897119909

119872119871

1205892[sign (

119871) cos3 120579

1198711

+ sign (119871)

sdot sin3 1205791198711

] 1205793

1198712

le minus12058921205792

1198712

120582119871

(36)

Now the following analysis has been conducted for (33) andthe load movement for the longitudinal motion is describedby

119871= 0minus 1198971199091205791198712

cos 1205791198711

119871= 0minus 1198971199091205791198712

sin 1205791198711

(37)

Around the hover condition1198830= 0 and 119885

0= 0 the signs for

119871and

119871are opposite to the one of 120579

1198712

provided that 1205791198711

ispositive

Considering the term (sign (119871) cos3 120579

1198711

+

sign (119871) sin3 120579

1198711

)1205793

1198712

for all possible combinations ofthe signs of 120579

1198712

and 1205791198711

the analysis follows below

(i) If 1205791198711

gt 0 and 1205791198712

gt 0 then sign (119871) = minus1

sign (119871) = minus1 cos3 120579

1198711

gt 0 and sin3 1205791198711

gt 0 There-fore (sign (

119871) cos3 120579

1198711

+ sign (119871) sin3 120579

1198711

)1205793

1198712

lt 0

(ii) If 1205791198711

lt 0 and 1205791198712

gt 0 then sign (119871) = minus1

sign (119871) = +1 cos3 120579

1198711

gt 0 and sin31205791198711

lt 0 So(sign (

119871) cos3 120579

1198711

+ sign (119871) sin3 120579

1198711

)1205793

1198712

lt 0

(iii) If 1205791198711

gt 0 and 1205791198712

lt 0 then sign (119871) = +1

and sign (119871) = +1 cos3 120579

1198711

gt 0 and sin3 1205791198711

gt

0 So the following is true (sign (119871) cos3 120579

1198711

+

sign (119871) sin3 120579

1198711

)1205793

1198712

lt 0

(iv) If 1205791198711

lt 0 and 1205791198712

lt 0 then sign (119871) = +1 and

sign (119871) = minus1 cos3 120579

1198711

gt 0 and sin3 1205791198711

lt 0 So(sign (

119871) cos3 120579

1198711

+ sign (119871) sin3 120579

1198711

)1205793

1198712

lt 0

Therefore

minus 11989612057912058921205792

1198712

+119896119863119897119909

119872119871

sdot 1205892[sign (

119871) cos3 120579

1198711

+ sign (119871) sin3 120579

1198711

] 1205793

1198712

le minus11989612057912058921205792

1198712

(38)

Now examining the rest of the terms of (1198711198911

V1)(120579119871) then

1205893[1205893120579 minus 1205894119902] 119902

+ (minus12058931205894120579 + (120589

2

4+ 1205895) 119902) [119883

21120579 + 119883

22119902] = 120589

2

3120579119902

minus 120589312058941199022minus 12058931205894119883211205792minus 1205893120589411988322120579119902

+ (1205892

4+ 1205895)11988321120579119902 + (120589

2

4+ 1205895)119883221199022

= minus [12058931205894minus (1205892

4+ 1205895)11988321] 1199022minus 12058931205894119883211205792

+ [1205892

3minus 1205893120589411988322+ (1205892

4+ 1205895)11988321] 120579119902

(39)

For the system 120579 and 1205791198711

always have different signs 120579 and 119902

always have different signs and 119902 and 1205791198711

have the same signsSo

[1205892

3minus 1205893120589411988322+ (1205892

4+ 1205895)11988321] 120579119902 le 0

if [1205892

3minus 1205893120589411988322+ (1205892

4+ 1205895)11988321] ge 0

(40)

Therefore

1205893[1205893120579 minus 1205894119902] 119902

+ (minus12058931205894120579 + (120589

2

4+ 1205895) 119902) [119883

21120579 + 119883

22119902]

le minus [12058931205894minus (1205892

4+ 1205895)11988321] 1199022minus 12058931205894119883211205792

(41)

Choose the design parameters to satisfy

[12058931205894minus (1205892

4+ 1205895)11988321] gt 0 (42)

If11988321

gt 0 then

minus [12058931205894minus (1205892

4+ 1205895)11988321] 1199022minus 12058931205894119883211205792lt 0 (43)

10 Journal of Control Science and Engineering

If11988321

lt 0 then

minus [12058931205894minus (1205892

4+ 1205895)11988321] 1199022minus 12058931205894119883211205792lt 0 (44)

in the region ofminus[12058931205894minus(1205892

4+1205895)11988321]1199022lt 12058931205894119883211205792 Since 120579 is

caused by the loadmotion here the value of 120579 should bemuchsmaller than 119902 The inequality (44) is easy to be satisfied

In summary of the above analysis and by defining

Ψ1(120579119871) = 11989612057912058921205792

1198712

+ [12058931205894minus (1205892

4+ 1205895)11988321] 1199022

+ 12058931205894119883211205792

Ψ2(120579119871) = [120589

31205894minus (1205892

4+ 1205895)11988321] 1199022+ 12058931205894119883211205792

(45)

the following lemma can be derived

Lemma 1 Defining a Lyapunov function (28) and choosingthe design parameters to satisfy 120589

1= (2119892119897

119909120587)1205892 [12058931205894minus (1205892

4+

1205895)11988321] gt 0 and [1205892

3minus1205893120589411988322+(1205892

4+1205895)11988321] ge 0 then within

the region specified by minus[12058931205894minus (1205892

4+ 1205895)11988321]1199022lt 12058931205894119883211205792

(i) V1(0) = 0 and V

1(120579119871) gt 0 forall120579

119871= 0

(ii) V1(120579119871) rarr infin as 120579

119871 rarr infin

(iii) (1198711198911

V1)(120579119871) le minusΨ

1(120579119871) forall120579119871 around the hover condi-

tion or (1198711198911

V1)(120579119871) le minusΨ

2(120579119871) forall120579119871 if (35) holds

Both functions Ψ1and Ψ

2are nonnegative

Recall the control term 119901(119909119867(119905)) in the first subsystem

of (11a) and (11b) it can be seen that [(119863119901)(119909119867)]minus1 exists

for all 119909119867(119905) The unknown vector fields 119902(120579

119871(119905) 119909119867) and

119867(119905 120579119871(119905) 119909119867) model the uncertainties imposed onto the

system Since 119902(120579119871(119905) 119909119867) is directly mapped into the ldquocon-

trolrdquo space of 119909119867(119905) it can be considered as a matched

uncertainty119867(119905 120579119871(119905) 119909119867) is unknown and it does belong to

the control space of 119909119867(119905) so it represents the mismatched

uncertainty in the system [12]Generally the range of the (longitudinal) load suspension

angle is within minus1205872 lt 120579119871lt 1205872 The helicopter velocities

and load suspension angle havemaximumoperational valuestherefore the uncertainties in the system are bounded Withthe Lyapunov function defined in (28)

(1198711198921

V1) (120579119871) = ⟨nablaV

1(120579119871) 1198921⟩ =

119896119863

119872119871119897119909

12058921205791198712

(46)

In 1198661(120579119871) the columns 119892

1and 119892

2are the same therefore

(1198711198922

V1) (120579119871) =

119896119863

119872119871119897119909

12058921205791198712

(47)

By considering the maximum values of cos (sdot) and sin (sdot)functions the bounding values relating to the uncertainty119902(120579119871(119905) 119909119867) can be estimated as follows

10038171003817100381710038171003817119902 (120579119871 119909119867)10038171003817100381710038171003817le

10038161003816100381610038161003816100381610038161003816

1

119897119909

( + )

+1

119872119871

(minus119896119863(2 +

120581

119897119909

) (119906 + 119908) +119896120579

119897119909

)1205791198712

10038161003816100381610038161003816100381610038161003816

le

10038161003816100381610038161003816100381610038161003816

119896120579

119897119909

1205791198712

10038161003816100381610038161003816100381610038161003816+

10038161003816100381610038161003816100381610038161003816

119896119863

119872119871

(2 +120581

119897119909

) (119906 + 119908)

10038161003816100381610038161003816100381610038161003816+

10038161003816100381610038161003816100381610038161003816

1

119897119909

( + )

10038161003816100381610038161003816100381610038161003816

le119896120579

119897119909

100381710038171003817100381710038171205791198712

10038171003817100381710038171003817+

119896119863

119872119871

(2 +120581

119897119909

)1003817100381710038171003817119901 (119909119867)1003817100381710038171003817 + 1205722 (119905)

le 1205831

10038161003816100381610038161003816(1198711198922

V1) (120579119871)10038161003816100381610038161003816+ 1205721

1003817100381710038171003817119901 (119909119867)1003817100381710038171003817 + 1205722 (119905)

(48)

where 1205831

= 119872119871119896120579119896119863 1205721

= (119896119863119872119871)(2 + 120581119897

119909) (120581 =

max [1205811 1205812]) and 120572

2(119905) is defined by 120572

2(119905) = max |(1119897

119909)( +

)| gt 0For the mismatched uncertainty 119867(119905 120579

119871(119905) 119909119867) the fol-

lowing analysis is conducted to obtain its bounding functionThe mismatched uncertainty can be rewritten as

119867(119905 120579119871 (119905) 119909119867) =

[[[[[

[

0

0

(11988611119906 + 11988612119908)

(11988323119906 + 119883

24119908)

]]]]]

]

=

[[[[[

[

0 0

0 0

11988611

11988612

11988323

11988324

]]]]]

]

[119906

119908]

(49)

Let 119860 = [1198861111988612

1198832311988324] then

10038171003817100381710038171003817119867 (119905 120579

119871 119909119867)10038171003817100381710038171003817le1003817100381710038171003817100381711986010038171003817100381710038171003817

1003817100381710038171003817119909119867 (119905)1003817100381710038171003817 (50)

Define a positive function as

1006704120579 (120579119871) = radic(1205792

1198711

+ 12057921198712

+ 1205792 + 1199022+ 120576) (51)

where 120576 is a very small positive constant Choosing1205731

= (1198721198711198971199091198961198631205892)119860 then the mismatched uncertainty

119867(119905 120579119871(119905) 119909119867) is bounded by the following inequality

10038171003817100381710038171003817119867 (119905 120579

119871 119909119867)10038171003817100381710038171003817

le 1006704120579minus1

(120579119871) 1205731

2

sum

119894=1

(10038161003816100381610038161003816(119871119892119894

V1) (120579119871)10038161003816100381610038161003816)1003816100381610038161003816119901119894 (119909119867)

1003816100381610038161003816

(52)

where 119892119894denotes the 119894th column of the matrix function

1198661(120579119871) and 119901

119894is the 119894th component of 119901(119909

119867) respectively As

1205731has a design parameter 120589

2involved it is easy to have 120573

1to

lead (52) to be true For the matched uncertainty we have

10038171003817100381710038171003817119902 (120579119871 119909119867)10038171003817100381710038171003817le

119896120579

119897119909

100381610038161003816100381610038161205791198712

10038161003816100381610038161003816+ 1205721

1003817100381710038171003817119901 (119909119867)1003817100381710038171003817 + 1205722 (119905) (53)

Journal of Control Science and Engineering 11

Following the above analysis the following lemma can bederived

Lemma 2 The uncertainties (52) and (53) are bounded andsatisfy if 120572

1 1205731ge 0 and 120572

1+ 1205731lt 1 for 120572

1= (119896119863119872119871)(2 +

120581119897119909) and 120573

1= (1198721198711198971199091198961198631205892)119860

4 Numerical Results

Using a simulation software the linearized model for a UH-60 helicopter is obtained with the following state and inputmatrices

119860 =

[[[[[[[[[[[[[[[[[[[

[

minus00000 00003 minus00000 minus00000 00000 minus00000 10000 minus00023 00503

minus00003 minus00000 minus00000 minus00000 minus00000 minus00000 minus00000 09989 00463

minus00001 00000 minus00000 minus00000 00000 00000 minus00000 minus00464 10002

minus19386 minus28593 minus00030 minus00195 minus00126 00168 minus12000 57076 02683

328694 minus03640 00027 00126 minus00311 minus00013 minus41348 minus22350 03713

03138 minus14034 00004 00150 00034 minus03340 02866 06307 23410

minus09144 minus04783 00017 00226 minus00246 00007 minus65893 minus21301 minus00598

00631 minus06744 00008 00029 00046 00025 02059 minus14255 minus01636

minus07549 minus00119 minus00002 00009 00028 00006 minus00749 minus01032 minus02214

]]]]]]]]]]]]]]]]]]]

]

119861 =

[[[[[[[[[[[[[[[[[[[

[

minus00000 00000 minus00000 00000

00000 00000 minus00000 minus00000

minus00000 00000 minus00000 minus00000

minus22190 05734 03170 minus00019

35381 minus00453 minus00963 02007

minus15629 00237 minus60740 minus01014

19943 minus00714 minus00541 00731

00446 minus01097 00553 minus00281

minus08197 minus00082 01573 minus00635

]]]]]]]]]]]]]]]]]]]

]

(54)

The desired condition 1205721+ 1205731lt 1 can be verified by consid-

ering a typical case that the UH-60 helicopter carries a loadweight 1000 lb by a 15 ft slung length and let 119896

119863= 75 lbft

For the UH-60 helicopter model with reference to thegeneral mathematical model presented in Section 2 we have1198831205791119904

= 05734 1198831205790

= 03170 1198851205791119904

= 00237 1198851205790

= minus607401198721205791119904

= minus01097 and 1198721205790

= 00553 So the followingparameters can be calculated to have the values 119886

11= minus02

and 11988612

= minus002 For 119883119906= minus00195 119885

119906= 00150 119872

119906=

00029 119872119902= minus14255 (119883

119902minus 119908119890) = 57076 and (119885

119902+ 119906119890) =

06307 we have 11988323

= 006 And also for 119883119908

= 00168119885119908

= minus03340 and 119872119908

= 00025 we have 11988324

= 001Based on all the above parameters 120590max(119860) = 00041 Inthis case let 120581 = 001 120572

1= (119896119863119872119871)(2 + 120581119897

119909) asymp 015

and 1205731= (119872

119871119897119909119896119863)120590max(119860) = 082 Therefore 120572

1+ 1205731=

015 + 082 = 097 which satisfies 1205721+ 1205731

lt 1 Anotherexample considers that the UH-60 helicopter carries a loadweight 500 lb by a 15 ft slung length and let 119896

119863= 50 lbft then

1205721= 02 and 120573

1= 062 therefore (120572

1+ 1205731) = 082 Therefore

this assumption 1205721+ 1205731

lt 1 is realistic for the systemdiscussed in here The numerical values vary with respectto the slung load configuration It may represent the system

dynamic variations To investigate the stability characteristicsof dynamic variations two series of simulation studies werecarried out

Firstly the slung length was kept constant at 15 ft andthe load varied between the limits 500ndash2000 lb with 119896

119863=

75 lbft producing the system poles listed in Table 1 Thecorresponding root loci are sketched in Figures 3 and 4Figure 4 clearly shows that the positions of the system polescrossed the imaginary axis as the load weight increased

Secondly load weight was kept constant and the slunglength varied between the limit of 10ndash20 ft Table 2 sum-marises the results of variation of the system poles and thecorresponding root loci are illustrated in Figures 5 and 6

The pole locus indicates that the system poles have thevariation trends of moving further towards right direction onthe 119904-plane with increase of load weight and slung lengthsFrom the simulation results it is clearly shown that thesystem becomes less stable when the load andor the slunglength increases The simulation results are compared withthe stability analysis presented above

Now consider the case of fixed slung length of 15 ft withdifferent load weights In the cases of a UH-60 helicopter that

12 Journal of Control Science and Engineering

Table 1 System poles locations for a fixed slung length (119897 = 15ft) but with different load weights

Helicopter 500 lb 1000 lb 1500 lb 2000 lbminus10919 minus11159 minus07848 minus06221 minus09369minus63938 minus64301 minus92203 minus79848 minus75613minus00032 minus00048 minus00034 minus00052 minus00052minus00977 minus00818 minus00888 minus00972 minus00694minus03045 minus03036 minus03394 minus03318 minus03333minus03159 plusmn 04363119894 minus03199 plusmn 04628119894 minus02295 plusmn 05050119894 minus02366 plusmn 05974119894 minus01834 plusmn 04968119894minus00489 plusmn 03898119894 minus00211 plusmn 03868119894 00074 plusmn 03445119894 00119 plusmn 04406119894 00264 plusmn 03199119894mdash minus00316 plusmn 15043119894 minus01020 plusmn 17380119894 minus00368 plusmn 18753119894 minus01160 plusmn 20003119894mdash minus00736 plusmn 15019119894 minus07662 plusmn 11680119894 minus07349 plusmn 19064119894 minus12367 plusmn 08995119894

Table 2 System poles for a fixed load (1000 lb) but with different slung lengths

Helicopter 10 ft 15 ft 20 ftminus10919 minus10560 minus07848 minus08687minus63938 minus99252 minus92203 minus85565minus00032 minus00061 minus00034 minus00114minus00977 minus02012 minus00888 minus00953minus03045 minus02606 minus03394 minus03479minus03159 plusmn 04363119894 minus01809 plusmn 06219119894 minus02295 plusmn 05050119894 minus01636 plusmn 06092119894minus00489 plusmn 03898119894 00643 plusmn 04577119894 00074 plusmn 03445119894 00799 plusmn 04594119894mdash minus01055 plusmn 19883119894 minus01020 plusmn 17380119894 minus01247 plusmn 14470119894mdash minus08847 plusmn 12464119894 minus07662 plusmn 11680119894 minus06352 plusmn 10384119894

Re(poles)

Im(poles)

minus10 minus5 0 5minus08minus06minus04minus02

002040608

Figure 3 System root-locus diagram showing pole locus forconstant slung length (15 ft) as load weight is increased

minus04 minus035 minus03 minus025 minus02 minus015 minus01 minus005 0 005 01

Re(poles)

Im(poles)

minus08minus06minus04minus02

002040608

Figure 4 Enlarged view of Figure 3 showing poles locus near toRe(poles) = 0 axis

minus10 minus8 minus6 minus4 minus2 0 2

Re(poles)

Im(poles)

minus08minus06minus04minus02

002040608

Figure 5 System root-locus showing pole locus for constant loadweight (1000 lb) as slung length is increased

minus04 minus035 minus03 minus025 minus02 minus015 minus01 minus005 0 005 01

Re(poles)

Im(poles)

minus08minus06minus04minus02

002040608

Figure 6 Enlarged view of Figure 5 showing poles locus near toRe(poles) = 0 axis

Journal of Control Science and Engineering 13

carries a loadweight 500 lb and 1000 lb then1205721= 03 and 041

with 1205731= 041 and 082 respectively therefore 120572

1+ 1205731lt 1

thus the system is stable But when the load weight is 1500 lband 2000 lb then 120572

1= 01 and 0075 and also 120573

1= 123 and

164 respectively therefore 1205721+ 1205731gt 1 thus the system is

unstable For the case of the constant load weight (1000 lb) asslung length is increased then for a 10 ft and 15 ft slung length1205721= 015 and 120573

1= 055 and 082 respectively therefore

1205721+ 1205731lt 1 thus the system is stable But when the slung

length increased to 20 ft then 1205721

= 015 and 1205731

= 109therefore (120572

1+ 1205731) = 124 that is 120572

1+ 1205731gt 1 thus the

system is unstable However the system can be marginallystable if the UH-60 helicopter carries a load weight 725 lb bya 20 ft slung length then 120572

1= 02 and 120573

1= 08 therefore

(1205721+ 1205731) = 1 For this set of analysis 119896

119863is assumed to

be 119896119863

= 75 lbft However 119896119863can vary depending on the

weather condition and an example of 119896119863= 50 lbft is shown

above for illustrative purpose This analysis suggests that themaximum value of 20 ft slung length may be used for slungload operation However the load weight limit can vary withrespect to the capacity of the load carrying hooks helicopterweight environmental conditions and so forth

The examples presented here show that the simulationresults are correlated with the stability analysis Thus desiredcondition for the stability is met and it can locate the load atthe specified position or its neighbourhood

5 Conclusions

In this paper stability analysis of a helicopter with an under-slung load system is investigatedThe investigation identifiedthe conditions for the stabilisation of the system and thepositioning of the underslung load at hover condition Sta-bility analysis and numerical results proved that the desiredcondition for the stability is met then it is possible to locatethe load at the specified position or its neighbourhood Anillustration example is also given in the paper The methodresults in a guaranteed load positioning accuracy whichdepends on the design parameters and the method can beapplicable to any individual helicopter with external loadsystem

Nomenclature

119871119872119873 Overall helicopter rolling pitching andyawing moments

119901 119902 119903 Helicopter roll pitch and yaw rates aboutbody reference axes

119883119884 119885 Overall helicopter force components120601 120579 120595 Roll pitch and yaw angles119906 V 119908 Helicopter velocity components at centre

of gravity1205790 Main rotor collective

1205791119904 Longitudinal cyclic commands

1205791119888 Lateral cyclic commands

1205790119879 Tail rotor collective

1198830 1198840 1198850 Location of the suspension point withrespect to earth referenced 119909 119910 and 119911

directions

0 0 0 The helicopter velocity in the 119909 119910 and 119911

directions0 0 0 The helicopter acceleration in the 119909 119910 and119911 directions

120579119871 Load suspension angle in the119883-119885 plane

with respect to 119885-axis120601119871 Load suspension angle in the 119884-119885 plane

with respect to 119885-axis119872119871 Mass of the suspended load

119892 Acceleration due to gravity119897 Slung length119896119863 Aerodynamic drag force constant

(119896119863= (12)120588119878119862

119863)

120588 Air density119878 The load area presented to the airflow119862119863 Drag coefficient for the load

Competing Interests

The author declares that there are no competing interestsregarding the publication of this paper

References

[1] K Thanapalan and F Zhang ldquoModelling and simulation studyof a helicopter with an external slung load systemrdquo I-ManagersJournal on Instrumentation amp Control Engineering vol 1 no 1pp 9ndash17 2013

[2] M Bisgaard J D Bendtsen and A La Cour-Harbo ldquoModelingof generic slung load systemrdquo Journal of Guidance Control andDynamics vol 32 no 2 pp 573ndash585 2009

[3] Handling Qualities Requirements for Military Rotorcraft ADS-33-D US Army Aviation and Troop Command St Louis MissUSA 1996

[4] K Thanapalan and T M Wong ldquoModeing of a helicopter withan under-slung load systemrdquo in Proceedings of the 29th ChineseControl Conference (CCC rsquo10) pp 1451ndash1456 Beijing China July2010

[5] B Feng ldquoRobust control for lateral and longitudinal channels ofsmall-scale unmanned helicoptersrdquo Journal of Control Scienceand Engineering vol 2015 Article ID 483096 8 pages 2015

[6] J J Gribble ldquoLinear quadratic Gaussianloop transfer recoverydesign for a helicopter in low-speed flightrdquo Journal of GuidanceControl and Dynamics vol 16 no 4 pp 754ndash761 1993

[7] M A Manness J J Gribble and D J M Smith ldquoMultivariablemethods for helicopter flight control law designrdquo in Proceedingsof the 16th Annual Society European Rotorcraft Forum PaperNoIII52 Glasgow UK September 1990

[8] W L Garrard E Low and S Prouty ldquoDesign of attitude andrate command systems for helicopters using eigenstructureassignmentrdquo Journal of Guidance Control and Dynamics vol12 no 6 pp 783ndash791 1989

[9] Y B Shtessel and I A Shkolnikov ldquoAeronautical and spacevehicle control in dynamic sliding manifoldsrdquo InternationalJournal of Control vol 76 no 9-10 pp 1000ndash1017 2003

[10] D FThompson J S Pruyn andA Shukla ldquoFeedback design forrobust tracking and robust stiffness in flight control actuatorsusing a modified QFT techniquerdquo International Journal ofControl vol 72 no 16 pp 1480ndash1497 1999

14 Journal of Control Science and Engineering

[11] K Thanapalan ldquoModelling of a helicopter systemrdquo in Proceed-ings of the 1st Virtual Control Conference (VCC rsquo10) pp 45ndash52Aalborg University 2010

[12] D P Goodall and J Wang ldquoStabilization of a class of uncertainnonlinear affine systems subject to control constraintsrdquo Interna-tional Journal of Robust and Nonlinear Control vol 11 no 9 pp797ndash818 2001

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

4 Journal of Control Science and Engineering

11986121

= (1198721205791119904

minus 119886111198831205791119904

minus 119886121198851205791119904

)

11986122

= (1198721205790

minus 119886111198831205790

minus 119886111198851205790

)

(9)

Using (8) the system model can be rearranged to include thevariables 120579 and 119902 into the loadmodel For the stability analysispurpose an extra term is introduced into the system modelwhich is zero with the expression

1205811119896119863sign (

119871) cos 120579

1198711

119906

119872119871119897119909

+1205812119896119863sign (

119871) cos 120579

1198711

119908

119872119871119897119909

minus1205811119896119863sign (

119871) cos 120579

1198711

119906

119872119871119897119909

minus1205812119896119863sign (

119871) cos 120579

1198711

119908

119872119871119897119909

(10)

where 120581119894gt 0 (119894 = 1 2) are small positive constants

With this arrangement for the longitudinal motion of thehelicopter with an underslung load combined system modelcan be written as follows

120579119871 (119905)

= 1198911(120579119871 (119905))

+ 1198661(120579119871 (119905)) [119901 (119909

119867 (119905)) + 119902 (120579119871 (119905) 119909119867 (119905))]

+ 119867(119905 120579119871 119909119867 (119905))

(11a)

119867 (119905) = 119891

2(120579119871 (119905) 119909119867 (119905)) + 119866

2 (119905) (11b)

where

120579119871 (119905) = [120579119871

1

1205791198712

120579 119902]119879

119909119867= [119906 119908]

119879

(119905) = [1205791119904 1205790]119879

119901 (119909119867) = [119906

2+ 1205811119906 1199082+ 1205812119908]119879

1198911(120579119871 (119905)) =

[[[[[[[

[

1205791198712

minus119892

119897119909

sin 1205791198711

+119896119863119897119909

119872119871

(sign (119871) cos3 120579

1198711

+ sign (119871) sin3 120579

1198711

) 1205792

1198712

minus 1198961205791205791198712

119902

(11988321120579 + 119883

22119902)

]]]]]]]

]

1198661(120579119871 (119905)) =

[[[[[[[[[

[

0 0

119896119863sign (

119871) cos 120579

1198711

119872119871119897119909

119896119863sign (

119871) sin 120579

1198711

119872119871119897119909

0 0

0 0

]]]]]]]]]

]

119902 (120579119871 (119905) 119909119867 (119905))

=[[

[

0

cos 1205791198711

119897119909

+sin 1205791198711

119897119909

minus2119896119863

119872119871

(sign (119871) cos2 120579

1198711

119906 + sign (119871) sin2 120579

1198711

119908) 1205791198712

minus1205811119896119863sign (

119871) cos 120579

1198711

119906

119872119871119897119909

minus1205812119896119863sign (

119871) cos 120579

1198711

119908

119872119871119897119909

]]

]

119867(119905 120579119871 (119905) 119909119867 (119905)) =

[[[[[

[

0

0

(11988611119906 + 11988612119908)

11988323119906 + 119883

24119908

]]]]]

]

1198912(120579119871 (119905) 119909119867 (119905)) = [

(11988331120579 + 119883

32119902 + 119883

33119906 + 119883

34119908)

(11988341120579 + 119883

42119902 + 119883

43119906 + 119883

44119908)

]

1198662= [

1198831205791119904

1198831205790

1198851205791119904

1198851205790

]

(12)

Journal of Control Science and Engineering 5

Y

Z

ly

(XL YL ZL)

(X0 Y0 Z0)

120593L

Figure 2 Coordinate system for the lateralmotion in the119884-119885 plane

It is assumed that the longitudinal motion is primarilycontrolled by longitudinal cyclic commands (120579

1119904) and main

rotor collective 1205790

For the case of the lateral motion in the 119884-119885 plane withthe coordinate system described in Figure 2 the load modelis

119871119910119911

= minus119892

119897119910

sin120593119871119910119911

+

cos120593119871119910119911

119897119910

0+

sin120593119871119910119911

119897119910

0

+

119896119863sign (

119871) cos120593

119871119910119911

119872119871119897119910

2

0

+

119896119863sign (

119871) sin120593

119871119910119911

119872119871119897119910

2

0

minus2119896119863

119872119871

(sign (119871) cos2 120593

119871119910119911

0

+ sign (119871) sin2 120593

119871119910119911

0) 119871minus 119896120593119871

+119896119863119897119910

119872119871

[sign (119871) cos3 120593

119871119910119911

+ sign (119871) sin3 120593

119871119910119911

]

sdot 2

119871119910119911

(13)

Define 119871= [120593119871

119871]119879

= [1205931198711

1205931198712]119879 then the model can be

written as follows

1198711

= 1205931198712

(14a)

1198712

=minus119892

119897119910

sin1205931198711

+119896119863119897119910

119872119871

[sign (119871) cos3 120593

1198711

+ sign (119871) sin3 120593

1198711

] 1205932

1198712

minus 1198961205931205931198712

+ (119896119863sign (

119871) cos120593

1198711

119872119871119897119910

) 2

0

+ (119896119863sign (

119871) sin120593

1198711

119872119871119897119910

) 2

0+cos1205931198711

119897119910

0

+sin1205931198711

119897119910

0minus2119896119863

119872119871

(sign (119871) cos2 120593

1198711

0

+ sign (119871) sin2 120593

1198711

0) 1205931198712

(14b)

Now for the helicopter model the lateral rotational motion isdescribed by the roll angle 120601 and roll rate 119901 together with thetranslationmotion components V 119908 so the equation of lateralmotion can be written as follows

[[[[[

[

V

]]]]]

]

=

[[[[[[

[

0 1 0 0

0 119871119901

119871V 119871119908

119892 (119884119901minus 119908119890) 119884V 119884

119908

0 (119885119901minus V119890) 119885V 119885

119908

]]]]]]

]

[[[[[

[

120601

119901

V

119908

]]]]]

]

+

[[[[[

[

0 0

1198711205791119888

1198711205790119879

1198841205791119888

1198841205790119879

1198851205791119888

1198851205790119879

]]]]]

]

[1205791119888

1205790119879

]

(15)

now using a linear transformation 1198791such that 119909

119867(119905) =

1198791119911(119905) and 119879

1is defined by

1198791=

[[[[[

[

1 0 0 0

0 1 11988711

11988712

0 0 1 0

0 0 0 1

]]]]]

]

(16)

where

11988711

= minus1198851205791119888

1198711205790119879

minus 1198851205790119879

1198711205791119888

1198841205791119888

1198851205790119879

minus 1198851205791119888

1198841205790119879

11988712

= minus1198841205790119879

1198711205791119888

minus 1198841205791119888

1198711205790119879

1198841205791119888

1198851205790119879

minus 1198851205791119888

1198841205790119879

(17)

6 Journal of Control Science and Engineering

Let 119901 = (119901 minus 11988711V minus 11988712119908) then (15) can be written in the

following form

[[[[[[

[

V

]]]]]]

]

=

[[[[[

[

11988411

11988412

11988413

11988414

11988421

11988422

11988423

11988424

11988431

11988432

11988433

11988434

11988441

11988442

11988443

11988444

]]]]]

]

[[[[[

[

120601

119901

V

119908

]]]]]

]

+

[[[[[

[

0 0

11987321

11987322

1198841205791119888

1198841205790119879

1198851205791119888

1198851205790119879

]]]]]

]

[1205791119888

1205790119879

]

(18)

where

11988411

= 0

11988412

= 1

11988413

= 11988711

11988414

= 11988712

11988421

= minus (11988711119892)

11988422

= (119871119901minus 11988711(119884119901+ 119908119890) minus 11988712(119885119901minus V119890))

11988423

= (119871119901minus 11988711(119884119901+ 119908119890) minus 11988712(119885119901minus V119890)) 11988711

+ (119871119901minus 11988711119884V minus 119887

12119885V)

11988424

= (119871119901minus 11988711(119884119901+ 119908119890) minus 11988712(119885119901minus V119890)) 11988712

+ (119871119901minus 11988711119884119908minus 11988712119885119908)

11988431

= 119892

11988432

= (119884119901+ 119908119890)

11988433

= ((119884119901+ 119908119890) 11988711+ 119884V)

11988434

= ((119884119901+ 119908119890) 11988712+ 119884119908)

11988441

= 0

11988442

= (119885119901minus V119890)

11988443

= ((119885119901minus V119890) 11988711+ 119885V)

11988444

= ((119885119901minus V119890) 11988712+ 119885119908)

11987321

= (1198711205791119888

minus 119887111198841205791119888

minus 119887121198851205791119888

)

11987322

= (1198711205790119879

minus 119887111198841205790119879

minus 119887121198851205790119879

)

(19)

Using (18) the helicopter with underslung load for lateralmotion in the 119884-119885 plane can be described by

119871(119905)

= 1198911(120593119871(119905))

+ 1198661(120593119871(119905)) [119901 (119909

119867 (119905)) + 119902 (120593119871(119905) 119909119867 (119905))]

+ 119867 (119905 120593119871 119909119867 (119905))

(20a)

119867 (119905) = 119891

2(120593119871(119905) 119909119867 (119905)) + 119866

2 (119905) (20b)

where

119871(119905) = [

1198711

1198712

]119879

119901 (119909119867 (119905)) = [V2 119908

2]119879

1198911(120593119871(119905)) =

[[[[[[[

[

1205931198712

minus119892

119897119910

sin1205931198711

+119896119863119897119910

119872119871

(sign (119871) cos3 120593

1198711

+ sin (119871) sin3 120593

1198711

) 1205932

1198712

minus 1198961205931205931198712

119901

(minus11988711119892120601 + (119871

119901minus 11988711(119884119901+ 119908119890) minus 11988712(119885119901minus V119890)) 119901)

]]]]]]]

]

1198661(120593119871(119905)) =

[[[[[[[[

[

0 0

(119896119863sign (

119871) cos120593

1198711

119872119871119897119910

) (119896119863sign (

119871) sin120593

1198711

119872119871119897119910

)

0 0

0 0

]]]]]]]]

]

119902 (120593119871(119905) 119909119867 (119905)) =

[[

[

0

cos1205931198711

119897119910

V +sin1205931198711

119897119910

minus2119896119863

119872119871

(sign (119871) cos2 120593

1198711

V + sign (119871) sin2 120593

1198711

119908)1205931198712

]]

]

Journal of Control Science and Engineering 7

119867(119905 120593119871(119905) 119909119867 (119905)) =

[[[[[

[

0

0

( minus 119901)

(11988423V + 11988424119908)

]]]]]

]

(119905) = [1205791119888 1205790119879]119879

1198912(120593119871(119905) 119909119867 (119905)) = [

(11988431120601 + 11988432119901 + 11988433V + 11988434119908)

(11988442119901 + 11988443V + 11988444119908)

]

1198662= [

1198841205791119888

1198841205790119879

1198851205791119888

1198851205790119879

]

(21)

It is assumed that the lateral motion is primarily controlledby lateral cyclic commands (120579

1119888) and the tail rotor collective

1205790119879

3 Stability Analysis

The goal is to analyse the stability of the combined systemto ensure that it is possible to stabilise the helicopter withunderslung load system modelled by (11a) (11b) (20a) and(20b) in a real environment with uncertainties In this paperfor the stability analysis the Lyapunov second method isapplied for the helicopter with an underslung external loadsystem The analysis for the longitudinal motion is discussedfirst The system equations can be considered to have twomain parts that is known and unknown (or partly known)The known terms formed the nominal part of the systemmodelThe unknown or partly known part can be consideredas the uncertainty to the system The whole system is thenmodelled by a nominal part with the addition of uncertaintyIn fact the known elements in the subsystem (11a) arecharacterised by the prescribed triple (119891

1 1198661 119901) and it is

desired that the nominal part of the system is stableThe basic notations and concepts required for the analysis

are described first The state space is denoted by119883 fl R119899 andthe control space by119906 fl R119898 where 1 le 119898 le 119899TheEuclideaninner product (on 119883 or 119906 as appropriate) and induced normare denoted by ⟨sdot sdot⟩ and sdot respectively Let 119862(R119901R119902) and1198621(R119901R119902) denote the space of all continuous functions and

the space of continuous functions with continuous first-orderpartial derivatives respectively and let 119862infin(R119901R119902) denotethe space of functions whose partial derivatives of any orderexist and are continuous mapping R119901 rarr R119902 For a real-valued continuous scalar function 119909 rarr V(119909) defined onR119899 nabla V rarr nablaV isin R119899 denotes the gradient map The Liederivative of V along a vector field 119891 R119899 rarr R119899 is denoted by119871119891V R119899 rarr R which is defined by

(119871119891V) (119909) = ⟨nablaV (119909) 119891 (119909)⟩ (22)

The Lie bracket of vector fields 119891 119892 isin 119862infin(R119899R119899) is the

vector field [119891 119892] isin 119862infin(R119899R119899) defined by [119891 119892] = (119863119892)119891 minus

(119863119891)119892 where (119863119891)denotes the Jacobianmatrix of119891 and (119863119892)denotes the Jacobian matrix of 119892

In this paper nonlinear systemswith the following formatare considered

(119905) = 119891 (119909 (119905)) + 119866 (119909 (119905)) (119905) (23)

where 119909(119905) isin R119899 isin R119898 In general mathematical modelsof dynamical systems are usually imprecise due to modellingerrors and exogenous disturbances [12] Equation (23) can beconsidered as the nominal part of the system model and theuncertainty can be modelled as an additive perturbation tothe nominal systemmodel more specifically the structure ofthe system has the form

(119905) = 119891 (119909 (119905)) + 119866 (119909 (119905)) (119905) + 120599 (119909 (119905) 119906 (119905)) (24)

where 120599(119909(119905) 119906(119905))models the uncertainty in the systemSystem (24) is globally asymptotically stable to the zero

state if the system exhibits the following properties

(i) Existence and Continuation of Solutions For each 119909 isin R119899there exists a local solution119909 [0 119905

1) rarr R119899 (ie an absolutely

continuous function satisfying (24) almost everywhere (ae)and 119909(0) = 119909

0) and every such solution can be extended intoa solution on [0infin)

(ii) Boundedness of Solutions For each ℏ gt 0 there exists119903(ℏ) gt 0 such that 119909(119905) isin 119903(ℏ)119861

119899 for all 119905 ge 0 on every

solution 119909 [0infin) rarr R119899 with 1199090isin ℏ119861119899 where 119861

119899denote

the open unit ball centred at the origin in R119899

(iii) Stability of the State Origin For each 120575 gt 0 there exists119889(120575) gt 0 such that 119909(119905) isin 120575119861

119899for all 119905 ge 0 on every solution

119909 [0infin) rarr R119899 with 1199090isin 119889(120575)119861

119899

(iv) Global Attractivity of the State Origin For each ℏ gt 0 and120576 gt 0 there exists 119879(ℏ 120576) ge 0 such that 119909(119905) isin 119861

119899for all

119905 ge 119879(ℏ 120576) on every solution 119909 [0infin) rarr R119899 with 1199090isin ℏ119861119899

8 Journal of Control Science and Engineering

Consider a nonlinear system described by the ordinarydifferential equation as follows

(119905) = 119891 (119905 119909 (119905))

119909 (1199050) = 1199090

(25)

where 119891 R times 119883 rarr 119883 and 119891(119905 0) = 0 for all 119905To analyse the stability of (25) Lyapunovrsquos second stabilityanalysis method is applicable The Lyapunov approach is toshow that a candidate ldquoLyapunov functionrdquo is nonincreasingalong all solution to (25) by means that do not requireexplicit knowledge of solutions to (25) From this appropriateconclusion can be drawn regarding stability concepts relatingto solutions of the differential equation (25) An essentialpart of Lyapunovrsquos method is the determination of the timederivative of the candidate ldquoLyapunov functionrdquo along allsolution of the dynamical system

Consider a Lyapunov candidate (119905 119909) rarr V(119905 119909) Rtimes119883 rarr

R which satisfies the condition V isin 1198621(R times 119883) in which case

its time derivative along solutions to (25) is given by

V (119905 119909 (119905)) =120597V (119905 119909 (119905))

120597119905+ ⟨nablaV (119905 119909 (119905)) 119891 (119905 119909 (119905))⟩ (26)

for almost all 119905 isin RLet 119882(119909(119905)) denote a positive definite function If

V(119905 119909(119905)) satisfies

(i) V(119905 0) = 0 for all 119905 ge 0

(ii) 119882(119909(119905)) le V(119905 119909(119905)) for all 119909(119905) isin Φ 0 sub Φ sub 119883 andall 119905 ge 0

(iii) V(119905 119909(119905)) lt 0 in Φ

then V(119905 119909(119905)) is said to be a Lyapunov function in Φ IfV(119905 119909(119905)) in Φ then V(119905 119909(119905)) is said to be a weak Lyapunovfunction

A set Ε is said to be an invariant set with respect to thedynamical system = 119891(119909) if

119909 (0) isin Ε 997891997888rarr 119909 (119905) isin Ε forall119905 isin R+ (27)

In other words Ε is the set of points such that if a solution of = 119891(119909) belongs to Ε at some instant initialized points at119905 = 0 then it belongs to Ε for all future time

Now a setΕ sub 119909 is said to be a local invariantmanifold for(25) if for any 1199090 isin Ε 119909(119905) with 119909(0) = 119909

0 is in Ε for |119905| lt 119879

where 119879 gt 0 If 119879 = infin then Ε is said to be an invariantmanifold

Now considering the longitudinal motion (Figure 1) forthe helicopter with an underslung external load systemchoose a Lyapunov function candidate for the first subsystemV1as

V1(120579119871) =

1

2[12058911205792

1198711

+ 12058921205792

1198712

+ (1205893120579 minus 1205894119902)2+ 12058951199022] (28)

where 120589119894(119894 = 1 2 3 4 5) are design parameters to be

determined Then

V1(120579119871) = ⟨nablaV

1(120579119871) 1198911(120579119871) + 1198661(120579119871 (119905))

sdot [119901 (119909119867 (119905)) + 119902 (120579

119871 (119905) 119909119867 (119905))]

+ 119867(119905 120579119871 119909119867 (119905))⟩ = (119871

1198911

V1) (120579119871)

+ ⟨nablaV1(120579119871) 1198661(120579119871 (119905))

sdot [119901 (119909119867 (119905)) + 119902 (120579

119871 (119905) 119909119867 (119905))]

+ 119867(119905 120579119871 119909119867 (119905))⟩

(29)

From (28) nablaV1(120579119871) can be obtained as

nablaV1(120579119871)

= [12058911205791198711

12058921205791198712

1205893(1205893120579 minus 1205894119902) minus120589

4(1205893120579 minus 1205894119902) + 120589

5119902]119879

(30)

Substituting nablaV1(120579119871) and 119891

1(120579119871) into the derivative of

(1198711198911

V1)(120579119871) we have

(1198711198911

V1) (120579119871) = 12058911205791198711

1205791198712

minus 11989612057912058921205792

1198712

minus 1205892

119892

119897119909

sin 1205791198711

1205791198712

+119896119863119897119909

119872119871

1205892[sign (

119871) cos3 120579

1198711

+ sign (119871) sin3 120579

1198711

]

sdot 1205793

1198712

+ 1205893[1205893120579 minus 1205894119902] 119902 + (minus120589

31205894120579 + (120589

2

4+ 1205895) 119902)

sdot [11988321120579 + 119883

22119902]

(31)

Checking the term of minus sin (1205791198711

)1205791198712

it can be seen thatminus sin (120579

1198711

)1205791198712

le minus(2120587)1205791198711

1205791198712

when 1205791198711

and 1205791198712

are bothpositive or negative The situation of 120579

1198711

and 1205791198712

havingdifferent signs will help with the system stability Then

(1198711198911

V1) (120579119871) le 12058911205791198711

1205791198712

minus 11989612057912058921205792

1198712

minus 1205892

119892

119897119909

2

1205871205791198711

1205791198712

+119896119863119897119909

119872119871

sdot 1205892[sign (

119871) cos3 120579

1198711

+ sign (119871) sin3 120579

1198711

] 1205793

1198712

+ 1205893[1205893120579 minus 1205894119902] 119902 + (minus120589

31205894120579 + (120589

2

4+ 1205895) 119902)

sdot [11988321120579 + 119883

22119902]

(32)

If the design parameter 1205891is chosen as 120589

1= (2119892119897

119909120587)1205892

(1198711198911

V1)(120579119871) satisfies the following

(1198711198911

V1) (120579119871) le minus119896

12057912058921205792

1198712

+119896119863119897119909

119872119871

sdot 1205892[sign (

119871) cos3 120579

1198711

+ sign (119871) sin3 120579

1198711

] 1205793

1198712

+ 1205893[1205893120579 minus 1205894119902] 119902 + (minus120589

31205894120579 + (120589

2

4+ 1205895) 119902)

sdot [11988321120579 + 119883

22119902]

(33)

Journal of Control Science and Engineering 9

Further analysis on (33) will start from examining the firsttwo terms Rewrite these two terms So

minus 11989612057912058921205792

1198712

+119896119863119897119909

119872119871

1205892[sign (

119871) cos3 120579

1198711

+ sign (119871)

sdot sin3 1205791198711

] 1205793

1198712

= minus11989612057912058921205792

1198712

+119896119863119897119909

119872119871

12058921205791198712

[sign (119871)

sdot cos3 1205791198711

+ sign (119871) sin3 120579

1198711

] 1205792

1198712

le minus11989612057912058921205792

1198712

+119896119863119897119909

119872119871

1205892

10038161003816100381610038161003816max (120579

1198712

) (sign (119871) cos3 120579

1198711

+ sign (119871) sin3 120579

1198711

)100381610038161003816100381610038161205792

1198712

= minus12058921205792

1198712

119896120579

minus119896119863119897119909

119872119871

10038161003816100381610038161003816max (120579

1198712

)

sdot (sign (119871) cos3 120579

1198711

+ sign (119871) sin3 120579

1198711

)10038161003816100381610038161003816

(34)

If the hinge friction is big enough to satisfy the following

119896120579gt

119896119863119897119909

119872119871

10038161003816100381610038161003816max (120579

1198712

)

sdot (sign (119871) cos3 120579

1198711

+ sign (119871) sin3 120579

1198711

)10038161003816100381610038161003816

(35)

then

120582119871= 119896120579minus119896119863119897119909

119872119871

10038161003816100381610038161003816max (120579

1198712

)

sdot (sign (119871) cos3 120579

1198711

+ sign (119871) sin3 120579

1198711

)10038161003816100381610038161003816 ge 0

minus 11989612057912058921205792

1198712

+119896119863119897119909

119872119871

1205892[sign (

119871) cos3 120579

1198711

+ sign (119871)

sdot sin3 1205791198711

] 1205793

1198712

le minus12058921205792

1198712

120582119871

(36)

Now the following analysis has been conducted for (33) andthe load movement for the longitudinal motion is describedby

119871= 0minus 1198971199091205791198712

cos 1205791198711

119871= 0minus 1198971199091205791198712

sin 1205791198711

(37)

Around the hover condition1198830= 0 and 119885

0= 0 the signs for

119871and

119871are opposite to the one of 120579

1198712

provided that 1205791198711

ispositive

Considering the term (sign (119871) cos3 120579

1198711

+

sign (119871) sin3 120579

1198711

)1205793

1198712

for all possible combinations ofthe signs of 120579

1198712

and 1205791198711

the analysis follows below

(i) If 1205791198711

gt 0 and 1205791198712

gt 0 then sign (119871) = minus1

sign (119871) = minus1 cos3 120579

1198711

gt 0 and sin3 1205791198711

gt 0 There-fore (sign (

119871) cos3 120579

1198711

+ sign (119871) sin3 120579

1198711

)1205793

1198712

lt 0

(ii) If 1205791198711

lt 0 and 1205791198712

gt 0 then sign (119871) = minus1

sign (119871) = +1 cos3 120579

1198711

gt 0 and sin31205791198711

lt 0 So(sign (

119871) cos3 120579

1198711

+ sign (119871) sin3 120579

1198711

)1205793

1198712

lt 0

(iii) If 1205791198711

gt 0 and 1205791198712

lt 0 then sign (119871) = +1

and sign (119871) = +1 cos3 120579

1198711

gt 0 and sin3 1205791198711

gt

0 So the following is true (sign (119871) cos3 120579

1198711

+

sign (119871) sin3 120579

1198711

)1205793

1198712

lt 0

(iv) If 1205791198711

lt 0 and 1205791198712

lt 0 then sign (119871) = +1 and

sign (119871) = minus1 cos3 120579

1198711

gt 0 and sin3 1205791198711

lt 0 So(sign (

119871) cos3 120579

1198711

+ sign (119871) sin3 120579

1198711

)1205793

1198712

lt 0

Therefore

minus 11989612057912058921205792

1198712

+119896119863119897119909

119872119871

sdot 1205892[sign (

119871) cos3 120579

1198711

+ sign (119871) sin3 120579

1198711

] 1205793

1198712

le minus11989612057912058921205792

1198712

(38)

Now examining the rest of the terms of (1198711198911

V1)(120579119871) then

1205893[1205893120579 minus 1205894119902] 119902

+ (minus12058931205894120579 + (120589

2

4+ 1205895) 119902) [119883

21120579 + 119883

22119902] = 120589

2

3120579119902

minus 120589312058941199022minus 12058931205894119883211205792minus 1205893120589411988322120579119902

+ (1205892

4+ 1205895)11988321120579119902 + (120589

2

4+ 1205895)119883221199022

= minus [12058931205894minus (1205892

4+ 1205895)11988321] 1199022minus 12058931205894119883211205792

+ [1205892

3minus 1205893120589411988322+ (1205892

4+ 1205895)11988321] 120579119902

(39)

For the system 120579 and 1205791198711

always have different signs 120579 and 119902

always have different signs and 119902 and 1205791198711

have the same signsSo

[1205892

3minus 1205893120589411988322+ (1205892

4+ 1205895)11988321] 120579119902 le 0

if [1205892

3minus 1205893120589411988322+ (1205892

4+ 1205895)11988321] ge 0

(40)

Therefore

1205893[1205893120579 minus 1205894119902] 119902

+ (minus12058931205894120579 + (120589

2

4+ 1205895) 119902) [119883

21120579 + 119883

22119902]

le minus [12058931205894minus (1205892

4+ 1205895)11988321] 1199022minus 12058931205894119883211205792

(41)

Choose the design parameters to satisfy

[12058931205894minus (1205892

4+ 1205895)11988321] gt 0 (42)

If11988321

gt 0 then

minus [12058931205894minus (1205892

4+ 1205895)11988321] 1199022minus 12058931205894119883211205792lt 0 (43)

10 Journal of Control Science and Engineering

If11988321

lt 0 then

minus [12058931205894minus (1205892

4+ 1205895)11988321] 1199022minus 12058931205894119883211205792lt 0 (44)

in the region ofminus[12058931205894minus(1205892

4+1205895)11988321]1199022lt 12058931205894119883211205792 Since 120579 is

caused by the loadmotion here the value of 120579 should bemuchsmaller than 119902 The inequality (44) is easy to be satisfied

In summary of the above analysis and by defining

Ψ1(120579119871) = 11989612057912058921205792

1198712

+ [12058931205894minus (1205892

4+ 1205895)11988321] 1199022

+ 12058931205894119883211205792

Ψ2(120579119871) = [120589

31205894minus (1205892

4+ 1205895)11988321] 1199022+ 12058931205894119883211205792

(45)

the following lemma can be derived

Lemma 1 Defining a Lyapunov function (28) and choosingthe design parameters to satisfy 120589

1= (2119892119897

119909120587)1205892 [12058931205894minus (1205892

4+

1205895)11988321] gt 0 and [1205892

3minus1205893120589411988322+(1205892

4+1205895)11988321] ge 0 then within

the region specified by minus[12058931205894minus (1205892

4+ 1205895)11988321]1199022lt 12058931205894119883211205792

(i) V1(0) = 0 and V

1(120579119871) gt 0 forall120579

119871= 0

(ii) V1(120579119871) rarr infin as 120579

119871 rarr infin

(iii) (1198711198911

V1)(120579119871) le minusΨ

1(120579119871) forall120579119871 around the hover condi-

tion or (1198711198911

V1)(120579119871) le minusΨ

2(120579119871) forall120579119871 if (35) holds

Both functions Ψ1and Ψ

2are nonnegative

Recall the control term 119901(119909119867(119905)) in the first subsystem

of (11a) and (11b) it can be seen that [(119863119901)(119909119867)]minus1 exists

for all 119909119867(119905) The unknown vector fields 119902(120579

119871(119905) 119909119867) and

119867(119905 120579119871(119905) 119909119867) model the uncertainties imposed onto the

system Since 119902(120579119871(119905) 119909119867) is directly mapped into the ldquocon-

trolrdquo space of 119909119867(119905) it can be considered as a matched

uncertainty119867(119905 120579119871(119905) 119909119867) is unknown and it does belong to

the control space of 119909119867(119905) so it represents the mismatched

uncertainty in the system [12]Generally the range of the (longitudinal) load suspension

angle is within minus1205872 lt 120579119871lt 1205872 The helicopter velocities

and load suspension angle havemaximumoperational valuestherefore the uncertainties in the system are bounded Withthe Lyapunov function defined in (28)

(1198711198921

V1) (120579119871) = ⟨nablaV

1(120579119871) 1198921⟩ =

119896119863

119872119871119897119909

12058921205791198712

(46)

In 1198661(120579119871) the columns 119892

1and 119892

2are the same therefore

(1198711198922

V1) (120579119871) =

119896119863

119872119871119897119909

12058921205791198712

(47)

By considering the maximum values of cos (sdot) and sin (sdot)functions the bounding values relating to the uncertainty119902(120579119871(119905) 119909119867) can be estimated as follows

10038171003817100381710038171003817119902 (120579119871 119909119867)10038171003817100381710038171003817le

10038161003816100381610038161003816100381610038161003816

1

119897119909

( + )

+1

119872119871

(minus119896119863(2 +

120581

119897119909

) (119906 + 119908) +119896120579

119897119909

)1205791198712

10038161003816100381610038161003816100381610038161003816

le

10038161003816100381610038161003816100381610038161003816

119896120579

119897119909

1205791198712

10038161003816100381610038161003816100381610038161003816+

10038161003816100381610038161003816100381610038161003816

119896119863

119872119871

(2 +120581

119897119909

) (119906 + 119908)

10038161003816100381610038161003816100381610038161003816+

10038161003816100381610038161003816100381610038161003816

1

119897119909

( + )

10038161003816100381610038161003816100381610038161003816

le119896120579

119897119909

100381710038171003817100381710038171205791198712

10038171003817100381710038171003817+

119896119863

119872119871

(2 +120581

119897119909

)1003817100381710038171003817119901 (119909119867)1003817100381710038171003817 + 1205722 (119905)

le 1205831

10038161003816100381610038161003816(1198711198922

V1) (120579119871)10038161003816100381610038161003816+ 1205721

1003817100381710038171003817119901 (119909119867)1003817100381710038171003817 + 1205722 (119905)

(48)

where 1205831

= 119872119871119896120579119896119863 1205721

= (119896119863119872119871)(2 + 120581119897

119909) (120581 =

max [1205811 1205812]) and 120572

2(119905) is defined by 120572

2(119905) = max |(1119897

119909)( +

)| gt 0For the mismatched uncertainty 119867(119905 120579

119871(119905) 119909119867) the fol-

lowing analysis is conducted to obtain its bounding functionThe mismatched uncertainty can be rewritten as

119867(119905 120579119871 (119905) 119909119867) =

[[[[[

[

0

0

(11988611119906 + 11988612119908)

(11988323119906 + 119883

24119908)

]]]]]

]

=

[[[[[

[

0 0

0 0

11988611

11988612

11988323

11988324

]]]]]

]

[119906

119908]

(49)

Let 119860 = [1198861111988612

1198832311988324] then

10038171003817100381710038171003817119867 (119905 120579

119871 119909119867)10038171003817100381710038171003817le1003817100381710038171003817100381711986010038171003817100381710038171003817

1003817100381710038171003817119909119867 (119905)1003817100381710038171003817 (50)

Define a positive function as

1006704120579 (120579119871) = radic(1205792

1198711

+ 12057921198712

+ 1205792 + 1199022+ 120576) (51)

where 120576 is a very small positive constant Choosing1205731

= (1198721198711198971199091198961198631205892)119860 then the mismatched uncertainty

119867(119905 120579119871(119905) 119909119867) is bounded by the following inequality

10038171003817100381710038171003817119867 (119905 120579

119871 119909119867)10038171003817100381710038171003817

le 1006704120579minus1

(120579119871) 1205731

2

sum

119894=1

(10038161003816100381610038161003816(119871119892119894

V1) (120579119871)10038161003816100381610038161003816)1003816100381610038161003816119901119894 (119909119867)

1003816100381610038161003816

(52)

where 119892119894denotes the 119894th column of the matrix function

1198661(120579119871) and 119901

119894is the 119894th component of 119901(119909

119867) respectively As

1205731has a design parameter 120589

2involved it is easy to have 120573

1to

lead (52) to be true For the matched uncertainty we have

10038171003817100381710038171003817119902 (120579119871 119909119867)10038171003817100381710038171003817le

119896120579

119897119909

100381610038161003816100381610038161205791198712

10038161003816100381610038161003816+ 1205721

1003817100381710038171003817119901 (119909119867)1003817100381710038171003817 + 1205722 (119905) (53)

Journal of Control Science and Engineering 11

Following the above analysis the following lemma can bederived

Lemma 2 The uncertainties (52) and (53) are bounded andsatisfy if 120572

1 1205731ge 0 and 120572

1+ 1205731lt 1 for 120572

1= (119896119863119872119871)(2 +

120581119897119909) and 120573

1= (1198721198711198971199091198961198631205892)119860

4 Numerical Results

Using a simulation software the linearized model for a UH-60 helicopter is obtained with the following state and inputmatrices

119860 =

[[[[[[[[[[[[[[[[[[[

[

minus00000 00003 minus00000 minus00000 00000 minus00000 10000 minus00023 00503

minus00003 minus00000 minus00000 minus00000 minus00000 minus00000 minus00000 09989 00463

minus00001 00000 minus00000 minus00000 00000 00000 minus00000 minus00464 10002

minus19386 minus28593 minus00030 minus00195 minus00126 00168 minus12000 57076 02683

328694 minus03640 00027 00126 minus00311 minus00013 minus41348 minus22350 03713

03138 minus14034 00004 00150 00034 minus03340 02866 06307 23410

minus09144 minus04783 00017 00226 minus00246 00007 minus65893 minus21301 minus00598

00631 minus06744 00008 00029 00046 00025 02059 minus14255 minus01636

minus07549 minus00119 minus00002 00009 00028 00006 minus00749 minus01032 minus02214

]]]]]]]]]]]]]]]]]]]

]

119861 =

[[[[[[[[[[[[[[[[[[[

[

minus00000 00000 minus00000 00000

00000 00000 minus00000 minus00000

minus00000 00000 minus00000 minus00000

minus22190 05734 03170 minus00019

35381 minus00453 minus00963 02007

minus15629 00237 minus60740 minus01014

19943 minus00714 minus00541 00731

00446 minus01097 00553 minus00281

minus08197 minus00082 01573 minus00635

]]]]]]]]]]]]]]]]]]]

]

(54)

The desired condition 1205721+ 1205731lt 1 can be verified by consid-

ering a typical case that the UH-60 helicopter carries a loadweight 1000 lb by a 15 ft slung length and let 119896

119863= 75 lbft

For the UH-60 helicopter model with reference to thegeneral mathematical model presented in Section 2 we have1198831205791119904

= 05734 1198831205790

= 03170 1198851205791119904

= 00237 1198851205790

= minus607401198721205791119904

= minus01097 and 1198721205790

= 00553 So the followingparameters can be calculated to have the values 119886

11= minus02

and 11988612

= minus002 For 119883119906= minus00195 119885

119906= 00150 119872

119906=

00029 119872119902= minus14255 (119883

119902minus 119908119890) = 57076 and (119885

119902+ 119906119890) =

06307 we have 11988323

= 006 And also for 119883119908

= 00168119885119908

= minus03340 and 119872119908

= 00025 we have 11988324

= 001Based on all the above parameters 120590max(119860) = 00041 Inthis case let 120581 = 001 120572

1= (119896119863119872119871)(2 + 120581119897

119909) asymp 015

and 1205731= (119872

119871119897119909119896119863)120590max(119860) = 082 Therefore 120572

1+ 1205731=

015 + 082 = 097 which satisfies 1205721+ 1205731

lt 1 Anotherexample considers that the UH-60 helicopter carries a loadweight 500 lb by a 15 ft slung length and let 119896

119863= 50 lbft then

1205721= 02 and 120573

1= 062 therefore (120572

1+ 1205731) = 082 Therefore

this assumption 1205721+ 1205731

lt 1 is realistic for the systemdiscussed in here The numerical values vary with respectto the slung load configuration It may represent the system

dynamic variations To investigate the stability characteristicsof dynamic variations two series of simulation studies werecarried out

Firstly the slung length was kept constant at 15 ft andthe load varied between the limits 500ndash2000 lb with 119896

119863=

75 lbft producing the system poles listed in Table 1 Thecorresponding root loci are sketched in Figures 3 and 4Figure 4 clearly shows that the positions of the system polescrossed the imaginary axis as the load weight increased

Secondly load weight was kept constant and the slunglength varied between the limit of 10ndash20 ft Table 2 sum-marises the results of variation of the system poles and thecorresponding root loci are illustrated in Figures 5 and 6

The pole locus indicates that the system poles have thevariation trends of moving further towards right direction onthe 119904-plane with increase of load weight and slung lengthsFrom the simulation results it is clearly shown that thesystem becomes less stable when the load andor the slunglength increases The simulation results are compared withthe stability analysis presented above

Now consider the case of fixed slung length of 15 ft withdifferent load weights In the cases of a UH-60 helicopter that

12 Journal of Control Science and Engineering

Table 1 System poles locations for a fixed slung length (119897 = 15ft) but with different load weights

Helicopter 500 lb 1000 lb 1500 lb 2000 lbminus10919 minus11159 minus07848 minus06221 minus09369minus63938 minus64301 minus92203 minus79848 minus75613minus00032 minus00048 minus00034 minus00052 minus00052minus00977 minus00818 minus00888 minus00972 minus00694minus03045 minus03036 minus03394 minus03318 minus03333minus03159 plusmn 04363119894 minus03199 plusmn 04628119894 minus02295 plusmn 05050119894 minus02366 plusmn 05974119894 minus01834 plusmn 04968119894minus00489 plusmn 03898119894 minus00211 plusmn 03868119894 00074 plusmn 03445119894 00119 plusmn 04406119894 00264 plusmn 03199119894mdash minus00316 plusmn 15043119894 minus01020 plusmn 17380119894 minus00368 plusmn 18753119894 minus01160 plusmn 20003119894mdash minus00736 plusmn 15019119894 minus07662 plusmn 11680119894 minus07349 plusmn 19064119894 minus12367 plusmn 08995119894

Table 2 System poles for a fixed load (1000 lb) but with different slung lengths

Helicopter 10 ft 15 ft 20 ftminus10919 minus10560 minus07848 minus08687minus63938 minus99252 minus92203 minus85565minus00032 minus00061 minus00034 minus00114minus00977 minus02012 minus00888 minus00953minus03045 minus02606 minus03394 minus03479minus03159 plusmn 04363119894 minus01809 plusmn 06219119894 minus02295 plusmn 05050119894 minus01636 plusmn 06092119894minus00489 plusmn 03898119894 00643 plusmn 04577119894 00074 plusmn 03445119894 00799 plusmn 04594119894mdash minus01055 plusmn 19883119894 minus01020 plusmn 17380119894 minus01247 plusmn 14470119894mdash minus08847 plusmn 12464119894 minus07662 plusmn 11680119894 minus06352 plusmn 10384119894

Re(poles)

Im(poles)

minus10 minus5 0 5minus08minus06minus04minus02

002040608

Figure 3 System root-locus diagram showing pole locus forconstant slung length (15 ft) as load weight is increased

minus04 minus035 minus03 minus025 minus02 minus015 minus01 minus005 0 005 01

Re(poles)

Im(poles)

minus08minus06minus04minus02

002040608

Figure 4 Enlarged view of Figure 3 showing poles locus near toRe(poles) = 0 axis

minus10 minus8 minus6 minus4 minus2 0 2

Re(poles)

Im(poles)

minus08minus06minus04minus02

002040608

Figure 5 System root-locus showing pole locus for constant loadweight (1000 lb) as slung length is increased

minus04 minus035 minus03 minus025 minus02 minus015 minus01 minus005 0 005 01

Re(poles)

Im(poles)

minus08minus06minus04minus02

002040608

Figure 6 Enlarged view of Figure 5 showing poles locus near toRe(poles) = 0 axis

Journal of Control Science and Engineering 13

carries a loadweight 500 lb and 1000 lb then1205721= 03 and 041

with 1205731= 041 and 082 respectively therefore 120572

1+ 1205731lt 1

thus the system is stable But when the load weight is 1500 lband 2000 lb then 120572

1= 01 and 0075 and also 120573

1= 123 and

164 respectively therefore 1205721+ 1205731gt 1 thus the system is

unstable For the case of the constant load weight (1000 lb) asslung length is increased then for a 10 ft and 15 ft slung length1205721= 015 and 120573

1= 055 and 082 respectively therefore

1205721+ 1205731lt 1 thus the system is stable But when the slung

length increased to 20 ft then 1205721

= 015 and 1205731

= 109therefore (120572

1+ 1205731) = 124 that is 120572

1+ 1205731gt 1 thus the

system is unstable However the system can be marginallystable if the UH-60 helicopter carries a load weight 725 lb bya 20 ft slung length then 120572

1= 02 and 120573

1= 08 therefore

(1205721+ 1205731) = 1 For this set of analysis 119896

119863is assumed to

be 119896119863

= 75 lbft However 119896119863can vary depending on the

weather condition and an example of 119896119863= 50 lbft is shown

above for illustrative purpose This analysis suggests that themaximum value of 20 ft slung length may be used for slungload operation However the load weight limit can vary withrespect to the capacity of the load carrying hooks helicopterweight environmental conditions and so forth

The examples presented here show that the simulationresults are correlated with the stability analysis Thus desiredcondition for the stability is met and it can locate the load atthe specified position or its neighbourhood

5 Conclusions

In this paper stability analysis of a helicopter with an under-slung load system is investigatedThe investigation identifiedthe conditions for the stabilisation of the system and thepositioning of the underslung load at hover condition Sta-bility analysis and numerical results proved that the desiredcondition for the stability is met then it is possible to locatethe load at the specified position or its neighbourhood Anillustration example is also given in the paper The methodresults in a guaranteed load positioning accuracy whichdepends on the design parameters and the method can beapplicable to any individual helicopter with external loadsystem

Nomenclature

119871119872119873 Overall helicopter rolling pitching andyawing moments

119901 119902 119903 Helicopter roll pitch and yaw rates aboutbody reference axes

119883119884 119885 Overall helicopter force components120601 120579 120595 Roll pitch and yaw angles119906 V 119908 Helicopter velocity components at centre

of gravity1205790 Main rotor collective

1205791119904 Longitudinal cyclic commands

1205791119888 Lateral cyclic commands

1205790119879 Tail rotor collective

1198830 1198840 1198850 Location of the suspension point withrespect to earth referenced 119909 119910 and 119911

directions

0 0 0 The helicopter velocity in the 119909 119910 and 119911

directions0 0 0 The helicopter acceleration in the 119909 119910 and119911 directions

120579119871 Load suspension angle in the119883-119885 plane

with respect to 119885-axis120601119871 Load suspension angle in the 119884-119885 plane

with respect to 119885-axis119872119871 Mass of the suspended load

119892 Acceleration due to gravity119897 Slung length119896119863 Aerodynamic drag force constant

(119896119863= (12)120588119878119862

119863)

120588 Air density119878 The load area presented to the airflow119862119863 Drag coefficient for the load

Competing Interests

The author declares that there are no competing interestsregarding the publication of this paper

References

[1] K Thanapalan and F Zhang ldquoModelling and simulation studyof a helicopter with an external slung load systemrdquo I-ManagersJournal on Instrumentation amp Control Engineering vol 1 no 1pp 9ndash17 2013

[2] M Bisgaard J D Bendtsen and A La Cour-Harbo ldquoModelingof generic slung load systemrdquo Journal of Guidance Control andDynamics vol 32 no 2 pp 573ndash585 2009

[3] Handling Qualities Requirements for Military Rotorcraft ADS-33-D US Army Aviation and Troop Command St Louis MissUSA 1996

[4] K Thanapalan and T M Wong ldquoModeing of a helicopter withan under-slung load systemrdquo in Proceedings of the 29th ChineseControl Conference (CCC rsquo10) pp 1451ndash1456 Beijing China July2010

[5] B Feng ldquoRobust control for lateral and longitudinal channels ofsmall-scale unmanned helicoptersrdquo Journal of Control Scienceand Engineering vol 2015 Article ID 483096 8 pages 2015

[6] J J Gribble ldquoLinear quadratic Gaussianloop transfer recoverydesign for a helicopter in low-speed flightrdquo Journal of GuidanceControl and Dynamics vol 16 no 4 pp 754ndash761 1993

[7] M A Manness J J Gribble and D J M Smith ldquoMultivariablemethods for helicopter flight control law designrdquo in Proceedingsof the 16th Annual Society European Rotorcraft Forum PaperNoIII52 Glasgow UK September 1990

[8] W L Garrard E Low and S Prouty ldquoDesign of attitude andrate command systems for helicopters using eigenstructureassignmentrdquo Journal of Guidance Control and Dynamics vol12 no 6 pp 783ndash791 1989

[9] Y B Shtessel and I A Shkolnikov ldquoAeronautical and spacevehicle control in dynamic sliding manifoldsrdquo InternationalJournal of Control vol 76 no 9-10 pp 1000ndash1017 2003

[10] D FThompson J S Pruyn andA Shukla ldquoFeedback design forrobust tracking and robust stiffness in flight control actuatorsusing a modified QFT techniquerdquo International Journal ofControl vol 72 no 16 pp 1480ndash1497 1999

14 Journal of Control Science and Engineering

[11] K Thanapalan ldquoModelling of a helicopter systemrdquo in Proceed-ings of the 1st Virtual Control Conference (VCC rsquo10) pp 45ndash52Aalborg University 2010

[12] D P Goodall and J Wang ldquoStabilization of a class of uncertainnonlinear affine systems subject to control constraintsrdquo Interna-tional Journal of Robust and Nonlinear Control vol 11 no 9 pp797ndash818 2001

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Journal of Control Science and Engineering 5

Y

Z

ly

(XL YL ZL)

(X0 Y0 Z0)

120593L

Figure 2 Coordinate system for the lateralmotion in the119884-119885 plane

It is assumed that the longitudinal motion is primarilycontrolled by longitudinal cyclic commands (120579

1119904) and main

rotor collective 1205790

For the case of the lateral motion in the 119884-119885 plane withthe coordinate system described in Figure 2 the load modelis

119871119910119911

= minus119892

119897119910

sin120593119871119910119911

+

cos120593119871119910119911

119897119910

0+

sin120593119871119910119911

119897119910

0

+

119896119863sign (

119871) cos120593

119871119910119911

119872119871119897119910

2

0

+

119896119863sign (

119871) sin120593

119871119910119911

119872119871119897119910

2

0

minus2119896119863

119872119871

(sign (119871) cos2 120593

119871119910119911

0

+ sign (119871) sin2 120593

119871119910119911

0) 119871minus 119896120593119871

+119896119863119897119910

119872119871

[sign (119871) cos3 120593

119871119910119911

+ sign (119871) sin3 120593

119871119910119911

]

sdot 2

119871119910119911

(13)

Define 119871= [120593119871

119871]119879

= [1205931198711

1205931198712]119879 then the model can be

written as follows

1198711

= 1205931198712

(14a)

1198712

=minus119892

119897119910

sin1205931198711

+119896119863119897119910

119872119871

[sign (119871) cos3 120593

1198711

+ sign (119871) sin3 120593

1198711

] 1205932

1198712

minus 1198961205931205931198712

+ (119896119863sign (

119871) cos120593

1198711

119872119871119897119910

) 2

0

+ (119896119863sign (

119871) sin120593

1198711

119872119871119897119910

) 2

0+cos1205931198711

119897119910

0

+sin1205931198711

119897119910

0minus2119896119863

119872119871

(sign (119871) cos2 120593

1198711

0

+ sign (119871) sin2 120593

1198711

0) 1205931198712

(14b)

Now for the helicopter model the lateral rotational motion isdescribed by the roll angle 120601 and roll rate 119901 together with thetranslationmotion components V 119908 so the equation of lateralmotion can be written as follows

[[[[[

[

V

]]]]]

]

=

[[[[[[

[

0 1 0 0

0 119871119901

119871V 119871119908

119892 (119884119901minus 119908119890) 119884V 119884

119908

0 (119885119901minus V119890) 119885V 119885

119908

]]]]]]

]

[[[[[

[

120601

119901

V

119908

]]]]]

]

+

[[[[[

[

0 0

1198711205791119888

1198711205790119879

1198841205791119888

1198841205790119879

1198851205791119888

1198851205790119879

]]]]]

]

[1205791119888

1205790119879

]

(15)

now using a linear transformation 1198791such that 119909

119867(119905) =

1198791119911(119905) and 119879

1is defined by

1198791=

[[[[[

[

1 0 0 0

0 1 11988711

11988712

0 0 1 0

0 0 0 1

]]]]]

]

(16)

where

11988711

= minus1198851205791119888

1198711205790119879

minus 1198851205790119879

1198711205791119888

1198841205791119888

1198851205790119879

minus 1198851205791119888

1198841205790119879

11988712

= minus1198841205790119879

1198711205791119888

minus 1198841205791119888

1198711205790119879

1198841205791119888

1198851205790119879

minus 1198851205791119888

1198841205790119879

(17)

6 Journal of Control Science and Engineering

Let 119901 = (119901 minus 11988711V minus 11988712119908) then (15) can be written in the

following form

[[[[[[

[

V

]]]]]]

]

=

[[[[[

[

11988411

11988412

11988413

11988414

11988421

11988422

11988423

11988424

11988431

11988432

11988433

11988434

11988441

11988442

11988443

11988444

]]]]]

]

[[[[[

[

120601

119901

V

119908

]]]]]

]

+

[[[[[

[

0 0

11987321

11987322

1198841205791119888

1198841205790119879

1198851205791119888

1198851205790119879

]]]]]

]

[1205791119888

1205790119879

]

(18)

where

11988411

= 0

11988412

= 1

11988413

= 11988711

11988414

= 11988712

11988421

= minus (11988711119892)

11988422

= (119871119901minus 11988711(119884119901+ 119908119890) minus 11988712(119885119901minus V119890))

11988423

= (119871119901minus 11988711(119884119901+ 119908119890) minus 11988712(119885119901minus V119890)) 11988711

+ (119871119901minus 11988711119884V minus 119887

12119885V)

11988424

= (119871119901minus 11988711(119884119901+ 119908119890) minus 11988712(119885119901minus V119890)) 11988712

+ (119871119901minus 11988711119884119908minus 11988712119885119908)

11988431

= 119892

11988432

= (119884119901+ 119908119890)

11988433

= ((119884119901+ 119908119890) 11988711+ 119884V)

11988434

= ((119884119901+ 119908119890) 11988712+ 119884119908)

11988441

= 0

11988442

= (119885119901minus V119890)

11988443

= ((119885119901minus V119890) 11988711+ 119885V)

11988444

= ((119885119901minus V119890) 11988712+ 119885119908)

11987321

= (1198711205791119888

minus 119887111198841205791119888

minus 119887121198851205791119888

)

11987322

= (1198711205790119879

minus 119887111198841205790119879

minus 119887121198851205790119879

)

(19)

Using (18) the helicopter with underslung load for lateralmotion in the 119884-119885 plane can be described by

119871(119905)

= 1198911(120593119871(119905))

+ 1198661(120593119871(119905)) [119901 (119909

119867 (119905)) + 119902 (120593119871(119905) 119909119867 (119905))]

+ 119867 (119905 120593119871 119909119867 (119905))

(20a)

119867 (119905) = 119891

2(120593119871(119905) 119909119867 (119905)) + 119866

2 (119905) (20b)

where

119871(119905) = [

1198711

1198712

]119879

119901 (119909119867 (119905)) = [V2 119908

2]119879

1198911(120593119871(119905)) =

[[[[[[[

[

1205931198712

minus119892

119897119910

sin1205931198711

+119896119863119897119910

119872119871

(sign (119871) cos3 120593

1198711

+ sin (119871) sin3 120593

1198711

) 1205932

1198712

minus 1198961205931205931198712

119901

(minus11988711119892120601 + (119871

119901minus 11988711(119884119901+ 119908119890) minus 11988712(119885119901minus V119890)) 119901)

]]]]]]]

]

1198661(120593119871(119905)) =

[[[[[[[[

[

0 0

(119896119863sign (

119871) cos120593

1198711

119872119871119897119910

) (119896119863sign (

119871) sin120593

1198711

119872119871119897119910

)

0 0

0 0

]]]]]]]]

]

119902 (120593119871(119905) 119909119867 (119905)) =

[[

[

0

cos1205931198711

119897119910

V +sin1205931198711

119897119910

minus2119896119863

119872119871

(sign (119871) cos2 120593

1198711

V + sign (119871) sin2 120593

1198711

119908)1205931198712

]]

]

Journal of Control Science and Engineering 7

119867(119905 120593119871(119905) 119909119867 (119905)) =

[[[[[

[

0

0

( minus 119901)

(11988423V + 11988424119908)

]]]]]

]

(119905) = [1205791119888 1205790119879]119879

1198912(120593119871(119905) 119909119867 (119905)) = [

(11988431120601 + 11988432119901 + 11988433V + 11988434119908)

(11988442119901 + 11988443V + 11988444119908)

]

1198662= [

1198841205791119888

1198841205790119879

1198851205791119888

1198851205790119879

]

(21)

It is assumed that the lateral motion is primarily controlledby lateral cyclic commands (120579

1119888) and the tail rotor collective

1205790119879

3 Stability Analysis

The goal is to analyse the stability of the combined systemto ensure that it is possible to stabilise the helicopter withunderslung load system modelled by (11a) (11b) (20a) and(20b) in a real environment with uncertainties In this paperfor the stability analysis the Lyapunov second method isapplied for the helicopter with an underslung external loadsystem The analysis for the longitudinal motion is discussedfirst The system equations can be considered to have twomain parts that is known and unknown (or partly known)The known terms formed the nominal part of the systemmodelThe unknown or partly known part can be consideredas the uncertainty to the system The whole system is thenmodelled by a nominal part with the addition of uncertaintyIn fact the known elements in the subsystem (11a) arecharacterised by the prescribed triple (119891

1 1198661 119901) and it is

desired that the nominal part of the system is stableThe basic notations and concepts required for the analysis

are described first The state space is denoted by119883 fl R119899 andthe control space by119906 fl R119898 where 1 le 119898 le 119899TheEuclideaninner product (on 119883 or 119906 as appropriate) and induced normare denoted by ⟨sdot sdot⟩ and sdot respectively Let 119862(R119901R119902) and1198621(R119901R119902) denote the space of all continuous functions and

the space of continuous functions with continuous first-orderpartial derivatives respectively and let 119862infin(R119901R119902) denotethe space of functions whose partial derivatives of any orderexist and are continuous mapping R119901 rarr R119902 For a real-valued continuous scalar function 119909 rarr V(119909) defined onR119899 nabla V rarr nablaV isin R119899 denotes the gradient map The Liederivative of V along a vector field 119891 R119899 rarr R119899 is denoted by119871119891V R119899 rarr R which is defined by

(119871119891V) (119909) = ⟨nablaV (119909) 119891 (119909)⟩ (22)

The Lie bracket of vector fields 119891 119892 isin 119862infin(R119899R119899) is the

vector field [119891 119892] isin 119862infin(R119899R119899) defined by [119891 119892] = (119863119892)119891 minus

(119863119891)119892 where (119863119891)denotes the Jacobianmatrix of119891 and (119863119892)denotes the Jacobian matrix of 119892

In this paper nonlinear systemswith the following formatare considered

(119905) = 119891 (119909 (119905)) + 119866 (119909 (119905)) (119905) (23)

where 119909(119905) isin R119899 isin R119898 In general mathematical modelsof dynamical systems are usually imprecise due to modellingerrors and exogenous disturbances [12] Equation (23) can beconsidered as the nominal part of the system model and theuncertainty can be modelled as an additive perturbation tothe nominal systemmodel more specifically the structure ofthe system has the form

(119905) = 119891 (119909 (119905)) + 119866 (119909 (119905)) (119905) + 120599 (119909 (119905) 119906 (119905)) (24)

where 120599(119909(119905) 119906(119905))models the uncertainty in the systemSystem (24) is globally asymptotically stable to the zero

state if the system exhibits the following properties

(i) Existence and Continuation of Solutions For each 119909 isin R119899there exists a local solution119909 [0 119905

1) rarr R119899 (ie an absolutely

continuous function satisfying (24) almost everywhere (ae)and 119909(0) = 119909

0) and every such solution can be extended intoa solution on [0infin)

(ii) Boundedness of Solutions For each ℏ gt 0 there exists119903(ℏ) gt 0 such that 119909(119905) isin 119903(ℏ)119861

119899 for all 119905 ge 0 on every

solution 119909 [0infin) rarr R119899 with 1199090isin ℏ119861119899 where 119861

119899denote

the open unit ball centred at the origin in R119899

(iii) Stability of the State Origin For each 120575 gt 0 there exists119889(120575) gt 0 such that 119909(119905) isin 120575119861

119899for all 119905 ge 0 on every solution

119909 [0infin) rarr R119899 with 1199090isin 119889(120575)119861

119899

(iv) Global Attractivity of the State Origin For each ℏ gt 0 and120576 gt 0 there exists 119879(ℏ 120576) ge 0 such that 119909(119905) isin 119861

119899for all

119905 ge 119879(ℏ 120576) on every solution 119909 [0infin) rarr R119899 with 1199090isin ℏ119861119899

8 Journal of Control Science and Engineering

Consider a nonlinear system described by the ordinarydifferential equation as follows

(119905) = 119891 (119905 119909 (119905))

119909 (1199050) = 1199090

(25)

where 119891 R times 119883 rarr 119883 and 119891(119905 0) = 0 for all 119905To analyse the stability of (25) Lyapunovrsquos second stabilityanalysis method is applicable The Lyapunov approach is toshow that a candidate ldquoLyapunov functionrdquo is nonincreasingalong all solution to (25) by means that do not requireexplicit knowledge of solutions to (25) From this appropriateconclusion can be drawn regarding stability concepts relatingto solutions of the differential equation (25) An essentialpart of Lyapunovrsquos method is the determination of the timederivative of the candidate ldquoLyapunov functionrdquo along allsolution of the dynamical system

Consider a Lyapunov candidate (119905 119909) rarr V(119905 119909) Rtimes119883 rarr

R which satisfies the condition V isin 1198621(R times 119883) in which case

its time derivative along solutions to (25) is given by

V (119905 119909 (119905)) =120597V (119905 119909 (119905))

120597119905+ ⟨nablaV (119905 119909 (119905)) 119891 (119905 119909 (119905))⟩ (26)

for almost all 119905 isin RLet 119882(119909(119905)) denote a positive definite function If

V(119905 119909(119905)) satisfies

(i) V(119905 0) = 0 for all 119905 ge 0

(ii) 119882(119909(119905)) le V(119905 119909(119905)) for all 119909(119905) isin Φ 0 sub Φ sub 119883 andall 119905 ge 0

(iii) V(119905 119909(119905)) lt 0 in Φ

then V(119905 119909(119905)) is said to be a Lyapunov function in Φ IfV(119905 119909(119905)) in Φ then V(119905 119909(119905)) is said to be a weak Lyapunovfunction

A set Ε is said to be an invariant set with respect to thedynamical system = 119891(119909) if

119909 (0) isin Ε 997891997888rarr 119909 (119905) isin Ε forall119905 isin R+ (27)

In other words Ε is the set of points such that if a solution of = 119891(119909) belongs to Ε at some instant initialized points at119905 = 0 then it belongs to Ε for all future time

Now a setΕ sub 119909 is said to be a local invariantmanifold for(25) if for any 1199090 isin Ε 119909(119905) with 119909(0) = 119909

0 is in Ε for |119905| lt 119879

where 119879 gt 0 If 119879 = infin then Ε is said to be an invariantmanifold

Now considering the longitudinal motion (Figure 1) forthe helicopter with an underslung external load systemchoose a Lyapunov function candidate for the first subsystemV1as

V1(120579119871) =

1

2[12058911205792

1198711

+ 12058921205792

1198712

+ (1205893120579 minus 1205894119902)2+ 12058951199022] (28)

where 120589119894(119894 = 1 2 3 4 5) are design parameters to be

determined Then

V1(120579119871) = ⟨nablaV

1(120579119871) 1198911(120579119871) + 1198661(120579119871 (119905))

sdot [119901 (119909119867 (119905)) + 119902 (120579

119871 (119905) 119909119867 (119905))]

+ 119867(119905 120579119871 119909119867 (119905))⟩ = (119871

1198911

V1) (120579119871)

+ ⟨nablaV1(120579119871) 1198661(120579119871 (119905))

sdot [119901 (119909119867 (119905)) + 119902 (120579

119871 (119905) 119909119867 (119905))]

+ 119867(119905 120579119871 119909119867 (119905))⟩

(29)

From (28) nablaV1(120579119871) can be obtained as

nablaV1(120579119871)

= [12058911205791198711

12058921205791198712

1205893(1205893120579 minus 1205894119902) minus120589

4(1205893120579 minus 1205894119902) + 120589

5119902]119879

(30)

Substituting nablaV1(120579119871) and 119891

1(120579119871) into the derivative of

(1198711198911

V1)(120579119871) we have

(1198711198911

V1) (120579119871) = 12058911205791198711

1205791198712

minus 11989612057912058921205792

1198712

minus 1205892

119892

119897119909

sin 1205791198711

1205791198712

+119896119863119897119909

119872119871

1205892[sign (

119871) cos3 120579

1198711

+ sign (119871) sin3 120579

1198711

]

sdot 1205793

1198712

+ 1205893[1205893120579 minus 1205894119902] 119902 + (minus120589

31205894120579 + (120589

2

4+ 1205895) 119902)

sdot [11988321120579 + 119883

22119902]

(31)

Checking the term of minus sin (1205791198711

)1205791198712

it can be seen thatminus sin (120579

1198711

)1205791198712

le minus(2120587)1205791198711

1205791198712

when 1205791198711

and 1205791198712

are bothpositive or negative The situation of 120579

1198711

and 1205791198712

havingdifferent signs will help with the system stability Then

(1198711198911

V1) (120579119871) le 12058911205791198711

1205791198712

minus 11989612057912058921205792

1198712

minus 1205892

119892

119897119909

2

1205871205791198711

1205791198712

+119896119863119897119909

119872119871

sdot 1205892[sign (

119871) cos3 120579

1198711

+ sign (119871) sin3 120579

1198711

] 1205793

1198712

+ 1205893[1205893120579 minus 1205894119902] 119902 + (minus120589

31205894120579 + (120589

2

4+ 1205895) 119902)

sdot [11988321120579 + 119883

22119902]

(32)

If the design parameter 1205891is chosen as 120589

1= (2119892119897

119909120587)1205892

(1198711198911

V1)(120579119871) satisfies the following

(1198711198911

V1) (120579119871) le minus119896

12057912058921205792

1198712

+119896119863119897119909

119872119871

sdot 1205892[sign (

119871) cos3 120579

1198711

+ sign (119871) sin3 120579

1198711

] 1205793

1198712

+ 1205893[1205893120579 minus 1205894119902] 119902 + (minus120589

31205894120579 + (120589

2

4+ 1205895) 119902)

sdot [11988321120579 + 119883

22119902]

(33)

Journal of Control Science and Engineering 9

Further analysis on (33) will start from examining the firsttwo terms Rewrite these two terms So

minus 11989612057912058921205792

1198712

+119896119863119897119909

119872119871

1205892[sign (

119871) cos3 120579

1198711

+ sign (119871)

sdot sin3 1205791198711

] 1205793

1198712

= minus11989612057912058921205792

1198712

+119896119863119897119909

119872119871

12058921205791198712

[sign (119871)

sdot cos3 1205791198711

+ sign (119871) sin3 120579

1198711

] 1205792

1198712

le minus11989612057912058921205792

1198712

+119896119863119897119909

119872119871

1205892

10038161003816100381610038161003816max (120579

1198712

) (sign (119871) cos3 120579

1198711

+ sign (119871) sin3 120579

1198711

)100381610038161003816100381610038161205792

1198712

= minus12058921205792

1198712

119896120579

minus119896119863119897119909

119872119871

10038161003816100381610038161003816max (120579

1198712

)

sdot (sign (119871) cos3 120579

1198711

+ sign (119871) sin3 120579

1198711

)10038161003816100381610038161003816

(34)

If the hinge friction is big enough to satisfy the following

119896120579gt

119896119863119897119909

119872119871

10038161003816100381610038161003816max (120579

1198712

)

sdot (sign (119871) cos3 120579

1198711

+ sign (119871) sin3 120579

1198711

)10038161003816100381610038161003816

(35)

then

120582119871= 119896120579minus119896119863119897119909

119872119871

10038161003816100381610038161003816max (120579

1198712

)

sdot (sign (119871) cos3 120579

1198711

+ sign (119871) sin3 120579

1198711

)10038161003816100381610038161003816 ge 0

minus 11989612057912058921205792

1198712

+119896119863119897119909

119872119871

1205892[sign (

119871) cos3 120579

1198711

+ sign (119871)

sdot sin3 1205791198711

] 1205793

1198712

le minus12058921205792

1198712

120582119871

(36)

Now the following analysis has been conducted for (33) andthe load movement for the longitudinal motion is describedby

119871= 0minus 1198971199091205791198712

cos 1205791198711

119871= 0minus 1198971199091205791198712

sin 1205791198711

(37)

Around the hover condition1198830= 0 and 119885

0= 0 the signs for

119871and

119871are opposite to the one of 120579

1198712

provided that 1205791198711

ispositive

Considering the term (sign (119871) cos3 120579

1198711

+

sign (119871) sin3 120579

1198711

)1205793

1198712

for all possible combinations ofthe signs of 120579

1198712

and 1205791198711

the analysis follows below

(i) If 1205791198711

gt 0 and 1205791198712

gt 0 then sign (119871) = minus1

sign (119871) = minus1 cos3 120579

1198711

gt 0 and sin3 1205791198711

gt 0 There-fore (sign (

119871) cos3 120579

1198711

+ sign (119871) sin3 120579

1198711

)1205793

1198712

lt 0

(ii) If 1205791198711

lt 0 and 1205791198712

gt 0 then sign (119871) = minus1

sign (119871) = +1 cos3 120579

1198711

gt 0 and sin31205791198711

lt 0 So(sign (

119871) cos3 120579

1198711

+ sign (119871) sin3 120579

1198711

)1205793

1198712

lt 0

(iii) If 1205791198711

gt 0 and 1205791198712

lt 0 then sign (119871) = +1

and sign (119871) = +1 cos3 120579

1198711

gt 0 and sin3 1205791198711

gt

0 So the following is true (sign (119871) cos3 120579

1198711

+

sign (119871) sin3 120579

1198711

)1205793

1198712

lt 0

(iv) If 1205791198711

lt 0 and 1205791198712

lt 0 then sign (119871) = +1 and

sign (119871) = minus1 cos3 120579

1198711

gt 0 and sin3 1205791198711

lt 0 So(sign (

119871) cos3 120579

1198711

+ sign (119871) sin3 120579

1198711

)1205793

1198712

lt 0

Therefore

minus 11989612057912058921205792

1198712

+119896119863119897119909

119872119871

sdot 1205892[sign (

119871) cos3 120579

1198711

+ sign (119871) sin3 120579

1198711

] 1205793

1198712

le minus11989612057912058921205792

1198712

(38)

Now examining the rest of the terms of (1198711198911

V1)(120579119871) then

1205893[1205893120579 minus 1205894119902] 119902

+ (minus12058931205894120579 + (120589

2

4+ 1205895) 119902) [119883

21120579 + 119883

22119902] = 120589

2

3120579119902

minus 120589312058941199022minus 12058931205894119883211205792minus 1205893120589411988322120579119902

+ (1205892

4+ 1205895)11988321120579119902 + (120589

2

4+ 1205895)119883221199022

= minus [12058931205894minus (1205892

4+ 1205895)11988321] 1199022minus 12058931205894119883211205792

+ [1205892

3minus 1205893120589411988322+ (1205892

4+ 1205895)11988321] 120579119902

(39)

For the system 120579 and 1205791198711

always have different signs 120579 and 119902

always have different signs and 119902 and 1205791198711

have the same signsSo

[1205892

3minus 1205893120589411988322+ (1205892

4+ 1205895)11988321] 120579119902 le 0

if [1205892

3minus 1205893120589411988322+ (1205892

4+ 1205895)11988321] ge 0

(40)

Therefore

1205893[1205893120579 minus 1205894119902] 119902

+ (minus12058931205894120579 + (120589

2

4+ 1205895) 119902) [119883

21120579 + 119883

22119902]

le minus [12058931205894minus (1205892

4+ 1205895)11988321] 1199022minus 12058931205894119883211205792

(41)

Choose the design parameters to satisfy

[12058931205894minus (1205892

4+ 1205895)11988321] gt 0 (42)

If11988321

gt 0 then

minus [12058931205894minus (1205892

4+ 1205895)11988321] 1199022minus 12058931205894119883211205792lt 0 (43)

10 Journal of Control Science and Engineering

If11988321

lt 0 then

minus [12058931205894minus (1205892

4+ 1205895)11988321] 1199022minus 12058931205894119883211205792lt 0 (44)

in the region ofminus[12058931205894minus(1205892

4+1205895)11988321]1199022lt 12058931205894119883211205792 Since 120579 is

caused by the loadmotion here the value of 120579 should bemuchsmaller than 119902 The inequality (44) is easy to be satisfied

In summary of the above analysis and by defining

Ψ1(120579119871) = 11989612057912058921205792

1198712

+ [12058931205894minus (1205892

4+ 1205895)11988321] 1199022

+ 12058931205894119883211205792

Ψ2(120579119871) = [120589

31205894minus (1205892

4+ 1205895)11988321] 1199022+ 12058931205894119883211205792

(45)

the following lemma can be derived

Lemma 1 Defining a Lyapunov function (28) and choosingthe design parameters to satisfy 120589

1= (2119892119897

119909120587)1205892 [12058931205894minus (1205892

4+

1205895)11988321] gt 0 and [1205892

3minus1205893120589411988322+(1205892

4+1205895)11988321] ge 0 then within

the region specified by minus[12058931205894minus (1205892

4+ 1205895)11988321]1199022lt 12058931205894119883211205792

(i) V1(0) = 0 and V

1(120579119871) gt 0 forall120579

119871= 0

(ii) V1(120579119871) rarr infin as 120579

119871 rarr infin

(iii) (1198711198911

V1)(120579119871) le minusΨ

1(120579119871) forall120579119871 around the hover condi-

tion or (1198711198911

V1)(120579119871) le minusΨ

2(120579119871) forall120579119871 if (35) holds

Both functions Ψ1and Ψ

2are nonnegative

Recall the control term 119901(119909119867(119905)) in the first subsystem

of (11a) and (11b) it can be seen that [(119863119901)(119909119867)]minus1 exists

for all 119909119867(119905) The unknown vector fields 119902(120579

119871(119905) 119909119867) and

119867(119905 120579119871(119905) 119909119867) model the uncertainties imposed onto the

system Since 119902(120579119871(119905) 119909119867) is directly mapped into the ldquocon-

trolrdquo space of 119909119867(119905) it can be considered as a matched

uncertainty119867(119905 120579119871(119905) 119909119867) is unknown and it does belong to

the control space of 119909119867(119905) so it represents the mismatched

uncertainty in the system [12]Generally the range of the (longitudinal) load suspension

angle is within minus1205872 lt 120579119871lt 1205872 The helicopter velocities

and load suspension angle havemaximumoperational valuestherefore the uncertainties in the system are bounded Withthe Lyapunov function defined in (28)

(1198711198921

V1) (120579119871) = ⟨nablaV

1(120579119871) 1198921⟩ =

119896119863

119872119871119897119909

12058921205791198712

(46)

In 1198661(120579119871) the columns 119892

1and 119892

2are the same therefore

(1198711198922

V1) (120579119871) =

119896119863

119872119871119897119909

12058921205791198712

(47)

By considering the maximum values of cos (sdot) and sin (sdot)functions the bounding values relating to the uncertainty119902(120579119871(119905) 119909119867) can be estimated as follows

10038171003817100381710038171003817119902 (120579119871 119909119867)10038171003817100381710038171003817le

10038161003816100381610038161003816100381610038161003816

1

119897119909

( + )

+1

119872119871

(minus119896119863(2 +

120581

119897119909

) (119906 + 119908) +119896120579

119897119909

)1205791198712

10038161003816100381610038161003816100381610038161003816

le

10038161003816100381610038161003816100381610038161003816

119896120579

119897119909

1205791198712

10038161003816100381610038161003816100381610038161003816+

10038161003816100381610038161003816100381610038161003816

119896119863

119872119871

(2 +120581

119897119909

) (119906 + 119908)

10038161003816100381610038161003816100381610038161003816+

10038161003816100381610038161003816100381610038161003816

1

119897119909

( + )

10038161003816100381610038161003816100381610038161003816

le119896120579

119897119909

100381710038171003817100381710038171205791198712

10038171003817100381710038171003817+

119896119863

119872119871

(2 +120581

119897119909

)1003817100381710038171003817119901 (119909119867)1003817100381710038171003817 + 1205722 (119905)

le 1205831

10038161003816100381610038161003816(1198711198922

V1) (120579119871)10038161003816100381610038161003816+ 1205721

1003817100381710038171003817119901 (119909119867)1003817100381710038171003817 + 1205722 (119905)

(48)

where 1205831

= 119872119871119896120579119896119863 1205721

= (119896119863119872119871)(2 + 120581119897

119909) (120581 =

max [1205811 1205812]) and 120572

2(119905) is defined by 120572

2(119905) = max |(1119897

119909)( +

)| gt 0For the mismatched uncertainty 119867(119905 120579

119871(119905) 119909119867) the fol-

lowing analysis is conducted to obtain its bounding functionThe mismatched uncertainty can be rewritten as

119867(119905 120579119871 (119905) 119909119867) =

[[[[[

[

0

0

(11988611119906 + 11988612119908)

(11988323119906 + 119883

24119908)

]]]]]

]

=

[[[[[

[

0 0

0 0

11988611

11988612

11988323

11988324

]]]]]

]

[119906

119908]

(49)

Let 119860 = [1198861111988612

1198832311988324] then

10038171003817100381710038171003817119867 (119905 120579

119871 119909119867)10038171003817100381710038171003817le1003817100381710038171003817100381711986010038171003817100381710038171003817

1003817100381710038171003817119909119867 (119905)1003817100381710038171003817 (50)

Define a positive function as

1006704120579 (120579119871) = radic(1205792

1198711

+ 12057921198712

+ 1205792 + 1199022+ 120576) (51)

where 120576 is a very small positive constant Choosing1205731

= (1198721198711198971199091198961198631205892)119860 then the mismatched uncertainty

119867(119905 120579119871(119905) 119909119867) is bounded by the following inequality

10038171003817100381710038171003817119867 (119905 120579

119871 119909119867)10038171003817100381710038171003817

le 1006704120579minus1

(120579119871) 1205731

2

sum

119894=1

(10038161003816100381610038161003816(119871119892119894

V1) (120579119871)10038161003816100381610038161003816)1003816100381610038161003816119901119894 (119909119867)

1003816100381610038161003816

(52)

where 119892119894denotes the 119894th column of the matrix function

1198661(120579119871) and 119901

119894is the 119894th component of 119901(119909

119867) respectively As

1205731has a design parameter 120589

2involved it is easy to have 120573

1to

lead (52) to be true For the matched uncertainty we have

10038171003817100381710038171003817119902 (120579119871 119909119867)10038171003817100381710038171003817le

119896120579

119897119909

100381610038161003816100381610038161205791198712

10038161003816100381610038161003816+ 1205721

1003817100381710038171003817119901 (119909119867)1003817100381710038171003817 + 1205722 (119905) (53)

Journal of Control Science and Engineering 11

Following the above analysis the following lemma can bederived

Lemma 2 The uncertainties (52) and (53) are bounded andsatisfy if 120572

1 1205731ge 0 and 120572

1+ 1205731lt 1 for 120572

1= (119896119863119872119871)(2 +

120581119897119909) and 120573

1= (1198721198711198971199091198961198631205892)119860

4 Numerical Results

Using a simulation software the linearized model for a UH-60 helicopter is obtained with the following state and inputmatrices

119860 =

[[[[[[[[[[[[[[[[[[[

[

minus00000 00003 minus00000 minus00000 00000 minus00000 10000 minus00023 00503

minus00003 minus00000 minus00000 minus00000 minus00000 minus00000 minus00000 09989 00463

minus00001 00000 minus00000 minus00000 00000 00000 minus00000 minus00464 10002

minus19386 minus28593 minus00030 minus00195 minus00126 00168 minus12000 57076 02683

328694 minus03640 00027 00126 minus00311 minus00013 minus41348 minus22350 03713

03138 minus14034 00004 00150 00034 minus03340 02866 06307 23410

minus09144 minus04783 00017 00226 minus00246 00007 minus65893 minus21301 minus00598

00631 minus06744 00008 00029 00046 00025 02059 minus14255 minus01636

minus07549 minus00119 minus00002 00009 00028 00006 minus00749 minus01032 minus02214

]]]]]]]]]]]]]]]]]]]

]

119861 =

[[[[[[[[[[[[[[[[[[[

[

minus00000 00000 minus00000 00000

00000 00000 minus00000 minus00000

minus00000 00000 minus00000 minus00000

minus22190 05734 03170 minus00019

35381 minus00453 minus00963 02007

minus15629 00237 minus60740 minus01014

19943 minus00714 minus00541 00731

00446 minus01097 00553 minus00281

minus08197 minus00082 01573 minus00635

]]]]]]]]]]]]]]]]]]]

]

(54)

The desired condition 1205721+ 1205731lt 1 can be verified by consid-

ering a typical case that the UH-60 helicopter carries a loadweight 1000 lb by a 15 ft slung length and let 119896

119863= 75 lbft

For the UH-60 helicopter model with reference to thegeneral mathematical model presented in Section 2 we have1198831205791119904

= 05734 1198831205790

= 03170 1198851205791119904

= 00237 1198851205790

= minus607401198721205791119904

= minus01097 and 1198721205790

= 00553 So the followingparameters can be calculated to have the values 119886

11= minus02

and 11988612

= minus002 For 119883119906= minus00195 119885

119906= 00150 119872

119906=

00029 119872119902= minus14255 (119883

119902minus 119908119890) = 57076 and (119885

119902+ 119906119890) =

06307 we have 11988323

= 006 And also for 119883119908

= 00168119885119908

= minus03340 and 119872119908

= 00025 we have 11988324

= 001Based on all the above parameters 120590max(119860) = 00041 Inthis case let 120581 = 001 120572

1= (119896119863119872119871)(2 + 120581119897

119909) asymp 015

and 1205731= (119872

119871119897119909119896119863)120590max(119860) = 082 Therefore 120572

1+ 1205731=

015 + 082 = 097 which satisfies 1205721+ 1205731

lt 1 Anotherexample considers that the UH-60 helicopter carries a loadweight 500 lb by a 15 ft slung length and let 119896

119863= 50 lbft then

1205721= 02 and 120573

1= 062 therefore (120572

1+ 1205731) = 082 Therefore

this assumption 1205721+ 1205731

lt 1 is realistic for the systemdiscussed in here The numerical values vary with respectto the slung load configuration It may represent the system

dynamic variations To investigate the stability characteristicsof dynamic variations two series of simulation studies werecarried out

Firstly the slung length was kept constant at 15 ft andthe load varied between the limits 500ndash2000 lb with 119896

119863=

75 lbft producing the system poles listed in Table 1 Thecorresponding root loci are sketched in Figures 3 and 4Figure 4 clearly shows that the positions of the system polescrossed the imaginary axis as the load weight increased

Secondly load weight was kept constant and the slunglength varied between the limit of 10ndash20 ft Table 2 sum-marises the results of variation of the system poles and thecorresponding root loci are illustrated in Figures 5 and 6

The pole locus indicates that the system poles have thevariation trends of moving further towards right direction onthe 119904-plane with increase of load weight and slung lengthsFrom the simulation results it is clearly shown that thesystem becomes less stable when the load andor the slunglength increases The simulation results are compared withthe stability analysis presented above

Now consider the case of fixed slung length of 15 ft withdifferent load weights In the cases of a UH-60 helicopter that

12 Journal of Control Science and Engineering

Table 1 System poles locations for a fixed slung length (119897 = 15ft) but with different load weights

Helicopter 500 lb 1000 lb 1500 lb 2000 lbminus10919 minus11159 minus07848 minus06221 minus09369minus63938 minus64301 minus92203 minus79848 minus75613minus00032 minus00048 minus00034 minus00052 minus00052minus00977 minus00818 minus00888 minus00972 minus00694minus03045 minus03036 minus03394 minus03318 minus03333minus03159 plusmn 04363119894 minus03199 plusmn 04628119894 minus02295 plusmn 05050119894 minus02366 plusmn 05974119894 minus01834 plusmn 04968119894minus00489 plusmn 03898119894 minus00211 plusmn 03868119894 00074 plusmn 03445119894 00119 plusmn 04406119894 00264 plusmn 03199119894mdash minus00316 plusmn 15043119894 minus01020 plusmn 17380119894 minus00368 plusmn 18753119894 minus01160 plusmn 20003119894mdash minus00736 plusmn 15019119894 minus07662 plusmn 11680119894 minus07349 plusmn 19064119894 minus12367 plusmn 08995119894

Table 2 System poles for a fixed load (1000 lb) but with different slung lengths

Helicopter 10 ft 15 ft 20 ftminus10919 minus10560 minus07848 minus08687minus63938 minus99252 minus92203 minus85565minus00032 minus00061 minus00034 minus00114minus00977 minus02012 minus00888 minus00953minus03045 minus02606 minus03394 minus03479minus03159 plusmn 04363119894 minus01809 plusmn 06219119894 minus02295 plusmn 05050119894 minus01636 plusmn 06092119894minus00489 plusmn 03898119894 00643 plusmn 04577119894 00074 plusmn 03445119894 00799 plusmn 04594119894mdash minus01055 plusmn 19883119894 minus01020 plusmn 17380119894 minus01247 plusmn 14470119894mdash minus08847 plusmn 12464119894 minus07662 plusmn 11680119894 minus06352 plusmn 10384119894

Re(poles)

Im(poles)

minus10 minus5 0 5minus08minus06minus04minus02

002040608

Figure 3 System root-locus diagram showing pole locus forconstant slung length (15 ft) as load weight is increased

minus04 minus035 minus03 minus025 minus02 minus015 minus01 minus005 0 005 01

Re(poles)

Im(poles)

minus08minus06minus04minus02

002040608

Figure 4 Enlarged view of Figure 3 showing poles locus near toRe(poles) = 0 axis

minus10 minus8 minus6 minus4 minus2 0 2

Re(poles)

Im(poles)

minus08minus06minus04minus02

002040608

Figure 5 System root-locus showing pole locus for constant loadweight (1000 lb) as slung length is increased

minus04 minus035 minus03 minus025 minus02 minus015 minus01 minus005 0 005 01

Re(poles)

Im(poles)

minus08minus06minus04minus02

002040608

Figure 6 Enlarged view of Figure 5 showing poles locus near toRe(poles) = 0 axis

Journal of Control Science and Engineering 13

carries a loadweight 500 lb and 1000 lb then1205721= 03 and 041

with 1205731= 041 and 082 respectively therefore 120572

1+ 1205731lt 1

thus the system is stable But when the load weight is 1500 lband 2000 lb then 120572

1= 01 and 0075 and also 120573

1= 123 and

164 respectively therefore 1205721+ 1205731gt 1 thus the system is

unstable For the case of the constant load weight (1000 lb) asslung length is increased then for a 10 ft and 15 ft slung length1205721= 015 and 120573

1= 055 and 082 respectively therefore

1205721+ 1205731lt 1 thus the system is stable But when the slung

length increased to 20 ft then 1205721

= 015 and 1205731

= 109therefore (120572

1+ 1205731) = 124 that is 120572

1+ 1205731gt 1 thus the

system is unstable However the system can be marginallystable if the UH-60 helicopter carries a load weight 725 lb bya 20 ft slung length then 120572

1= 02 and 120573

1= 08 therefore

(1205721+ 1205731) = 1 For this set of analysis 119896

119863is assumed to

be 119896119863

= 75 lbft However 119896119863can vary depending on the

weather condition and an example of 119896119863= 50 lbft is shown

above for illustrative purpose This analysis suggests that themaximum value of 20 ft slung length may be used for slungload operation However the load weight limit can vary withrespect to the capacity of the load carrying hooks helicopterweight environmental conditions and so forth

The examples presented here show that the simulationresults are correlated with the stability analysis Thus desiredcondition for the stability is met and it can locate the load atthe specified position or its neighbourhood

5 Conclusions

In this paper stability analysis of a helicopter with an under-slung load system is investigatedThe investigation identifiedthe conditions for the stabilisation of the system and thepositioning of the underslung load at hover condition Sta-bility analysis and numerical results proved that the desiredcondition for the stability is met then it is possible to locatethe load at the specified position or its neighbourhood Anillustration example is also given in the paper The methodresults in a guaranteed load positioning accuracy whichdepends on the design parameters and the method can beapplicable to any individual helicopter with external loadsystem

Nomenclature

119871119872119873 Overall helicopter rolling pitching andyawing moments

119901 119902 119903 Helicopter roll pitch and yaw rates aboutbody reference axes

119883119884 119885 Overall helicopter force components120601 120579 120595 Roll pitch and yaw angles119906 V 119908 Helicopter velocity components at centre

of gravity1205790 Main rotor collective

1205791119904 Longitudinal cyclic commands

1205791119888 Lateral cyclic commands

1205790119879 Tail rotor collective

1198830 1198840 1198850 Location of the suspension point withrespect to earth referenced 119909 119910 and 119911

directions

0 0 0 The helicopter velocity in the 119909 119910 and 119911

directions0 0 0 The helicopter acceleration in the 119909 119910 and119911 directions

120579119871 Load suspension angle in the119883-119885 plane

with respect to 119885-axis120601119871 Load suspension angle in the 119884-119885 plane

with respect to 119885-axis119872119871 Mass of the suspended load

119892 Acceleration due to gravity119897 Slung length119896119863 Aerodynamic drag force constant

(119896119863= (12)120588119878119862

119863)

120588 Air density119878 The load area presented to the airflow119862119863 Drag coefficient for the load

Competing Interests

The author declares that there are no competing interestsregarding the publication of this paper

References

[1] K Thanapalan and F Zhang ldquoModelling and simulation studyof a helicopter with an external slung load systemrdquo I-ManagersJournal on Instrumentation amp Control Engineering vol 1 no 1pp 9ndash17 2013

[2] M Bisgaard J D Bendtsen and A La Cour-Harbo ldquoModelingof generic slung load systemrdquo Journal of Guidance Control andDynamics vol 32 no 2 pp 573ndash585 2009

[3] Handling Qualities Requirements for Military Rotorcraft ADS-33-D US Army Aviation and Troop Command St Louis MissUSA 1996

[4] K Thanapalan and T M Wong ldquoModeing of a helicopter withan under-slung load systemrdquo in Proceedings of the 29th ChineseControl Conference (CCC rsquo10) pp 1451ndash1456 Beijing China July2010

[5] B Feng ldquoRobust control for lateral and longitudinal channels ofsmall-scale unmanned helicoptersrdquo Journal of Control Scienceand Engineering vol 2015 Article ID 483096 8 pages 2015

[6] J J Gribble ldquoLinear quadratic Gaussianloop transfer recoverydesign for a helicopter in low-speed flightrdquo Journal of GuidanceControl and Dynamics vol 16 no 4 pp 754ndash761 1993

[7] M A Manness J J Gribble and D J M Smith ldquoMultivariablemethods for helicopter flight control law designrdquo in Proceedingsof the 16th Annual Society European Rotorcraft Forum PaperNoIII52 Glasgow UK September 1990

[8] W L Garrard E Low and S Prouty ldquoDesign of attitude andrate command systems for helicopters using eigenstructureassignmentrdquo Journal of Guidance Control and Dynamics vol12 no 6 pp 783ndash791 1989

[9] Y B Shtessel and I A Shkolnikov ldquoAeronautical and spacevehicle control in dynamic sliding manifoldsrdquo InternationalJournal of Control vol 76 no 9-10 pp 1000ndash1017 2003

[10] D FThompson J S Pruyn andA Shukla ldquoFeedback design forrobust tracking and robust stiffness in flight control actuatorsusing a modified QFT techniquerdquo International Journal ofControl vol 72 no 16 pp 1480ndash1497 1999

14 Journal of Control Science and Engineering

[11] K Thanapalan ldquoModelling of a helicopter systemrdquo in Proceed-ings of the 1st Virtual Control Conference (VCC rsquo10) pp 45ndash52Aalborg University 2010

[12] D P Goodall and J Wang ldquoStabilization of a class of uncertainnonlinear affine systems subject to control constraintsrdquo Interna-tional Journal of Robust and Nonlinear Control vol 11 no 9 pp797ndash818 2001

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

6 Journal of Control Science and Engineering

Let 119901 = (119901 minus 11988711V minus 11988712119908) then (15) can be written in the

following form

[[[[[[

[

V

]]]]]]

]

=

[[[[[

[

11988411

11988412

11988413

11988414

11988421

11988422

11988423

11988424

11988431

11988432

11988433

11988434

11988441

11988442

11988443

11988444

]]]]]

]

[[[[[

[

120601

119901

V

119908

]]]]]

]

+

[[[[[

[

0 0

11987321

11987322

1198841205791119888

1198841205790119879

1198851205791119888

1198851205790119879

]]]]]

]

[1205791119888

1205790119879

]

(18)

where

11988411

= 0

11988412

= 1

11988413

= 11988711

11988414

= 11988712

11988421

= minus (11988711119892)

11988422

= (119871119901minus 11988711(119884119901+ 119908119890) minus 11988712(119885119901minus V119890))

11988423

= (119871119901minus 11988711(119884119901+ 119908119890) minus 11988712(119885119901minus V119890)) 11988711

+ (119871119901minus 11988711119884V minus 119887

12119885V)

11988424

= (119871119901minus 11988711(119884119901+ 119908119890) minus 11988712(119885119901minus V119890)) 11988712

+ (119871119901minus 11988711119884119908minus 11988712119885119908)

11988431

= 119892

11988432

= (119884119901+ 119908119890)

11988433

= ((119884119901+ 119908119890) 11988711+ 119884V)

11988434

= ((119884119901+ 119908119890) 11988712+ 119884119908)

11988441

= 0

11988442

= (119885119901minus V119890)

11988443

= ((119885119901minus V119890) 11988711+ 119885V)

11988444

= ((119885119901minus V119890) 11988712+ 119885119908)

11987321

= (1198711205791119888

minus 119887111198841205791119888

minus 119887121198851205791119888

)

11987322

= (1198711205790119879

minus 119887111198841205790119879

minus 119887121198851205790119879

)

(19)

Using (18) the helicopter with underslung load for lateralmotion in the 119884-119885 plane can be described by

119871(119905)

= 1198911(120593119871(119905))

+ 1198661(120593119871(119905)) [119901 (119909

119867 (119905)) + 119902 (120593119871(119905) 119909119867 (119905))]

+ 119867 (119905 120593119871 119909119867 (119905))

(20a)

119867 (119905) = 119891

2(120593119871(119905) 119909119867 (119905)) + 119866

2 (119905) (20b)

where

119871(119905) = [

1198711

1198712

]119879

119901 (119909119867 (119905)) = [V2 119908

2]119879

1198911(120593119871(119905)) =

[[[[[[[

[

1205931198712

minus119892

119897119910

sin1205931198711

+119896119863119897119910

119872119871

(sign (119871) cos3 120593

1198711

+ sin (119871) sin3 120593

1198711

) 1205932

1198712

minus 1198961205931205931198712

119901

(minus11988711119892120601 + (119871

119901minus 11988711(119884119901+ 119908119890) minus 11988712(119885119901minus V119890)) 119901)

]]]]]]]

]

1198661(120593119871(119905)) =

[[[[[[[[

[

0 0

(119896119863sign (

119871) cos120593

1198711

119872119871119897119910

) (119896119863sign (

119871) sin120593

1198711

119872119871119897119910

)

0 0

0 0

]]]]]]]]

]

119902 (120593119871(119905) 119909119867 (119905)) =

[[

[

0

cos1205931198711

119897119910

V +sin1205931198711

119897119910

minus2119896119863

119872119871

(sign (119871) cos2 120593

1198711

V + sign (119871) sin2 120593

1198711

119908)1205931198712

]]

]

Journal of Control Science and Engineering 7

119867(119905 120593119871(119905) 119909119867 (119905)) =

[[[[[

[

0

0

( minus 119901)

(11988423V + 11988424119908)

]]]]]

]

(119905) = [1205791119888 1205790119879]119879

1198912(120593119871(119905) 119909119867 (119905)) = [

(11988431120601 + 11988432119901 + 11988433V + 11988434119908)

(11988442119901 + 11988443V + 11988444119908)

]

1198662= [

1198841205791119888

1198841205790119879

1198851205791119888

1198851205790119879

]

(21)

It is assumed that the lateral motion is primarily controlledby lateral cyclic commands (120579

1119888) and the tail rotor collective

1205790119879

3 Stability Analysis

The goal is to analyse the stability of the combined systemto ensure that it is possible to stabilise the helicopter withunderslung load system modelled by (11a) (11b) (20a) and(20b) in a real environment with uncertainties In this paperfor the stability analysis the Lyapunov second method isapplied for the helicopter with an underslung external loadsystem The analysis for the longitudinal motion is discussedfirst The system equations can be considered to have twomain parts that is known and unknown (or partly known)The known terms formed the nominal part of the systemmodelThe unknown or partly known part can be consideredas the uncertainty to the system The whole system is thenmodelled by a nominal part with the addition of uncertaintyIn fact the known elements in the subsystem (11a) arecharacterised by the prescribed triple (119891

1 1198661 119901) and it is

desired that the nominal part of the system is stableThe basic notations and concepts required for the analysis

are described first The state space is denoted by119883 fl R119899 andthe control space by119906 fl R119898 where 1 le 119898 le 119899TheEuclideaninner product (on 119883 or 119906 as appropriate) and induced normare denoted by ⟨sdot sdot⟩ and sdot respectively Let 119862(R119901R119902) and1198621(R119901R119902) denote the space of all continuous functions and

the space of continuous functions with continuous first-orderpartial derivatives respectively and let 119862infin(R119901R119902) denotethe space of functions whose partial derivatives of any orderexist and are continuous mapping R119901 rarr R119902 For a real-valued continuous scalar function 119909 rarr V(119909) defined onR119899 nabla V rarr nablaV isin R119899 denotes the gradient map The Liederivative of V along a vector field 119891 R119899 rarr R119899 is denoted by119871119891V R119899 rarr R which is defined by

(119871119891V) (119909) = ⟨nablaV (119909) 119891 (119909)⟩ (22)

The Lie bracket of vector fields 119891 119892 isin 119862infin(R119899R119899) is the

vector field [119891 119892] isin 119862infin(R119899R119899) defined by [119891 119892] = (119863119892)119891 minus

(119863119891)119892 where (119863119891)denotes the Jacobianmatrix of119891 and (119863119892)denotes the Jacobian matrix of 119892

In this paper nonlinear systemswith the following formatare considered

(119905) = 119891 (119909 (119905)) + 119866 (119909 (119905)) (119905) (23)

where 119909(119905) isin R119899 isin R119898 In general mathematical modelsof dynamical systems are usually imprecise due to modellingerrors and exogenous disturbances [12] Equation (23) can beconsidered as the nominal part of the system model and theuncertainty can be modelled as an additive perturbation tothe nominal systemmodel more specifically the structure ofthe system has the form

(119905) = 119891 (119909 (119905)) + 119866 (119909 (119905)) (119905) + 120599 (119909 (119905) 119906 (119905)) (24)

where 120599(119909(119905) 119906(119905))models the uncertainty in the systemSystem (24) is globally asymptotically stable to the zero

state if the system exhibits the following properties

(i) Existence and Continuation of Solutions For each 119909 isin R119899there exists a local solution119909 [0 119905

1) rarr R119899 (ie an absolutely

continuous function satisfying (24) almost everywhere (ae)and 119909(0) = 119909

0) and every such solution can be extended intoa solution on [0infin)

(ii) Boundedness of Solutions For each ℏ gt 0 there exists119903(ℏ) gt 0 such that 119909(119905) isin 119903(ℏ)119861

119899 for all 119905 ge 0 on every

solution 119909 [0infin) rarr R119899 with 1199090isin ℏ119861119899 where 119861

119899denote

the open unit ball centred at the origin in R119899

(iii) Stability of the State Origin For each 120575 gt 0 there exists119889(120575) gt 0 such that 119909(119905) isin 120575119861

119899for all 119905 ge 0 on every solution

119909 [0infin) rarr R119899 with 1199090isin 119889(120575)119861

119899

(iv) Global Attractivity of the State Origin For each ℏ gt 0 and120576 gt 0 there exists 119879(ℏ 120576) ge 0 such that 119909(119905) isin 119861

119899for all

119905 ge 119879(ℏ 120576) on every solution 119909 [0infin) rarr R119899 with 1199090isin ℏ119861119899

8 Journal of Control Science and Engineering

Consider a nonlinear system described by the ordinarydifferential equation as follows

(119905) = 119891 (119905 119909 (119905))

119909 (1199050) = 1199090

(25)

where 119891 R times 119883 rarr 119883 and 119891(119905 0) = 0 for all 119905To analyse the stability of (25) Lyapunovrsquos second stabilityanalysis method is applicable The Lyapunov approach is toshow that a candidate ldquoLyapunov functionrdquo is nonincreasingalong all solution to (25) by means that do not requireexplicit knowledge of solutions to (25) From this appropriateconclusion can be drawn regarding stability concepts relatingto solutions of the differential equation (25) An essentialpart of Lyapunovrsquos method is the determination of the timederivative of the candidate ldquoLyapunov functionrdquo along allsolution of the dynamical system

Consider a Lyapunov candidate (119905 119909) rarr V(119905 119909) Rtimes119883 rarr

R which satisfies the condition V isin 1198621(R times 119883) in which case

its time derivative along solutions to (25) is given by

V (119905 119909 (119905)) =120597V (119905 119909 (119905))

120597119905+ ⟨nablaV (119905 119909 (119905)) 119891 (119905 119909 (119905))⟩ (26)

for almost all 119905 isin RLet 119882(119909(119905)) denote a positive definite function If

V(119905 119909(119905)) satisfies

(i) V(119905 0) = 0 for all 119905 ge 0

(ii) 119882(119909(119905)) le V(119905 119909(119905)) for all 119909(119905) isin Φ 0 sub Φ sub 119883 andall 119905 ge 0

(iii) V(119905 119909(119905)) lt 0 in Φ

then V(119905 119909(119905)) is said to be a Lyapunov function in Φ IfV(119905 119909(119905)) in Φ then V(119905 119909(119905)) is said to be a weak Lyapunovfunction

A set Ε is said to be an invariant set with respect to thedynamical system = 119891(119909) if

119909 (0) isin Ε 997891997888rarr 119909 (119905) isin Ε forall119905 isin R+ (27)

In other words Ε is the set of points such that if a solution of = 119891(119909) belongs to Ε at some instant initialized points at119905 = 0 then it belongs to Ε for all future time

Now a setΕ sub 119909 is said to be a local invariantmanifold for(25) if for any 1199090 isin Ε 119909(119905) with 119909(0) = 119909

0 is in Ε for |119905| lt 119879

where 119879 gt 0 If 119879 = infin then Ε is said to be an invariantmanifold

Now considering the longitudinal motion (Figure 1) forthe helicopter with an underslung external load systemchoose a Lyapunov function candidate for the first subsystemV1as

V1(120579119871) =

1

2[12058911205792

1198711

+ 12058921205792

1198712

+ (1205893120579 minus 1205894119902)2+ 12058951199022] (28)

where 120589119894(119894 = 1 2 3 4 5) are design parameters to be

determined Then

V1(120579119871) = ⟨nablaV

1(120579119871) 1198911(120579119871) + 1198661(120579119871 (119905))

sdot [119901 (119909119867 (119905)) + 119902 (120579

119871 (119905) 119909119867 (119905))]

+ 119867(119905 120579119871 119909119867 (119905))⟩ = (119871

1198911

V1) (120579119871)

+ ⟨nablaV1(120579119871) 1198661(120579119871 (119905))

sdot [119901 (119909119867 (119905)) + 119902 (120579

119871 (119905) 119909119867 (119905))]

+ 119867(119905 120579119871 119909119867 (119905))⟩

(29)

From (28) nablaV1(120579119871) can be obtained as

nablaV1(120579119871)

= [12058911205791198711

12058921205791198712

1205893(1205893120579 minus 1205894119902) minus120589

4(1205893120579 minus 1205894119902) + 120589

5119902]119879

(30)

Substituting nablaV1(120579119871) and 119891

1(120579119871) into the derivative of

(1198711198911

V1)(120579119871) we have

(1198711198911

V1) (120579119871) = 12058911205791198711

1205791198712

minus 11989612057912058921205792

1198712

minus 1205892

119892

119897119909

sin 1205791198711

1205791198712

+119896119863119897119909

119872119871

1205892[sign (

119871) cos3 120579

1198711

+ sign (119871) sin3 120579

1198711

]

sdot 1205793

1198712

+ 1205893[1205893120579 minus 1205894119902] 119902 + (minus120589

31205894120579 + (120589

2

4+ 1205895) 119902)

sdot [11988321120579 + 119883

22119902]

(31)

Checking the term of minus sin (1205791198711

)1205791198712

it can be seen thatminus sin (120579

1198711

)1205791198712

le minus(2120587)1205791198711

1205791198712

when 1205791198711

and 1205791198712

are bothpositive or negative The situation of 120579

1198711

and 1205791198712

havingdifferent signs will help with the system stability Then

(1198711198911

V1) (120579119871) le 12058911205791198711

1205791198712

minus 11989612057912058921205792

1198712

minus 1205892

119892

119897119909

2

1205871205791198711

1205791198712

+119896119863119897119909

119872119871

sdot 1205892[sign (

119871) cos3 120579

1198711

+ sign (119871) sin3 120579

1198711

] 1205793

1198712

+ 1205893[1205893120579 minus 1205894119902] 119902 + (minus120589

31205894120579 + (120589

2

4+ 1205895) 119902)

sdot [11988321120579 + 119883

22119902]

(32)

If the design parameter 1205891is chosen as 120589

1= (2119892119897

119909120587)1205892

(1198711198911

V1)(120579119871) satisfies the following

(1198711198911

V1) (120579119871) le minus119896

12057912058921205792

1198712

+119896119863119897119909

119872119871

sdot 1205892[sign (

119871) cos3 120579

1198711

+ sign (119871) sin3 120579

1198711

] 1205793

1198712

+ 1205893[1205893120579 minus 1205894119902] 119902 + (minus120589

31205894120579 + (120589

2

4+ 1205895) 119902)

sdot [11988321120579 + 119883

22119902]

(33)

Journal of Control Science and Engineering 9

Further analysis on (33) will start from examining the firsttwo terms Rewrite these two terms So

minus 11989612057912058921205792

1198712

+119896119863119897119909

119872119871

1205892[sign (

119871) cos3 120579

1198711

+ sign (119871)

sdot sin3 1205791198711

] 1205793

1198712

= minus11989612057912058921205792

1198712

+119896119863119897119909

119872119871

12058921205791198712

[sign (119871)

sdot cos3 1205791198711

+ sign (119871) sin3 120579

1198711

] 1205792

1198712

le minus11989612057912058921205792

1198712

+119896119863119897119909

119872119871

1205892

10038161003816100381610038161003816max (120579

1198712

) (sign (119871) cos3 120579

1198711

+ sign (119871) sin3 120579

1198711

)100381610038161003816100381610038161205792

1198712

= minus12058921205792

1198712

119896120579

minus119896119863119897119909

119872119871

10038161003816100381610038161003816max (120579

1198712

)

sdot (sign (119871) cos3 120579

1198711

+ sign (119871) sin3 120579

1198711

)10038161003816100381610038161003816

(34)

If the hinge friction is big enough to satisfy the following

119896120579gt

119896119863119897119909

119872119871

10038161003816100381610038161003816max (120579

1198712

)

sdot (sign (119871) cos3 120579

1198711

+ sign (119871) sin3 120579

1198711

)10038161003816100381610038161003816

(35)

then

120582119871= 119896120579minus119896119863119897119909

119872119871

10038161003816100381610038161003816max (120579

1198712

)

sdot (sign (119871) cos3 120579

1198711

+ sign (119871) sin3 120579

1198711

)10038161003816100381610038161003816 ge 0

minus 11989612057912058921205792

1198712

+119896119863119897119909

119872119871

1205892[sign (

119871) cos3 120579

1198711

+ sign (119871)

sdot sin3 1205791198711

] 1205793

1198712

le minus12058921205792

1198712

120582119871

(36)

Now the following analysis has been conducted for (33) andthe load movement for the longitudinal motion is describedby

119871= 0minus 1198971199091205791198712

cos 1205791198711

119871= 0minus 1198971199091205791198712

sin 1205791198711

(37)

Around the hover condition1198830= 0 and 119885

0= 0 the signs for

119871and

119871are opposite to the one of 120579

1198712

provided that 1205791198711

ispositive

Considering the term (sign (119871) cos3 120579

1198711

+

sign (119871) sin3 120579

1198711

)1205793

1198712

for all possible combinations ofthe signs of 120579

1198712

and 1205791198711

the analysis follows below

(i) If 1205791198711

gt 0 and 1205791198712

gt 0 then sign (119871) = minus1

sign (119871) = minus1 cos3 120579

1198711

gt 0 and sin3 1205791198711

gt 0 There-fore (sign (

119871) cos3 120579

1198711

+ sign (119871) sin3 120579

1198711

)1205793

1198712

lt 0

(ii) If 1205791198711

lt 0 and 1205791198712

gt 0 then sign (119871) = minus1

sign (119871) = +1 cos3 120579

1198711

gt 0 and sin31205791198711

lt 0 So(sign (

119871) cos3 120579

1198711

+ sign (119871) sin3 120579

1198711

)1205793

1198712

lt 0

(iii) If 1205791198711

gt 0 and 1205791198712

lt 0 then sign (119871) = +1

and sign (119871) = +1 cos3 120579

1198711

gt 0 and sin3 1205791198711

gt

0 So the following is true (sign (119871) cos3 120579

1198711

+

sign (119871) sin3 120579

1198711

)1205793

1198712

lt 0

(iv) If 1205791198711

lt 0 and 1205791198712

lt 0 then sign (119871) = +1 and

sign (119871) = minus1 cos3 120579

1198711

gt 0 and sin3 1205791198711

lt 0 So(sign (

119871) cos3 120579

1198711

+ sign (119871) sin3 120579

1198711

)1205793

1198712

lt 0

Therefore

minus 11989612057912058921205792

1198712

+119896119863119897119909

119872119871

sdot 1205892[sign (

119871) cos3 120579

1198711

+ sign (119871) sin3 120579

1198711

] 1205793

1198712

le minus11989612057912058921205792

1198712

(38)

Now examining the rest of the terms of (1198711198911

V1)(120579119871) then

1205893[1205893120579 minus 1205894119902] 119902

+ (minus12058931205894120579 + (120589

2

4+ 1205895) 119902) [119883

21120579 + 119883

22119902] = 120589

2

3120579119902

minus 120589312058941199022minus 12058931205894119883211205792minus 1205893120589411988322120579119902

+ (1205892

4+ 1205895)11988321120579119902 + (120589

2

4+ 1205895)119883221199022

= minus [12058931205894minus (1205892

4+ 1205895)11988321] 1199022minus 12058931205894119883211205792

+ [1205892

3minus 1205893120589411988322+ (1205892

4+ 1205895)11988321] 120579119902

(39)

For the system 120579 and 1205791198711

always have different signs 120579 and 119902

always have different signs and 119902 and 1205791198711

have the same signsSo

[1205892

3minus 1205893120589411988322+ (1205892

4+ 1205895)11988321] 120579119902 le 0

if [1205892

3minus 1205893120589411988322+ (1205892

4+ 1205895)11988321] ge 0

(40)

Therefore

1205893[1205893120579 minus 1205894119902] 119902

+ (minus12058931205894120579 + (120589

2

4+ 1205895) 119902) [119883

21120579 + 119883

22119902]

le minus [12058931205894minus (1205892

4+ 1205895)11988321] 1199022minus 12058931205894119883211205792

(41)

Choose the design parameters to satisfy

[12058931205894minus (1205892

4+ 1205895)11988321] gt 0 (42)

If11988321

gt 0 then

minus [12058931205894minus (1205892

4+ 1205895)11988321] 1199022minus 12058931205894119883211205792lt 0 (43)

10 Journal of Control Science and Engineering

If11988321

lt 0 then

minus [12058931205894minus (1205892

4+ 1205895)11988321] 1199022minus 12058931205894119883211205792lt 0 (44)

in the region ofminus[12058931205894minus(1205892

4+1205895)11988321]1199022lt 12058931205894119883211205792 Since 120579 is

caused by the loadmotion here the value of 120579 should bemuchsmaller than 119902 The inequality (44) is easy to be satisfied

In summary of the above analysis and by defining

Ψ1(120579119871) = 11989612057912058921205792

1198712

+ [12058931205894minus (1205892

4+ 1205895)11988321] 1199022

+ 12058931205894119883211205792

Ψ2(120579119871) = [120589

31205894minus (1205892

4+ 1205895)11988321] 1199022+ 12058931205894119883211205792

(45)

the following lemma can be derived

Lemma 1 Defining a Lyapunov function (28) and choosingthe design parameters to satisfy 120589

1= (2119892119897

119909120587)1205892 [12058931205894minus (1205892

4+

1205895)11988321] gt 0 and [1205892

3minus1205893120589411988322+(1205892

4+1205895)11988321] ge 0 then within

the region specified by minus[12058931205894minus (1205892

4+ 1205895)11988321]1199022lt 12058931205894119883211205792

(i) V1(0) = 0 and V

1(120579119871) gt 0 forall120579

119871= 0

(ii) V1(120579119871) rarr infin as 120579

119871 rarr infin

(iii) (1198711198911

V1)(120579119871) le minusΨ

1(120579119871) forall120579119871 around the hover condi-

tion or (1198711198911

V1)(120579119871) le minusΨ

2(120579119871) forall120579119871 if (35) holds

Both functions Ψ1and Ψ

2are nonnegative

Recall the control term 119901(119909119867(119905)) in the first subsystem

of (11a) and (11b) it can be seen that [(119863119901)(119909119867)]minus1 exists

for all 119909119867(119905) The unknown vector fields 119902(120579

119871(119905) 119909119867) and

119867(119905 120579119871(119905) 119909119867) model the uncertainties imposed onto the

system Since 119902(120579119871(119905) 119909119867) is directly mapped into the ldquocon-

trolrdquo space of 119909119867(119905) it can be considered as a matched

uncertainty119867(119905 120579119871(119905) 119909119867) is unknown and it does belong to

the control space of 119909119867(119905) so it represents the mismatched

uncertainty in the system [12]Generally the range of the (longitudinal) load suspension

angle is within minus1205872 lt 120579119871lt 1205872 The helicopter velocities

and load suspension angle havemaximumoperational valuestherefore the uncertainties in the system are bounded Withthe Lyapunov function defined in (28)

(1198711198921

V1) (120579119871) = ⟨nablaV

1(120579119871) 1198921⟩ =

119896119863

119872119871119897119909

12058921205791198712

(46)

In 1198661(120579119871) the columns 119892

1and 119892

2are the same therefore

(1198711198922

V1) (120579119871) =

119896119863

119872119871119897119909

12058921205791198712

(47)

By considering the maximum values of cos (sdot) and sin (sdot)functions the bounding values relating to the uncertainty119902(120579119871(119905) 119909119867) can be estimated as follows

10038171003817100381710038171003817119902 (120579119871 119909119867)10038171003817100381710038171003817le

10038161003816100381610038161003816100381610038161003816

1

119897119909

( + )

+1

119872119871

(minus119896119863(2 +

120581

119897119909

) (119906 + 119908) +119896120579

119897119909

)1205791198712

10038161003816100381610038161003816100381610038161003816

le

10038161003816100381610038161003816100381610038161003816

119896120579

119897119909

1205791198712

10038161003816100381610038161003816100381610038161003816+

10038161003816100381610038161003816100381610038161003816

119896119863

119872119871

(2 +120581

119897119909

) (119906 + 119908)

10038161003816100381610038161003816100381610038161003816+

10038161003816100381610038161003816100381610038161003816

1

119897119909

( + )

10038161003816100381610038161003816100381610038161003816

le119896120579

119897119909

100381710038171003817100381710038171205791198712

10038171003817100381710038171003817+

119896119863

119872119871

(2 +120581

119897119909

)1003817100381710038171003817119901 (119909119867)1003817100381710038171003817 + 1205722 (119905)

le 1205831

10038161003816100381610038161003816(1198711198922

V1) (120579119871)10038161003816100381610038161003816+ 1205721

1003817100381710038171003817119901 (119909119867)1003817100381710038171003817 + 1205722 (119905)

(48)

where 1205831

= 119872119871119896120579119896119863 1205721

= (119896119863119872119871)(2 + 120581119897

119909) (120581 =

max [1205811 1205812]) and 120572

2(119905) is defined by 120572

2(119905) = max |(1119897

119909)( +

)| gt 0For the mismatched uncertainty 119867(119905 120579

119871(119905) 119909119867) the fol-

lowing analysis is conducted to obtain its bounding functionThe mismatched uncertainty can be rewritten as

119867(119905 120579119871 (119905) 119909119867) =

[[[[[

[

0

0

(11988611119906 + 11988612119908)

(11988323119906 + 119883

24119908)

]]]]]

]

=

[[[[[

[

0 0

0 0

11988611

11988612

11988323

11988324

]]]]]

]

[119906

119908]

(49)

Let 119860 = [1198861111988612

1198832311988324] then

10038171003817100381710038171003817119867 (119905 120579

119871 119909119867)10038171003817100381710038171003817le1003817100381710038171003817100381711986010038171003817100381710038171003817

1003817100381710038171003817119909119867 (119905)1003817100381710038171003817 (50)

Define a positive function as

1006704120579 (120579119871) = radic(1205792

1198711

+ 12057921198712

+ 1205792 + 1199022+ 120576) (51)

where 120576 is a very small positive constant Choosing1205731

= (1198721198711198971199091198961198631205892)119860 then the mismatched uncertainty

119867(119905 120579119871(119905) 119909119867) is bounded by the following inequality

10038171003817100381710038171003817119867 (119905 120579

119871 119909119867)10038171003817100381710038171003817

le 1006704120579minus1

(120579119871) 1205731

2

sum

119894=1

(10038161003816100381610038161003816(119871119892119894

V1) (120579119871)10038161003816100381610038161003816)1003816100381610038161003816119901119894 (119909119867)

1003816100381610038161003816

(52)

where 119892119894denotes the 119894th column of the matrix function

1198661(120579119871) and 119901

119894is the 119894th component of 119901(119909

119867) respectively As

1205731has a design parameter 120589

2involved it is easy to have 120573

1to

lead (52) to be true For the matched uncertainty we have

10038171003817100381710038171003817119902 (120579119871 119909119867)10038171003817100381710038171003817le

119896120579

119897119909

100381610038161003816100381610038161205791198712

10038161003816100381610038161003816+ 1205721

1003817100381710038171003817119901 (119909119867)1003817100381710038171003817 + 1205722 (119905) (53)

Journal of Control Science and Engineering 11

Following the above analysis the following lemma can bederived

Lemma 2 The uncertainties (52) and (53) are bounded andsatisfy if 120572

1 1205731ge 0 and 120572

1+ 1205731lt 1 for 120572

1= (119896119863119872119871)(2 +

120581119897119909) and 120573

1= (1198721198711198971199091198961198631205892)119860

4 Numerical Results

Using a simulation software the linearized model for a UH-60 helicopter is obtained with the following state and inputmatrices

119860 =

[[[[[[[[[[[[[[[[[[[

[

minus00000 00003 minus00000 minus00000 00000 minus00000 10000 minus00023 00503

minus00003 minus00000 minus00000 minus00000 minus00000 minus00000 minus00000 09989 00463

minus00001 00000 minus00000 minus00000 00000 00000 minus00000 minus00464 10002

minus19386 minus28593 minus00030 minus00195 minus00126 00168 minus12000 57076 02683

328694 minus03640 00027 00126 minus00311 minus00013 minus41348 minus22350 03713

03138 minus14034 00004 00150 00034 minus03340 02866 06307 23410

minus09144 minus04783 00017 00226 minus00246 00007 minus65893 minus21301 minus00598

00631 minus06744 00008 00029 00046 00025 02059 minus14255 minus01636

minus07549 minus00119 minus00002 00009 00028 00006 minus00749 minus01032 minus02214

]]]]]]]]]]]]]]]]]]]

]

119861 =

[[[[[[[[[[[[[[[[[[[

[

minus00000 00000 minus00000 00000

00000 00000 minus00000 minus00000

minus00000 00000 minus00000 minus00000

minus22190 05734 03170 minus00019

35381 minus00453 minus00963 02007

minus15629 00237 minus60740 minus01014

19943 minus00714 minus00541 00731

00446 minus01097 00553 minus00281

minus08197 minus00082 01573 minus00635

]]]]]]]]]]]]]]]]]]]

]

(54)

The desired condition 1205721+ 1205731lt 1 can be verified by consid-

ering a typical case that the UH-60 helicopter carries a loadweight 1000 lb by a 15 ft slung length and let 119896

119863= 75 lbft

For the UH-60 helicopter model with reference to thegeneral mathematical model presented in Section 2 we have1198831205791119904

= 05734 1198831205790

= 03170 1198851205791119904

= 00237 1198851205790

= minus607401198721205791119904

= minus01097 and 1198721205790

= 00553 So the followingparameters can be calculated to have the values 119886

11= minus02

and 11988612

= minus002 For 119883119906= minus00195 119885

119906= 00150 119872

119906=

00029 119872119902= minus14255 (119883

119902minus 119908119890) = 57076 and (119885

119902+ 119906119890) =

06307 we have 11988323

= 006 And also for 119883119908

= 00168119885119908

= minus03340 and 119872119908

= 00025 we have 11988324

= 001Based on all the above parameters 120590max(119860) = 00041 Inthis case let 120581 = 001 120572

1= (119896119863119872119871)(2 + 120581119897

119909) asymp 015

and 1205731= (119872

119871119897119909119896119863)120590max(119860) = 082 Therefore 120572

1+ 1205731=

015 + 082 = 097 which satisfies 1205721+ 1205731

lt 1 Anotherexample considers that the UH-60 helicopter carries a loadweight 500 lb by a 15 ft slung length and let 119896

119863= 50 lbft then

1205721= 02 and 120573

1= 062 therefore (120572

1+ 1205731) = 082 Therefore

this assumption 1205721+ 1205731

lt 1 is realistic for the systemdiscussed in here The numerical values vary with respectto the slung load configuration It may represent the system

dynamic variations To investigate the stability characteristicsof dynamic variations two series of simulation studies werecarried out

Firstly the slung length was kept constant at 15 ft andthe load varied between the limits 500ndash2000 lb with 119896

119863=

75 lbft producing the system poles listed in Table 1 Thecorresponding root loci are sketched in Figures 3 and 4Figure 4 clearly shows that the positions of the system polescrossed the imaginary axis as the load weight increased

Secondly load weight was kept constant and the slunglength varied between the limit of 10ndash20 ft Table 2 sum-marises the results of variation of the system poles and thecorresponding root loci are illustrated in Figures 5 and 6

The pole locus indicates that the system poles have thevariation trends of moving further towards right direction onthe 119904-plane with increase of load weight and slung lengthsFrom the simulation results it is clearly shown that thesystem becomes less stable when the load andor the slunglength increases The simulation results are compared withthe stability analysis presented above

Now consider the case of fixed slung length of 15 ft withdifferent load weights In the cases of a UH-60 helicopter that

12 Journal of Control Science and Engineering

Table 1 System poles locations for a fixed slung length (119897 = 15ft) but with different load weights

Helicopter 500 lb 1000 lb 1500 lb 2000 lbminus10919 minus11159 minus07848 minus06221 minus09369minus63938 minus64301 minus92203 minus79848 minus75613minus00032 minus00048 minus00034 minus00052 minus00052minus00977 minus00818 minus00888 minus00972 minus00694minus03045 minus03036 minus03394 minus03318 minus03333minus03159 plusmn 04363119894 minus03199 plusmn 04628119894 minus02295 plusmn 05050119894 minus02366 plusmn 05974119894 minus01834 plusmn 04968119894minus00489 plusmn 03898119894 minus00211 plusmn 03868119894 00074 plusmn 03445119894 00119 plusmn 04406119894 00264 plusmn 03199119894mdash minus00316 plusmn 15043119894 minus01020 plusmn 17380119894 minus00368 plusmn 18753119894 minus01160 plusmn 20003119894mdash minus00736 plusmn 15019119894 minus07662 plusmn 11680119894 minus07349 plusmn 19064119894 minus12367 plusmn 08995119894

Table 2 System poles for a fixed load (1000 lb) but with different slung lengths

Helicopter 10 ft 15 ft 20 ftminus10919 minus10560 minus07848 minus08687minus63938 minus99252 minus92203 minus85565minus00032 minus00061 minus00034 minus00114minus00977 minus02012 minus00888 minus00953minus03045 minus02606 minus03394 minus03479minus03159 plusmn 04363119894 minus01809 plusmn 06219119894 minus02295 plusmn 05050119894 minus01636 plusmn 06092119894minus00489 plusmn 03898119894 00643 plusmn 04577119894 00074 plusmn 03445119894 00799 plusmn 04594119894mdash minus01055 plusmn 19883119894 minus01020 plusmn 17380119894 minus01247 plusmn 14470119894mdash minus08847 plusmn 12464119894 minus07662 plusmn 11680119894 minus06352 plusmn 10384119894

Re(poles)

Im(poles)

minus10 minus5 0 5minus08minus06minus04minus02

002040608

Figure 3 System root-locus diagram showing pole locus forconstant slung length (15 ft) as load weight is increased

minus04 minus035 minus03 minus025 minus02 minus015 minus01 minus005 0 005 01

Re(poles)

Im(poles)

minus08minus06minus04minus02

002040608

Figure 4 Enlarged view of Figure 3 showing poles locus near toRe(poles) = 0 axis

minus10 minus8 minus6 minus4 minus2 0 2

Re(poles)

Im(poles)

minus08minus06minus04minus02

002040608

Figure 5 System root-locus showing pole locus for constant loadweight (1000 lb) as slung length is increased

minus04 minus035 minus03 minus025 minus02 minus015 minus01 minus005 0 005 01

Re(poles)

Im(poles)

minus08minus06minus04minus02

002040608

Figure 6 Enlarged view of Figure 5 showing poles locus near toRe(poles) = 0 axis

Journal of Control Science and Engineering 13

carries a loadweight 500 lb and 1000 lb then1205721= 03 and 041

with 1205731= 041 and 082 respectively therefore 120572

1+ 1205731lt 1

thus the system is stable But when the load weight is 1500 lband 2000 lb then 120572

1= 01 and 0075 and also 120573

1= 123 and

164 respectively therefore 1205721+ 1205731gt 1 thus the system is

unstable For the case of the constant load weight (1000 lb) asslung length is increased then for a 10 ft and 15 ft slung length1205721= 015 and 120573

1= 055 and 082 respectively therefore

1205721+ 1205731lt 1 thus the system is stable But when the slung

length increased to 20 ft then 1205721

= 015 and 1205731

= 109therefore (120572

1+ 1205731) = 124 that is 120572

1+ 1205731gt 1 thus the

system is unstable However the system can be marginallystable if the UH-60 helicopter carries a load weight 725 lb bya 20 ft slung length then 120572

1= 02 and 120573

1= 08 therefore

(1205721+ 1205731) = 1 For this set of analysis 119896

119863is assumed to

be 119896119863

= 75 lbft However 119896119863can vary depending on the

weather condition and an example of 119896119863= 50 lbft is shown

above for illustrative purpose This analysis suggests that themaximum value of 20 ft slung length may be used for slungload operation However the load weight limit can vary withrespect to the capacity of the load carrying hooks helicopterweight environmental conditions and so forth

The examples presented here show that the simulationresults are correlated with the stability analysis Thus desiredcondition for the stability is met and it can locate the load atthe specified position or its neighbourhood

5 Conclusions

In this paper stability analysis of a helicopter with an under-slung load system is investigatedThe investigation identifiedthe conditions for the stabilisation of the system and thepositioning of the underslung load at hover condition Sta-bility analysis and numerical results proved that the desiredcondition for the stability is met then it is possible to locatethe load at the specified position or its neighbourhood Anillustration example is also given in the paper The methodresults in a guaranteed load positioning accuracy whichdepends on the design parameters and the method can beapplicable to any individual helicopter with external loadsystem

Nomenclature

119871119872119873 Overall helicopter rolling pitching andyawing moments

119901 119902 119903 Helicopter roll pitch and yaw rates aboutbody reference axes

119883119884 119885 Overall helicopter force components120601 120579 120595 Roll pitch and yaw angles119906 V 119908 Helicopter velocity components at centre

of gravity1205790 Main rotor collective

1205791119904 Longitudinal cyclic commands

1205791119888 Lateral cyclic commands

1205790119879 Tail rotor collective

1198830 1198840 1198850 Location of the suspension point withrespect to earth referenced 119909 119910 and 119911

directions

0 0 0 The helicopter velocity in the 119909 119910 and 119911

directions0 0 0 The helicopter acceleration in the 119909 119910 and119911 directions

120579119871 Load suspension angle in the119883-119885 plane

with respect to 119885-axis120601119871 Load suspension angle in the 119884-119885 plane

with respect to 119885-axis119872119871 Mass of the suspended load

119892 Acceleration due to gravity119897 Slung length119896119863 Aerodynamic drag force constant

(119896119863= (12)120588119878119862

119863)

120588 Air density119878 The load area presented to the airflow119862119863 Drag coefficient for the load

Competing Interests

The author declares that there are no competing interestsregarding the publication of this paper

References

[1] K Thanapalan and F Zhang ldquoModelling and simulation studyof a helicopter with an external slung load systemrdquo I-ManagersJournal on Instrumentation amp Control Engineering vol 1 no 1pp 9ndash17 2013

[2] M Bisgaard J D Bendtsen and A La Cour-Harbo ldquoModelingof generic slung load systemrdquo Journal of Guidance Control andDynamics vol 32 no 2 pp 573ndash585 2009

[3] Handling Qualities Requirements for Military Rotorcraft ADS-33-D US Army Aviation and Troop Command St Louis MissUSA 1996

[4] K Thanapalan and T M Wong ldquoModeing of a helicopter withan under-slung load systemrdquo in Proceedings of the 29th ChineseControl Conference (CCC rsquo10) pp 1451ndash1456 Beijing China July2010

[5] B Feng ldquoRobust control for lateral and longitudinal channels ofsmall-scale unmanned helicoptersrdquo Journal of Control Scienceand Engineering vol 2015 Article ID 483096 8 pages 2015

[6] J J Gribble ldquoLinear quadratic Gaussianloop transfer recoverydesign for a helicopter in low-speed flightrdquo Journal of GuidanceControl and Dynamics vol 16 no 4 pp 754ndash761 1993

[7] M A Manness J J Gribble and D J M Smith ldquoMultivariablemethods for helicopter flight control law designrdquo in Proceedingsof the 16th Annual Society European Rotorcraft Forum PaperNoIII52 Glasgow UK September 1990

[8] W L Garrard E Low and S Prouty ldquoDesign of attitude andrate command systems for helicopters using eigenstructureassignmentrdquo Journal of Guidance Control and Dynamics vol12 no 6 pp 783ndash791 1989

[9] Y B Shtessel and I A Shkolnikov ldquoAeronautical and spacevehicle control in dynamic sliding manifoldsrdquo InternationalJournal of Control vol 76 no 9-10 pp 1000ndash1017 2003

[10] D FThompson J S Pruyn andA Shukla ldquoFeedback design forrobust tracking and robust stiffness in flight control actuatorsusing a modified QFT techniquerdquo International Journal ofControl vol 72 no 16 pp 1480ndash1497 1999

14 Journal of Control Science and Engineering

[11] K Thanapalan ldquoModelling of a helicopter systemrdquo in Proceed-ings of the 1st Virtual Control Conference (VCC rsquo10) pp 45ndash52Aalborg University 2010

[12] D P Goodall and J Wang ldquoStabilization of a class of uncertainnonlinear affine systems subject to control constraintsrdquo Interna-tional Journal of Robust and Nonlinear Control vol 11 no 9 pp797ndash818 2001

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Journal of Control Science and Engineering 7

119867(119905 120593119871(119905) 119909119867 (119905)) =

[[[[[

[

0

0

( minus 119901)

(11988423V + 11988424119908)

]]]]]

]

(119905) = [1205791119888 1205790119879]119879

1198912(120593119871(119905) 119909119867 (119905)) = [

(11988431120601 + 11988432119901 + 11988433V + 11988434119908)

(11988442119901 + 11988443V + 11988444119908)

]

1198662= [

1198841205791119888

1198841205790119879

1198851205791119888

1198851205790119879

]

(21)

It is assumed that the lateral motion is primarily controlledby lateral cyclic commands (120579

1119888) and the tail rotor collective

1205790119879

3 Stability Analysis

The goal is to analyse the stability of the combined systemto ensure that it is possible to stabilise the helicopter withunderslung load system modelled by (11a) (11b) (20a) and(20b) in a real environment with uncertainties In this paperfor the stability analysis the Lyapunov second method isapplied for the helicopter with an underslung external loadsystem The analysis for the longitudinal motion is discussedfirst The system equations can be considered to have twomain parts that is known and unknown (or partly known)The known terms formed the nominal part of the systemmodelThe unknown or partly known part can be consideredas the uncertainty to the system The whole system is thenmodelled by a nominal part with the addition of uncertaintyIn fact the known elements in the subsystem (11a) arecharacterised by the prescribed triple (119891

1 1198661 119901) and it is

desired that the nominal part of the system is stableThe basic notations and concepts required for the analysis

are described first The state space is denoted by119883 fl R119899 andthe control space by119906 fl R119898 where 1 le 119898 le 119899TheEuclideaninner product (on 119883 or 119906 as appropriate) and induced normare denoted by ⟨sdot sdot⟩ and sdot respectively Let 119862(R119901R119902) and1198621(R119901R119902) denote the space of all continuous functions and

the space of continuous functions with continuous first-orderpartial derivatives respectively and let 119862infin(R119901R119902) denotethe space of functions whose partial derivatives of any orderexist and are continuous mapping R119901 rarr R119902 For a real-valued continuous scalar function 119909 rarr V(119909) defined onR119899 nabla V rarr nablaV isin R119899 denotes the gradient map The Liederivative of V along a vector field 119891 R119899 rarr R119899 is denoted by119871119891V R119899 rarr R which is defined by

(119871119891V) (119909) = ⟨nablaV (119909) 119891 (119909)⟩ (22)

The Lie bracket of vector fields 119891 119892 isin 119862infin(R119899R119899) is the

vector field [119891 119892] isin 119862infin(R119899R119899) defined by [119891 119892] = (119863119892)119891 minus

(119863119891)119892 where (119863119891)denotes the Jacobianmatrix of119891 and (119863119892)denotes the Jacobian matrix of 119892

In this paper nonlinear systemswith the following formatare considered

(119905) = 119891 (119909 (119905)) + 119866 (119909 (119905)) (119905) (23)

where 119909(119905) isin R119899 isin R119898 In general mathematical modelsof dynamical systems are usually imprecise due to modellingerrors and exogenous disturbances [12] Equation (23) can beconsidered as the nominal part of the system model and theuncertainty can be modelled as an additive perturbation tothe nominal systemmodel more specifically the structure ofthe system has the form

(119905) = 119891 (119909 (119905)) + 119866 (119909 (119905)) (119905) + 120599 (119909 (119905) 119906 (119905)) (24)

where 120599(119909(119905) 119906(119905))models the uncertainty in the systemSystem (24) is globally asymptotically stable to the zero

state if the system exhibits the following properties

(i) Existence and Continuation of Solutions For each 119909 isin R119899there exists a local solution119909 [0 119905

1) rarr R119899 (ie an absolutely

continuous function satisfying (24) almost everywhere (ae)and 119909(0) = 119909

0) and every such solution can be extended intoa solution on [0infin)

(ii) Boundedness of Solutions For each ℏ gt 0 there exists119903(ℏ) gt 0 such that 119909(119905) isin 119903(ℏ)119861

119899 for all 119905 ge 0 on every

solution 119909 [0infin) rarr R119899 with 1199090isin ℏ119861119899 where 119861

119899denote

the open unit ball centred at the origin in R119899

(iii) Stability of the State Origin For each 120575 gt 0 there exists119889(120575) gt 0 such that 119909(119905) isin 120575119861

119899for all 119905 ge 0 on every solution

119909 [0infin) rarr R119899 with 1199090isin 119889(120575)119861

119899

(iv) Global Attractivity of the State Origin For each ℏ gt 0 and120576 gt 0 there exists 119879(ℏ 120576) ge 0 such that 119909(119905) isin 119861

119899for all

119905 ge 119879(ℏ 120576) on every solution 119909 [0infin) rarr R119899 with 1199090isin ℏ119861119899

8 Journal of Control Science and Engineering

Consider a nonlinear system described by the ordinarydifferential equation as follows

(119905) = 119891 (119905 119909 (119905))

119909 (1199050) = 1199090

(25)

where 119891 R times 119883 rarr 119883 and 119891(119905 0) = 0 for all 119905To analyse the stability of (25) Lyapunovrsquos second stabilityanalysis method is applicable The Lyapunov approach is toshow that a candidate ldquoLyapunov functionrdquo is nonincreasingalong all solution to (25) by means that do not requireexplicit knowledge of solutions to (25) From this appropriateconclusion can be drawn regarding stability concepts relatingto solutions of the differential equation (25) An essentialpart of Lyapunovrsquos method is the determination of the timederivative of the candidate ldquoLyapunov functionrdquo along allsolution of the dynamical system

Consider a Lyapunov candidate (119905 119909) rarr V(119905 119909) Rtimes119883 rarr

R which satisfies the condition V isin 1198621(R times 119883) in which case

its time derivative along solutions to (25) is given by

V (119905 119909 (119905)) =120597V (119905 119909 (119905))

120597119905+ ⟨nablaV (119905 119909 (119905)) 119891 (119905 119909 (119905))⟩ (26)

for almost all 119905 isin RLet 119882(119909(119905)) denote a positive definite function If

V(119905 119909(119905)) satisfies

(i) V(119905 0) = 0 for all 119905 ge 0

(ii) 119882(119909(119905)) le V(119905 119909(119905)) for all 119909(119905) isin Φ 0 sub Φ sub 119883 andall 119905 ge 0

(iii) V(119905 119909(119905)) lt 0 in Φ

then V(119905 119909(119905)) is said to be a Lyapunov function in Φ IfV(119905 119909(119905)) in Φ then V(119905 119909(119905)) is said to be a weak Lyapunovfunction

A set Ε is said to be an invariant set with respect to thedynamical system = 119891(119909) if

119909 (0) isin Ε 997891997888rarr 119909 (119905) isin Ε forall119905 isin R+ (27)

In other words Ε is the set of points such that if a solution of = 119891(119909) belongs to Ε at some instant initialized points at119905 = 0 then it belongs to Ε for all future time

Now a setΕ sub 119909 is said to be a local invariantmanifold for(25) if for any 1199090 isin Ε 119909(119905) with 119909(0) = 119909

0 is in Ε for |119905| lt 119879

where 119879 gt 0 If 119879 = infin then Ε is said to be an invariantmanifold

Now considering the longitudinal motion (Figure 1) forthe helicopter with an underslung external load systemchoose a Lyapunov function candidate for the first subsystemV1as

V1(120579119871) =

1

2[12058911205792

1198711

+ 12058921205792

1198712

+ (1205893120579 minus 1205894119902)2+ 12058951199022] (28)

where 120589119894(119894 = 1 2 3 4 5) are design parameters to be

determined Then

V1(120579119871) = ⟨nablaV

1(120579119871) 1198911(120579119871) + 1198661(120579119871 (119905))

sdot [119901 (119909119867 (119905)) + 119902 (120579

119871 (119905) 119909119867 (119905))]

+ 119867(119905 120579119871 119909119867 (119905))⟩ = (119871

1198911

V1) (120579119871)

+ ⟨nablaV1(120579119871) 1198661(120579119871 (119905))

sdot [119901 (119909119867 (119905)) + 119902 (120579

119871 (119905) 119909119867 (119905))]

+ 119867(119905 120579119871 119909119867 (119905))⟩

(29)

From (28) nablaV1(120579119871) can be obtained as

nablaV1(120579119871)

= [12058911205791198711

12058921205791198712

1205893(1205893120579 minus 1205894119902) minus120589

4(1205893120579 minus 1205894119902) + 120589

5119902]119879

(30)

Substituting nablaV1(120579119871) and 119891

1(120579119871) into the derivative of

(1198711198911

V1)(120579119871) we have

(1198711198911

V1) (120579119871) = 12058911205791198711

1205791198712

minus 11989612057912058921205792

1198712

minus 1205892

119892

119897119909

sin 1205791198711

1205791198712

+119896119863119897119909

119872119871

1205892[sign (

119871) cos3 120579

1198711

+ sign (119871) sin3 120579

1198711

]

sdot 1205793

1198712

+ 1205893[1205893120579 minus 1205894119902] 119902 + (minus120589

31205894120579 + (120589

2

4+ 1205895) 119902)

sdot [11988321120579 + 119883

22119902]

(31)

Checking the term of minus sin (1205791198711

)1205791198712

it can be seen thatminus sin (120579

1198711

)1205791198712

le minus(2120587)1205791198711

1205791198712

when 1205791198711

and 1205791198712

are bothpositive or negative The situation of 120579

1198711

and 1205791198712

havingdifferent signs will help with the system stability Then

(1198711198911

V1) (120579119871) le 12058911205791198711

1205791198712

minus 11989612057912058921205792

1198712

minus 1205892

119892

119897119909

2

1205871205791198711

1205791198712

+119896119863119897119909

119872119871

sdot 1205892[sign (

119871) cos3 120579

1198711

+ sign (119871) sin3 120579

1198711

] 1205793

1198712

+ 1205893[1205893120579 minus 1205894119902] 119902 + (minus120589

31205894120579 + (120589

2

4+ 1205895) 119902)

sdot [11988321120579 + 119883

22119902]

(32)

If the design parameter 1205891is chosen as 120589

1= (2119892119897

119909120587)1205892

(1198711198911

V1)(120579119871) satisfies the following

(1198711198911

V1) (120579119871) le minus119896

12057912058921205792

1198712

+119896119863119897119909

119872119871

sdot 1205892[sign (

119871) cos3 120579

1198711

+ sign (119871) sin3 120579

1198711

] 1205793

1198712

+ 1205893[1205893120579 minus 1205894119902] 119902 + (minus120589

31205894120579 + (120589

2

4+ 1205895) 119902)

sdot [11988321120579 + 119883

22119902]

(33)

Journal of Control Science and Engineering 9

Further analysis on (33) will start from examining the firsttwo terms Rewrite these two terms So

minus 11989612057912058921205792

1198712

+119896119863119897119909

119872119871

1205892[sign (

119871) cos3 120579

1198711

+ sign (119871)

sdot sin3 1205791198711

] 1205793

1198712

= minus11989612057912058921205792

1198712

+119896119863119897119909

119872119871

12058921205791198712

[sign (119871)

sdot cos3 1205791198711

+ sign (119871) sin3 120579

1198711

] 1205792

1198712

le minus11989612057912058921205792

1198712

+119896119863119897119909

119872119871

1205892

10038161003816100381610038161003816max (120579

1198712

) (sign (119871) cos3 120579

1198711

+ sign (119871) sin3 120579

1198711

)100381610038161003816100381610038161205792

1198712

= minus12058921205792

1198712

119896120579

minus119896119863119897119909

119872119871

10038161003816100381610038161003816max (120579

1198712

)

sdot (sign (119871) cos3 120579

1198711

+ sign (119871) sin3 120579

1198711

)10038161003816100381610038161003816

(34)

If the hinge friction is big enough to satisfy the following

119896120579gt

119896119863119897119909

119872119871

10038161003816100381610038161003816max (120579

1198712

)

sdot (sign (119871) cos3 120579

1198711

+ sign (119871) sin3 120579

1198711

)10038161003816100381610038161003816

(35)

then

120582119871= 119896120579minus119896119863119897119909

119872119871

10038161003816100381610038161003816max (120579

1198712

)

sdot (sign (119871) cos3 120579

1198711

+ sign (119871) sin3 120579

1198711

)10038161003816100381610038161003816 ge 0

minus 11989612057912058921205792

1198712

+119896119863119897119909

119872119871

1205892[sign (

119871) cos3 120579

1198711

+ sign (119871)

sdot sin3 1205791198711

] 1205793

1198712

le minus12058921205792

1198712

120582119871

(36)

Now the following analysis has been conducted for (33) andthe load movement for the longitudinal motion is describedby

119871= 0minus 1198971199091205791198712

cos 1205791198711

119871= 0minus 1198971199091205791198712

sin 1205791198711

(37)

Around the hover condition1198830= 0 and 119885

0= 0 the signs for

119871and

119871are opposite to the one of 120579

1198712

provided that 1205791198711

ispositive

Considering the term (sign (119871) cos3 120579

1198711

+

sign (119871) sin3 120579

1198711

)1205793

1198712

for all possible combinations ofthe signs of 120579

1198712

and 1205791198711

the analysis follows below

(i) If 1205791198711

gt 0 and 1205791198712

gt 0 then sign (119871) = minus1

sign (119871) = minus1 cos3 120579

1198711

gt 0 and sin3 1205791198711

gt 0 There-fore (sign (

119871) cos3 120579

1198711

+ sign (119871) sin3 120579

1198711

)1205793

1198712

lt 0

(ii) If 1205791198711

lt 0 and 1205791198712

gt 0 then sign (119871) = minus1

sign (119871) = +1 cos3 120579

1198711

gt 0 and sin31205791198711

lt 0 So(sign (

119871) cos3 120579

1198711

+ sign (119871) sin3 120579

1198711

)1205793

1198712

lt 0

(iii) If 1205791198711

gt 0 and 1205791198712

lt 0 then sign (119871) = +1

and sign (119871) = +1 cos3 120579

1198711

gt 0 and sin3 1205791198711

gt

0 So the following is true (sign (119871) cos3 120579

1198711

+

sign (119871) sin3 120579

1198711

)1205793

1198712

lt 0

(iv) If 1205791198711

lt 0 and 1205791198712

lt 0 then sign (119871) = +1 and

sign (119871) = minus1 cos3 120579

1198711

gt 0 and sin3 1205791198711

lt 0 So(sign (

119871) cos3 120579

1198711

+ sign (119871) sin3 120579

1198711

)1205793

1198712

lt 0

Therefore

minus 11989612057912058921205792

1198712

+119896119863119897119909

119872119871

sdot 1205892[sign (

119871) cos3 120579

1198711

+ sign (119871) sin3 120579

1198711

] 1205793

1198712

le minus11989612057912058921205792

1198712

(38)

Now examining the rest of the terms of (1198711198911

V1)(120579119871) then

1205893[1205893120579 minus 1205894119902] 119902

+ (minus12058931205894120579 + (120589

2

4+ 1205895) 119902) [119883

21120579 + 119883

22119902] = 120589

2

3120579119902

minus 120589312058941199022minus 12058931205894119883211205792minus 1205893120589411988322120579119902

+ (1205892

4+ 1205895)11988321120579119902 + (120589

2

4+ 1205895)119883221199022

= minus [12058931205894minus (1205892

4+ 1205895)11988321] 1199022minus 12058931205894119883211205792

+ [1205892

3minus 1205893120589411988322+ (1205892

4+ 1205895)11988321] 120579119902

(39)

For the system 120579 and 1205791198711

always have different signs 120579 and 119902

always have different signs and 119902 and 1205791198711

have the same signsSo

[1205892

3minus 1205893120589411988322+ (1205892

4+ 1205895)11988321] 120579119902 le 0

if [1205892

3minus 1205893120589411988322+ (1205892

4+ 1205895)11988321] ge 0

(40)

Therefore

1205893[1205893120579 minus 1205894119902] 119902

+ (minus12058931205894120579 + (120589

2

4+ 1205895) 119902) [119883

21120579 + 119883

22119902]

le minus [12058931205894minus (1205892

4+ 1205895)11988321] 1199022minus 12058931205894119883211205792

(41)

Choose the design parameters to satisfy

[12058931205894minus (1205892

4+ 1205895)11988321] gt 0 (42)

If11988321

gt 0 then

minus [12058931205894minus (1205892

4+ 1205895)11988321] 1199022minus 12058931205894119883211205792lt 0 (43)

10 Journal of Control Science and Engineering

If11988321

lt 0 then

minus [12058931205894minus (1205892

4+ 1205895)11988321] 1199022minus 12058931205894119883211205792lt 0 (44)

in the region ofminus[12058931205894minus(1205892

4+1205895)11988321]1199022lt 12058931205894119883211205792 Since 120579 is

caused by the loadmotion here the value of 120579 should bemuchsmaller than 119902 The inequality (44) is easy to be satisfied

In summary of the above analysis and by defining

Ψ1(120579119871) = 11989612057912058921205792

1198712

+ [12058931205894minus (1205892

4+ 1205895)11988321] 1199022

+ 12058931205894119883211205792

Ψ2(120579119871) = [120589

31205894minus (1205892

4+ 1205895)11988321] 1199022+ 12058931205894119883211205792

(45)

the following lemma can be derived

Lemma 1 Defining a Lyapunov function (28) and choosingthe design parameters to satisfy 120589

1= (2119892119897

119909120587)1205892 [12058931205894minus (1205892

4+

1205895)11988321] gt 0 and [1205892

3minus1205893120589411988322+(1205892

4+1205895)11988321] ge 0 then within

the region specified by minus[12058931205894minus (1205892

4+ 1205895)11988321]1199022lt 12058931205894119883211205792

(i) V1(0) = 0 and V

1(120579119871) gt 0 forall120579

119871= 0

(ii) V1(120579119871) rarr infin as 120579

119871 rarr infin

(iii) (1198711198911

V1)(120579119871) le minusΨ

1(120579119871) forall120579119871 around the hover condi-

tion or (1198711198911

V1)(120579119871) le minusΨ

2(120579119871) forall120579119871 if (35) holds

Both functions Ψ1and Ψ

2are nonnegative

Recall the control term 119901(119909119867(119905)) in the first subsystem

of (11a) and (11b) it can be seen that [(119863119901)(119909119867)]minus1 exists

for all 119909119867(119905) The unknown vector fields 119902(120579

119871(119905) 119909119867) and

119867(119905 120579119871(119905) 119909119867) model the uncertainties imposed onto the

system Since 119902(120579119871(119905) 119909119867) is directly mapped into the ldquocon-

trolrdquo space of 119909119867(119905) it can be considered as a matched

uncertainty119867(119905 120579119871(119905) 119909119867) is unknown and it does belong to

the control space of 119909119867(119905) so it represents the mismatched

uncertainty in the system [12]Generally the range of the (longitudinal) load suspension

angle is within minus1205872 lt 120579119871lt 1205872 The helicopter velocities

and load suspension angle havemaximumoperational valuestherefore the uncertainties in the system are bounded Withthe Lyapunov function defined in (28)

(1198711198921

V1) (120579119871) = ⟨nablaV

1(120579119871) 1198921⟩ =

119896119863

119872119871119897119909

12058921205791198712

(46)

In 1198661(120579119871) the columns 119892

1and 119892

2are the same therefore

(1198711198922

V1) (120579119871) =

119896119863

119872119871119897119909

12058921205791198712

(47)

By considering the maximum values of cos (sdot) and sin (sdot)functions the bounding values relating to the uncertainty119902(120579119871(119905) 119909119867) can be estimated as follows

10038171003817100381710038171003817119902 (120579119871 119909119867)10038171003817100381710038171003817le

10038161003816100381610038161003816100381610038161003816

1

119897119909

( + )

+1

119872119871

(minus119896119863(2 +

120581

119897119909

) (119906 + 119908) +119896120579

119897119909

)1205791198712

10038161003816100381610038161003816100381610038161003816

le

10038161003816100381610038161003816100381610038161003816

119896120579

119897119909

1205791198712

10038161003816100381610038161003816100381610038161003816+

10038161003816100381610038161003816100381610038161003816

119896119863

119872119871

(2 +120581

119897119909

) (119906 + 119908)

10038161003816100381610038161003816100381610038161003816+

10038161003816100381610038161003816100381610038161003816

1

119897119909

( + )

10038161003816100381610038161003816100381610038161003816

le119896120579

119897119909

100381710038171003817100381710038171205791198712

10038171003817100381710038171003817+

119896119863

119872119871

(2 +120581

119897119909

)1003817100381710038171003817119901 (119909119867)1003817100381710038171003817 + 1205722 (119905)

le 1205831

10038161003816100381610038161003816(1198711198922

V1) (120579119871)10038161003816100381610038161003816+ 1205721

1003817100381710038171003817119901 (119909119867)1003817100381710038171003817 + 1205722 (119905)

(48)

where 1205831

= 119872119871119896120579119896119863 1205721

= (119896119863119872119871)(2 + 120581119897

119909) (120581 =

max [1205811 1205812]) and 120572

2(119905) is defined by 120572

2(119905) = max |(1119897

119909)( +

)| gt 0For the mismatched uncertainty 119867(119905 120579

119871(119905) 119909119867) the fol-

lowing analysis is conducted to obtain its bounding functionThe mismatched uncertainty can be rewritten as

119867(119905 120579119871 (119905) 119909119867) =

[[[[[

[

0

0

(11988611119906 + 11988612119908)

(11988323119906 + 119883

24119908)

]]]]]

]

=

[[[[[

[

0 0

0 0

11988611

11988612

11988323

11988324

]]]]]

]

[119906

119908]

(49)

Let 119860 = [1198861111988612

1198832311988324] then

10038171003817100381710038171003817119867 (119905 120579

119871 119909119867)10038171003817100381710038171003817le1003817100381710038171003817100381711986010038171003817100381710038171003817

1003817100381710038171003817119909119867 (119905)1003817100381710038171003817 (50)

Define a positive function as

1006704120579 (120579119871) = radic(1205792

1198711

+ 12057921198712

+ 1205792 + 1199022+ 120576) (51)

where 120576 is a very small positive constant Choosing1205731

= (1198721198711198971199091198961198631205892)119860 then the mismatched uncertainty

119867(119905 120579119871(119905) 119909119867) is bounded by the following inequality

10038171003817100381710038171003817119867 (119905 120579

119871 119909119867)10038171003817100381710038171003817

le 1006704120579minus1

(120579119871) 1205731

2

sum

119894=1

(10038161003816100381610038161003816(119871119892119894

V1) (120579119871)10038161003816100381610038161003816)1003816100381610038161003816119901119894 (119909119867)

1003816100381610038161003816

(52)

where 119892119894denotes the 119894th column of the matrix function

1198661(120579119871) and 119901

119894is the 119894th component of 119901(119909

119867) respectively As

1205731has a design parameter 120589

2involved it is easy to have 120573

1to

lead (52) to be true For the matched uncertainty we have

10038171003817100381710038171003817119902 (120579119871 119909119867)10038171003817100381710038171003817le

119896120579

119897119909

100381610038161003816100381610038161205791198712

10038161003816100381610038161003816+ 1205721

1003817100381710038171003817119901 (119909119867)1003817100381710038171003817 + 1205722 (119905) (53)

Journal of Control Science and Engineering 11

Following the above analysis the following lemma can bederived

Lemma 2 The uncertainties (52) and (53) are bounded andsatisfy if 120572

1 1205731ge 0 and 120572

1+ 1205731lt 1 for 120572

1= (119896119863119872119871)(2 +

120581119897119909) and 120573

1= (1198721198711198971199091198961198631205892)119860

4 Numerical Results

Using a simulation software the linearized model for a UH-60 helicopter is obtained with the following state and inputmatrices

119860 =

[[[[[[[[[[[[[[[[[[[

[

minus00000 00003 minus00000 minus00000 00000 minus00000 10000 minus00023 00503

minus00003 minus00000 minus00000 minus00000 minus00000 minus00000 minus00000 09989 00463

minus00001 00000 minus00000 minus00000 00000 00000 minus00000 minus00464 10002

minus19386 minus28593 minus00030 minus00195 minus00126 00168 minus12000 57076 02683

328694 minus03640 00027 00126 minus00311 minus00013 minus41348 minus22350 03713

03138 minus14034 00004 00150 00034 minus03340 02866 06307 23410

minus09144 minus04783 00017 00226 minus00246 00007 minus65893 minus21301 minus00598

00631 minus06744 00008 00029 00046 00025 02059 minus14255 minus01636

minus07549 minus00119 minus00002 00009 00028 00006 minus00749 minus01032 minus02214

]]]]]]]]]]]]]]]]]]]

]

119861 =

[[[[[[[[[[[[[[[[[[[

[

minus00000 00000 minus00000 00000

00000 00000 minus00000 minus00000

minus00000 00000 minus00000 minus00000

minus22190 05734 03170 minus00019

35381 minus00453 minus00963 02007

minus15629 00237 minus60740 minus01014

19943 minus00714 minus00541 00731

00446 minus01097 00553 minus00281

minus08197 minus00082 01573 minus00635

]]]]]]]]]]]]]]]]]]]

]

(54)

The desired condition 1205721+ 1205731lt 1 can be verified by consid-

ering a typical case that the UH-60 helicopter carries a loadweight 1000 lb by a 15 ft slung length and let 119896

119863= 75 lbft

For the UH-60 helicopter model with reference to thegeneral mathematical model presented in Section 2 we have1198831205791119904

= 05734 1198831205790

= 03170 1198851205791119904

= 00237 1198851205790

= minus607401198721205791119904

= minus01097 and 1198721205790

= 00553 So the followingparameters can be calculated to have the values 119886

11= minus02

and 11988612

= minus002 For 119883119906= minus00195 119885

119906= 00150 119872

119906=

00029 119872119902= minus14255 (119883

119902minus 119908119890) = 57076 and (119885

119902+ 119906119890) =

06307 we have 11988323

= 006 And also for 119883119908

= 00168119885119908

= minus03340 and 119872119908

= 00025 we have 11988324

= 001Based on all the above parameters 120590max(119860) = 00041 Inthis case let 120581 = 001 120572

1= (119896119863119872119871)(2 + 120581119897

119909) asymp 015

and 1205731= (119872

119871119897119909119896119863)120590max(119860) = 082 Therefore 120572

1+ 1205731=

015 + 082 = 097 which satisfies 1205721+ 1205731

lt 1 Anotherexample considers that the UH-60 helicopter carries a loadweight 500 lb by a 15 ft slung length and let 119896

119863= 50 lbft then

1205721= 02 and 120573

1= 062 therefore (120572

1+ 1205731) = 082 Therefore

this assumption 1205721+ 1205731

lt 1 is realistic for the systemdiscussed in here The numerical values vary with respectto the slung load configuration It may represent the system

dynamic variations To investigate the stability characteristicsof dynamic variations two series of simulation studies werecarried out

Firstly the slung length was kept constant at 15 ft andthe load varied between the limits 500ndash2000 lb with 119896

119863=

75 lbft producing the system poles listed in Table 1 Thecorresponding root loci are sketched in Figures 3 and 4Figure 4 clearly shows that the positions of the system polescrossed the imaginary axis as the load weight increased

Secondly load weight was kept constant and the slunglength varied between the limit of 10ndash20 ft Table 2 sum-marises the results of variation of the system poles and thecorresponding root loci are illustrated in Figures 5 and 6

The pole locus indicates that the system poles have thevariation trends of moving further towards right direction onthe 119904-plane with increase of load weight and slung lengthsFrom the simulation results it is clearly shown that thesystem becomes less stable when the load andor the slunglength increases The simulation results are compared withthe stability analysis presented above

Now consider the case of fixed slung length of 15 ft withdifferent load weights In the cases of a UH-60 helicopter that

12 Journal of Control Science and Engineering

Table 1 System poles locations for a fixed slung length (119897 = 15ft) but with different load weights

Helicopter 500 lb 1000 lb 1500 lb 2000 lbminus10919 minus11159 minus07848 minus06221 minus09369minus63938 minus64301 minus92203 minus79848 minus75613minus00032 minus00048 minus00034 minus00052 minus00052minus00977 minus00818 minus00888 minus00972 minus00694minus03045 minus03036 minus03394 minus03318 minus03333minus03159 plusmn 04363119894 minus03199 plusmn 04628119894 minus02295 plusmn 05050119894 minus02366 plusmn 05974119894 minus01834 plusmn 04968119894minus00489 plusmn 03898119894 minus00211 plusmn 03868119894 00074 plusmn 03445119894 00119 plusmn 04406119894 00264 plusmn 03199119894mdash minus00316 plusmn 15043119894 minus01020 plusmn 17380119894 minus00368 plusmn 18753119894 minus01160 plusmn 20003119894mdash minus00736 plusmn 15019119894 minus07662 plusmn 11680119894 minus07349 plusmn 19064119894 minus12367 plusmn 08995119894

Table 2 System poles for a fixed load (1000 lb) but with different slung lengths

Helicopter 10 ft 15 ft 20 ftminus10919 minus10560 minus07848 minus08687minus63938 minus99252 minus92203 minus85565minus00032 minus00061 minus00034 minus00114minus00977 minus02012 minus00888 minus00953minus03045 minus02606 minus03394 minus03479minus03159 plusmn 04363119894 minus01809 plusmn 06219119894 minus02295 plusmn 05050119894 minus01636 plusmn 06092119894minus00489 plusmn 03898119894 00643 plusmn 04577119894 00074 plusmn 03445119894 00799 plusmn 04594119894mdash minus01055 plusmn 19883119894 minus01020 plusmn 17380119894 minus01247 plusmn 14470119894mdash minus08847 plusmn 12464119894 minus07662 plusmn 11680119894 minus06352 plusmn 10384119894

Re(poles)

Im(poles)

minus10 minus5 0 5minus08minus06minus04minus02

002040608

Figure 3 System root-locus diagram showing pole locus forconstant slung length (15 ft) as load weight is increased

minus04 minus035 minus03 minus025 minus02 minus015 minus01 minus005 0 005 01

Re(poles)

Im(poles)

minus08minus06minus04minus02

002040608

Figure 4 Enlarged view of Figure 3 showing poles locus near toRe(poles) = 0 axis

minus10 minus8 minus6 minus4 minus2 0 2

Re(poles)

Im(poles)

minus08minus06minus04minus02

002040608

Figure 5 System root-locus showing pole locus for constant loadweight (1000 lb) as slung length is increased

minus04 minus035 minus03 minus025 minus02 minus015 minus01 minus005 0 005 01

Re(poles)

Im(poles)

minus08minus06minus04minus02

002040608

Figure 6 Enlarged view of Figure 5 showing poles locus near toRe(poles) = 0 axis

Journal of Control Science and Engineering 13

carries a loadweight 500 lb and 1000 lb then1205721= 03 and 041

with 1205731= 041 and 082 respectively therefore 120572

1+ 1205731lt 1

thus the system is stable But when the load weight is 1500 lband 2000 lb then 120572

1= 01 and 0075 and also 120573

1= 123 and

164 respectively therefore 1205721+ 1205731gt 1 thus the system is

unstable For the case of the constant load weight (1000 lb) asslung length is increased then for a 10 ft and 15 ft slung length1205721= 015 and 120573

1= 055 and 082 respectively therefore

1205721+ 1205731lt 1 thus the system is stable But when the slung

length increased to 20 ft then 1205721

= 015 and 1205731

= 109therefore (120572

1+ 1205731) = 124 that is 120572

1+ 1205731gt 1 thus the

system is unstable However the system can be marginallystable if the UH-60 helicopter carries a load weight 725 lb bya 20 ft slung length then 120572

1= 02 and 120573

1= 08 therefore

(1205721+ 1205731) = 1 For this set of analysis 119896

119863is assumed to

be 119896119863

= 75 lbft However 119896119863can vary depending on the

weather condition and an example of 119896119863= 50 lbft is shown

above for illustrative purpose This analysis suggests that themaximum value of 20 ft slung length may be used for slungload operation However the load weight limit can vary withrespect to the capacity of the load carrying hooks helicopterweight environmental conditions and so forth

The examples presented here show that the simulationresults are correlated with the stability analysis Thus desiredcondition for the stability is met and it can locate the load atthe specified position or its neighbourhood

5 Conclusions

In this paper stability analysis of a helicopter with an under-slung load system is investigatedThe investigation identifiedthe conditions for the stabilisation of the system and thepositioning of the underslung load at hover condition Sta-bility analysis and numerical results proved that the desiredcondition for the stability is met then it is possible to locatethe load at the specified position or its neighbourhood Anillustration example is also given in the paper The methodresults in a guaranteed load positioning accuracy whichdepends on the design parameters and the method can beapplicable to any individual helicopter with external loadsystem

Nomenclature

119871119872119873 Overall helicopter rolling pitching andyawing moments

119901 119902 119903 Helicopter roll pitch and yaw rates aboutbody reference axes

119883119884 119885 Overall helicopter force components120601 120579 120595 Roll pitch and yaw angles119906 V 119908 Helicopter velocity components at centre

of gravity1205790 Main rotor collective

1205791119904 Longitudinal cyclic commands

1205791119888 Lateral cyclic commands

1205790119879 Tail rotor collective

1198830 1198840 1198850 Location of the suspension point withrespect to earth referenced 119909 119910 and 119911

directions

0 0 0 The helicopter velocity in the 119909 119910 and 119911

directions0 0 0 The helicopter acceleration in the 119909 119910 and119911 directions

120579119871 Load suspension angle in the119883-119885 plane

with respect to 119885-axis120601119871 Load suspension angle in the 119884-119885 plane

with respect to 119885-axis119872119871 Mass of the suspended load

119892 Acceleration due to gravity119897 Slung length119896119863 Aerodynamic drag force constant

(119896119863= (12)120588119878119862

119863)

120588 Air density119878 The load area presented to the airflow119862119863 Drag coefficient for the load

Competing Interests

The author declares that there are no competing interestsregarding the publication of this paper

References

[1] K Thanapalan and F Zhang ldquoModelling and simulation studyof a helicopter with an external slung load systemrdquo I-ManagersJournal on Instrumentation amp Control Engineering vol 1 no 1pp 9ndash17 2013

[2] M Bisgaard J D Bendtsen and A La Cour-Harbo ldquoModelingof generic slung load systemrdquo Journal of Guidance Control andDynamics vol 32 no 2 pp 573ndash585 2009

[3] Handling Qualities Requirements for Military Rotorcraft ADS-33-D US Army Aviation and Troop Command St Louis MissUSA 1996

[4] K Thanapalan and T M Wong ldquoModeing of a helicopter withan under-slung load systemrdquo in Proceedings of the 29th ChineseControl Conference (CCC rsquo10) pp 1451ndash1456 Beijing China July2010

[5] B Feng ldquoRobust control for lateral and longitudinal channels ofsmall-scale unmanned helicoptersrdquo Journal of Control Scienceand Engineering vol 2015 Article ID 483096 8 pages 2015

[6] J J Gribble ldquoLinear quadratic Gaussianloop transfer recoverydesign for a helicopter in low-speed flightrdquo Journal of GuidanceControl and Dynamics vol 16 no 4 pp 754ndash761 1993

[7] M A Manness J J Gribble and D J M Smith ldquoMultivariablemethods for helicopter flight control law designrdquo in Proceedingsof the 16th Annual Society European Rotorcraft Forum PaperNoIII52 Glasgow UK September 1990

[8] W L Garrard E Low and S Prouty ldquoDesign of attitude andrate command systems for helicopters using eigenstructureassignmentrdquo Journal of Guidance Control and Dynamics vol12 no 6 pp 783ndash791 1989

[9] Y B Shtessel and I A Shkolnikov ldquoAeronautical and spacevehicle control in dynamic sliding manifoldsrdquo InternationalJournal of Control vol 76 no 9-10 pp 1000ndash1017 2003

[10] D FThompson J S Pruyn andA Shukla ldquoFeedback design forrobust tracking and robust stiffness in flight control actuatorsusing a modified QFT techniquerdquo International Journal ofControl vol 72 no 16 pp 1480ndash1497 1999

14 Journal of Control Science and Engineering

[11] K Thanapalan ldquoModelling of a helicopter systemrdquo in Proceed-ings of the 1st Virtual Control Conference (VCC rsquo10) pp 45ndash52Aalborg University 2010

[12] D P Goodall and J Wang ldquoStabilization of a class of uncertainnonlinear affine systems subject to control constraintsrdquo Interna-tional Journal of Robust and Nonlinear Control vol 11 no 9 pp797ndash818 2001

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

8 Journal of Control Science and Engineering

Consider a nonlinear system described by the ordinarydifferential equation as follows

(119905) = 119891 (119905 119909 (119905))

119909 (1199050) = 1199090

(25)

where 119891 R times 119883 rarr 119883 and 119891(119905 0) = 0 for all 119905To analyse the stability of (25) Lyapunovrsquos second stabilityanalysis method is applicable The Lyapunov approach is toshow that a candidate ldquoLyapunov functionrdquo is nonincreasingalong all solution to (25) by means that do not requireexplicit knowledge of solutions to (25) From this appropriateconclusion can be drawn regarding stability concepts relatingto solutions of the differential equation (25) An essentialpart of Lyapunovrsquos method is the determination of the timederivative of the candidate ldquoLyapunov functionrdquo along allsolution of the dynamical system

Consider a Lyapunov candidate (119905 119909) rarr V(119905 119909) Rtimes119883 rarr

R which satisfies the condition V isin 1198621(R times 119883) in which case

its time derivative along solutions to (25) is given by

V (119905 119909 (119905)) =120597V (119905 119909 (119905))

120597119905+ ⟨nablaV (119905 119909 (119905)) 119891 (119905 119909 (119905))⟩ (26)

for almost all 119905 isin RLet 119882(119909(119905)) denote a positive definite function If

V(119905 119909(119905)) satisfies

(i) V(119905 0) = 0 for all 119905 ge 0

(ii) 119882(119909(119905)) le V(119905 119909(119905)) for all 119909(119905) isin Φ 0 sub Φ sub 119883 andall 119905 ge 0

(iii) V(119905 119909(119905)) lt 0 in Φ

then V(119905 119909(119905)) is said to be a Lyapunov function in Φ IfV(119905 119909(119905)) in Φ then V(119905 119909(119905)) is said to be a weak Lyapunovfunction

A set Ε is said to be an invariant set with respect to thedynamical system = 119891(119909) if

119909 (0) isin Ε 997891997888rarr 119909 (119905) isin Ε forall119905 isin R+ (27)

In other words Ε is the set of points such that if a solution of = 119891(119909) belongs to Ε at some instant initialized points at119905 = 0 then it belongs to Ε for all future time

Now a setΕ sub 119909 is said to be a local invariantmanifold for(25) if for any 1199090 isin Ε 119909(119905) with 119909(0) = 119909

0 is in Ε for |119905| lt 119879

where 119879 gt 0 If 119879 = infin then Ε is said to be an invariantmanifold

Now considering the longitudinal motion (Figure 1) forthe helicopter with an underslung external load systemchoose a Lyapunov function candidate for the first subsystemV1as

V1(120579119871) =

1

2[12058911205792

1198711

+ 12058921205792

1198712

+ (1205893120579 minus 1205894119902)2+ 12058951199022] (28)

where 120589119894(119894 = 1 2 3 4 5) are design parameters to be

determined Then

V1(120579119871) = ⟨nablaV

1(120579119871) 1198911(120579119871) + 1198661(120579119871 (119905))

sdot [119901 (119909119867 (119905)) + 119902 (120579

119871 (119905) 119909119867 (119905))]

+ 119867(119905 120579119871 119909119867 (119905))⟩ = (119871

1198911

V1) (120579119871)

+ ⟨nablaV1(120579119871) 1198661(120579119871 (119905))

sdot [119901 (119909119867 (119905)) + 119902 (120579

119871 (119905) 119909119867 (119905))]

+ 119867(119905 120579119871 119909119867 (119905))⟩

(29)

From (28) nablaV1(120579119871) can be obtained as

nablaV1(120579119871)

= [12058911205791198711

12058921205791198712

1205893(1205893120579 minus 1205894119902) minus120589

4(1205893120579 minus 1205894119902) + 120589

5119902]119879

(30)

Substituting nablaV1(120579119871) and 119891

1(120579119871) into the derivative of

(1198711198911

V1)(120579119871) we have

(1198711198911

V1) (120579119871) = 12058911205791198711

1205791198712

minus 11989612057912058921205792

1198712

minus 1205892

119892

119897119909

sin 1205791198711

1205791198712

+119896119863119897119909

119872119871

1205892[sign (

119871) cos3 120579

1198711

+ sign (119871) sin3 120579

1198711

]

sdot 1205793

1198712

+ 1205893[1205893120579 minus 1205894119902] 119902 + (minus120589

31205894120579 + (120589

2

4+ 1205895) 119902)

sdot [11988321120579 + 119883

22119902]

(31)

Checking the term of minus sin (1205791198711

)1205791198712

it can be seen thatminus sin (120579

1198711

)1205791198712

le minus(2120587)1205791198711

1205791198712

when 1205791198711

and 1205791198712

are bothpositive or negative The situation of 120579

1198711

and 1205791198712

havingdifferent signs will help with the system stability Then

(1198711198911

V1) (120579119871) le 12058911205791198711

1205791198712

minus 11989612057912058921205792

1198712

minus 1205892

119892

119897119909

2

1205871205791198711

1205791198712

+119896119863119897119909

119872119871

sdot 1205892[sign (

119871) cos3 120579

1198711

+ sign (119871) sin3 120579

1198711

] 1205793

1198712

+ 1205893[1205893120579 minus 1205894119902] 119902 + (minus120589

31205894120579 + (120589

2

4+ 1205895) 119902)

sdot [11988321120579 + 119883

22119902]

(32)

If the design parameter 1205891is chosen as 120589

1= (2119892119897

119909120587)1205892

(1198711198911

V1)(120579119871) satisfies the following

(1198711198911

V1) (120579119871) le minus119896

12057912058921205792

1198712

+119896119863119897119909

119872119871

sdot 1205892[sign (

119871) cos3 120579

1198711

+ sign (119871) sin3 120579

1198711

] 1205793

1198712

+ 1205893[1205893120579 minus 1205894119902] 119902 + (minus120589

31205894120579 + (120589

2

4+ 1205895) 119902)

sdot [11988321120579 + 119883

22119902]

(33)

Journal of Control Science and Engineering 9

Further analysis on (33) will start from examining the firsttwo terms Rewrite these two terms So

minus 11989612057912058921205792

1198712

+119896119863119897119909

119872119871

1205892[sign (

119871) cos3 120579

1198711

+ sign (119871)

sdot sin3 1205791198711

] 1205793

1198712

= minus11989612057912058921205792

1198712

+119896119863119897119909

119872119871

12058921205791198712

[sign (119871)

sdot cos3 1205791198711

+ sign (119871) sin3 120579

1198711

] 1205792

1198712

le minus11989612057912058921205792

1198712

+119896119863119897119909

119872119871

1205892

10038161003816100381610038161003816max (120579

1198712

) (sign (119871) cos3 120579

1198711

+ sign (119871) sin3 120579

1198711

)100381610038161003816100381610038161205792

1198712

= minus12058921205792

1198712

119896120579

minus119896119863119897119909

119872119871

10038161003816100381610038161003816max (120579

1198712

)

sdot (sign (119871) cos3 120579

1198711

+ sign (119871) sin3 120579

1198711

)10038161003816100381610038161003816

(34)

If the hinge friction is big enough to satisfy the following

119896120579gt

119896119863119897119909

119872119871

10038161003816100381610038161003816max (120579

1198712

)

sdot (sign (119871) cos3 120579

1198711

+ sign (119871) sin3 120579

1198711

)10038161003816100381610038161003816

(35)

then

120582119871= 119896120579minus119896119863119897119909

119872119871

10038161003816100381610038161003816max (120579

1198712

)

sdot (sign (119871) cos3 120579

1198711

+ sign (119871) sin3 120579

1198711

)10038161003816100381610038161003816 ge 0

minus 11989612057912058921205792

1198712

+119896119863119897119909

119872119871

1205892[sign (

119871) cos3 120579

1198711

+ sign (119871)

sdot sin3 1205791198711

] 1205793

1198712

le minus12058921205792

1198712

120582119871

(36)

Now the following analysis has been conducted for (33) andthe load movement for the longitudinal motion is describedby

119871= 0minus 1198971199091205791198712

cos 1205791198711

119871= 0minus 1198971199091205791198712

sin 1205791198711

(37)

Around the hover condition1198830= 0 and 119885

0= 0 the signs for

119871and

119871are opposite to the one of 120579

1198712

provided that 1205791198711

ispositive

Considering the term (sign (119871) cos3 120579

1198711

+

sign (119871) sin3 120579

1198711

)1205793

1198712

for all possible combinations ofthe signs of 120579

1198712

and 1205791198711

the analysis follows below

(i) If 1205791198711

gt 0 and 1205791198712

gt 0 then sign (119871) = minus1

sign (119871) = minus1 cos3 120579

1198711

gt 0 and sin3 1205791198711

gt 0 There-fore (sign (

119871) cos3 120579

1198711

+ sign (119871) sin3 120579

1198711

)1205793

1198712

lt 0

(ii) If 1205791198711

lt 0 and 1205791198712

gt 0 then sign (119871) = minus1

sign (119871) = +1 cos3 120579

1198711

gt 0 and sin31205791198711

lt 0 So(sign (

119871) cos3 120579

1198711

+ sign (119871) sin3 120579

1198711

)1205793

1198712

lt 0

(iii) If 1205791198711

gt 0 and 1205791198712

lt 0 then sign (119871) = +1

and sign (119871) = +1 cos3 120579

1198711

gt 0 and sin3 1205791198711

gt

0 So the following is true (sign (119871) cos3 120579

1198711

+

sign (119871) sin3 120579

1198711

)1205793

1198712

lt 0

(iv) If 1205791198711

lt 0 and 1205791198712

lt 0 then sign (119871) = +1 and

sign (119871) = minus1 cos3 120579

1198711

gt 0 and sin3 1205791198711

lt 0 So(sign (

119871) cos3 120579

1198711

+ sign (119871) sin3 120579

1198711

)1205793

1198712

lt 0

Therefore

minus 11989612057912058921205792

1198712

+119896119863119897119909

119872119871

sdot 1205892[sign (

119871) cos3 120579

1198711

+ sign (119871) sin3 120579

1198711

] 1205793

1198712

le minus11989612057912058921205792

1198712

(38)

Now examining the rest of the terms of (1198711198911

V1)(120579119871) then

1205893[1205893120579 minus 1205894119902] 119902

+ (minus12058931205894120579 + (120589

2

4+ 1205895) 119902) [119883

21120579 + 119883

22119902] = 120589

2

3120579119902

minus 120589312058941199022minus 12058931205894119883211205792minus 1205893120589411988322120579119902

+ (1205892

4+ 1205895)11988321120579119902 + (120589

2

4+ 1205895)119883221199022

= minus [12058931205894minus (1205892

4+ 1205895)11988321] 1199022minus 12058931205894119883211205792

+ [1205892

3minus 1205893120589411988322+ (1205892

4+ 1205895)11988321] 120579119902

(39)

For the system 120579 and 1205791198711

always have different signs 120579 and 119902

always have different signs and 119902 and 1205791198711

have the same signsSo

[1205892

3minus 1205893120589411988322+ (1205892

4+ 1205895)11988321] 120579119902 le 0

if [1205892

3minus 1205893120589411988322+ (1205892

4+ 1205895)11988321] ge 0

(40)

Therefore

1205893[1205893120579 minus 1205894119902] 119902

+ (minus12058931205894120579 + (120589

2

4+ 1205895) 119902) [119883

21120579 + 119883

22119902]

le minus [12058931205894minus (1205892

4+ 1205895)11988321] 1199022minus 12058931205894119883211205792

(41)

Choose the design parameters to satisfy

[12058931205894minus (1205892

4+ 1205895)11988321] gt 0 (42)

If11988321

gt 0 then

minus [12058931205894minus (1205892

4+ 1205895)11988321] 1199022minus 12058931205894119883211205792lt 0 (43)

10 Journal of Control Science and Engineering

If11988321

lt 0 then

minus [12058931205894minus (1205892

4+ 1205895)11988321] 1199022minus 12058931205894119883211205792lt 0 (44)

in the region ofminus[12058931205894minus(1205892

4+1205895)11988321]1199022lt 12058931205894119883211205792 Since 120579 is

caused by the loadmotion here the value of 120579 should bemuchsmaller than 119902 The inequality (44) is easy to be satisfied

In summary of the above analysis and by defining

Ψ1(120579119871) = 11989612057912058921205792

1198712

+ [12058931205894minus (1205892

4+ 1205895)11988321] 1199022

+ 12058931205894119883211205792

Ψ2(120579119871) = [120589

31205894minus (1205892

4+ 1205895)11988321] 1199022+ 12058931205894119883211205792

(45)

the following lemma can be derived

Lemma 1 Defining a Lyapunov function (28) and choosingthe design parameters to satisfy 120589

1= (2119892119897

119909120587)1205892 [12058931205894minus (1205892

4+

1205895)11988321] gt 0 and [1205892

3minus1205893120589411988322+(1205892

4+1205895)11988321] ge 0 then within

the region specified by minus[12058931205894minus (1205892

4+ 1205895)11988321]1199022lt 12058931205894119883211205792

(i) V1(0) = 0 and V

1(120579119871) gt 0 forall120579

119871= 0

(ii) V1(120579119871) rarr infin as 120579

119871 rarr infin

(iii) (1198711198911

V1)(120579119871) le minusΨ

1(120579119871) forall120579119871 around the hover condi-

tion or (1198711198911

V1)(120579119871) le minusΨ

2(120579119871) forall120579119871 if (35) holds

Both functions Ψ1and Ψ

2are nonnegative

Recall the control term 119901(119909119867(119905)) in the first subsystem

of (11a) and (11b) it can be seen that [(119863119901)(119909119867)]minus1 exists

for all 119909119867(119905) The unknown vector fields 119902(120579

119871(119905) 119909119867) and

119867(119905 120579119871(119905) 119909119867) model the uncertainties imposed onto the

system Since 119902(120579119871(119905) 119909119867) is directly mapped into the ldquocon-

trolrdquo space of 119909119867(119905) it can be considered as a matched

uncertainty119867(119905 120579119871(119905) 119909119867) is unknown and it does belong to

the control space of 119909119867(119905) so it represents the mismatched

uncertainty in the system [12]Generally the range of the (longitudinal) load suspension

angle is within minus1205872 lt 120579119871lt 1205872 The helicopter velocities

and load suspension angle havemaximumoperational valuestherefore the uncertainties in the system are bounded Withthe Lyapunov function defined in (28)

(1198711198921

V1) (120579119871) = ⟨nablaV

1(120579119871) 1198921⟩ =

119896119863

119872119871119897119909

12058921205791198712

(46)

In 1198661(120579119871) the columns 119892

1and 119892

2are the same therefore

(1198711198922

V1) (120579119871) =

119896119863

119872119871119897119909

12058921205791198712

(47)

By considering the maximum values of cos (sdot) and sin (sdot)functions the bounding values relating to the uncertainty119902(120579119871(119905) 119909119867) can be estimated as follows

10038171003817100381710038171003817119902 (120579119871 119909119867)10038171003817100381710038171003817le

10038161003816100381610038161003816100381610038161003816

1

119897119909

( + )

+1

119872119871

(minus119896119863(2 +

120581

119897119909

) (119906 + 119908) +119896120579

119897119909

)1205791198712

10038161003816100381610038161003816100381610038161003816

le

10038161003816100381610038161003816100381610038161003816

119896120579

119897119909

1205791198712

10038161003816100381610038161003816100381610038161003816+

10038161003816100381610038161003816100381610038161003816

119896119863

119872119871

(2 +120581

119897119909

) (119906 + 119908)

10038161003816100381610038161003816100381610038161003816+

10038161003816100381610038161003816100381610038161003816

1

119897119909

( + )

10038161003816100381610038161003816100381610038161003816

le119896120579

119897119909

100381710038171003817100381710038171205791198712

10038171003817100381710038171003817+

119896119863

119872119871

(2 +120581

119897119909

)1003817100381710038171003817119901 (119909119867)1003817100381710038171003817 + 1205722 (119905)

le 1205831

10038161003816100381610038161003816(1198711198922

V1) (120579119871)10038161003816100381610038161003816+ 1205721

1003817100381710038171003817119901 (119909119867)1003817100381710038171003817 + 1205722 (119905)

(48)

where 1205831

= 119872119871119896120579119896119863 1205721

= (119896119863119872119871)(2 + 120581119897

119909) (120581 =

max [1205811 1205812]) and 120572

2(119905) is defined by 120572

2(119905) = max |(1119897

119909)( +

)| gt 0For the mismatched uncertainty 119867(119905 120579

119871(119905) 119909119867) the fol-

lowing analysis is conducted to obtain its bounding functionThe mismatched uncertainty can be rewritten as

119867(119905 120579119871 (119905) 119909119867) =

[[[[[

[

0

0

(11988611119906 + 11988612119908)

(11988323119906 + 119883

24119908)

]]]]]

]

=

[[[[[

[

0 0

0 0

11988611

11988612

11988323

11988324

]]]]]

]

[119906

119908]

(49)

Let 119860 = [1198861111988612

1198832311988324] then

10038171003817100381710038171003817119867 (119905 120579

119871 119909119867)10038171003817100381710038171003817le1003817100381710038171003817100381711986010038171003817100381710038171003817

1003817100381710038171003817119909119867 (119905)1003817100381710038171003817 (50)

Define a positive function as

1006704120579 (120579119871) = radic(1205792

1198711

+ 12057921198712

+ 1205792 + 1199022+ 120576) (51)

where 120576 is a very small positive constant Choosing1205731

= (1198721198711198971199091198961198631205892)119860 then the mismatched uncertainty

119867(119905 120579119871(119905) 119909119867) is bounded by the following inequality

10038171003817100381710038171003817119867 (119905 120579

119871 119909119867)10038171003817100381710038171003817

le 1006704120579minus1

(120579119871) 1205731

2

sum

119894=1

(10038161003816100381610038161003816(119871119892119894

V1) (120579119871)10038161003816100381610038161003816)1003816100381610038161003816119901119894 (119909119867)

1003816100381610038161003816

(52)

where 119892119894denotes the 119894th column of the matrix function

1198661(120579119871) and 119901

119894is the 119894th component of 119901(119909

119867) respectively As

1205731has a design parameter 120589

2involved it is easy to have 120573

1to

lead (52) to be true For the matched uncertainty we have

10038171003817100381710038171003817119902 (120579119871 119909119867)10038171003817100381710038171003817le

119896120579

119897119909

100381610038161003816100381610038161205791198712

10038161003816100381610038161003816+ 1205721

1003817100381710038171003817119901 (119909119867)1003817100381710038171003817 + 1205722 (119905) (53)

Journal of Control Science and Engineering 11

Following the above analysis the following lemma can bederived

Lemma 2 The uncertainties (52) and (53) are bounded andsatisfy if 120572

1 1205731ge 0 and 120572

1+ 1205731lt 1 for 120572

1= (119896119863119872119871)(2 +

120581119897119909) and 120573

1= (1198721198711198971199091198961198631205892)119860

4 Numerical Results

Using a simulation software the linearized model for a UH-60 helicopter is obtained with the following state and inputmatrices

119860 =

[[[[[[[[[[[[[[[[[[[

[

minus00000 00003 minus00000 minus00000 00000 minus00000 10000 minus00023 00503

minus00003 minus00000 minus00000 minus00000 minus00000 minus00000 minus00000 09989 00463

minus00001 00000 minus00000 minus00000 00000 00000 minus00000 minus00464 10002

minus19386 minus28593 minus00030 minus00195 minus00126 00168 minus12000 57076 02683

328694 minus03640 00027 00126 minus00311 minus00013 minus41348 minus22350 03713

03138 minus14034 00004 00150 00034 minus03340 02866 06307 23410

minus09144 minus04783 00017 00226 minus00246 00007 minus65893 minus21301 minus00598

00631 minus06744 00008 00029 00046 00025 02059 minus14255 minus01636

minus07549 minus00119 minus00002 00009 00028 00006 minus00749 minus01032 minus02214

]]]]]]]]]]]]]]]]]]]

]

119861 =

[[[[[[[[[[[[[[[[[[[

[

minus00000 00000 minus00000 00000

00000 00000 minus00000 minus00000

minus00000 00000 minus00000 minus00000

minus22190 05734 03170 minus00019

35381 minus00453 minus00963 02007

minus15629 00237 minus60740 minus01014

19943 minus00714 minus00541 00731

00446 minus01097 00553 minus00281

minus08197 minus00082 01573 minus00635

]]]]]]]]]]]]]]]]]]]

]

(54)

The desired condition 1205721+ 1205731lt 1 can be verified by consid-

ering a typical case that the UH-60 helicopter carries a loadweight 1000 lb by a 15 ft slung length and let 119896

119863= 75 lbft

For the UH-60 helicopter model with reference to thegeneral mathematical model presented in Section 2 we have1198831205791119904

= 05734 1198831205790

= 03170 1198851205791119904

= 00237 1198851205790

= minus607401198721205791119904

= minus01097 and 1198721205790

= 00553 So the followingparameters can be calculated to have the values 119886

11= minus02

and 11988612

= minus002 For 119883119906= minus00195 119885

119906= 00150 119872

119906=

00029 119872119902= minus14255 (119883

119902minus 119908119890) = 57076 and (119885

119902+ 119906119890) =

06307 we have 11988323

= 006 And also for 119883119908

= 00168119885119908

= minus03340 and 119872119908

= 00025 we have 11988324

= 001Based on all the above parameters 120590max(119860) = 00041 Inthis case let 120581 = 001 120572

1= (119896119863119872119871)(2 + 120581119897

119909) asymp 015

and 1205731= (119872

119871119897119909119896119863)120590max(119860) = 082 Therefore 120572

1+ 1205731=

015 + 082 = 097 which satisfies 1205721+ 1205731

lt 1 Anotherexample considers that the UH-60 helicopter carries a loadweight 500 lb by a 15 ft slung length and let 119896

119863= 50 lbft then

1205721= 02 and 120573

1= 062 therefore (120572

1+ 1205731) = 082 Therefore

this assumption 1205721+ 1205731

lt 1 is realistic for the systemdiscussed in here The numerical values vary with respectto the slung load configuration It may represent the system

dynamic variations To investigate the stability characteristicsof dynamic variations two series of simulation studies werecarried out

Firstly the slung length was kept constant at 15 ft andthe load varied between the limits 500ndash2000 lb with 119896

119863=

75 lbft producing the system poles listed in Table 1 Thecorresponding root loci are sketched in Figures 3 and 4Figure 4 clearly shows that the positions of the system polescrossed the imaginary axis as the load weight increased

Secondly load weight was kept constant and the slunglength varied between the limit of 10ndash20 ft Table 2 sum-marises the results of variation of the system poles and thecorresponding root loci are illustrated in Figures 5 and 6

The pole locus indicates that the system poles have thevariation trends of moving further towards right direction onthe 119904-plane with increase of load weight and slung lengthsFrom the simulation results it is clearly shown that thesystem becomes less stable when the load andor the slunglength increases The simulation results are compared withthe stability analysis presented above

Now consider the case of fixed slung length of 15 ft withdifferent load weights In the cases of a UH-60 helicopter that

12 Journal of Control Science and Engineering

Table 1 System poles locations for a fixed slung length (119897 = 15ft) but with different load weights

Helicopter 500 lb 1000 lb 1500 lb 2000 lbminus10919 minus11159 minus07848 minus06221 minus09369minus63938 minus64301 minus92203 minus79848 minus75613minus00032 minus00048 minus00034 minus00052 minus00052minus00977 minus00818 minus00888 minus00972 minus00694minus03045 minus03036 minus03394 minus03318 minus03333minus03159 plusmn 04363119894 minus03199 plusmn 04628119894 minus02295 plusmn 05050119894 minus02366 plusmn 05974119894 minus01834 plusmn 04968119894minus00489 plusmn 03898119894 minus00211 plusmn 03868119894 00074 plusmn 03445119894 00119 plusmn 04406119894 00264 plusmn 03199119894mdash minus00316 plusmn 15043119894 minus01020 plusmn 17380119894 minus00368 plusmn 18753119894 minus01160 plusmn 20003119894mdash minus00736 plusmn 15019119894 minus07662 plusmn 11680119894 minus07349 plusmn 19064119894 minus12367 plusmn 08995119894

Table 2 System poles for a fixed load (1000 lb) but with different slung lengths

Helicopter 10 ft 15 ft 20 ftminus10919 minus10560 minus07848 minus08687minus63938 minus99252 minus92203 minus85565minus00032 minus00061 minus00034 minus00114minus00977 minus02012 minus00888 minus00953minus03045 minus02606 minus03394 minus03479minus03159 plusmn 04363119894 minus01809 plusmn 06219119894 minus02295 plusmn 05050119894 minus01636 plusmn 06092119894minus00489 plusmn 03898119894 00643 plusmn 04577119894 00074 plusmn 03445119894 00799 plusmn 04594119894mdash minus01055 plusmn 19883119894 minus01020 plusmn 17380119894 minus01247 plusmn 14470119894mdash minus08847 plusmn 12464119894 minus07662 plusmn 11680119894 minus06352 plusmn 10384119894

Re(poles)

Im(poles)

minus10 minus5 0 5minus08minus06minus04minus02

002040608

Figure 3 System root-locus diagram showing pole locus forconstant slung length (15 ft) as load weight is increased

minus04 minus035 minus03 minus025 minus02 minus015 minus01 minus005 0 005 01

Re(poles)

Im(poles)

minus08minus06minus04minus02

002040608

Figure 4 Enlarged view of Figure 3 showing poles locus near toRe(poles) = 0 axis

minus10 minus8 minus6 minus4 minus2 0 2

Re(poles)

Im(poles)

minus08minus06minus04minus02

002040608

Figure 5 System root-locus showing pole locus for constant loadweight (1000 lb) as slung length is increased

minus04 minus035 minus03 minus025 minus02 minus015 minus01 minus005 0 005 01

Re(poles)

Im(poles)

minus08minus06minus04minus02

002040608

Figure 6 Enlarged view of Figure 5 showing poles locus near toRe(poles) = 0 axis

Journal of Control Science and Engineering 13

carries a loadweight 500 lb and 1000 lb then1205721= 03 and 041

with 1205731= 041 and 082 respectively therefore 120572

1+ 1205731lt 1

thus the system is stable But when the load weight is 1500 lband 2000 lb then 120572

1= 01 and 0075 and also 120573

1= 123 and

164 respectively therefore 1205721+ 1205731gt 1 thus the system is

unstable For the case of the constant load weight (1000 lb) asslung length is increased then for a 10 ft and 15 ft slung length1205721= 015 and 120573

1= 055 and 082 respectively therefore

1205721+ 1205731lt 1 thus the system is stable But when the slung

length increased to 20 ft then 1205721

= 015 and 1205731

= 109therefore (120572

1+ 1205731) = 124 that is 120572

1+ 1205731gt 1 thus the

system is unstable However the system can be marginallystable if the UH-60 helicopter carries a load weight 725 lb bya 20 ft slung length then 120572

1= 02 and 120573

1= 08 therefore

(1205721+ 1205731) = 1 For this set of analysis 119896

119863is assumed to

be 119896119863

= 75 lbft However 119896119863can vary depending on the

weather condition and an example of 119896119863= 50 lbft is shown

above for illustrative purpose This analysis suggests that themaximum value of 20 ft slung length may be used for slungload operation However the load weight limit can vary withrespect to the capacity of the load carrying hooks helicopterweight environmental conditions and so forth

The examples presented here show that the simulationresults are correlated with the stability analysis Thus desiredcondition for the stability is met and it can locate the load atthe specified position or its neighbourhood

5 Conclusions

In this paper stability analysis of a helicopter with an under-slung load system is investigatedThe investigation identifiedthe conditions for the stabilisation of the system and thepositioning of the underslung load at hover condition Sta-bility analysis and numerical results proved that the desiredcondition for the stability is met then it is possible to locatethe load at the specified position or its neighbourhood Anillustration example is also given in the paper The methodresults in a guaranteed load positioning accuracy whichdepends on the design parameters and the method can beapplicable to any individual helicopter with external loadsystem

Nomenclature

119871119872119873 Overall helicopter rolling pitching andyawing moments

119901 119902 119903 Helicopter roll pitch and yaw rates aboutbody reference axes

119883119884 119885 Overall helicopter force components120601 120579 120595 Roll pitch and yaw angles119906 V 119908 Helicopter velocity components at centre

of gravity1205790 Main rotor collective

1205791119904 Longitudinal cyclic commands

1205791119888 Lateral cyclic commands

1205790119879 Tail rotor collective

1198830 1198840 1198850 Location of the suspension point withrespect to earth referenced 119909 119910 and 119911

directions

0 0 0 The helicopter velocity in the 119909 119910 and 119911

directions0 0 0 The helicopter acceleration in the 119909 119910 and119911 directions

120579119871 Load suspension angle in the119883-119885 plane

with respect to 119885-axis120601119871 Load suspension angle in the 119884-119885 plane

with respect to 119885-axis119872119871 Mass of the suspended load

119892 Acceleration due to gravity119897 Slung length119896119863 Aerodynamic drag force constant

(119896119863= (12)120588119878119862

119863)

120588 Air density119878 The load area presented to the airflow119862119863 Drag coefficient for the load

Competing Interests

The author declares that there are no competing interestsregarding the publication of this paper

References

[1] K Thanapalan and F Zhang ldquoModelling and simulation studyof a helicopter with an external slung load systemrdquo I-ManagersJournal on Instrumentation amp Control Engineering vol 1 no 1pp 9ndash17 2013

[2] M Bisgaard J D Bendtsen and A La Cour-Harbo ldquoModelingof generic slung load systemrdquo Journal of Guidance Control andDynamics vol 32 no 2 pp 573ndash585 2009

[3] Handling Qualities Requirements for Military Rotorcraft ADS-33-D US Army Aviation and Troop Command St Louis MissUSA 1996

[4] K Thanapalan and T M Wong ldquoModeing of a helicopter withan under-slung load systemrdquo in Proceedings of the 29th ChineseControl Conference (CCC rsquo10) pp 1451ndash1456 Beijing China July2010

[5] B Feng ldquoRobust control for lateral and longitudinal channels ofsmall-scale unmanned helicoptersrdquo Journal of Control Scienceand Engineering vol 2015 Article ID 483096 8 pages 2015

[6] J J Gribble ldquoLinear quadratic Gaussianloop transfer recoverydesign for a helicopter in low-speed flightrdquo Journal of GuidanceControl and Dynamics vol 16 no 4 pp 754ndash761 1993

[7] M A Manness J J Gribble and D J M Smith ldquoMultivariablemethods for helicopter flight control law designrdquo in Proceedingsof the 16th Annual Society European Rotorcraft Forum PaperNoIII52 Glasgow UK September 1990

[8] W L Garrard E Low and S Prouty ldquoDesign of attitude andrate command systems for helicopters using eigenstructureassignmentrdquo Journal of Guidance Control and Dynamics vol12 no 6 pp 783ndash791 1989

[9] Y B Shtessel and I A Shkolnikov ldquoAeronautical and spacevehicle control in dynamic sliding manifoldsrdquo InternationalJournal of Control vol 76 no 9-10 pp 1000ndash1017 2003

[10] D FThompson J S Pruyn andA Shukla ldquoFeedback design forrobust tracking and robust stiffness in flight control actuatorsusing a modified QFT techniquerdquo International Journal ofControl vol 72 no 16 pp 1480ndash1497 1999

14 Journal of Control Science and Engineering

[11] K Thanapalan ldquoModelling of a helicopter systemrdquo in Proceed-ings of the 1st Virtual Control Conference (VCC rsquo10) pp 45ndash52Aalborg University 2010

[12] D P Goodall and J Wang ldquoStabilization of a class of uncertainnonlinear affine systems subject to control constraintsrdquo Interna-tional Journal of Robust and Nonlinear Control vol 11 no 9 pp797ndash818 2001

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Journal of Control Science and Engineering 9

Further analysis on (33) will start from examining the firsttwo terms Rewrite these two terms So

minus 11989612057912058921205792

1198712

+119896119863119897119909

119872119871

1205892[sign (

119871) cos3 120579

1198711

+ sign (119871)

sdot sin3 1205791198711

] 1205793

1198712

= minus11989612057912058921205792

1198712

+119896119863119897119909

119872119871

12058921205791198712

[sign (119871)

sdot cos3 1205791198711

+ sign (119871) sin3 120579

1198711

] 1205792

1198712

le minus11989612057912058921205792

1198712

+119896119863119897119909

119872119871

1205892

10038161003816100381610038161003816max (120579

1198712

) (sign (119871) cos3 120579

1198711

+ sign (119871) sin3 120579

1198711

)100381610038161003816100381610038161205792

1198712

= minus12058921205792

1198712

119896120579

minus119896119863119897119909

119872119871

10038161003816100381610038161003816max (120579

1198712

)

sdot (sign (119871) cos3 120579

1198711

+ sign (119871) sin3 120579

1198711

)10038161003816100381610038161003816

(34)

If the hinge friction is big enough to satisfy the following

119896120579gt

119896119863119897119909

119872119871

10038161003816100381610038161003816max (120579

1198712

)

sdot (sign (119871) cos3 120579

1198711

+ sign (119871) sin3 120579

1198711

)10038161003816100381610038161003816

(35)

then

120582119871= 119896120579minus119896119863119897119909

119872119871

10038161003816100381610038161003816max (120579

1198712

)

sdot (sign (119871) cos3 120579

1198711

+ sign (119871) sin3 120579

1198711

)10038161003816100381610038161003816 ge 0

minus 11989612057912058921205792

1198712

+119896119863119897119909

119872119871

1205892[sign (

119871) cos3 120579

1198711

+ sign (119871)

sdot sin3 1205791198711

] 1205793

1198712

le minus12058921205792

1198712

120582119871

(36)

Now the following analysis has been conducted for (33) andthe load movement for the longitudinal motion is describedby

119871= 0minus 1198971199091205791198712

cos 1205791198711

119871= 0minus 1198971199091205791198712

sin 1205791198711

(37)

Around the hover condition1198830= 0 and 119885

0= 0 the signs for

119871and

119871are opposite to the one of 120579

1198712

provided that 1205791198711

ispositive

Considering the term (sign (119871) cos3 120579

1198711

+

sign (119871) sin3 120579

1198711

)1205793

1198712

for all possible combinations ofthe signs of 120579

1198712

and 1205791198711

the analysis follows below

(i) If 1205791198711

gt 0 and 1205791198712

gt 0 then sign (119871) = minus1

sign (119871) = minus1 cos3 120579

1198711

gt 0 and sin3 1205791198711

gt 0 There-fore (sign (

119871) cos3 120579

1198711

+ sign (119871) sin3 120579

1198711

)1205793

1198712

lt 0

(ii) If 1205791198711

lt 0 and 1205791198712

gt 0 then sign (119871) = minus1

sign (119871) = +1 cos3 120579

1198711

gt 0 and sin31205791198711

lt 0 So(sign (

119871) cos3 120579

1198711

+ sign (119871) sin3 120579

1198711

)1205793

1198712

lt 0

(iii) If 1205791198711

gt 0 and 1205791198712

lt 0 then sign (119871) = +1

and sign (119871) = +1 cos3 120579

1198711

gt 0 and sin3 1205791198711

gt

0 So the following is true (sign (119871) cos3 120579

1198711

+

sign (119871) sin3 120579

1198711

)1205793

1198712

lt 0

(iv) If 1205791198711

lt 0 and 1205791198712

lt 0 then sign (119871) = +1 and

sign (119871) = minus1 cos3 120579

1198711

gt 0 and sin3 1205791198711

lt 0 So(sign (

119871) cos3 120579

1198711

+ sign (119871) sin3 120579

1198711

)1205793

1198712

lt 0

Therefore

minus 11989612057912058921205792

1198712

+119896119863119897119909

119872119871

sdot 1205892[sign (

119871) cos3 120579

1198711

+ sign (119871) sin3 120579

1198711

] 1205793

1198712

le minus11989612057912058921205792

1198712

(38)

Now examining the rest of the terms of (1198711198911

V1)(120579119871) then

1205893[1205893120579 minus 1205894119902] 119902

+ (minus12058931205894120579 + (120589

2

4+ 1205895) 119902) [119883

21120579 + 119883

22119902] = 120589

2

3120579119902

minus 120589312058941199022minus 12058931205894119883211205792minus 1205893120589411988322120579119902

+ (1205892

4+ 1205895)11988321120579119902 + (120589

2

4+ 1205895)119883221199022

= minus [12058931205894minus (1205892

4+ 1205895)11988321] 1199022minus 12058931205894119883211205792

+ [1205892

3minus 1205893120589411988322+ (1205892

4+ 1205895)11988321] 120579119902

(39)

For the system 120579 and 1205791198711

always have different signs 120579 and 119902

always have different signs and 119902 and 1205791198711

have the same signsSo

[1205892

3minus 1205893120589411988322+ (1205892

4+ 1205895)11988321] 120579119902 le 0

if [1205892

3minus 1205893120589411988322+ (1205892

4+ 1205895)11988321] ge 0

(40)

Therefore

1205893[1205893120579 minus 1205894119902] 119902

+ (minus12058931205894120579 + (120589

2

4+ 1205895) 119902) [119883

21120579 + 119883

22119902]

le minus [12058931205894minus (1205892

4+ 1205895)11988321] 1199022minus 12058931205894119883211205792

(41)

Choose the design parameters to satisfy

[12058931205894minus (1205892

4+ 1205895)11988321] gt 0 (42)

If11988321

gt 0 then

minus [12058931205894minus (1205892

4+ 1205895)11988321] 1199022minus 12058931205894119883211205792lt 0 (43)

10 Journal of Control Science and Engineering

If11988321

lt 0 then

minus [12058931205894minus (1205892

4+ 1205895)11988321] 1199022minus 12058931205894119883211205792lt 0 (44)

in the region ofminus[12058931205894minus(1205892

4+1205895)11988321]1199022lt 12058931205894119883211205792 Since 120579 is

caused by the loadmotion here the value of 120579 should bemuchsmaller than 119902 The inequality (44) is easy to be satisfied

In summary of the above analysis and by defining

Ψ1(120579119871) = 11989612057912058921205792

1198712

+ [12058931205894minus (1205892

4+ 1205895)11988321] 1199022

+ 12058931205894119883211205792

Ψ2(120579119871) = [120589

31205894minus (1205892

4+ 1205895)11988321] 1199022+ 12058931205894119883211205792

(45)

the following lemma can be derived

Lemma 1 Defining a Lyapunov function (28) and choosingthe design parameters to satisfy 120589

1= (2119892119897

119909120587)1205892 [12058931205894minus (1205892

4+

1205895)11988321] gt 0 and [1205892

3minus1205893120589411988322+(1205892

4+1205895)11988321] ge 0 then within

the region specified by minus[12058931205894minus (1205892

4+ 1205895)11988321]1199022lt 12058931205894119883211205792

(i) V1(0) = 0 and V

1(120579119871) gt 0 forall120579

119871= 0

(ii) V1(120579119871) rarr infin as 120579

119871 rarr infin

(iii) (1198711198911

V1)(120579119871) le minusΨ

1(120579119871) forall120579119871 around the hover condi-

tion or (1198711198911

V1)(120579119871) le minusΨ

2(120579119871) forall120579119871 if (35) holds

Both functions Ψ1and Ψ

2are nonnegative

Recall the control term 119901(119909119867(119905)) in the first subsystem

of (11a) and (11b) it can be seen that [(119863119901)(119909119867)]minus1 exists

for all 119909119867(119905) The unknown vector fields 119902(120579

119871(119905) 119909119867) and

119867(119905 120579119871(119905) 119909119867) model the uncertainties imposed onto the

system Since 119902(120579119871(119905) 119909119867) is directly mapped into the ldquocon-

trolrdquo space of 119909119867(119905) it can be considered as a matched

uncertainty119867(119905 120579119871(119905) 119909119867) is unknown and it does belong to

the control space of 119909119867(119905) so it represents the mismatched

uncertainty in the system [12]Generally the range of the (longitudinal) load suspension

angle is within minus1205872 lt 120579119871lt 1205872 The helicopter velocities

and load suspension angle havemaximumoperational valuestherefore the uncertainties in the system are bounded Withthe Lyapunov function defined in (28)

(1198711198921

V1) (120579119871) = ⟨nablaV

1(120579119871) 1198921⟩ =

119896119863

119872119871119897119909

12058921205791198712

(46)

In 1198661(120579119871) the columns 119892

1and 119892

2are the same therefore

(1198711198922

V1) (120579119871) =

119896119863

119872119871119897119909

12058921205791198712

(47)

By considering the maximum values of cos (sdot) and sin (sdot)functions the bounding values relating to the uncertainty119902(120579119871(119905) 119909119867) can be estimated as follows

10038171003817100381710038171003817119902 (120579119871 119909119867)10038171003817100381710038171003817le

10038161003816100381610038161003816100381610038161003816

1

119897119909

( + )

+1

119872119871

(minus119896119863(2 +

120581

119897119909

) (119906 + 119908) +119896120579

119897119909

)1205791198712

10038161003816100381610038161003816100381610038161003816

le

10038161003816100381610038161003816100381610038161003816

119896120579

119897119909

1205791198712

10038161003816100381610038161003816100381610038161003816+

10038161003816100381610038161003816100381610038161003816

119896119863

119872119871

(2 +120581

119897119909

) (119906 + 119908)

10038161003816100381610038161003816100381610038161003816+

10038161003816100381610038161003816100381610038161003816

1

119897119909

( + )

10038161003816100381610038161003816100381610038161003816

le119896120579

119897119909

100381710038171003817100381710038171205791198712

10038171003817100381710038171003817+

119896119863

119872119871

(2 +120581

119897119909

)1003817100381710038171003817119901 (119909119867)1003817100381710038171003817 + 1205722 (119905)

le 1205831

10038161003816100381610038161003816(1198711198922

V1) (120579119871)10038161003816100381610038161003816+ 1205721

1003817100381710038171003817119901 (119909119867)1003817100381710038171003817 + 1205722 (119905)

(48)

where 1205831

= 119872119871119896120579119896119863 1205721

= (119896119863119872119871)(2 + 120581119897

119909) (120581 =

max [1205811 1205812]) and 120572

2(119905) is defined by 120572

2(119905) = max |(1119897

119909)( +

)| gt 0For the mismatched uncertainty 119867(119905 120579

119871(119905) 119909119867) the fol-

lowing analysis is conducted to obtain its bounding functionThe mismatched uncertainty can be rewritten as

119867(119905 120579119871 (119905) 119909119867) =

[[[[[

[

0

0

(11988611119906 + 11988612119908)

(11988323119906 + 119883

24119908)

]]]]]

]

=

[[[[[

[

0 0

0 0

11988611

11988612

11988323

11988324

]]]]]

]

[119906

119908]

(49)

Let 119860 = [1198861111988612

1198832311988324] then

10038171003817100381710038171003817119867 (119905 120579

119871 119909119867)10038171003817100381710038171003817le1003817100381710038171003817100381711986010038171003817100381710038171003817

1003817100381710038171003817119909119867 (119905)1003817100381710038171003817 (50)

Define a positive function as

1006704120579 (120579119871) = radic(1205792

1198711

+ 12057921198712

+ 1205792 + 1199022+ 120576) (51)

where 120576 is a very small positive constant Choosing1205731

= (1198721198711198971199091198961198631205892)119860 then the mismatched uncertainty

119867(119905 120579119871(119905) 119909119867) is bounded by the following inequality

10038171003817100381710038171003817119867 (119905 120579

119871 119909119867)10038171003817100381710038171003817

le 1006704120579minus1

(120579119871) 1205731

2

sum

119894=1

(10038161003816100381610038161003816(119871119892119894

V1) (120579119871)10038161003816100381610038161003816)1003816100381610038161003816119901119894 (119909119867)

1003816100381610038161003816

(52)

where 119892119894denotes the 119894th column of the matrix function

1198661(120579119871) and 119901

119894is the 119894th component of 119901(119909

119867) respectively As

1205731has a design parameter 120589

2involved it is easy to have 120573

1to

lead (52) to be true For the matched uncertainty we have

10038171003817100381710038171003817119902 (120579119871 119909119867)10038171003817100381710038171003817le

119896120579

119897119909

100381610038161003816100381610038161205791198712

10038161003816100381610038161003816+ 1205721

1003817100381710038171003817119901 (119909119867)1003817100381710038171003817 + 1205722 (119905) (53)

Journal of Control Science and Engineering 11

Following the above analysis the following lemma can bederived

Lemma 2 The uncertainties (52) and (53) are bounded andsatisfy if 120572

1 1205731ge 0 and 120572

1+ 1205731lt 1 for 120572

1= (119896119863119872119871)(2 +

120581119897119909) and 120573

1= (1198721198711198971199091198961198631205892)119860

4 Numerical Results

Using a simulation software the linearized model for a UH-60 helicopter is obtained with the following state and inputmatrices

119860 =

[[[[[[[[[[[[[[[[[[[

[

minus00000 00003 minus00000 minus00000 00000 minus00000 10000 minus00023 00503

minus00003 minus00000 minus00000 minus00000 minus00000 minus00000 minus00000 09989 00463

minus00001 00000 minus00000 minus00000 00000 00000 minus00000 minus00464 10002

minus19386 minus28593 minus00030 minus00195 minus00126 00168 minus12000 57076 02683

328694 minus03640 00027 00126 minus00311 minus00013 minus41348 minus22350 03713

03138 minus14034 00004 00150 00034 minus03340 02866 06307 23410

minus09144 minus04783 00017 00226 minus00246 00007 minus65893 minus21301 minus00598

00631 minus06744 00008 00029 00046 00025 02059 minus14255 minus01636

minus07549 minus00119 minus00002 00009 00028 00006 minus00749 minus01032 minus02214

]]]]]]]]]]]]]]]]]]]

]

119861 =

[[[[[[[[[[[[[[[[[[[

[

minus00000 00000 minus00000 00000

00000 00000 minus00000 minus00000

minus00000 00000 minus00000 minus00000

minus22190 05734 03170 minus00019

35381 minus00453 minus00963 02007

minus15629 00237 minus60740 minus01014

19943 minus00714 minus00541 00731

00446 minus01097 00553 minus00281

minus08197 minus00082 01573 minus00635

]]]]]]]]]]]]]]]]]]]

]

(54)

The desired condition 1205721+ 1205731lt 1 can be verified by consid-

ering a typical case that the UH-60 helicopter carries a loadweight 1000 lb by a 15 ft slung length and let 119896

119863= 75 lbft

For the UH-60 helicopter model with reference to thegeneral mathematical model presented in Section 2 we have1198831205791119904

= 05734 1198831205790

= 03170 1198851205791119904

= 00237 1198851205790

= minus607401198721205791119904

= minus01097 and 1198721205790

= 00553 So the followingparameters can be calculated to have the values 119886

11= minus02

and 11988612

= minus002 For 119883119906= minus00195 119885

119906= 00150 119872

119906=

00029 119872119902= minus14255 (119883

119902minus 119908119890) = 57076 and (119885

119902+ 119906119890) =

06307 we have 11988323

= 006 And also for 119883119908

= 00168119885119908

= minus03340 and 119872119908

= 00025 we have 11988324

= 001Based on all the above parameters 120590max(119860) = 00041 Inthis case let 120581 = 001 120572

1= (119896119863119872119871)(2 + 120581119897

119909) asymp 015

and 1205731= (119872

119871119897119909119896119863)120590max(119860) = 082 Therefore 120572

1+ 1205731=

015 + 082 = 097 which satisfies 1205721+ 1205731

lt 1 Anotherexample considers that the UH-60 helicopter carries a loadweight 500 lb by a 15 ft slung length and let 119896

119863= 50 lbft then

1205721= 02 and 120573

1= 062 therefore (120572

1+ 1205731) = 082 Therefore

this assumption 1205721+ 1205731

lt 1 is realistic for the systemdiscussed in here The numerical values vary with respectto the slung load configuration It may represent the system

dynamic variations To investigate the stability characteristicsof dynamic variations two series of simulation studies werecarried out

Firstly the slung length was kept constant at 15 ft andthe load varied between the limits 500ndash2000 lb with 119896

119863=

75 lbft producing the system poles listed in Table 1 Thecorresponding root loci are sketched in Figures 3 and 4Figure 4 clearly shows that the positions of the system polescrossed the imaginary axis as the load weight increased

Secondly load weight was kept constant and the slunglength varied between the limit of 10ndash20 ft Table 2 sum-marises the results of variation of the system poles and thecorresponding root loci are illustrated in Figures 5 and 6

The pole locus indicates that the system poles have thevariation trends of moving further towards right direction onthe 119904-plane with increase of load weight and slung lengthsFrom the simulation results it is clearly shown that thesystem becomes less stable when the load andor the slunglength increases The simulation results are compared withthe stability analysis presented above

Now consider the case of fixed slung length of 15 ft withdifferent load weights In the cases of a UH-60 helicopter that

12 Journal of Control Science and Engineering

Table 1 System poles locations for a fixed slung length (119897 = 15ft) but with different load weights

Helicopter 500 lb 1000 lb 1500 lb 2000 lbminus10919 minus11159 minus07848 minus06221 minus09369minus63938 minus64301 minus92203 minus79848 minus75613minus00032 minus00048 minus00034 minus00052 minus00052minus00977 minus00818 minus00888 minus00972 minus00694minus03045 minus03036 minus03394 minus03318 minus03333minus03159 plusmn 04363119894 minus03199 plusmn 04628119894 minus02295 plusmn 05050119894 minus02366 plusmn 05974119894 minus01834 plusmn 04968119894minus00489 plusmn 03898119894 minus00211 plusmn 03868119894 00074 plusmn 03445119894 00119 plusmn 04406119894 00264 plusmn 03199119894mdash minus00316 plusmn 15043119894 minus01020 plusmn 17380119894 minus00368 plusmn 18753119894 minus01160 plusmn 20003119894mdash minus00736 plusmn 15019119894 minus07662 plusmn 11680119894 minus07349 plusmn 19064119894 minus12367 plusmn 08995119894

Table 2 System poles for a fixed load (1000 lb) but with different slung lengths

Helicopter 10 ft 15 ft 20 ftminus10919 minus10560 minus07848 minus08687minus63938 minus99252 minus92203 minus85565minus00032 minus00061 minus00034 minus00114minus00977 minus02012 minus00888 minus00953minus03045 minus02606 minus03394 minus03479minus03159 plusmn 04363119894 minus01809 plusmn 06219119894 minus02295 plusmn 05050119894 minus01636 plusmn 06092119894minus00489 plusmn 03898119894 00643 plusmn 04577119894 00074 plusmn 03445119894 00799 plusmn 04594119894mdash minus01055 plusmn 19883119894 minus01020 plusmn 17380119894 minus01247 plusmn 14470119894mdash minus08847 plusmn 12464119894 minus07662 plusmn 11680119894 minus06352 plusmn 10384119894

Re(poles)

Im(poles)

minus10 minus5 0 5minus08minus06minus04minus02

002040608

Figure 3 System root-locus diagram showing pole locus forconstant slung length (15 ft) as load weight is increased

minus04 minus035 minus03 minus025 minus02 minus015 minus01 minus005 0 005 01

Re(poles)

Im(poles)

minus08minus06minus04minus02

002040608

Figure 4 Enlarged view of Figure 3 showing poles locus near toRe(poles) = 0 axis

minus10 minus8 minus6 minus4 minus2 0 2

Re(poles)

Im(poles)

minus08minus06minus04minus02

002040608

Figure 5 System root-locus showing pole locus for constant loadweight (1000 lb) as slung length is increased

minus04 minus035 minus03 minus025 minus02 minus015 minus01 minus005 0 005 01

Re(poles)

Im(poles)

minus08minus06minus04minus02

002040608

Figure 6 Enlarged view of Figure 5 showing poles locus near toRe(poles) = 0 axis

Journal of Control Science and Engineering 13

carries a loadweight 500 lb and 1000 lb then1205721= 03 and 041

with 1205731= 041 and 082 respectively therefore 120572

1+ 1205731lt 1

thus the system is stable But when the load weight is 1500 lband 2000 lb then 120572

1= 01 and 0075 and also 120573

1= 123 and

164 respectively therefore 1205721+ 1205731gt 1 thus the system is

unstable For the case of the constant load weight (1000 lb) asslung length is increased then for a 10 ft and 15 ft slung length1205721= 015 and 120573

1= 055 and 082 respectively therefore

1205721+ 1205731lt 1 thus the system is stable But when the slung

length increased to 20 ft then 1205721

= 015 and 1205731

= 109therefore (120572

1+ 1205731) = 124 that is 120572

1+ 1205731gt 1 thus the

system is unstable However the system can be marginallystable if the UH-60 helicopter carries a load weight 725 lb bya 20 ft slung length then 120572

1= 02 and 120573

1= 08 therefore

(1205721+ 1205731) = 1 For this set of analysis 119896

119863is assumed to

be 119896119863

= 75 lbft However 119896119863can vary depending on the

weather condition and an example of 119896119863= 50 lbft is shown

above for illustrative purpose This analysis suggests that themaximum value of 20 ft slung length may be used for slungload operation However the load weight limit can vary withrespect to the capacity of the load carrying hooks helicopterweight environmental conditions and so forth

The examples presented here show that the simulationresults are correlated with the stability analysis Thus desiredcondition for the stability is met and it can locate the load atthe specified position or its neighbourhood

5 Conclusions

In this paper stability analysis of a helicopter with an under-slung load system is investigatedThe investigation identifiedthe conditions for the stabilisation of the system and thepositioning of the underslung load at hover condition Sta-bility analysis and numerical results proved that the desiredcondition for the stability is met then it is possible to locatethe load at the specified position or its neighbourhood Anillustration example is also given in the paper The methodresults in a guaranteed load positioning accuracy whichdepends on the design parameters and the method can beapplicable to any individual helicopter with external loadsystem

Nomenclature

119871119872119873 Overall helicopter rolling pitching andyawing moments

119901 119902 119903 Helicopter roll pitch and yaw rates aboutbody reference axes

119883119884 119885 Overall helicopter force components120601 120579 120595 Roll pitch and yaw angles119906 V 119908 Helicopter velocity components at centre

of gravity1205790 Main rotor collective

1205791119904 Longitudinal cyclic commands

1205791119888 Lateral cyclic commands

1205790119879 Tail rotor collective

1198830 1198840 1198850 Location of the suspension point withrespect to earth referenced 119909 119910 and 119911

directions

0 0 0 The helicopter velocity in the 119909 119910 and 119911

directions0 0 0 The helicopter acceleration in the 119909 119910 and119911 directions

120579119871 Load suspension angle in the119883-119885 plane

with respect to 119885-axis120601119871 Load suspension angle in the 119884-119885 plane

with respect to 119885-axis119872119871 Mass of the suspended load

119892 Acceleration due to gravity119897 Slung length119896119863 Aerodynamic drag force constant

(119896119863= (12)120588119878119862

119863)

120588 Air density119878 The load area presented to the airflow119862119863 Drag coefficient for the load

Competing Interests

The author declares that there are no competing interestsregarding the publication of this paper

References

[1] K Thanapalan and F Zhang ldquoModelling and simulation studyof a helicopter with an external slung load systemrdquo I-ManagersJournal on Instrumentation amp Control Engineering vol 1 no 1pp 9ndash17 2013

[2] M Bisgaard J D Bendtsen and A La Cour-Harbo ldquoModelingof generic slung load systemrdquo Journal of Guidance Control andDynamics vol 32 no 2 pp 573ndash585 2009

[3] Handling Qualities Requirements for Military Rotorcraft ADS-33-D US Army Aviation and Troop Command St Louis MissUSA 1996

[4] K Thanapalan and T M Wong ldquoModeing of a helicopter withan under-slung load systemrdquo in Proceedings of the 29th ChineseControl Conference (CCC rsquo10) pp 1451ndash1456 Beijing China July2010

[5] B Feng ldquoRobust control for lateral and longitudinal channels ofsmall-scale unmanned helicoptersrdquo Journal of Control Scienceand Engineering vol 2015 Article ID 483096 8 pages 2015

[6] J J Gribble ldquoLinear quadratic Gaussianloop transfer recoverydesign for a helicopter in low-speed flightrdquo Journal of GuidanceControl and Dynamics vol 16 no 4 pp 754ndash761 1993

[7] M A Manness J J Gribble and D J M Smith ldquoMultivariablemethods for helicopter flight control law designrdquo in Proceedingsof the 16th Annual Society European Rotorcraft Forum PaperNoIII52 Glasgow UK September 1990

[8] W L Garrard E Low and S Prouty ldquoDesign of attitude andrate command systems for helicopters using eigenstructureassignmentrdquo Journal of Guidance Control and Dynamics vol12 no 6 pp 783ndash791 1989

[9] Y B Shtessel and I A Shkolnikov ldquoAeronautical and spacevehicle control in dynamic sliding manifoldsrdquo InternationalJournal of Control vol 76 no 9-10 pp 1000ndash1017 2003

[10] D FThompson J S Pruyn andA Shukla ldquoFeedback design forrobust tracking and robust stiffness in flight control actuatorsusing a modified QFT techniquerdquo International Journal ofControl vol 72 no 16 pp 1480ndash1497 1999

14 Journal of Control Science and Engineering

[11] K Thanapalan ldquoModelling of a helicopter systemrdquo in Proceed-ings of the 1st Virtual Control Conference (VCC rsquo10) pp 45ndash52Aalborg University 2010

[12] D P Goodall and J Wang ldquoStabilization of a class of uncertainnonlinear affine systems subject to control constraintsrdquo Interna-tional Journal of Robust and Nonlinear Control vol 11 no 9 pp797ndash818 2001

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

10 Journal of Control Science and Engineering

If11988321

lt 0 then

minus [12058931205894minus (1205892

4+ 1205895)11988321] 1199022minus 12058931205894119883211205792lt 0 (44)

in the region ofminus[12058931205894minus(1205892

4+1205895)11988321]1199022lt 12058931205894119883211205792 Since 120579 is

caused by the loadmotion here the value of 120579 should bemuchsmaller than 119902 The inequality (44) is easy to be satisfied

In summary of the above analysis and by defining

Ψ1(120579119871) = 11989612057912058921205792

1198712

+ [12058931205894minus (1205892

4+ 1205895)11988321] 1199022

+ 12058931205894119883211205792

Ψ2(120579119871) = [120589

31205894minus (1205892

4+ 1205895)11988321] 1199022+ 12058931205894119883211205792

(45)

the following lemma can be derived

Lemma 1 Defining a Lyapunov function (28) and choosingthe design parameters to satisfy 120589

1= (2119892119897

119909120587)1205892 [12058931205894minus (1205892

4+

1205895)11988321] gt 0 and [1205892

3minus1205893120589411988322+(1205892

4+1205895)11988321] ge 0 then within

the region specified by minus[12058931205894minus (1205892

4+ 1205895)11988321]1199022lt 12058931205894119883211205792

(i) V1(0) = 0 and V

1(120579119871) gt 0 forall120579

119871= 0

(ii) V1(120579119871) rarr infin as 120579

119871 rarr infin

(iii) (1198711198911

V1)(120579119871) le minusΨ

1(120579119871) forall120579119871 around the hover condi-

tion or (1198711198911

V1)(120579119871) le minusΨ

2(120579119871) forall120579119871 if (35) holds

Both functions Ψ1and Ψ

2are nonnegative

Recall the control term 119901(119909119867(119905)) in the first subsystem

of (11a) and (11b) it can be seen that [(119863119901)(119909119867)]minus1 exists

for all 119909119867(119905) The unknown vector fields 119902(120579

119871(119905) 119909119867) and

119867(119905 120579119871(119905) 119909119867) model the uncertainties imposed onto the

system Since 119902(120579119871(119905) 119909119867) is directly mapped into the ldquocon-

trolrdquo space of 119909119867(119905) it can be considered as a matched

uncertainty119867(119905 120579119871(119905) 119909119867) is unknown and it does belong to

the control space of 119909119867(119905) so it represents the mismatched

uncertainty in the system [12]Generally the range of the (longitudinal) load suspension

angle is within minus1205872 lt 120579119871lt 1205872 The helicopter velocities

and load suspension angle havemaximumoperational valuestherefore the uncertainties in the system are bounded Withthe Lyapunov function defined in (28)

(1198711198921

V1) (120579119871) = ⟨nablaV

1(120579119871) 1198921⟩ =

119896119863

119872119871119897119909

12058921205791198712

(46)

In 1198661(120579119871) the columns 119892

1and 119892

2are the same therefore

(1198711198922

V1) (120579119871) =

119896119863

119872119871119897119909

12058921205791198712

(47)

By considering the maximum values of cos (sdot) and sin (sdot)functions the bounding values relating to the uncertainty119902(120579119871(119905) 119909119867) can be estimated as follows

10038171003817100381710038171003817119902 (120579119871 119909119867)10038171003817100381710038171003817le

10038161003816100381610038161003816100381610038161003816

1

119897119909

( + )

+1

119872119871

(minus119896119863(2 +

120581

119897119909

) (119906 + 119908) +119896120579

119897119909

)1205791198712

10038161003816100381610038161003816100381610038161003816

le

10038161003816100381610038161003816100381610038161003816

119896120579

119897119909

1205791198712

10038161003816100381610038161003816100381610038161003816+

10038161003816100381610038161003816100381610038161003816

119896119863

119872119871

(2 +120581

119897119909

) (119906 + 119908)

10038161003816100381610038161003816100381610038161003816+

10038161003816100381610038161003816100381610038161003816

1

119897119909

( + )

10038161003816100381610038161003816100381610038161003816

le119896120579

119897119909

100381710038171003817100381710038171205791198712

10038171003817100381710038171003817+

119896119863

119872119871

(2 +120581

119897119909

)1003817100381710038171003817119901 (119909119867)1003817100381710038171003817 + 1205722 (119905)

le 1205831

10038161003816100381610038161003816(1198711198922

V1) (120579119871)10038161003816100381610038161003816+ 1205721

1003817100381710038171003817119901 (119909119867)1003817100381710038171003817 + 1205722 (119905)

(48)

where 1205831

= 119872119871119896120579119896119863 1205721

= (119896119863119872119871)(2 + 120581119897

119909) (120581 =

max [1205811 1205812]) and 120572

2(119905) is defined by 120572

2(119905) = max |(1119897

119909)( +

)| gt 0For the mismatched uncertainty 119867(119905 120579

119871(119905) 119909119867) the fol-

lowing analysis is conducted to obtain its bounding functionThe mismatched uncertainty can be rewritten as

119867(119905 120579119871 (119905) 119909119867) =

[[[[[

[

0

0

(11988611119906 + 11988612119908)

(11988323119906 + 119883

24119908)

]]]]]

]

=

[[[[[

[

0 0

0 0

11988611

11988612

11988323

11988324

]]]]]

]

[119906

119908]

(49)

Let 119860 = [1198861111988612

1198832311988324] then

10038171003817100381710038171003817119867 (119905 120579

119871 119909119867)10038171003817100381710038171003817le1003817100381710038171003817100381711986010038171003817100381710038171003817

1003817100381710038171003817119909119867 (119905)1003817100381710038171003817 (50)

Define a positive function as

1006704120579 (120579119871) = radic(1205792

1198711

+ 12057921198712

+ 1205792 + 1199022+ 120576) (51)

where 120576 is a very small positive constant Choosing1205731

= (1198721198711198971199091198961198631205892)119860 then the mismatched uncertainty

119867(119905 120579119871(119905) 119909119867) is bounded by the following inequality

10038171003817100381710038171003817119867 (119905 120579

119871 119909119867)10038171003817100381710038171003817

le 1006704120579minus1

(120579119871) 1205731

2

sum

119894=1

(10038161003816100381610038161003816(119871119892119894

V1) (120579119871)10038161003816100381610038161003816)1003816100381610038161003816119901119894 (119909119867)

1003816100381610038161003816

(52)

where 119892119894denotes the 119894th column of the matrix function

1198661(120579119871) and 119901

119894is the 119894th component of 119901(119909

119867) respectively As

1205731has a design parameter 120589

2involved it is easy to have 120573

1to

lead (52) to be true For the matched uncertainty we have

10038171003817100381710038171003817119902 (120579119871 119909119867)10038171003817100381710038171003817le

119896120579

119897119909

100381610038161003816100381610038161205791198712

10038161003816100381610038161003816+ 1205721

1003817100381710038171003817119901 (119909119867)1003817100381710038171003817 + 1205722 (119905) (53)

Journal of Control Science and Engineering 11

Following the above analysis the following lemma can bederived

Lemma 2 The uncertainties (52) and (53) are bounded andsatisfy if 120572

1 1205731ge 0 and 120572

1+ 1205731lt 1 for 120572

1= (119896119863119872119871)(2 +

120581119897119909) and 120573

1= (1198721198711198971199091198961198631205892)119860

4 Numerical Results

Using a simulation software the linearized model for a UH-60 helicopter is obtained with the following state and inputmatrices

119860 =

[[[[[[[[[[[[[[[[[[[

[

minus00000 00003 minus00000 minus00000 00000 minus00000 10000 minus00023 00503

minus00003 minus00000 minus00000 minus00000 minus00000 minus00000 minus00000 09989 00463

minus00001 00000 minus00000 minus00000 00000 00000 minus00000 minus00464 10002

minus19386 minus28593 minus00030 minus00195 minus00126 00168 minus12000 57076 02683

328694 minus03640 00027 00126 minus00311 minus00013 minus41348 minus22350 03713

03138 minus14034 00004 00150 00034 minus03340 02866 06307 23410

minus09144 minus04783 00017 00226 minus00246 00007 minus65893 minus21301 minus00598

00631 minus06744 00008 00029 00046 00025 02059 minus14255 minus01636

minus07549 minus00119 minus00002 00009 00028 00006 minus00749 minus01032 minus02214

]]]]]]]]]]]]]]]]]]]

]

119861 =

[[[[[[[[[[[[[[[[[[[

[

minus00000 00000 minus00000 00000

00000 00000 minus00000 minus00000

minus00000 00000 minus00000 minus00000

minus22190 05734 03170 minus00019

35381 minus00453 minus00963 02007

minus15629 00237 minus60740 minus01014

19943 minus00714 minus00541 00731

00446 minus01097 00553 minus00281

minus08197 minus00082 01573 minus00635

]]]]]]]]]]]]]]]]]]]

]

(54)

The desired condition 1205721+ 1205731lt 1 can be verified by consid-

ering a typical case that the UH-60 helicopter carries a loadweight 1000 lb by a 15 ft slung length and let 119896

119863= 75 lbft

For the UH-60 helicopter model with reference to thegeneral mathematical model presented in Section 2 we have1198831205791119904

= 05734 1198831205790

= 03170 1198851205791119904

= 00237 1198851205790

= minus607401198721205791119904

= minus01097 and 1198721205790

= 00553 So the followingparameters can be calculated to have the values 119886

11= minus02

and 11988612

= minus002 For 119883119906= minus00195 119885

119906= 00150 119872

119906=

00029 119872119902= minus14255 (119883

119902minus 119908119890) = 57076 and (119885

119902+ 119906119890) =

06307 we have 11988323

= 006 And also for 119883119908

= 00168119885119908

= minus03340 and 119872119908

= 00025 we have 11988324

= 001Based on all the above parameters 120590max(119860) = 00041 Inthis case let 120581 = 001 120572

1= (119896119863119872119871)(2 + 120581119897

119909) asymp 015

and 1205731= (119872

119871119897119909119896119863)120590max(119860) = 082 Therefore 120572

1+ 1205731=

015 + 082 = 097 which satisfies 1205721+ 1205731

lt 1 Anotherexample considers that the UH-60 helicopter carries a loadweight 500 lb by a 15 ft slung length and let 119896

119863= 50 lbft then

1205721= 02 and 120573

1= 062 therefore (120572

1+ 1205731) = 082 Therefore

this assumption 1205721+ 1205731

lt 1 is realistic for the systemdiscussed in here The numerical values vary with respectto the slung load configuration It may represent the system

dynamic variations To investigate the stability characteristicsof dynamic variations two series of simulation studies werecarried out

Firstly the slung length was kept constant at 15 ft andthe load varied between the limits 500ndash2000 lb with 119896

119863=

75 lbft producing the system poles listed in Table 1 Thecorresponding root loci are sketched in Figures 3 and 4Figure 4 clearly shows that the positions of the system polescrossed the imaginary axis as the load weight increased

Secondly load weight was kept constant and the slunglength varied between the limit of 10ndash20 ft Table 2 sum-marises the results of variation of the system poles and thecorresponding root loci are illustrated in Figures 5 and 6

The pole locus indicates that the system poles have thevariation trends of moving further towards right direction onthe 119904-plane with increase of load weight and slung lengthsFrom the simulation results it is clearly shown that thesystem becomes less stable when the load andor the slunglength increases The simulation results are compared withthe stability analysis presented above

Now consider the case of fixed slung length of 15 ft withdifferent load weights In the cases of a UH-60 helicopter that

12 Journal of Control Science and Engineering

Table 1 System poles locations for a fixed slung length (119897 = 15ft) but with different load weights

Helicopter 500 lb 1000 lb 1500 lb 2000 lbminus10919 minus11159 minus07848 minus06221 minus09369minus63938 minus64301 minus92203 minus79848 minus75613minus00032 minus00048 minus00034 minus00052 minus00052minus00977 minus00818 minus00888 minus00972 minus00694minus03045 minus03036 minus03394 minus03318 minus03333minus03159 plusmn 04363119894 minus03199 plusmn 04628119894 minus02295 plusmn 05050119894 minus02366 plusmn 05974119894 minus01834 plusmn 04968119894minus00489 plusmn 03898119894 minus00211 plusmn 03868119894 00074 plusmn 03445119894 00119 plusmn 04406119894 00264 plusmn 03199119894mdash minus00316 plusmn 15043119894 minus01020 plusmn 17380119894 minus00368 plusmn 18753119894 minus01160 plusmn 20003119894mdash minus00736 plusmn 15019119894 minus07662 plusmn 11680119894 minus07349 plusmn 19064119894 minus12367 plusmn 08995119894

Table 2 System poles for a fixed load (1000 lb) but with different slung lengths

Helicopter 10 ft 15 ft 20 ftminus10919 minus10560 minus07848 minus08687minus63938 minus99252 minus92203 minus85565minus00032 minus00061 minus00034 minus00114minus00977 minus02012 minus00888 minus00953minus03045 minus02606 minus03394 minus03479minus03159 plusmn 04363119894 minus01809 plusmn 06219119894 minus02295 plusmn 05050119894 minus01636 plusmn 06092119894minus00489 plusmn 03898119894 00643 plusmn 04577119894 00074 plusmn 03445119894 00799 plusmn 04594119894mdash minus01055 plusmn 19883119894 minus01020 plusmn 17380119894 minus01247 plusmn 14470119894mdash minus08847 plusmn 12464119894 minus07662 plusmn 11680119894 minus06352 plusmn 10384119894

Re(poles)

Im(poles)

minus10 minus5 0 5minus08minus06minus04minus02

002040608

Figure 3 System root-locus diagram showing pole locus forconstant slung length (15 ft) as load weight is increased

minus04 minus035 minus03 minus025 minus02 minus015 minus01 minus005 0 005 01

Re(poles)

Im(poles)

minus08minus06minus04minus02

002040608

Figure 4 Enlarged view of Figure 3 showing poles locus near toRe(poles) = 0 axis

minus10 minus8 minus6 minus4 minus2 0 2

Re(poles)

Im(poles)

minus08minus06minus04minus02

002040608

Figure 5 System root-locus showing pole locus for constant loadweight (1000 lb) as slung length is increased

minus04 minus035 minus03 minus025 minus02 minus015 minus01 minus005 0 005 01

Re(poles)

Im(poles)

minus08minus06minus04minus02

002040608

Figure 6 Enlarged view of Figure 5 showing poles locus near toRe(poles) = 0 axis

Journal of Control Science and Engineering 13

carries a loadweight 500 lb and 1000 lb then1205721= 03 and 041

with 1205731= 041 and 082 respectively therefore 120572

1+ 1205731lt 1

thus the system is stable But when the load weight is 1500 lband 2000 lb then 120572

1= 01 and 0075 and also 120573

1= 123 and

164 respectively therefore 1205721+ 1205731gt 1 thus the system is

unstable For the case of the constant load weight (1000 lb) asslung length is increased then for a 10 ft and 15 ft slung length1205721= 015 and 120573

1= 055 and 082 respectively therefore

1205721+ 1205731lt 1 thus the system is stable But when the slung

length increased to 20 ft then 1205721

= 015 and 1205731

= 109therefore (120572

1+ 1205731) = 124 that is 120572

1+ 1205731gt 1 thus the

system is unstable However the system can be marginallystable if the UH-60 helicopter carries a load weight 725 lb bya 20 ft slung length then 120572

1= 02 and 120573

1= 08 therefore

(1205721+ 1205731) = 1 For this set of analysis 119896

119863is assumed to

be 119896119863

= 75 lbft However 119896119863can vary depending on the

weather condition and an example of 119896119863= 50 lbft is shown

above for illustrative purpose This analysis suggests that themaximum value of 20 ft slung length may be used for slungload operation However the load weight limit can vary withrespect to the capacity of the load carrying hooks helicopterweight environmental conditions and so forth

The examples presented here show that the simulationresults are correlated with the stability analysis Thus desiredcondition for the stability is met and it can locate the load atthe specified position or its neighbourhood

5 Conclusions

In this paper stability analysis of a helicopter with an under-slung load system is investigatedThe investigation identifiedthe conditions for the stabilisation of the system and thepositioning of the underslung load at hover condition Sta-bility analysis and numerical results proved that the desiredcondition for the stability is met then it is possible to locatethe load at the specified position or its neighbourhood Anillustration example is also given in the paper The methodresults in a guaranteed load positioning accuracy whichdepends on the design parameters and the method can beapplicable to any individual helicopter with external loadsystem

Nomenclature

119871119872119873 Overall helicopter rolling pitching andyawing moments

119901 119902 119903 Helicopter roll pitch and yaw rates aboutbody reference axes

119883119884 119885 Overall helicopter force components120601 120579 120595 Roll pitch and yaw angles119906 V 119908 Helicopter velocity components at centre

of gravity1205790 Main rotor collective

1205791119904 Longitudinal cyclic commands

1205791119888 Lateral cyclic commands

1205790119879 Tail rotor collective

1198830 1198840 1198850 Location of the suspension point withrespect to earth referenced 119909 119910 and 119911

directions

0 0 0 The helicopter velocity in the 119909 119910 and 119911

directions0 0 0 The helicopter acceleration in the 119909 119910 and119911 directions

120579119871 Load suspension angle in the119883-119885 plane

with respect to 119885-axis120601119871 Load suspension angle in the 119884-119885 plane

with respect to 119885-axis119872119871 Mass of the suspended load

119892 Acceleration due to gravity119897 Slung length119896119863 Aerodynamic drag force constant

(119896119863= (12)120588119878119862

119863)

120588 Air density119878 The load area presented to the airflow119862119863 Drag coefficient for the load

Competing Interests

The author declares that there are no competing interestsregarding the publication of this paper

References

[1] K Thanapalan and F Zhang ldquoModelling and simulation studyof a helicopter with an external slung load systemrdquo I-ManagersJournal on Instrumentation amp Control Engineering vol 1 no 1pp 9ndash17 2013

[2] M Bisgaard J D Bendtsen and A La Cour-Harbo ldquoModelingof generic slung load systemrdquo Journal of Guidance Control andDynamics vol 32 no 2 pp 573ndash585 2009

[3] Handling Qualities Requirements for Military Rotorcraft ADS-33-D US Army Aviation and Troop Command St Louis MissUSA 1996

[4] K Thanapalan and T M Wong ldquoModeing of a helicopter withan under-slung load systemrdquo in Proceedings of the 29th ChineseControl Conference (CCC rsquo10) pp 1451ndash1456 Beijing China July2010

[5] B Feng ldquoRobust control for lateral and longitudinal channels ofsmall-scale unmanned helicoptersrdquo Journal of Control Scienceand Engineering vol 2015 Article ID 483096 8 pages 2015

[6] J J Gribble ldquoLinear quadratic Gaussianloop transfer recoverydesign for a helicopter in low-speed flightrdquo Journal of GuidanceControl and Dynamics vol 16 no 4 pp 754ndash761 1993

[7] M A Manness J J Gribble and D J M Smith ldquoMultivariablemethods for helicopter flight control law designrdquo in Proceedingsof the 16th Annual Society European Rotorcraft Forum PaperNoIII52 Glasgow UK September 1990

[8] W L Garrard E Low and S Prouty ldquoDesign of attitude andrate command systems for helicopters using eigenstructureassignmentrdquo Journal of Guidance Control and Dynamics vol12 no 6 pp 783ndash791 1989

[9] Y B Shtessel and I A Shkolnikov ldquoAeronautical and spacevehicle control in dynamic sliding manifoldsrdquo InternationalJournal of Control vol 76 no 9-10 pp 1000ndash1017 2003

[10] D FThompson J S Pruyn andA Shukla ldquoFeedback design forrobust tracking and robust stiffness in flight control actuatorsusing a modified QFT techniquerdquo International Journal ofControl vol 72 no 16 pp 1480ndash1497 1999

14 Journal of Control Science and Engineering

[11] K Thanapalan ldquoModelling of a helicopter systemrdquo in Proceed-ings of the 1st Virtual Control Conference (VCC rsquo10) pp 45ndash52Aalborg University 2010

[12] D P Goodall and J Wang ldquoStabilization of a class of uncertainnonlinear affine systems subject to control constraintsrdquo Interna-tional Journal of Robust and Nonlinear Control vol 11 no 9 pp797ndash818 2001

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Journal of Control Science and Engineering 11

Following the above analysis the following lemma can bederived

Lemma 2 The uncertainties (52) and (53) are bounded andsatisfy if 120572

1 1205731ge 0 and 120572

1+ 1205731lt 1 for 120572

1= (119896119863119872119871)(2 +

120581119897119909) and 120573

1= (1198721198711198971199091198961198631205892)119860

4 Numerical Results

Using a simulation software the linearized model for a UH-60 helicopter is obtained with the following state and inputmatrices

119860 =

[[[[[[[[[[[[[[[[[[[

[

minus00000 00003 minus00000 minus00000 00000 minus00000 10000 minus00023 00503

minus00003 minus00000 minus00000 minus00000 minus00000 minus00000 minus00000 09989 00463

minus00001 00000 minus00000 minus00000 00000 00000 minus00000 minus00464 10002

minus19386 minus28593 minus00030 minus00195 minus00126 00168 minus12000 57076 02683

328694 minus03640 00027 00126 minus00311 minus00013 minus41348 minus22350 03713

03138 minus14034 00004 00150 00034 minus03340 02866 06307 23410

minus09144 minus04783 00017 00226 minus00246 00007 minus65893 minus21301 minus00598

00631 minus06744 00008 00029 00046 00025 02059 minus14255 minus01636

minus07549 minus00119 minus00002 00009 00028 00006 minus00749 minus01032 minus02214

]]]]]]]]]]]]]]]]]]]

]

119861 =

[[[[[[[[[[[[[[[[[[[

[

minus00000 00000 minus00000 00000

00000 00000 minus00000 minus00000

minus00000 00000 minus00000 minus00000

minus22190 05734 03170 minus00019

35381 minus00453 minus00963 02007

minus15629 00237 minus60740 minus01014

19943 minus00714 minus00541 00731

00446 minus01097 00553 minus00281

minus08197 minus00082 01573 minus00635

]]]]]]]]]]]]]]]]]]]

]

(54)

The desired condition 1205721+ 1205731lt 1 can be verified by consid-

ering a typical case that the UH-60 helicopter carries a loadweight 1000 lb by a 15 ft slung length and let 119896

119863= 75 lbft

For the UH-60 helicopter model with reference to thegeneral mathematical model presented in Section 2 we have1198831205791119904

= 05734 1198831205790

= 03170 1198851205791119904

= 00237 1198851205790

= minus607401198721205791119904

= minus01097 and 1198721205790

= 00553 So the followingparameters can be calculated to have the values 119886

11= minus02

and 11988612

= minus002 For 119883119906= minus00195 119885

119906= 00150 119872

119906=

00029 119872119902= minus14255 (119883

119902minus 119908119890) = 57076 and (119885

119902+ 119906119890) =

06307 we have 11988323

= 006 And also for 119883119908

= 00168119885119908

= minus03340 and 119872119908

= 00025 we have 11988324

= 001Based on all the above parameters 120590max(119860) = 00041 Inthis case let 120581 = 001 120572

1= (119896119863119872119871)(2 + 120581119897

119909) asymp 015

and 1205731= (119872

119871119897119909119896119863)120590max(119860) = 082 Therefore 120572

1+ 1205731=

015 + 082 = 097 which satisfies 1205721+ 1205731

lt 1 Anotherexample considers that the UH-60 helicopter carries a loadweight 500 lb by a 15 ft slung length and let 119896

119863= 50 lbft then

1205721= 02 and 120573

1= 062 therefore (120572

1+ 1205731) = 082 Therefore

this assumption 1205721+ 1205731

lt 1 is realistic for the systemdiscussed in here The numerical values vary with respectto the slung load configuration It may represent the system

dynamic variations To investigate the stability characteristicsof dynamic variations two series of simulation studies werecarried out

Firstly the slung length was kept constant at 15 ft andthe load varied between the limits 500ndash2000 lb with 119896

119863=

75 lbft producing the system poles listed in Table 1 Thecorresponding root loci are sketched in Figures 3 and 4Figure 4 clearly shows that the positions of the system polescrossed the imaginary axis as the load weight increased

Secondly load weight was kept constant and the slunglength varied between the limit of 10ndash20 ft Table 2 sum-marises the results of variation of the system poles and thecorresponding root loci are illustrated in Figures 5 and 6

The pole locus indicates that the system poles have thevariation trends of moving further towards right direction onthe 119904-plane with increase of load weight and slung lengthsFrom the simulation results it is clearly shown that thesystem becomes less stable when the load andor the slunglength increases The simulation results are compared withthe stability analysis presented above

Now consider the case of fixed slung length of 15 ft withdifferent load weights In the cases of a UH-60 helicopter that

12 Journal of Control Science and Engineering

Table 1 System poles locations for a fixed slung length (119897 = 15ft) but with different load weights

Helicopter 500 lb 1000 lb 1500 lb 2000 lbminus10919 minus11159 minus07848 minus06221 minus09369minus63938 minus64301 minus92203 minus79848 minus75613minus00032 minus00048 minus00034 minus00052 minus00052minus00977 minus00818 minus00888 minus00972 minus00694minus03045 minus03036 minus03394 minus03318 minus03333minus03159 plusmn 04363119894 minus03199 plusmn 04628119894 minus02295 plusmn 05050119894 minus02366 plusmn 05974119894 minus01834 plusmn 04968119894minus00489 plusmn 03898119894 minus00211 plusmn 03868119894 00074 plusmn 03445119894 00119 plusmn 04406119894 00264 plusmn 03199119894mdash minus00316 plusmn 15043119894 minus01020 plusmn 17380119894 minus00368 plusmn 18753119894 minus01160 plusmn 20003119894mdash minus00736 plusmn 15019119894 minus07662 plusmn 11680119894 minus07349 plusmn 19064119894 minus12367 plusmn 08995119894

Table 2 System poles for a fixed load (1000 lb) but with different slung lengths

Helicopter 10 ft 15 ft 20 ftminus10919 minus10560 minus07848 minus08687minus63938 minus99252 minus92203 minus85565minus00032 minus00061 minus00034 minus00114minus00977 minus02012 minus00888 minus00953minus03045 minus02606 minus03394 minus03479minus03159 plusmn 04363119894 minus01809 plusmn 06219119894 minus02295 plusmn 05050119894 minus01636 plusmn 06092119894minus00489 plusmn 03898119894 00643 plusmn 04577119894 00074 plusmn 03445119894 00799 plusmn 04594119894mdash minus01055 plusmn 19883119894 minus01020 plusmn 17380119894 minus01247 plusmn 14470119894mdash minus08847 plusmn 12464119894 minus07662 plusmn 11680119894 minus06352 plusmn 10384119894

Re(poles)

Im(poles)

minus10 minus5 0 5minus08minus06minus04minus02

002040608

Figure 3 System root-locus diagram showing pole locus forconstant slung length (15 ft) as load weight is increased

minus04 minus035 minus03 minus025 minus02 minus015 minus01 minus005 0 005 01

Re(poles)

Im(poles)

minus08minus06minus04minus02

002040608

Figure 4 Enlarged view of Figure 3 showing poles locus near toRe(poles) = 0 axis

minus10 minus8 minus6 minus4 minus2 0 2

Re(poles)

Im(poles)

minus08minus06minus04minus02

002040608

Figure 5 System root-locus showing pole locus for constant loadweight (1000 lb) as slung length is increased

minus04 minus035 minus03 minus025 minus02 minus015 minus01 minus005 0 005 01

Re(poles)

Im(poles)

minus08minus06minus04minus02

002040608

Figure 6 Enlarged view of Figure 5 showing poles locus near toRe(poles) = 0 axis

Journal of Control Science and Engineering 13

carries a loadweight 500 lb and 1000 lb then1205721= 03 and 041

with 1205731= 041 and 082 respectively therefore 120572

1+ 1205731lt 1

thus the system is stable But when the load weight is 1500 lband 2000 lb then 120572

1= 01 and 0075 and also 120573

1= 123 and

164 respectively therefore 1205721+ 1205731gt 1 thus the system is

unstable For the case of the constant load weight (1000 lb) asslung length is increased then for a 10 ft and 15 ft slung length1205721= 015 and 120573

1= 055 and 082 respectively therefore

1205721+ 1205731lt 1 thus the system is stable But when the slung

length increased to 20 ft then 1205721

= 015 and 1205731

= 109therefore (120572

1+ 1205731) = 124 that is 120572

1+ 1205731gt 1 thus the

system is unstable However the system can be marginallystable if the UH-60 helicopter carries a load weight 725 lb bya 20 ft slung length then 120572

1= 02 and 120573

1= 08 therefore

(1205721+ 1205731) = 1 For this set of analysis 119896

119863is assumed to

be 119896119863

= 75 lbft However 119896119863can vary depending on the

weather condition and an example of 119896119863= 50 lbft is shown

above for illustrative purpose This analysis suggests that themaximum value of 20 ft slung length may be used for slungload operation However the load weight limit can vary withrespect to the capacity of the load carrying hooks helicopterweight environmental conditions and so forth

The examples presented here show that the simulationresults are correlated with the stability analysis Thus desiredcondition for the stability is met and it can locate the load atthe specified position or its neighbourhood

5 Conclusions

In this paper stability analysis of a helicopter with an under-slung load system is investigatedThe investigation identifiedthe conditions for the stabilisation of the system and thepositioning of the underslung load at hover condition Sta-bility analysis and numerical results proved that the desiredcondition for the stability is met then it is possible to locatethe load at the specified position or its neighbourhood Anillustration example is also given in the paper The methodresults in a guaranteed load positioning accuracy whichdepends on the design parameters and the method can beapplicable to any individual helicopter with external loadsystem

Nomenclature

119871119872119873 Overall helicopter rolling pitching andyawing moments

119901 119902 119903 Helicopter roll pitch and yaw rates aboutbody reference axes

119883119884 119885 Overall helicopter force components120601 120579 120595 Roll pitch and yaw angles119906 V 119908 Helicopter velocity components at centre

of gravity1205790 Main rotor collective

1205791119904 Longitudinal cyclic commands

1205791119888 Lateral cyclic commands

1205790119879 Tail rotor collective

1198830 1198840 1198850 Location of the suspension point withrespect to earth referenced 119909 119910 and 119911

directions

0 0 0 The helicopter velocity in the 119909 119910 and 119911

directions0 0 0 The helicopter acceleration in the 119909 119910 and119911 directions

120579119871 Load suspension angle in the119883-119885 plane

with respect to 119885-axis120601119871 Load suspension angle in the 119884-119885 plane

with respect to 119885-axis119872119871 Mass of the suspended load

119892 Acceleration due to gravity119897 Slung length119896119863 Aerodynamic drag force constant

(119896119863= (12)120588119878119862

119863)

120588 Air density119878 The load area presented to the airflow119862119863 Drag coefficient for the load

Competing Interests

The author declares that there are no competing interestsregarding the publication of this paper

References

[1] K Thanapalan and F Zhang ldquoModelling and simulation studyof a helicopter with an external slung load systemrdquo I-ManagersJournal on Instrumentation amp Control Engineering vol 1 no 1pp 9ndash17 2013

[2] M Bisgaard J D Bendtsen and A La Cour-Harbo ldquoModelingof generic slung load systemrdquo Journal of Guidance Control andDynamics vol 32 no 2 pp 573ndash585 2009

[3] Handling Qualities Requirements for Military Rotorcraft ADS-33-D US Army Aviation and Troop Command St Louis MissUSA 1996

[4] K Thanapalan and T M Wong ldquoModeing of a helicopter withan under-slung load systemrdquo in Proceedings of the 29th ChineseControl Conference (CCC rsquo10) pp 1451ndash1456 Beijing China July2010

[5] B Feng ldquoRobust control for lateral and longitudinal channels ofsmall-scale unmanned helicoptersrdquo Journal of Control Scienceand Engineering vol 2015 Article ID 483096 8 pages 2015

[6] J J Gribble ldquoLinear quadratic Gaussianloop transfer recoverydesign for a helicopter in low-speed flightrdquo Journal of GuidanceControl and Dynamics vol 16 no 4 pp 754ndash761 1993

[7] M A Manness J J Gribble and D J M Smith ldquoMultivariablemethods for helicopter flight control law designrdquo in Proceedingsof the 16th Annual Society European Rotorcraft Forum PaperNoIII52 Glasgow UK September 1990

[8] W L Garrard E Low and S Prouty ldquoDesign of attitude andrate command systems for helicopters using eigenstructureassignmentrdquo Journal of Guidance Control and Dynamics vol12 no 6 pp 783ndash791 1989

[9] Y B Shtessel and I A Shkolnikov ldquoAeronautical and spacevehicle control in dynamic sliding manifoldsrdquo InternationalJournal of Control vol 76 no 9-10 pp 1000ndash1017 2003

[10] D FThompson J S Pruyn andA Shukla ldquoFeedback design forrobust tracking and robust stiffness in flight control actuatorsusing a modified QFT techniquerdquo International Journal ofControl vol 72 no 16 pp 1480ndash1497 1999

14 Journal of Control Science and Engineering

[11] K Thanapalan ldquoModelling of a helicopter systemrdquo in Proceed-ings of the 1st Virtual Control Conference (VCC rsquo10) pp 45ndash52Aalborg University 2010

[12] D P Goodall and J Wang ldquoStabilization of a class of uncertainnonlinear affine systems subject to control constraintsrdquo Interna-tional Journal of Robust and Nonlinear Control vol 11 no 9 pp797ndash818 2001

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

12 Journal of Control Science and Engineering

Table 1 System poles locations for a fixed slung length (119897 = 15ft) but with different load weights

Helicopter 500 lb 1000 lb 1500 lb 2000 lbminus10919 minus11159 minus07848 minus06221 minus09369minus63938 minus64301 minus92203 minus79848 minus75613minus00032 minus00048 minus00034 minus00052 minus00052minus00977 minus00818 minus00888 minus00972 minus00694minus03045 minus03036 minus03394 minus03318 minus03333minus03159 plusmn 04363119894 minus03199 plusmn 04628119894 minus02295 plusmn 05050119894 minus02366 plusmn 05974119894 minus01834 plusmn 04968119894minus00489 plusmn 03898119894 minus00211 plusmn 03868119894 00074 plusmn 03445119894 00119 plusmn 04406119894 00264 plusmn 03199119894mdash minus00316 plusmn 15043119894 minus01020 plusmn 17380119894 minus00368 plusmn 18753119894 minus01160 plusmn 20003119894mdash minus00736 plusmn 15019119894 minus07662 plusmn 11680119894 minus07349 plusmn 19064119894 minus12367 plusmn 08995119894

Table 2 System poles for a fixed load (1000 lb) but with different slung lengths

Helicopter 10 ft 15 ft 20 ftminus10919 minus10560 minus07848 minus08687minus63938 minus99252 minus92203 minus85565minus00032 minus00061 minus00034 minus00114minus00977 minus02012 minus00888 minus00953minus03045 minus02606 minus03394 minus03479minus03159 plusmn 04363119894 minus01809 plusmn 06219119894 minus02295 plusmn 05050119894 minus01636 plusmn 06092119894minus00489 plusmn 03898119894 00643 plusmn 04577119894 00074 plusmn 03445119894 00799 plusmn 04594119894mdash minus01055 plusmn 19883119894 minus01020 plusmn 17380119894 minus01247 plusmn 14470119894mdash minus08847 plusmn 12464119894 minus07662 plusmn 11680119894 minus06352 plusmn 10384119894

Re(poles)

Im(poles)

minus10 minus5 0 5minus08minus06minus04minus02

002040608

Figure 3 System root-locus diagram showing pole locus forconstant slung length (15 ft) as load weight is increased

minus04 minus035 minus03 minus025 minus02 minus015 minus01 minus005 0 005 01

Re(poles)

Im(poles)

minus08minus06minus04minus02

002040608

Figure 4 Enlarged view of Figure 3 showing poles locus near toRe(poles) = 0 axis

minus10 minus8 minus6 minus4 minus2 0 2

Re(poles)

Im(poles)

minus08minus06minus04minus02

002040608

Figure 5 System root-locus showing pole locus for constant loadweight (1000 lb) as slung length is increased

minus04 minus035 minus03 minus025 minus02 minus015 minus01 minus005 0 005 01

Re(poles)

Im(poles)

minus08minus06minus04minus02

002040608

Figure 6 Enlarged view of Figure 5 showing poles locus near toRe(poles) = 0 axis

Journal of Control Science and Engineering 13

carries a loadweight 500 lb and 1000 lb then1205721= 03 and 041

with 1205731= 041 and 082 respectively therefore 120572

1+ 1205731lt 1

thus the system is stable But when the load weight is 1500 lband 2000 lb then 120572

1= 01 and 0075 and also 120573

1= 123 and

164 respectively therefore 1205721+ 1205731gt 1 thus the system is

unstable For the case of the constant load weight (1000 lb) asslung length is increased then for a 10 ft and 15 ft slung length1205721= 015 and 120573

1= 055 and 082 respectively therefore

1205721+ 1205731lt 1 thus the system is stable But when the slung

length increased to 20 ft then 1205721

= 015 and 1205731

= 109therefore (120572

1+ 1205731) = 124 that is 120572

1+ 1205731gt 1 thus the

system is unstable However the system can be marginallystable if the UH-60 helicopter carries a load weight 725 lb bya 20 ft slung length then 120572

1= 02 and 120573

1= 08 therefore

(1205721+ 1205731) = 1 For this set of analysis 119896

119863is assumed to

be 119896119863

= 75 lbft However 119896119863can vary depending on the

weather condition and an example of 119896119863= 50 lbft is shown

above for illustrative purpose This analysis suggests that themaximum value of 20 ft slung length may be used for slungload operation However the load weight limit can vary withrespect to the capacity of the load carrying hooks helicopterweight environmental conditions and so forth

The examples presented here show that the simulationresults are correlated with the stability analysis Thus desiredcondition for the stability is met and it can locate the load atthe specified position or its neighbourhood

5 Conclusions

In this paper stability analysis of a helicopter with an under-slung load system is investigatedThe investigation identifiedthe conditions for the stabilisation of the system and thepositioning of the underslung load at hover condition Sta-bility analysis and numerical results proved that the desiredcondition for the stability is met then it is possible to locatethe load at the specified position or its neighbourhood Anillustration example is also given in the paper The methodresults in a guaranteed load positioning accuracy whichdepends on the design parameters and the method can beapplicable to any individual helicopter with external loadsystem

Nomenclature

119871119872119873 Overall helicopter rolling pitching andyawing moments

119901 119902 119903 Helicopter roll pitch and yaw rates aboutbody reference axes

119883119884 119885 Overall helicopter force components120601 120579 120595 Roll pitch and yaw angles119906 V 119908 Helicopter velocity components at centre

of gravity1205790 Main rotor collective

1205791119904 Longitudinal cyclic commands

1205791119888 Lateral cyclic commands

1205790119879 Tail rotor collective

1198830 1198840 1198850 Location of the suspension point withrespect to earth referenced 119909 119910 and 119911

directions

0 0 0 The helicopter velocity in the 119909 119910 and 119911

directions0 0 0 The helicopter acceleration in the 119909 119910 and119911 directions

120579119871 Load suspension angle in the119883-119885 plane

with respect to 119885-axis120601119871 Load suspension angle in the 119884-119885 plane

with respect to 119885-axis119872119871 Mass of the suspended load

119892 Acceleration due to gravity119897 Slung length119896119863 Aerodynamic drag force constant

(119896119863= (12)120588119878119862

119863)

120588 Air density119878 The load area presented to the airflow119862119863 Drag coefficient for the load

Competing Interests

The author declares that there are no competing interestsregarding the publication of this paper

References

[1] K Thanapalan and F Zhang ldquoModelling and simulation studyof a helicopter with an external slung load systemrdquo I-ManagersJournal on Instrumentation amp Control Engineering vol 1 no 1pp 9ndash17 2013

[2] M Bisgaard J D Bendtsen and A La Cour-Harbo ldquoModelingof generic slung load systemrdquo Journal of Guidance Control andDynamics vol 32 no 2 pp 573ndash585 2009

[3] Handling Qualities Requirements for Military Rotorcraft ADS-33-D US Army Aviation and Troop Command St Louis MissUSA 1996

[4] K Thanapalan and T M Wong ldquoModeing of a helicopter withan under-slung load systemrdquo in Proceedings of the 29th ChineseControl Conference (CCC rsquo10) pp 1451ndash1456 Beijing China July2010

[5] B Feng ldquoRobust control for lateral and longitudinal channels ofsmall-scale unmanned helicoptersrdquo Journal of Control Scienceand Engineering vol 2015 Article ID 483096 8 pages 2015

[6] J J Gribble ldquoLinear quadratic Gaussianloop transfer recoverydesign for a helicopter in low-speed flightrdquo Journal of GuidanceControl and Dynamics vol 16 no 4 pp 754ndash761 1993

[7] M A Manness J J Gribble and D J M Smith ldquoMultivariablemethods for helicopter flight control law designrdquo in Proceedingsof the 16th Annual Society European Rotorcraft Forum PaperNoIII52 Glasgow UK September 1990

[8] W L Garrard E Low and S Prouty ldquoDesign of attitude andrate command systems for helicopters using eigenstructureassignmentrdquo Journal of Guidance Control and Dynamics vol12 no 6 pp 783ndash791 1989

[9] Y B Shtessel and I A Shkolnikov ldquoAeronautical and spacevehicle control in dynamic sliding manifoldsrdquo InternationalJournal of Control vol 76 no 9-10 pp 1000ndash1017 2003

[10] D FThompson J S Pruyn andA Shukla ldquoFeedback design forrobust tracking and robust stiffness in flight control actuatorsusing a modified QFT techniquerdquo International Journal ofControl vol 72 no 16 pp 1480ndash1497 1999

14 Journal of Control Science and Engineering

[11] K Thanapalan ldquoModelling of a helicopter systemrdquo in Proceed-ings of the 1st Virtual Control Conference (VCC rsquo10) pp 45ndash52Aalborg University 2010

[12] D P Goodall and J Wang ldquoStabilization of a class of uncertainnonlinear affine systems subject to control constraintsrdquo Interna-tional Journal of Robust and Nonlinear Control vol 11 no 9 pp797ndash818 2001

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Journal of Control Science and Engineering 13

carries a loadweight 500 lb and 1000 lb then1205721= 03 and 041

with 1205731= 041 and 082 respectively therefore 120572

1+ 1205731lt 1

thus the system is stable But when the load weight is 1500 lband 2000 lb then 120572

1= 01 and 0075 and also 120573

1= 123 and

164 respectively therefore 1205721+ 1205731gt 1 thus the system is

unstable For the case of the constant load weight (1000 lb) asslung length is increased then for a 10 ft and 15 ft slung length1205721= 015 and 120573

1= 055 and 082 respectively therefore

1205721+ 1205731lt 1 thus the system is stable But when the slung

length increased to 20 ft then 1205721

= 015 and 1205731

= 109therefore (120572

1+ 1205731) = 124 that is 120572

1+ 1205731gt 1 thus the

system is unstable However the system can be marginallystable if the UH-60 helicopter carries a load weight 725 lb bya 20 ft slung length then 120572

1= 02 and 120573

1= 08 therefore

(1205721+ 1205731) = 1 For this set of analysis 119896

119863is assumed to

be 119896119863

= 75 lbft However 119896119863can vary depending on the

weather condition and an example of 119896119863= 50 lbft is shown

above for illustrative purpose This analysis suggests that themaximum value of 20 ft slung length may be used for slungload operation However the load weight limit can vary withrespect to the capacity of the load carrying hooks helicopterweight environmental conditions and so forth

The examples presented here show that the simulationresults are correlated with the stability analysis Thus desiredcondition for the stability is met and it can locate the load atthe specified position or its neighbourhood

5 Conclusions

In this paper stability analysis of a helicopter with an under-slung load system is investigatedThe investigation identifiedthe conditions for the stabilisation of the system and thepositioning of the underslung load at hover condition Sta-bility analysis and numerical results proved that the desiredcondition for the stability is met then it is possible to locatethe load at the specified position or its neighbourhood Anillustration example is also given in the paper The methodresults in a guaranteed load positioning accuracy whichdepends on the design parameters and the method can beapplicable to any individual helicopter with external loadsystem

Nomenclature

119871119872119873 Overall helicopter rolling pitching andyawing moments

119901 119902 119903 Helicopter roll pitch and yaw rates aboutbody reference axes

119883119884 119885 Overall helicopter force components120601 120579 120595 Roll pitch and yaw angles119906 V 119908 Helicopter velocity components at centre

of gravity1205790 Main rotor collective

1205791119904 Longitudinal cyclic commands

1205791119888 Lateral cyclic commands

1205790119879 Tail rotor collective

1198830 1198840 1198850 Location of the suspension point withrespect to earth referenced 119909 119910 and 119911

directions

0 0 0 The helicopter velocity in the 119909 119910 and 119911

directions0 0 0 The helicopter acceleration in the 119909 119910 and119911 directions

120579119871 Load suspension angle in the119883-119885 plane

with respect to 119885-axis120601119871 Load suspension angle in the 119884-119885 plane

with respect to 119885-axis119872119871 Mass of the suspended load

119892 Acceleration due to gravity119897 Slung length119896119863 Aerodynamic drag force constant

(119896119863= (12)120588119878119862

119863)

120588 Air density119878 The load area presented to the airflow119862119863 Drag coefficient for the load

Competing Interests

The author declares that there are no competing interestsregarding the publication of this paper

References

[1] K Thanapalan and F Zhang ldquoModelling and simulation studyof a helicopter with an external slung load systemrdquo I-ManagersJournal on Instrumentation amp Control Engineering vol 1 no 1pp 9ndash17 2013

[2] M Bisgaard J D Bendtsen and A La Cour-Harbo ldquoModelingof generic slung load systemrdquo Journal of Guidance Control andDynamics vol 32 no 2 pp 573ndash585 2009

[3] Handling Qualities Requirements for Military Rotorcraft ADS-33-D US Army Aviation and Troop Command St Louis MissUSA 1996

[4] K Thanapalan and T M Wong ldquoModeing of a helicopter withan under-slung load systemrdquo in Proceedings of the 29th ChineseControl Conference (CCC rsquo10) pp 1451ndash1456 Beijing China July2010

[5] B Feng ldquoRobust control for lateral and longitudinal channels ofsmall-scale unmanned helicoptersrdquo Journal of Control Scienceand Engineering vol 2015 Article ID 483096 8 pages 2015

[6] J J Gribble ldquoLinear quadratic Gaussianloop transfer recoverydesign for a helicopter in low-speed flightrdquo Journal of GuidanceControl and Dynamics vol 16 no 4 pp 754ndash761 1993

[7] M A Manness J J Gribble and D J M Smith ldquoMultivariablemethods for helicopter flight control law designrdquo in Proceedingsof the 16th Annual Society European Rotorcraft Forum PaperNoIII52 Glasgow UK September 1990

[8] W L Garrard E Low and S Prouty ldquoDesign of attitude andrate command systems for helicopters using eigenstructureassignmentrdquo Journal of Guidance Control and Dynamics vol12 no 6 pp 783ndash791 1989

[9] Y B Shtessel and I A Shkolnikov ldquoAeronautical and spacevehicle control in dynamic sliding manifoldsrdquo InternationalJournal of Control vol 76 no 9-10 pp 1000ndash1017 2003

[10] D FThompson J S Pruyn andA Shukla ldquoFeedback design forrobust tracking and robust stiffness in flight control actuatorsusing a modified QFT techniquerdquo International Journal ofControl vol 72 no 16 pp 1480ndash1497 1999

14 Journal of Control Science and Engineering

[11] K Thanapalan ldquoModelling of a helicopter systemrdquo in Proceed-ings of the 1st Virtual Control Conference (VCC rsquo10) pp 45ndash52Aalborg University 2010

[12] D P Goodall and J Wang ldquoStabilization of a class of uncertainnonlinear affine systems subject to control constraintsrdquo Interna-tional Journal of Robust and Nonlinear Control vol 11 no 9 pp797ndash818 2001

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

14 Journal of Control Science and Engineering

[11] K Thanapalan ldquoModelling of a helicopter systemrdquo in Proceed-ings of the 1st Virtual Control Conference (VCC rsquo10) pp 45ndash52Aalborg University 2010

[12] D P Goodall and J Wang ldquoStabilization of a class of uncertainnonlinear affine systems subject to control constraintsrdquo Interna-tional Journal of Robust and Nonlinear Control vol 11 no 9 pp797ndash818 2001

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of