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Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2013 Article ID 589267 16 pageshttpdxdoiorg1011552013589267

Research ArticleNonlinear Control for Trajectory Tracking ofa Nonholonomic RC-Hovercraft with Discrete Inputs

Dictino Chaos David Moreno-Salinas Rociacuteo Muntildeoz-Mansilla and Joaquiacuten Aranda

Departament of Computer Science and Automatic Control UNED Madrid Spain

Correspondence should be addressed to Dictino Chaos dchaosdiaunedes

Received 17 June 2013 Accepted 23 October 2013

Academic Editor Guanghui Sun

Copyright copy 2013 Dictino Chaos et al 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 work studies the problem of trajectory tracking for an underactuated RC-hovercraft the control of which must be done bymeans of discrete inputs Thus the aim is to control a vehicle with very simple propellers that produce only a discrete set of controlcommands and with minimal information about the dynamics of the actuators The control problem is approached as a cascadecontrol problem where the outer loop stabilizes the position error and the inner loop stabilizes the orientation of the vehicleStability of the controller is theoretically demonstrated and the robustness of the control law against disturbances and noise isestablished Simulation examples and experiments on a real setup validate the control law showing the real system to be robustagainst disturbances noise and outdated dynamics

1 Introduction

Autonomous vehicles (AV) in general and autonomoussurface vehicles (ASV) in particular are becoming ubiquitousin many fields of engineering research due in part to theflexibility and versatility that a number of them display inthe execution of individual and cooperative tasks Thesecharacteristics coupled with the fact that their use avoidsplacing human lives at risk makes them quite attractivein multiple missions Central to the operation of theseautonomous vehicles is the availability of good trackingcontrollers that allow the AVs to execute demanding tasksin different operational scenarios Moreover in the presenteconomical scenario in which many research projects sufferwithdrawal of funds it is of the utmost importance to offerreliable low cost systems

The three main problems that arise in control theory ofautonomous vehicles according to the classification of [1] arepath following [2] point stabilization [3 4] and trajectorytracking As part of this trend the tracking control problemfor underactuated vehicles represents a very challengingresearch topic being thus the problem studied in the presentwork

Underactuated vehicles are of special interest due to theirreduced cost and the relative easiness of their construction

furthermore a fully actuated vehicle is only needed forsome special operations making its usage unnecessary fora wide number of actions In addition the possibility ofusing the same vehicle for terrestrial and maritime opera-tions put underactuated hybrid vehicles in a very attractivesituation In this sense the most common hybrid vehicleis the hovercraft Hovercraft is usually used as test vehiclefor control algorithms of underactuated marine systems dueto its dynamic properties the possibility of being tested onground and its relative (and mission dependent) low cost asin the experimental testbed developed on [5] or [6]

In addition to the low cost and the possible hybrid natureof the vehicle at hand the nonholonomic condition of theunderactuated vehicles is by itself interesting enough andcomplex for a detailed study An usual solution for fullyactuated vehicles consists in using sliding mode controllerssuch as the solution described in [7] The main limitation ofthis control strategy is the impossibility of acting over all thestates of the vehicle at the same time when an underactuatedvehicle as an hovercraft is considered The control problemof underactuated vehicles is much more challenging than thefully actuated one because of the limitation on the controllaws that can be implemented This is a consequence of thefamous result stated by Brockett [8] that states that some

2 Mathematical Problems in Engineering

kind of underactuated vehicles (that includes the hovercraft)cannot be stabilized to a certain pose with a continuous andtime invariant control law The reader is referred to [9] foran interesting survey in control problems for nonholonomicvehicles

Representative examples of tracking controllers applied tounderactuated hovercrafts are stated in [10] where the track-ing control problem is solved using backstepping techniquesachieving practical stability results in [11] that solve the sameproblem exploiting the differential flatness properties of avessel with respect to its position and in [12] where theauthors extend previous results for the tracking and pathfollowing problems to more general marine vehicles takinginto account the parametric uncertainty on the models

Other remarkable examples are found in [13] where thetracking problem of general underactuated ships is studiedin [14] that explains how to select outputs when generalizedforces act on the vehicle in [15] where Barbalatrsquos lemma andbackstepping techniques are combined to achieve asymptotictracking in [16] that solves the tracking problemof awheeledrobot in [17] where the authors address the tracking problemfor a surface craft and in [18] in which the challengingproblem of solving both point stabilization and trackingproblem at the same time is studied

It must be remarked that the control strategies describedabove achieve a good control performance assuming thatforce and torque imparted by propellers can be perfectlycontrolled Nevertheless thrust allocation is in general avery complex task in surface crafts (see eg [19]) andmoreover expensive equipments are usually required Thusthe availability of low cost vehicles equipped with simpleactuators and with a limited knowledge of their dynamicsis of the utmost importance for the problem studied inthe present work and a very interesting alternative from apractical point of view

In the last few years the interest on the control of under-actuated vehicles where only a discrete set of control inputs isavailable has increased significantly See for example [20 21]for a description on the challenges that arise in quantizedcontrol of mobile vehicles and remote control of systemsover networks with state quantization respectively Anotherremarkable example is found in the works [22 23] where anonholonomic underactuated hovercraft is controlled by adiscrete set of inputs In these works the problem of pointstabilization is solved using receding horizon predictive con-trollers and approaching the continuous output by the closestallowed control action This strategy works adequately whenthe optimal control law is bang-bang but it is not suitablewhen the control signals take intermediate values betweentwo allowed ones something impossible to achieve withall-nothing-reverse signals Unfortunately this is the actualtracking problem scenario because forces and torques shouldbe close to a continuous reference far away from saturation

In order to solve the control problem of underactuatedvehicles some authors consider second order sliding-modecontrol laws See for example [24] where this technique isapplied to the tracking problem of a surface vessel and [11]where the trajectory tracking problem for an underactuatedhovercraft is solved These control laws tend to produce

control actions that are similar to bang-bang actions There-fore they are good candidates to be approximated with all-nothing-reverse thrusters The later are the kind of thrustersused in the vehicle (hovercraft) dealt in this work due to theirsimplicity of control and implementation and also their lowcost

This kind of problem has been partially studied in theground robotic field Some interesting results on the field ofwheeled robots show that in the case of Dubinrsquos vehicles it ispossible to design a tracking controller that makes the vehicleconverge to a reference trajectory using only a discrete setof control actions (turn right and left accelerate and brake)See for example [25] that uses a sliding-mode control and[16] where a controller has been developed using optimalsynthesis

An interesting practical problem is the extension of theabove results for an underactuated low cost vehicle as ahovercraft The main difference between a wheeledgroundunderactuated vehicle and a marine underactuated vehicle isthe nonholonomic restriction on the trajectories that can befollowed by each vehicle The nonholonomic restriction in avehicle is the relationship between the acceleration and theorientation of the vehicle The sliding condition for marinevehicles (second-order nonholonomic restriction) is morecomplex than the nonsliding condition (first-order nonholo-nomic restriction) for wheeled robots Thus the aim of thiswork is to control a vehicle with very simple propellers thatproduce only a discrete set of control commands and witha minimal information about the dynamics of the actuatorsThe availability simplicity and low cost of these kind ofsystems even the hybrid nature of the hovercraft makethem very attractive to be used in a number of operationalscenarios Furthermore a group of these vehicles can be usedfor cooperative andor coordinated tasks with relatively lowprice

The sole conditions that are imposed to the actuators usedon this work can be summarized as follows

(1) The available control actions can turn right and leftwhile the vehicle go forward and backwards

(2) The available force and torque are bigger than theforce and torque necessary to exactly track the ref-erence (forces and torques can be dominated by thecontrol actions)

Notice that the first condition is important to ensure localcontrollability (see [26]) while second condition is necessaryto ensure feasibility of the trajectory

Thekey contribution of the paper is two-fold (i) we reflectthe fact that the inputs are discrete in the controller designprocess and (ii) we take explicitly into account the existenceof unknown bounded input disturbances and measurementnoise Theoretical stability proofs are obtained and in addi-tion the control law is successfully tested on a real low costvehicle In the literature to the best knowledge of the authorsthere are no attempts to solve the trajectory tracking problemfor a second order nonholonomic vehicle as the hovercraftwith a discrete set of inputs Furthermore in the few caseswhere a discrete set of inputs is considered in a vehicle of this

Mathematical Problems in Engineering 3

120595

r

Fp

Fs

o

X

Y

l(x y)

U

u

V = [x t]T

Figure 1 Inertial reference frame and body fixed frame

kind the effect of this practical limitation and the effect ofdisturbances and noise have never been studied in detail

The paper is organized as follows Section 2 describes themodel of the vehicle and the set of trajectories that can betracked with a discrete set of inputs The control problem isformulated in Section 3 In Section 4 a nonlinear controlleris designed Practical implementation and the effect of dis-turbances and noise are discussed in Section 5 The stabilityand robustness of the controller are analyzed in Section 6Simulation and experimental tests are shown in Section 7Finally conclusions are commented on in Section 8

2 Vehicle Model

The model of the vehicle operating over a solid surface isdefined in two reference frames one inertial global referenceframe attached to the floor where the RC-hovercraft movesand a body-fixed reference frame shown in Figure 1 Accord-ing to the model (the details of which can be found in [27])the dynamics of the system can be described in the globalreference frame by

= V119909 (1)

119910 = V119910 (2)

V119909= 119865 cos (120595) minus 119889

119906V119909+ 119901V119909 (3)

V119910= 119865 sin (120595) minus 119889

119906V119910+ 119901V119910 (4)

120595 = 119903 (5)

119903 = 120591 minus 119889

119903119903 + 119901

119903 (6)

The state of the vehicle is given by x = [119909 119910 V119909 V119910 120595 119903]

119879where 119909 and 119910 are the position coordinates of the centre ofmass of the RC-hovercraft 120595 is the orientation V

119909and V

119910

are the linear velocities and 119903 is the yaw rate The parameters119889

119906and 119889

119903of the model are the normalized drag coefficients

where 119889

119906= 119863

119906119898 119889

119903= 119863

119903119869 119898 is the mass of the

hovercraft 119869 the moment of inertia and 119863

119906and 119863

119903are

the linear and rotational drag coefficients respectively Thecontrol actions are the normalized force 119865 = (119865

119901+119865

119904)119898 and

Table 1 Model parameters

Parameter Value119898 0995Kg119869 0014Kgm2

119897 0075m119906max 0545N119906min 0347N119889

11990603588 sminus1

119889

11990317357 sminus1

Table 2 Force selection table

120591 gt 0 120591 lt 0

119865 gt 0 119865

119901= 119906max 119865119904 = minus119906min 119865

119901= minus119906min 119865119904 = 119906max

119865 lt 0 119865

119901= 0 119865

119904= 119906min 119865

119901= 119906min 119865119904 = 0

torque 120591 = 119897(119865

119901minus 119865

119904)119869 where 119865

119904and 119865

119901are the starboard

and port forces produced by the fans and 119897 is the distancebetween the propellers and the symmetry axis as Figure 1shows The dynamics of the vehicle are also subject to theunknown disturbance vector p = [119901V119909 119901V119910 119901119903]

119879 and theoutput of the system is corrupted by measurement noise n =

[119899

119909 119899

119910 119899V119909 119899V119910 119899120595 119899119903]

119879 such that the measured state is y =

x + n Noise and disturbances are unknown but bounded asfollows

sup119905ge0

p (119905) le 119901

119898 sup

119905ge0

1003817

1003817

1003817

1003817

1003817

1003817

1003817

1003817

119889p119889119905

1003817

1003817

1003817

1003817

1003817

1003817

1003817

1003817

le 119901

119898

sup119905ge0

n (119905) le 119899

119898

(7)

The value of the parameters for the experimental deviceare summarized in Table 1

21 Physical Constraints The experimental vehicle is undersome physical constraints as it is usual in any real deviceThecontrol inputs are forces produced by thrusters the valuesof which belong to a discrete set 119865

119904119901isin minus119906min 0 119906max

with 119906max gt 119906min gt 0 Under these considerations thecontrol actions 119865 and 120591 belong to the discrete set of ninecombinations of 119865

119904and 119865

119901shown in Figure 2 It is important

to notice that these control actions can move the hovercraftforward and backwards and right and left Moreover it willbe shown that it is possible to control the RC-hovercraft witha reduced set of control actions and without knowledge ofthe exact values of 119906min and 119906max The key point is that oncethe sign of the force and torque is known then there alwaysexists a proper selection of 119865

119904and 119865

119901such that the desired

values of sign(119865) and sign(120591) are obtained As Figure 2 showsthere exist more than one solution In Table 2 one possibleselection scheme is shown

22 Feasible Trajectories Dynamics of the RC-hovercraft areflat with respect to the position coordinates as demonstratedin [11] The reader is referred to [28] for further informationabout flat systems Thus when the reference spatial trajec-tories 119909

119903(119905) and 119910

119903(119905) are set then the whole state vector


[umax umax]

F =Fp minus Fs

m

[umax 0]

[umax minusumin]

120591 =l(Fp minus Fs)

J

[0 minusumin]

[minusumin minusumin]

[minusumin 0]

[minusumin umax]

[0 umax]

[0 0]

Figure 2 Discrete control inputs on the laboratory setup

and the control actions that must be applied to follow thereference can be computed For this purpose considering (3)and (4) in the absence of disturbances and defining 119865

119903as the

force needed to track the reference the ideal orientation 120595

119903

needed to exactly track the reference is given by the followingnonholonomic restrictions

119865

119903cos (120595

119903) = V119909119903+ 119889

119906V119909119903

119865

119903sin (120595

119903) = V119910119903+ 119889

119906V119910119903

(8)

The orientation is well defined if 119865119903

= 0 It is important tonotice that actually there exist two families of solutionsdepending on how the trajectory is followed by the vehiclegoing forward (119899 odd) or backwards (119899 even) as follows

120595

119903= atan2 ( 119910

119903+ 119889

119906119910

119903

119903+ 119889

119906

119903) + 119899120587

119865

119903= (minus1)

119899radic

(

119903+ 119889

119906

119903)

2

+ ( 119910

119903+ 119889

119906119910

119903)

2

(9)

Once the family of solutions of interest is selected theorientation120595

119903and its derivatives 119903

119903and 119903

119903 can be computed

Straightforward computations yield the requested torque120591

119903= 119903

119903+119889

119903119903

119903to track the reference Now we consider a spatial

trajectory x119903= [119909

119903(119905) 119910

119903(119905)]

119879 four times differentiable with119865

119903= 0 Two kinds of trajectories of interest can be defined as

follows

(1) A feasible trajectory where 119865

119903and 120591

119903are such that

119865

119904119901isin [minus119906min 119906max]

(2) A D-feasible trajectory where there always exist fourcombinations of allowed forces and torques (119865

1 120591

1)

(119865

2 120591

2) (1198653 120591

3) and (119865

4 120591

4) such that

1003816

1003816

1003816

1003816

119865

119903(119905)

1003816

1003816

1003816

1003816

lt min (1198651 119865

2 minus119865

3 minus119865

4)

1003816

1003816

1003816

1003816

120591

119903(119905)

1003816

1003816

1003816

1003816

lt min (minus1205911 120591

2 minus120591

3 120591

4)

(10)

On one hand we consider that a trajectory is feasible when itcan be followed by the vehicle that is when the necessaryforces and torques to track the trajectory can be producedby values of 119865

119904and 119865

119901smaller than the maximum available

Notice that if a trajectory is not feasible the only way to trackit is to increase the size of the thrusters in order to incrementthe values of 119865

119904and 119865

119901 allowing the trajectory to be feasible

againThe control actions 119865119903(119905) and 120591

119903(119905) that define a feasible

trajectory lie on the dotted region of Figure 3On the other hand we consider that a trajectory is D-

feasible when the forces and torques selected fromTable 2 aregreater in absolute value than the forces and torques neededto track the trajectory Notice that all the trajectories thatare D-feasible are also feasible but the opposite is not trueThe control actions 119865

119903(119905) and 120591

119903(119905) that define a D-feasible

trajectory lie on the dashed region of Figure 3


F =Fp + Fs

m

[umax umax]

[umax 0]

[umax minusumin]


J

[0 minusumin]


[minusumin 0]

[minusumin umax]

[0 umax]

1

2

3

4

5

6

7

8

9

(Fr 120591r)

F0

minus1205910 1205910

minusF0

[0 0]

A

Figure 3 Feasible (dotted) and D-feasible (dashed) regions A trajectory such as the green one is feasible because (119865119903(119905) 120591

119903(119905)) lies on the

dotted region but is not D-feasible because it lies outside the dashed region

3 Problem Formulation

The key objective of the present work is to design and toimplement a feedback control law for 119865

119904and 119865

119901to solve the

robust trajectory tracking problem for a vehicle commandedwith a discrete set of inputs In other words the aim is thedevelopment of a controller that is able to track a desiredtrajectory in the presence of disturbances and noise withonly a limited knowledge of the model of the vehicle and theactuators

Themain practical restriction is that the vehicle is under-actuated and the control law must only employ a discrete setof inputs produced by the thrusters 119865


Notice that the controller is oriented to be used on smallautonomous vehicles equipped with simple thrusters becausethe addition of more thrusters would add complexity to thesystem and would increase the power consumption

Consider the dynamics of the hovercraft describedby (1)ndash(6) a D-feasible spatial reference trajectoryx119890 and the spatial tracking error defined by e

119901=

[119909 minus 119909

119903 119910 minus 119910

119903 V119909minus V119909119903 V119910minus V119910119903]

119879 Then the requirementsfor a robust trajectory tracking controller are as follows

(1) The error e119901

is bounded in the presence of thebounded noise and disturbances described by (7)

(2) The final bound of e119901 can be made arbitrary small

if the noise 119899

119898and the disturbances 119901

119898are small

enough (or alternatively if force and torque are largeenough)

4 Nonlinear Control Design

The control design is based on the cascade structure illus-trated in Figure 4 In the outer loop the force and the

orientation are used as virtual control inputs for the positioncontroller to stabilize the position tracking errorThepositioncontroller defines an orientation reference 120595

119888to be tracked

by the inner loop the orientation controller that controlsthe orientation with the torque 120591 Therefore the positioncontroller determines the sign of the force 119865 to be appliedon the RC-hovercraft whereas the orientation controllerdetermines the sign of the torque 120591 Once sign(119865) and sign(120591)are computed the values of 119865

119904and 119865

119901are selected from

Table 2

41 Position Control The objective of this subsection is thedesign of a controller for the position of the hovercraft Theerror variables are defined by

119890

119909= 119909 minus 119909

119903

119890

119910= 119910 minus 119910

119903

119890V119909 = V119909minus V119909119903

119890V119910 = V119910minus V119910119903

(11)

And their dynamics can be written as

119890

119909= 119890V119909

119890

119910= 119890V119910

119890V119909 = 119865 cos (120595) minus 119889

119906119890V119909 minus 119865

119909119903+ 119901V119909

119890V119910 = 119865 sin (120595) minus 119889

119906119890V119910 minus 119865

119910119903+ 119901V119910

(12)

where 119865119909119903

= 119865

119903cos(120595

119903) and 119865

119910119903= 119865

119903sin(120595119903) are the nominal

forces needed to track exactly the reference in absence of


References

x

Positioncontrol

120595 r

120595c rc

Orientationcontrol Sign(120591)

Selector

minus25 minus2 minus15 minus1 minus05 0 05 1 15 2 25120591 = (Fb minus Fe)J

25

2

15

1

05

0

minus05

minus1

minus15

F=(F

b+Fe)m

Fe

Fb

Hovercraft

x

Sign(F)

[umax umax]

[umax 0]

[umax minusumin]

[0 minusumin]


[minusumin 0]

[minusumin umax]

[0 umax]

[0 0]

Figure 4 Control architecture

disturbances At this point a positive control constant 1198961 and

the variables 119904119909and 119904119910are introduced in order to proceedwith

the error stabilization as follows

119904

119909= 119890

119909+ 119896

1119890V119909

119904

119910= 119890

119910+ 119896

1119890V119910

(13)

It is easy to check from (13) that the tracking errorvariables converge exponentially to zero with rate 1119896

1if 119904119909

and 119904

119910are both equal to zero Furthermore the problem of

tracking error stabilization is reduced to the stabilization of119904

119909and 119904

119910 Computing their dynamics it is clear that 119865 and

the orientation 120595 are coupled Consider the following

119904

119909= 119896

1119865 cos (120595) + (1 minus 119889

119906119896

1) 119890V119909 minus 119896

1119865

119909119903+ 119896

1119901V119909

119904

119910= 119896

1119865 sin (120595) + (1 minus 119889

119906119896

1) 119890V119910 minus 119896

1119865

119910119903+ 119896

1119901V119910

(14)

Therefore the stabilization of both variables at the sametime is an interesting and difficult problem To overcomeit both control actions are decoupled by rotating the errorvector an angle defined by the orientation angle 120595 in thebody fixed frame of the vehicle providing the following newcoordinates

119911

1= 119904

119909cos (120595) + 119904

119910sin (120595)

119911

2= minus119904

119909sin (120595) + 119904

119910cos (120595)

(15)

Then simple computations yield

1= 119896

1119865 minus 119896

1119865

119903cos (120595 minus 120595

119903) + 119911

2119903

+ (1 minus 119889

119906119896

1) (119890V119909 cos (120595) + 119890V119910 sin (120595)) + 119896

1119901

1

(16)

2= 119896

1119865

119903sin (120595 minus 120595

119903) minus 119911

1119903

+ (1 minus 119889

119906119896

1) (minus119890V119909 sin (120595) + 119890V119910 cos (120595)) + 119896

1119901

2

(17)

where 1199011and 119901

2are bounded disturbances given by

119901

1= 119901V119909 cos (120595) + 119901V119910 sin (120595)

119901

2= minus119901V119909 sin (120595) + 119901V119910 cos (120595)

p =1003817

1003817

1003817

1003817

1003817

1003817

[119901

1 119901

2]

1198791003817

1003817

1003817

1003817

1003817

1003817

le 119901

119898

(18)

Analyzing the dynamics of 1199111and 119911

2 given by (16) and

(17) it must be noticed that (17) does not depend on 119865 so 119865

must be used to control 1199111 In order to solve this problem the

sign of 119865must be opposed to the sign of 1199111 thus we propose

sign (119865) =

minus sign (1199111) if 1003816

1003816

1003816

1003816

119911

1

1003816

1003816

1003816

1003816

ge 120576

sign (119865minus) if 1003816

1003816

1003816

1003816

119911

1

1003816

1003816

1003816

1003816

lt 120576

(19)

where 120576 is a small constant to avoid chattering by holding thesign of the force to its previous value sign(119865minus) when |119911

1|

is small Notice that this condition imposes a minimal timebetween switches because |119911

1|must change from minus120576 to +120576 (or

from +120576 to minus120576) before the force119865 changes its sign In additionthis control law tries to make |119911

1| lt 120576

In order to stabilize 1199112 the sign of 119865

119903sin(120595 minus 120595

119903)must be

the opposite of the sign of 1199112 For this purpose the following

reference for the orientation is defined as follows

120595

119888= 120595

119903minus 119896

2tanh (119911

2) sign (119865

119903) (20)

119903

119888=

120595

119888=

120595

119903minus

2119896

2(1 minus tanh (119911

2)

2

) sign (119865119903) (21)

The saturation of 1199112given by the operation tanh in (20) is

introduced to guarantee that when120595 = 120595

119888and 0 lt 119896

2lt 1205872

then 119865

119903sin(120595119888minus 120595

119903) = minus|119865

119903| sin(119896

2tanh(119911

2)) that is opposed

to 1199112In general the orientation reference 120595

119888is not perfectly

tracked by 120595 Thus the orientation errors are defined by

119890

120595= 120595 minus 120595

119888 (22)

119890

119903= 119903 minus 119903

119888 (23)

and the dynamics of z finally become

1= 119896

1119865 minus 119896

1119865

119903cos (119890

120595+ 120595

119888minus 120595

119903) + 119911

2119903 + 119896

1119901

1

+ (1 minus 119889

119906119896

1) (119890V119909 cos (119890120595 + 120595

119888) + 119890V119910 sin (119890120595 + 120595

119888))

(24)

2= minus 119896

1

1003816

1003816

1003816

1003816

119865

119903

1003816

1003816

1003816

1003816

sin (1198962tanh (119911

2)) minus 119911

1119903 + 119896

1119865

119903119908 + 119896

1119901

2

+ (1 minus 119889

119906119896

1) (119890V119910 cos (119890120595 + 120595

119888) minus 119890V119909 sin (119890120595 + 120595

119888))

(25)


where 119908 is the disturbance produced by the orientationtracking error 119908 = sin(120595 minus 120595

119903) minus sin(120595

119888minus 120595

119903) bounded by

|119908| le

1003816

1003816

1003816

1003816

120595 minus 120595

119888

1003816

1003816

1003816

1003816

=

1003816

1003816

1003816

1003816

1003816

119890

120595

1003816

1003816

1003816

1003816

1003816

(26)

Theorem 2 will show that for small 119890120595 (24)-(25) is a stable

system with disturbances 119890120595 1199011 and 119901

2

42 Orientation Control The objective of this subsection isthe design of an orientation controller so that 120595 can be ableto track the control reference 120595

119888 The error dynamics yields

119890

120595=

120595 minus

120595

119888= 119903 minus

120595

119888= 119890

119903

119890

119903= 120591 minus 120591

119888minus 119889

119903119890

119903+ 119901

119903

(27)

where 120591119888= 119889

119903

120595

119888minus

120595

119888is the control torque necessary to track

the orientation reference 120595119888 At this point a positive control

gain 119896

3is introduced in order to define the variable 119904

120595as

follows

119904

120595= 119890

120595+ 119896

3119890

119903 (28)

It is clear that the orientation error converges exponentiallyto zero with rate 1119896

3if 119904120595is equal to zero The computation

of its dynamics yields

119904

120595= 119896

3(120591 minus 120591

119888+ 119901

119903) + (1 minus 119896

3119889

119903) 119890

119903 (29)

Thus according to (29) in order to stabilize 119904120595 the sign of 120591

must be the opposite of the sign of 119904120595 This fact motivates the

following control law

sign (120591) =

minus sign (119904120595) if 1003816

1003816

1003816

1003816

1003816

119904

120595

1003816

1003816

1003816

1003816

1003816

ge 120576

sign (120591minus) if 1003816

1003816

1003816

1003816

1003816

119904

120595

1003816

1003816

1003816

1003816

1003816

lt 120576

(30)

where chattering is avoided in a similar way as in (19) holdingthe sign of the torque to its previous value sign(120591minus) when |119904

120595|

is small Notice that this control law tries to make |119904120595| lt 120576

5 Practical Implementation and Noise

The positions and velocities of the RC-hovercraft are mea-sured to compute the control law However due to thepresence of bounded noise n only the estimates of the abovevariables (119909 119910 120595 V

119909 V119910 and 119903) are available

Using the estimates as the actual values of the state theerror variables are computed with (11) then 119904

119909and 119904

119910are

computed using (13) and rotated with (15) to obtain theestimates

1= 119911

1+ 119899

1and

2= 119911

2+ 119899

2 where

1003816

1003816

1003816

1003816

119899

1

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119899

2

1003816

1003816

1003816

1003816

le (z + radic

2 (1 + 119896

1)) 119899

119898 (31)

The term radic2(1 + 119896

1) in (31) comes from the definition of 119904

119909

and 119904119910 (13) while the term z appears from (15) that involves

the estimate 120595

The computation of 2is carried out with (17) assuming

that the unknown disturbance 1199012is equal to 0 The estimated

orientation reference 120595

119888is computed using (20) and its

derivative

120595

119888is computed with (21) These values are finally

used in (28) to obtain the estimate 119904120595= 119904

120595+ 119891

3 where

1003816

1003816

1003816

1003816

119891

3

1003816

1003816

1003816

1003816

le 119896

3119896

2119896

1119901

119898+ 119899

119898(1 + 119896

3+ 119899

119898+ 119896

1

1003816

1003816

1003816

1003816

119865

119903

1003816

1003816

1003816

1003816

+ |119903|)

+ 119896

2119899

119898(1 + 119896

3

1003816

1003816

1003816

1003816

2

1003816

1003816

1003816

1003816

) (z + radic

2 (1 + 119896

1))

+ 2

radic

2119899

119898

1003816

1003816

1003816

1003816

1 minus 119889

119906119896

1

1003816

1003816

1003816

1003816

radic119890

2

V119909 + 119890

2

V119910

(32)

Thus the control law (19) and (30) become

sign (119865) =

minus sign (1) if 1003816

1003816

1003816

1003816

1

1003816

1003816

1003816

1003816

ge 120576

sign (119865minus) if 1003816

1003816

1003816

1003816

1

1003816

1003816

1003816

1003816

lt 120576

(33)

sign (120591) =

minus sign (119904120595) if 1003816

1003816

1003816

1003816

1003816

119904

120595

1003816

1003816

1003816

1003816

1003816

ge 120576

sign (120591minus) if 1003816

1003816

1003816

1003816

1003816

119904

120595

1003816

1003816

1003816

1003816

1003816

lt 120576

(34)

The control equations (33) and (34) define only the sign of119865 and 120591 In order to determine the actual control actions tobe applied to the vehicle 119865

119890and 119865

119901are selected according to

Table 2

6 Stability Analysis

In this section theoretical stability results and its practicalapplications are explained Detailed proofs of the theoremsare given in the Appendix

Notation In what follows we consider that a function 120572(119909) isa K class function if 120572(0) = 0 and 120572 is strictly increasing Afunction 120573(119909 119905) is classKL if it isK class with respect to 119909

and 120573 rarr 0 when 119905 rarr infin [29]

Lemma 1 The velocities of the vehicle are globally ultimatelybounded Proof in Section A1

The first result states that the velocities are boundedbecause the control actions and the disturbances that canincrease the kinetic energy are bounded and the terms thattend to reduce the kinetic energy are unbounded

61 Position Stability

Theorem 2 Consider the system (24)-(25) with control action(33) in the presence of bounded disturbances and noise (7)If disturbances and noise are small enough then for any D-feasible trajectory x

119903 there exist three positive constants 119896

1 1198962

and 119908

119898 and two functions 120573 and 120574 of class KL and K

respectively such that if |119908| le 119908

119898 then

z (119905) le 120573 (z (0) 119905) + 120574 (119908

119898+ 119899

119898+ 119901

119898+ 120576) (35)

Proof on Section A2

Corollary 3 Under the conditions of Theorem 2 there existtwo functions 120573

2and 1205742 of classKL andK respectively such

that for |119908| le 119908

119898

1003817

1003817

1003817

1003817

1003817

e119901(119905)

1003817

1003817

1003817

1003817

1003817

le 120573

2(

1003817

1003817

1003817

1003817

1003817

e119901(0)

1003817

1003817

1003817

1003817

1003817

119905) + 120574

2(119908

119898+ 119899

119898+ 119901

119898+ 120576) (36)

Proof in Section A3


Equation (35) means that for a small orientation error(and thus a small 119908) if noise and disturbances are not toolarge compared with the control action 119865 then z is boundedFurthermore as (36) states the steady state tracking error(when 120573

2vanishes) is a growing function depending on

the orientation error disturbances and noise Thus whilethe noise disturbances and orientation error increase thetracking performance decreases

62 Orientation Stability

Theorem 4 Consider the dynamics described by (29) withcontrol action (34) in the presence of bounded disturbances andnoise (7) If disturbances and noise are small enough then thereexist five positive constants 119911

119898 1198963 119886 119887 and 119888 and a K class

function 120574

3such that for any D-feasible trajectory x

119903and any

positive gain 119896

2 if z le 119911

119898 then

1003816

1003816

1003816

1003816

1003816

119904

120595(119905)

1003816

1003816

1003816

1003816

1003816

le max (1003816100381610038161003816

1003816

119904

120595(0)

1003816

1003816

1003816

1003816

1003816

minus 119886119905 120574

3(119887 + 119911

119898+ 119901

119898) 119899

119898+ 119888119901

119898120576)

(37)

Proof in Section A4

Corollary 5 Under conditions of Theorem 4 there existsanother constant 119889 such that for z(119905) le 119911

119898

1003816

1003816

1003816

1003816

1003816

119890

120595(119905)

1003816

1003816

1003816

1003816

1003816

le 119889 sdot 119890

minus1199051198963+ 120574

3(119887 + 119911

119898+ 119901

119898) 119899

119898+ 119888119901

119898+ 120576 (38)

Proof in Section A5

Equation (37) means that if the tracking error is boundedby 119911119898 and disturbances and noise are not too large compared

with the control input 120591 then the variable 119904120595is stabilized in

finite timeThus according to (38) the orientation error con-verges exponentially Notice that the performance decreaseswhile noise and disturbance levels increase as it occurs withthe position tracking

63 Overall Convergence Previous results have demon-strated that the orientation error 119890

120595 converges exponentially

to a neighborhood of the origin while z is bounded by 119911119898

Thus (26) implies that119908 converges to a neighborhood of theorigin also Furthermore when119908 is smaller than an arbitrarymaximum value 119908

119898 then z converges to a neighborhood of

the origin tooThe following result states that control laws (33) and (34)

can stabilize the tracking errorwhen both are used at the sametime

Theorem 6 Consider the dynamics of the hovercraft (1)ndash(6)in the presence of bounded disturbances and noise (7) andsubject to control actions (33)-(34) For any positive constant119872 if the initial condition of the hovercraft is such that z(0) lt119872 and the reference trajectory x

119903is D-feasible then it is

possible to choose the control constants 1198961 1198962 1198963 and 120576 such

that1003817

1003817

1003817

1003817

1003817

e119901(119905)

1003817

1003817

1003817

1003817

1003817

le 120573

3(

1003817

1003817

1003817

1003817

1003817

e119901(0)

1003817

1003817

1003817

1003817

1003817

119905) + 120574

4(119899

119898+ 119901

119898+ 120576) (39)

where 1205744is a classK function and 120573

3is a classKL function

Proof in Section A6

This result states that given any bound for the initialcondition 119872 and a trajectory that is D-Feasible then theconstants of the tracking controller can be set in order tostabilize the tracking error

There exists a compromise between the tracking perfor-mance and the control effort imposed by 120576 If 120576 is too largethen the tracking performance is poor as shown in (39) If 120576is too small then (33) and (34) will produce fast switches inthe control law

Remark 7 It must be noticed that the control gains andthus the tracking performance depend on the bounds ofthe initial conditions (see (A24) of the Appendix for moredetails) This result is conservative in the sense that if wewant to increase the region of attraction (increasing119872) thenthe gain 119896

2must be small and thus the effect of noise and

disturbances is larger The result is also conservative in thesense that it restricts its applicability toD-feasible trajectoriesbut in practice the system is stable from any initial conditionwith a good selection of the control gains

7 Results

In this section simulations and experiments on a real setuphave been developed in order to test and illustrate thepotential of the control design of Section 4

71 Simulation Results The forthcoming simulation exam-ples consist in tracking a circular reference with and withoutnoise and disturbances Then a more general trajectorytracking problem is studied to show that the control designcan overcome some of the limitations that are assumed onthe stability demonstrations The control parameters used inthe simulations are 119896

1= 6 s 119896

2= 1 s 119896

3= 1 s and 120576 = 001

711 Circular Trajectory without Noise and Disturbances Thereference trajectory is a circumference with its center at theorigin of the inertial coordinate frame and with a radius119877 of 2 meters The trajectory is tracked with a speed of119881 = 05ms The force and torque needed to track perfectlythis trajectory (computed as described in Section 22) areconstants (119865

119903= 02179ms2 and 120591

119903= 04339 rads2) so the

trajectory isD-feasible because they do not saturate actuatorsThe trajectory of the hovercraft is depicted in Figure 5The vehicle starts from the origin without disturbances andnoise This figure shows how the trajectory followed by thehovercraft converges and tracks almost perfectly the referencetrajectory

The evolution of the position is shown in Figure 6 andvelocities are shown in Figure 7 It must be noticed that theactual trajectory converges to the reference trajectory in atime close to 20 s

The orientation 120595 and its control reference 120595119888are shown

in Figure 8The orientation converges faster than the position(less than 5 seconds) The reason is intuitively clear becausethe position only converges when the orientation error 119890

120595 is

small enoughOnce 119890120595is smaller than119908

119898 z starts to converge

to zero and the control references 120595119888and 119903

119888converge to the

references 120595119903and 119903

119903 too


minus25 minus2 minus15 minus1 minus05 0 05 1 15 2 25

minus2

minus15

minus1

minus05

0

05

1

15

2

x(m

)

y (m)

Figure 5 Simulated trajectory (solid) and circular reference(dashed)

0 10 20 30 40 50 60minus3

minus2

minus1

0123

x(m

)

t (s)

(a)

0 10 20 30 40 50 60minus3

minus2

minus1

0123

y(m

)

t (s)

(b)

Figure 6 Simulated position (solid) versus references (dashed)

712 Effect of Noise and Disturbances Noise and distur-bances are introduced in the simulations to study theireffects on the control performance In order to make thesimulation more realistic the controller is simulated with asampling period of 117 s that corresponds to the frame rateof the camera in the real setup In addition a disturbancep = [0ms2 002ms2 002ms2] and random noise with anamplitude of 005 (meters for 119909 and 119910 ms for V

119909and V119910 rad

for 120595 and rads for 119903) are introduced in the simulationThe effect of noise and disturbances on the performance

depends strongly on the reference trajectory that must betracked To illustrate this effect Figure 9 shows the trajecto-ries followed by the system when the reference speed is fixedand the radius varies from 1 to 4 meters This figure shows

0 10 20 30 40 50 60

minus04minus02

0020406

x(m

s)

t (s)

(a)

0 10 20 30 40 50 60

minus04minus02

0020406

t (s)

y(m

s)

(b)

Figure 7 Simulated velocities (solid) versus references (dashed)

0 10 20 30 40 50 60minus4

minus2

0

2

4

6120595

(rad

)

t (s)

(a)

0 10 20 30 40 50 60minus08minus06minus04minus02

0020406

r(r

ads

)

t (s)

(b)

Figure 8 Simulated orientation and yaw rate (solid) vs references(dashed)

how the performance decreases for small radii for two rea-sons The first one is that the reduction of the radius reducesthe signalnoise ratio The second one deals with centripetalforces While the radius decreases centripetal forces increasewith a factor 1198812119877 This implies that forces increase and thetrajectories are pushed to the unfeasible region In this casethe impact of noise and disturbance becomes more evidentbecause the control authority is degraded

Figure 10 confirms the behavior predicted byTheorem 6First the trajectories experiment a transient that dependson the initial conditions This transient is of approximately25 s during which the tracking error decreases to the steady


minus5 minus4 minus3 minus2 minus1 0 1 2 3 4 5minus4

minus3

minus2

minus1

0

1

2

3

4

x(m

)

y (m)

Figure 9 Simulated trajectories with noise and disturbances fordifferent circle radius 119877 = 1 2 3 4 meters in red violet green andblue respectively

0 10 20 30 40 50 600

05

1

15

2

25

3

35

4

e p(m

)

t (s)

Figure 10 Tracking error for different circle radius 119877 = 1 2 3 4

meters in red violet green and blue respectively

state value After this time only a residual tracking errorremains It shows how the tracking error increases for smallradii because the trajectory is more difficult to track (close tobe unfeasible)

713 General Trajectory For this third scenario a new refer-ence trajectory is set 119909

119903(119905) = 2 sin(38119905) 119910

119903(119905) = 3 cos(28119905)

in order to test the control design The necessary forces andtorques to track the trajectory are depicted in Figure 11 It canbe noticed that the forces lie outside the D-feasible regionhowever the trajectory is feasible This example is used toillustrate that the D-feasible condition (10) is conservativeFurthermore Figure 12 shows that the same control designused for the circular trajectory converges to this non-D-feasible reference too


minus06

minus04

minus02

0

02

04

06

08

1

12

120591 (rads2)

F(m

s2)

Figure 11 Force map for a general trajectory that is feasible but notD-feasible

minus3 minus2 minus1 0 1 2minus25

minus2

minus15

minus1

minus05

0

05

1

15

2x

(m)

y (m)

Figure 12 General trajectory followed by the vehicle (blue) andreference (red)

This behavior can be explained from an intuitive pointof view D-feasible conditions (10) indicate that the referenceforce and torque must be smaller in absolute value that anyforce and torque than can be selected by the control law fromTable 2However this is a sufficient condition but not always anecessary condition For example suppose that the trajectoryis such that119865

119903is positive for all the time (the current scenario

as Figure 11 shows) then it is possible to select a positiveforce that is larger than 119865

119903 with a larger torque in absolute

value than 120591

119903 This is enough to stabilize the trajectory

because during the trajectory tracking a negative force isnever needed

72 Experimental Results

721 Experimental Setup The experimental setup consistsin an underactuated radio controlled RC-hovercraft a CCD


Figure 13 Experimental setup


minus06

minus04

minus02

0

02

04

06

08

x(m

)

y (m)

Figure 14 Actual tracking Trajectory (solid) and references(dashed)

camera to capture images from a zenithal point of viewand a computer to send the control commands via a radiocontrol interfaceThewhole system is shown in Figure 13Thelaboratory has similar capacities to the experimental testbeddeveloped on [5] or [6]

The CCD camera is equipped with wide angle optics thatcover a region of approximately 3m times 4m when the camerais placed at 4 meters over the floor The frame rate of thecomputer-camera system is 17 fps (frames per second)

The position of the vehicle is tracked by the camera bydetecting three LEDs that are attached to the RC-hovercraftThese 3 LEDs compose an isosceles triangle to determine thevehicle orientation The triangle size is 25 cm of length in itsmajor edge and 16 cm in its minor edge

The control signals are sent to the vehicle through acommunication interface that connects the parallel port ofthe computer with the radio controller The control lawis implemented in LabVIEW programming language Thecontrol algorithm works as follows

(1) Once a frame is captured the image is binarized andthe centroid of each individual LED is computed

0 5 10 15 20 250

01

02

03

04

05

06

07

08

09

1

Dist

ance

(m)

t

Figure 15 Tracking error

(2) The barrel distortion introduced by the lenses is cor-rected and the undistorted positions of the LEDs areobtained by solving the perspective transformation

(3) The centre of the mass of the vehicle and its ori-entation are computed from the positions of thetriangle vertexThe velocities are estimated by a finitedifference

(4) Finally the control law computes the forces 119865119904and

119865

119901 and the corresponding commands are sent to the

radio controller

For amore detailed description of the experimental setupsee [30]

722 Real Tracking Test The control design is now testedon the experimental setup described above To check therobustness of the controller a worst case analysis has beenrealized A trajectory with a very small radius 119877 = 06mis tracked with a constant speed of 119881 = 05ms Notice thatthis trajectory is very hard to track because of its small radiusas above mentioned in Section 712 Figure 14 shows thatthe vehicle is able to track this very challenging trajectorywith an acceptable tracking error As far as the authors knowthe performance achieved is comparable to the best resultsobtained with a laboratory hovercraft (see [10]) Notice theadded hard limitation imposed by the all-nothing actuatorswhose dynamics are unknown Figure 16 shows the evolutionof the position of the system and demonstrates that thecontrol law is able to track the trajectory Figure 17 shows thevelocities Notice that the measured velocity is noisy due tothe all-nothing-reverse action of the propellersThis problemcan be reduced by filtering the position measurements butthe aim of this example is to show that this controller candeal with these problems and the vehicle is able to trackthe velocity reference with great accuracy Figure 15 showsthat the final bound of the tracking error is about 04m andthus compatible with simulation results shown on Figure 10Finally Figure 18 shows the input thrust applied to the vehicle


0 5 10 15 20 25minus06minus04minus02

0020406

x(m

)

t (s)

(a)


002040608

y(m

)

t (s)

(b)

Figure 16 Actual positions (solid) versus references (dashed) in areal setup

t (s)0 5 10 15 20 25

minus1

minus05

0

05

1

x(m

s)

(a)

0 5 10 15 20 25minus1

minus05

0

05

1

y(m

s)

t (s)

(b)

Figure 17 Actual velocities (solid) versus references (dashed) in areal setup

8 Conclusions

In this paper a tracking control law for an underactuatednonholonomic vehicle with a discrete set of inputs has beendesigned and testedThe control lawhas been designed takinginto account the fact that the actuators of the vehicle can onlyproduce thrust in an all-nothing reverse fashion

Despite this hard limitation on the real setup this papershows how this control design can overcome these difficul-ties Furthermore this work demonstrates that a low cost

0 5 10 15 20 25minus04

minus02

0020406

Fs

(N)

t (s)

(a)

0 5 10 15 20 25minus04

minus02

0020406

t (s)

F p(N

)

(b)

Figure 18 Control actions 119865119904and 119865

119901applied in a real setup

underactuated vehicle with very simple actuators (capableof moving the vehicle forward and backwards and simul-taneously turn right and left) can be controlled in order totrack anyD-feasible trajectory under the sole assumption thatcontrol actions are large enough compared with noise anddisturbances

Theoretical proofs of stability and robustness againstnoise and disturbances have been obtained showing thatthe control law makes the position tracking error to besemiglobal finitely bounded if the vehicle starts with abounded tracking error and the trajectory is D-feasible

Simulations and experimental results state that the con-trol strategy is robust enough to be applied to a real environ-ment with disturbances noise and unmodeled dynamics Inaddition this control law is very suitable to be applied to lowcost and simple vehicleswhere precisemodels of the actuatorsand dynamics are not available

Appendix

A Proofs

This section contains the proofs of the theoretical results

A1 Proof of Lemma 1 Consider the functions 1198711= V21199092

119871

2= V21199102 and 119871

3= 119903

2

2 and compute the derivative alongthe trajectories of the vehicle

119871

1= minusV119909(119889

119906V119909minus119865 cos(120595)+119901V119909)

Defining 119865max and 120591max as the maximum values of |119865| and |120591|

that can be selected from Table 2 then

119871

1can be bounded by

119871

1lt minus119889

119906V2119909(1 minus

119865max + 119901

119898

119889

119906

1003816

1003816

1003816

1003816

V119909

1003816

1003816

1003816

1003816

) (A1)

Equation (A1) shows that

119871

1is strictly negative when

119889

119906|V119909| gt (119865max + 119901

119898) This implies that V

119909is ultimately


bounded by (119865max + 119901

119898)119889

119906 The same analysis can be done

with 119871

2and 119871

3showing that the set

Ω = x isin R6

1003816

1003816

1003816

1003816

V119909

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

V119910

1003816

1003816

1003816

1003816

1003816

le

119865max + 119901

119898

119889

119906

|119903| le

120591max + 119901

119898

119889

119903

(A2)

is a positively invariant set of the system (1)ndash(6)

A2 Proof of Theorem 2 Without loose of generality theanalysis is restricted to the region defined by Ω If noise anddisturbances are small enough then there exists a positiveconstant 120575 such that |119865| minus |119865

119903| minus 119901

119898ge 120575 by definition of

a D-feasible trajectory Then for any positive 1205761 it is always

possible to find a constant 1198961such that

radic

2

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

(1 minus 119889

119906119896

1)

119889

119906119896

1

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

(2119865max + 119901

119898) le 120575120576

1 (A3)

Moreover if disturbances are small enough then it is alsopossible to select 119908

119898so that

radic

2 (119901

119898+ 120575120576

1+

1003816

1003816

1003816

1003816

119865

119903

1003816

1003816

1003816

1003816

119908

119898) lt min (120575 100381610038161003816

1003816

119865

119903

1003816

1003816

1003816

1003816

sin (1198962)) (A4)

Consider the following Lyapunov candidate function 119881 =

(119911

2

1+ 119911

2

2)2 The computation of its derivative yields

119881 = 119896

1119911

1(119865 minus 119865

119903cos (120595 minus 120595

119903) + 119901

1)

minus 119896

1119911

2(

1003816

1003816

1003816

1003816

119865

119903

1003816

1003816

1003816

1003816

sin (1198962tanh (119911

2)) minus 119865

119903119908 minus 119901

2)

+ (1 minus 119889

119906119896

1) (119904

1119890V119909 + 119904

2119890V119910)

(A5)

For an appropriate stability analysis two cases must be takeninto account

Case A (|1199111| gt 119899

1+ 120576) In this analysis our aim is to bound all

the terms in (A5) The term |119865

119903cos(120595 minus 120595

119903)| can be bounded

by |119865119903| and |119911

1| can be bounded by z Then we define the

following function

119892 (119909) = min( 120575

radic2

119865

119903

radic2

sin(1198962tanh( 119909

radic2

))) (A6)

Condition (A3) in combination with (A6) implies thatminus120575|119911

1| minus |119865

119903119911

2| sin(119896

2tanh (|119911

2|)) le minusz119892(z) so

119881 le minus119896

1z (119892 (z) minus (120575120576

1+

1003816

1003816

1003816

1003816

119865

119903

1003816

1003816

1003816

1003816

|119908| + 119901

119898)) (A7)

Equation (A7) is negative when 119892(z) gt 120576

1+ |119865

119903|119908

119898+ 119901

119898

This condition holds when z gt 119911max 1 where 119911max 1 isdefined as

119911max 1 = radic

2 tanhminus1 ( 1

119896

2

sinminus1 (radic2(

119901max + 120575120576

1003816

1003816

1003816

1003816

119865

119903

1003816

1003816

1003816

1003816

+ 119908

119898)))

(A8)

Note that condition (A4) implies that 119911max 1 is finite and 119881

decreases when z gt 119911max 1

Case B (|1199111| le 119899

1+ 120576) In this case we have to establish the

bound of z On one hand we suppose that |1199111| gt |119911

2| and

with (31) we find

z le radic

2

1003816

1003816

1003816

1003816

119911

1

1003816

1003816

1003816

1003816

le

radic

2119899

1+ 120576 le

radic

2 (z + radic

2 (1 + 119896

1)) 119899

119898+ 120576

(A9)

So for any positive 1205762 if 119899119898is small enough it yields

z leradic2 (

radic2 (1 + 119896

1) 119899

119898+ 120576)

(1 minusradic2119899

119898)

= 120576

2 (A10)

On the other hand if |1199111| le |119911

2| then z le radic

2|119911

2| so we

find in both cases that z leradic2|119911

2| + 120576

2and |119911

1| le 2(|119911

2| +

1 minus 119896

1)119899

119898+ 120576 Thus the derivative of 119881 can be bounded as

119881 le 119896

1(2 (

1003816

1003816

1003816

1003816

119911

2

1003816

1003816

1003816

1003816

+ 1 minus 119896

1) 119899

119898+ 120576) (

1003816

1003816

1003816

1003816

119865max1003816

1003816

1003816

1003816

+

1003816

1003816

1003816

1003816

119865

119903

1003816

1003816

1003816

1003816

+ 119901

119898)

minus 119896

1

1003816

1003816

1003816

1003816

119911

2

1003816

1003816

1003816

1003816

(

1003816

1003816

1003816

1003816

119865

119903

1003816

1003816

1003816

1003816

sin (1198962tanh (100381610038161003816

1003816

119911

2

1003816

1003816

1003816

1003816

)) minus

1003816

1003816

1003816

1003816

119865

119903119908

1003816

1003816

1003816

1003816

minus 119901

119898)

+ 2 (

radic

2

1003816

1003816

1003816

1003816

119911

2

1003816

1003816

1003816

1003816

+ 120576

2) 119889

119906119896

1120575120576

1

(A11)

Grouping all terms that can be made arbitrary small byreducing noise and disturbances in 120576

3and 120576

4 (A11) can be

rewritten as

119881 le minus 119896

1(

1003816

1003816

1003816

1003816

119911

2

1003816

1003816

1003816

1003816

(

1003816

1003816

1003816

1003816

119865

119903

1003816

1003816

1003816

1003816

sin (1198962tanh (100381610038161003816

1003816

119911

2

1003816

1003816

1003816

1003816

))

minus

1003816

1003816

1003816

1003816

119865

119903119908

1003816

1003816

1003816

1003816

minus 119901

119898minus 120576

3) + 120576

4)

(A12)

Now we define

119911max 2 = tanhminus1 ( 1

119896

2

sinminus1 (119901

119898

1003816

1003816

1003816

1003816

119865

119903

1003816

1003816

1003816

1003816

+ 120576

3+ 119908

119898+ 120575)) (A13)

Combining (A13) with

119881 and grouping all the small termsin 1205765 then

119881 le 119896

1(minus|119911max 2|120575+1205765) that is negative if 1205765 is small

enough compared with 120575

Conclusion Defining 119911max = max(119911max 1 119911max 2) 119881 decreasesoutside the bounded region z le 119911max for small enough 119899

119898

Since 119881 is definite positive and radially unbounded then zis globally finitely bounded by 119911max 1 Furthermore as 119911max 1and 119911max 2 are class K functions of 119908

119898 119899119898 119901119898 and 120576 then

(35) holds

A3 Proof of Corollary 3 The bound of z implies that sis also bounded because (15) shows that s = z Then bydefinition of 119904

119909and 119904

119910

119896

1119890

119909= 119904

119909minus 119890

119909

119896

1119890

119910= 119904

119910minus 119890

119910

(A14)

This system is linear and stable thus it is ISS with respect to sthat is globally ultimately bounded since s = zThereforee119901is ISS with respect to 119899

119898 119901119898 119908119898 and 120576 then (36) holds


A4 Proof ofTheorem4 Without loose of generality considerthe initial state of the system (1)ndash(6) lying inside the setΩ From (23) |119890

119903| le |119903| + |119903

119903| le |120591

119898|119889

119903+ |119903

119903| The

velocities are bounded in Ω so (24)-(25) imply that thefirst derivatives of z can also be bounded and differentiating(25) it is straightforward to check that there exist positiveconstants 119888

1ndash1198886such that

z le 1003816

1003816

1003816

1003816

1

1003816

1003816

1003816

1003816

+

1003816

1003816

1003816

1003816

2

1003816

1003816

1003816

1003816

le 119888

1+ 119888

2119911

119898+ 119888

3119901

119898

1003816

1003816

1003816

1003816

2

1003816

1003816

1003816

1003816

le 119888

4+ 119888

5119911

119898+ 119888

6119901

119898

(A15)

By the definition of 120591119888 it yields

120591

119888= 120591

119903minus 119896

2sign (119865

119903) (1 minus tanh2 (119911

2)) (119889

119903

2+

2)

+ 2119896

2sign (119865

119903) (1 minus tanh2 (119911

2)) tanh (119911

2)

2

2

(A16)

Substituting (A15) into (A16) and (32) the effect of distur-bances and noise on the computation of 119904

120595and 120591

119888can be

bounded with a positive constant 1198887 and two polynomial

bounded functions 1205744and 120574

5 of classK as follows

1003816

1003816

1003816

1003816

120591

119888

1003816

1003816

1003816

1003816

le

1003816

1003816

1003816

1003816

120591

119903

1003816

1003816

1003816

1003816

+ 119896

2120574

4(119911

119898+ 119901

119898) (A17)

1003816

1003816

1003816

1003816

119891

3

1003816

1003816

1003816

1003816

le 120574

5(119888

7+ 119911

119898+ 119901

119898) 119899

119898+ 119896

3119896

2119896

1119901

119898 (A18)

Then |120591119903| lt |120591| because the reference trajectory is D-feasible

and thus there exists a positive constant 120575 such that |120591|minus|120591119903| le

120575 Now we define 119911119898as

119911

119898= minus

1

2

120574

4

minus1

(

120575

4119896

2

) (A19)

When z lt 119911

119898 the disturbances are small enough such that

119896

2120574

4(2119901

119898)+119901

119898lt 1205754 and with (A17) then |120591

119894|minus|120591

119888|minus|119901

119898| ge

120575 minus 1205754 minus 1205754 = 1205752 Now we select 1198963such that

1003816

1003816

1003816

1003816

1 minus 119896

3119889

119903

1003816

1003816

1003816

1003816

119896

3119889

119903

(

1003816

1003816

1003816

1003816

120591

119898

1003816

1003816

1003816

1003816

+ 119889

119903

1003816

1003816

1003816

1003816

119903

119903

1003816

1003816

1003816

1003816

) le

120575

4

(A20)

And we define the Lyapunov function119881

2= 119904

2

1205952 whose time

derivative is

119881

2= 119904

120595(119896

3(120591 minus 120591

119888+ 119901

119903) + (1 minus 119896

3119889

119903)119890

119903) Consider

first that |119904120595| ge 119891

3+120576 then the control law implies that the sign

of 120591 is the opposite of the sign of 119904120595 Condider the following

119881

2= 119904

120595119904

120595le minus120575119896

3(

1

2

minus

1

4

)

1003816

1003816

1003816

1003816

1003816

119904

120595

1003816

1003816

1003816

1003816

1003816

997888rarr 119904

120595le minus

120575119896

3

4

sign (119904120595)

(A21)

Solving (A21) for the equality and applying the comparisonlemma (see chapter 3 of [29]) we find that |119904

120595(119905)| le |119904

120595(0)| minus

120575119896

34119905 If 119904

120595lies inside the region |119904

120595| lt 119891

3+120576 (A21) does not

hold Nevertheless1198812decreases on the frontier of this region

preventing 119904120595from becoming larger than 119891

3+ 120576 Then in any

case |119904120595(119905)| le max(|119904

120595(0)| minus 120575119896

34119905 119891

3+ 120576) and substituting

the bound (A18) then (37) is obtained

A5 Proof of Corollary 5 By the definition of 119904120595in (28) the

orientation error dynamics become 1198963119890

120595= 119904

120595minus 119890

120595 whose

solution is

119890

120595(119905) = 119890

120595(0) 119890

minus1199051198963+

1

119896

3

int

119905

0

119890

minus(119905minus120591)1198963119904

120595(120591) 119889120591 (A22)

If the terms inside the integral are bounded by (37) then

1003816

1003816

1003816

1003816

1003816

119890

120595(119905)

1003816

1003816

1003816

1003816

1003816

le

1003816

1003816

1003816

1003816

1003816

119890

120595(0)

1003816

1003816

1003816

1003816

1003816

119890

minus1199051198963+

1003816

1003816

1003816

1003816

1003816

119904

120595(0)

1003816

1003816

1003816

1003816

1003816

(119890

|119904120595(0)|1198861198963minus 1) 119890

minus1199051198963

+ 120574

3(119887 + 119911

119898+ 119901

119898) 119899

119898+ 119888119901

119898+ 120576

(A23)

That takes the form of (38) with 119889 = supΩ|119890

120595| + |119904

120595|119890

|119904120595|1198861198963

A6 Proof of Theorem 6 The key point of the proof is that ifthe vehicle starts from a bounded initial condition z(0) lt

119872 where 119872 is an arbitrary constant it is always possible tochoose the control gain 119896

2such that during the time that 119908

needs to converge to119908 lt 119908

119898 z never reaches 119911

119898Therefore

the conditions of Theorem 4 always hold and the trackingerror is finitely bounded

Select 1198962so that (A19) of Theorem 4 is satisfied for 119911

119898=

2119872 as follows

119896

2=

120575

4120574

4(4119872)

(A24)

As z(0) lt 119872 it will take sometime for 119879 to go fromz = 119872 to z = 2119872 This time 119879 can be lower boundedby considering the maximum derivative of 119881

119881 le 119896

1

1003816

1003816

1003816

1003816

119911

2

1003816

1003816

1003816

1003816

(

1003816

1003816

1003816

1003816

119865

119903119908

1003816

1003816

1003816

1003816

+ 119901

2) + (1 minus 119889

119906119896

1) (119904

1119890V119909 + 119904

2119890V119910)

le 119887

1|z| + 119887

2

(A25)

for some constants 1198871and 119887

2 Select 119872 gt 119887

2119887

1then z gt

119887

2119887

1and thus

119881 = zz le 2119887

1z rarr z le 2119887

1rarr z le

z(0) + 2119887

1119905 Hence z can only grow with a finite rate of

2119887

1and therefore 119879 gt 1198722119887

2

Consider noise and disturbances small enough such that120574

3(119887 + 119872 + 119901

119898)119899

119898+ 119888119901

119898+ 120576 le 119889119890

minus1198791198963 From (38) it yieldsthat |119890

120595(119879)| le 2119889 sdot 119890

minus1198791198963 In addition for small enoughdisturbances compared with the control action consider thefollowing

max(119901

119898+ 120576

5

1003816

1003816

1003816

1003816

119865

119903

1003816

1003816

1003816

1003816

119901

119898+ 120575120576

1003816

1003816

1003816

1003816

119865

119903

1003816

1003816

1003816

1003816

) lt 119889 sdot 119890

minus1198791198963 (A26)

Substituting (A26) into (A8) and (A13) it yields

119911max (119879) = radic

2 tanhminus1 (4120574

4(4119872)

120575

sinminus1 (3radic2119889 sdot 119890

minus119872211988821198963))

(A27)

Function 120574

4is a polynomial order function in z bounded

by119872 Therefore when119872 rarr infin then 119911max rarr 0 and for alarge enough119872 then 119911max(119879) le 2119872 = 119911

119898


The result is now proved by contradiction Suppose thatthere exists some time 119905

1for which z(119905

1) gt 2119872 then there

must exist another time 119905

2for which z(119905

2) = 2119872 and 119881

grows in 119905

2 But 119881 only grows if z lt 119911max so according

to (26) 119911max(|119908(1199052)|) le 119911max(|119890120595(1199052)|) = 119872 However|119890

120595(119905

2)| le 2119889119890

1198791198963 and therefore z(1199052) lt 119911max(|119908(1199052)|) le

119911max(|119890120595(1199052)|) = 119872 because z lt 119911

119898for all 119879 le 119905

1le 119905

2 This

drives to a contradiction because z(1199052) = 2119872

Therefore z never reaches 119911119898and conditions of The-

orem 4 always hold This means that before some time 119905

2

conditions ofTheorem 2 holds thus we can conclude by (39)that e

119901 is finitely bounded

Acknowledgments

The authors would like to acknowledge the Spanish Min-istry of Economy and Competitiveness for its financialsupport under Grant DPI 2009-14552-C02-02 and theUNED University for the support under Research Project2012VPUNED0003

References

[1] P Encarnacao and A Pascoal ldquoCombined trajectory trackingand path following an application to the coordinated controlof autonomous marine craftrdquo in Proceedings of the 40th IEEEConference on Decision and Control (CDC rsquo01) vol 1 pp 964ndash969 December 2001

[2] K D Do and J Pan ldquoRobust and adaptive path following forunderactuated autonomous underwater vehiclesrdquo in Proceed-ings of the American Control Conference (ACC rsquo03) vol 3 pp1994ndash1999 June 2003

[3] K Y Pettersen and T I Fossen ldquoUnderactuated dynamicpositioning of a shipmdashexperimental resultsrdquo IEEE Transactionson Control Systems Technology vol 8 no 5 pp 856ndash863 2000

[4] W Oelen H Berghuis H Nijmeijer and C C de Wit ldquoHybridstabilizing control on a real mobile robotrdquo IEEE Robotics andAutomation Magazine vol 2 no 2 pp 16ndash23 1995

[5] R M Murray J Hickey E Klavins and S Low ldquoThe CaltechMulti-Vehicle Wireless Testbed (MVWT)rdquo Tech Rep Cali-fornia Institute of Technology Pasadena Calif USA 2003httpwwwcdscaltechedusimmurrayprojectsdurip01-mvwt

[6] V Vladimerou A Stubbs J Rubel A Fulford J Strickand G Dullerud ldquoA hovercraft testbed for decentralized andcooperative controlrdquo in Proceedings of the American ControlConference (AAC rsquo04) vol 6 pp 5332ndash5337 July 2004

[7] T Chatchanayuenyong and M Parnichkun ldquoNeural networkbased-time optimal sliding mode control for an autonomousunderwater robotrdquo Mechatronics vol 16 no 8 pp 471ndash4782006

[8] RWBrockett ldquoAsymptotic stability and feedback stabilizationrdquoin Differential Geometric Control Theory vol 27 pp 181ndash191Birkhaauser Boston Mass USA 1983

[9] I Kolmanovsky and N H McClamroch ldquoDevelopments innonholonomic control problemsrdquo IEEE Control Systems Mag-azine vol 15 no 6 pp 20ndash36 1995

[10] A P Aguiar L Cremean and J P Hespanha ldquoPosition trackingfor a nonlinear underactuated hovercraft controller designand experimental resultsrdquo in Proceedings of the 42nd IEEE

Conference on Decision and Control (CDC rsquo03) vol 4 pp 3858ndash3863 December 2003

[11] H Sira-Ramırez ldquoDynamic second-order sliding mode controlof the hovercraft vesselrdquo IEEE Transactions on Control SystemsTechnology vol 10 no 6 pp 860ndash865 2002

[12] A P Aguiar and J P Hespanha ldquoTrajectory-tracking and path-following of underactuated autonomous vehicles with para-metric modeling uncertaintyrdquo IEEE Transactions on AutomaticControl vol 52 no 8 pp 1362ndash1379 2007

[13] K Y Pettersen and H Nijmeijer ldquoUnderactuated ship trackingcontrol theory and experimentsrdquo International Journal of Con-trol vol 74 no 14 pp 1435ndash1446 2001

[14] G Toussaint T Basar and F Bullo ldquoTracking for nonlin-ear underactuated surface vessels with generalized forcesrdquo inProceedings of the IEEE International Conference on ControlApplications (CCA rsquo00) pp 355ndash3360 2000

[15] W Dong and Y Guo ldquoNonlinear tracking control of underac-tuated surface vesselrdquo in Proceedings of the American ControlConference (ACC rsquo05) vol 6 pp 4351ndash4356 June 2005

[16] A Balluchi A Bicchi B Piccoli and P Soueres ldquoStability androbustness of optimal synthesis for route tracking by Dubinsrsquovehiclesrdquo in Proceedings of the 39th IEEE Confernce on Decisionand Control (CDC rsquo00) vol 1 pp 581ndash586 December 2000

[17] E Lefeber K Y Pettersen and H Nijmeijer ldquoTracking controlof an underactuated shiprdquo IEEETransactions on Control SystemsTechnology vol 11 no 1 pp 52ndash61 2003

[18] K D Do Z P Jiang and J Pan ldquoUniversal controllers forstabilization and tracking of underactuated shipsrdquo Systems ampControl Letters vol 47 no 4 pp 299ndash317 2002

[19] T I Fossen andM Blanke ldquoNonlinear output feedback controlof underwater vehicle propellers using feedback form estimatedaxial flow velocityrdquo IEEE Journal of Oceanic Engineering vol 25no 2 pp 241ndash255 2000

[20] J Liu and N Elia ldquoQuantized control with applications tomobile vehiclesrdquo in Proceedings of the 41st IEEE Conferenceon Decision and Control (CDC rsquo02) vol 3 pp 2391ndash2396December 2002

[21] H Ishii and T Basar ldquoRemote control of LTI systems overnetworks with state quantizationrdquo Systems amp Control Lettersvol 54 no 1 pp 15ndash31 2005

[22] H Seguchi and T Ohtsuka ldquoNonlinear receding horizoncontrol of an RC hovercraftrdquo in Proceedings of the IEEEInternational Conference on Control Applications (ACC rsquo02) vol2 pp 1076ndash1081 September 2002

[23] H Seguchi and T Ohtsuka ldquoNonlinear receding horizoncontrol of an underactuated hovercraftrdquo International Journalof Robust and Nonlinear Control vol 13 no 3-4 pp 381ndash3982003

[24] H Ashrafiuon K R Muske L C McNinch and R ASoltan ldquoSliding-mode tracking control of surface vesselsrdquo IEEETransactions on Industrial Electronics vol 55 no 11 pp 4004ndash4012 2008

[25] D Chwa ldquoSliding-mode tracking control of nonholonomicwheeled mobile robots in polar coordinatesrdquo IEEE Transactionson Control Systems Technology vol 12 no 4 pp 637ndash644 2004

[26] A Megretski ldquoLocal controllabilityrdquo Dynamics of NonlinearSystems 2003 httpocwmiteducourseselectrical-engineer-ing-and-computer-science6-243j-dynamics-of-nonlinear-sys-tems-fall-2003lecture-noteslec12 6243 2003pdf

[27] L Cremean W B Dunbar D van Gogh et al ldquoThe Caltechmulti-vehicle wireless testbedrdquo in Proceedings of the 41st IEEE



[28] M Fliess J Levine P Martin and P Rouchon ldquoFlatnessand defect of non-linear systems introductory theory andexamplesrdquo International Journal of Control vol 61 no 6 pp1327ndash1361 1995

[29] H K Khalil Nonlinear Systems Prentice Hall New York NYUSA 3rd edition 2002

[30] D Chaos Control no lineal de vehculos subactuados marinosno-holonmicos [PhD thesis] Department of Informatica yAutomatica of UNED University Madrid Spain 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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OptimizationJournal of


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Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


kind of underactuated vehicles (that includes the hovercraft)cannot be stabilized to a certain pose with a continuous andtime invariant control law The reader is referred to [9] foran interesting survey in control problems for nonholonomicvehicles

Representative examples of tracking controllers applied tounderactuated hovercrafts are stated in [10] where the track-ing control problem is solved using backstepping techniquesachieving practical stability results in [11] that solve the sameproblem exploiting the differential flatness properties of avessel with respect to its position and in [12] where theauthors extend previous results for the tracking and pathfollowing problems to more general marine vehicles takinginto account the parametric uncertainty on the models

Other remarkable examples are found in [13] where thetracking problem of general underactuated ships is studiedin [14] that explains how to select outputs when generalizedforces act on the vehicle in [15] where Barbalatrsquos lemma andbackstepping techniques are combined to achieve asymptotictracking in [16] that solves the tracking problemof awheeledrobot in [17] where the authors address the tracking problemfor a surface craft and in [18] in which the challengingproblem of solving both point stabilization and trackingproblem at the same time is studied

It must be remarked that the control strategies describedabove achieve a good control performance assuming thatforce and torque imparted by propellers can be perfectlycontrolled Nevertheless thrust allocation is in general avery complex task in surface crafts (see eg [19]) andmoreover expensive equipments are usually required Thusthe availability of low cost vehicles equipped with simpleactuators and with a limited knowledge of their dynamicsis of the utmost importance for the problem studied inthe present work and a very interesting alternative from apractical point of view

In the last few years the interest on the control of under-actuated vehicles where only a discrete set of control inputs isavailable has increased significantly See for example [20 21]for a description on the challenges that arise in quantizedcontrol of mobile vehicles and remote control of systemsover networks with state quantization respectively Anotherremarkable example is found in the works [22 23] where anonholonomic underactuated hovercraft is controlled by adiscrete set of inputs In these works the problem of pointstabilization is solved using receding horizon predictive con-trollers and approaching the continuous output by the closestallowed control action This strategy works adequately whenthe optimal control law is bang-bang but it is not suitablewhen the control signals take intermediate values betweentwo allowed ones something impossible to achieve withall-nothing-reverse signals Unfortunately this is the actualtracking problem scenario because forces and torques shouldbe close to a continuous reference far away from saturation

In order to solve the control problem of underactuatedvehicles some authors consider second order sliding-modecontrol laws See for example [24] where this technique isapplied to the tracking problem of a surface vessel and [11]where the trajectory tracking problem for an underactuatedhovercraft is solved These control laws tend to produce

control actions that are similar to bang-bang actions There-fore they are good candidates to be approximated with all-nothing-reverse thrusters The later are the kind of thrustersused in the vehicle (hovercraft) dealt in this work due to theirsimplicity of control and implementation and also their lowcost

This kind of problem has been partially studied in theground robotic field Some interesting results on the field ofwheeled robots show that in the case of Dubinrsquos vehicles it ispossible to design a tracking controller that makes the vehicleconverge to a reference trajectory using only a discrete setof control actions (turn right and left accelerate and brake)See for example [25] that uses a sliding-mode control and[16] where a controller has been developed using optimalsynthesis

An interesting practical problem is the extension of theabove results for an underactuated low cost vehicle as ahovercraft The main difference between a wheeledgroundunderactuated vehicle and a marine underactuated vehicle isthe nonholonomic restriction on the trajectories that can befollowed by each vehicle The nonholonomic restriction in avehicle is the relationship between the acceleration and theorientation of the vehicle The sliding condition for marinevehicles (second-order nonholonomic restriction) is morecomplex than the nonsliding condition (first-order nonholo-nomic restriction) for wheeled robots Thus the aim of thiswork is to control a vehicle with very simple propellers thatproduce only a discrete set of control commands and witha minimal information about the dynamics of the actuatorsThe availability simplicity and low cost of these kind ofsystems even the hybrid nature of the hovercraft makethem very attractive to be used in a number of operationalscenarios Furthermore a group of these vehicles can be usedfor cooperative andor coordinated tasks with relatively lowprice

The sole conditions that are imposed to the actuators usedon this work can be summarized as follows

(1) The available control actions can turn right and leftwhile the vehicle go forward and backwards

(2) The available force and torque are bigger than theforce and torque necessary to exactly track the ref-erence (forces and torques can be dominated by thecontrol actions)

Notice that the first condition is important to ensure localcontrollability (see [26]) while second condition is necessaryto ensure feasibility of the trajectory

Thekey contribution of the paper is two-fold (i) we reflectthe fact that the inputs are discrete in the controller designprocess and (ii) we take explicitly into account the existenceof unknown bounded input disturbances and measurementnoise Theoretical stability proofs are obtained and in addi-tion the control law is successfully tested on a real low costvehicle In the literature to the best knowledge of the authorsthere are no attempts to solve the trajectory tracking problemfor a second order nonholonomic vehicle as the hovercraftwith a discrete set of inputs Furthermore in the few caseswhere a discrete set of inputs is considered in a vehicle of this


120595

r

Fp

Fs

o

X

Y

l(x y)

U

u

V = [x t]T




2 Vehicle Model


= V119909 (1)

119910 = V119910 (2)

V119909= 119865 cos (120595) minus 119889

119906V119909+ 119901V119909 (3)

V119910= 119865 sin (120595) minus 119889

119906V119910+ 119901V119910 (4)

120595 = 119903 (5)

119903 = 120591 minus 119889

119903119903 + 119901

119903 (6)



119909and V

119910


119906and 119889


where 119889

119906= 119863

119906119898 119889

119903= 119863

119903119869 119898 is the mass of the


119906and 119863

119903are


119901+119865

119904)119898 and



119897 0075m119906max 0545N119906min 0347N119889

11990603588 sminus1

119889

11990317357 sminus1


120591 gt 0 120591 lt 0

119865 gt 0 119865

119901= 119906max 119865119904 = minus119906min 119865

119901= minus119906min 119865119904 = 119906max

119865 lt 0 119865

119901= 0 119865

119904= 119906min 119865

119901= 119906min 119865119904 = 0

torque 120591 = 119897(119865

119901minus 119865

119904)119869 where 119865

119904and 119865




[119899

119909 119899

119910 119899V119909 119899V119910 119899120595 119899119903]



sup119905ge0

p (119905) le 119901

119898 sup

119905ge0

1003817

1003817

1003817

1003817

1003817

1003817

1003817

1003817

119889p119889119905

1003817

1003817

1003817

1003817

1003817

1003817

1003817

1003817

le 119901

119898

sup119905ge0

n (119905) le 119899

119898

(7)





119904and 119865



119904and 119865




119903(119905) and 119910



[umax umax]

F =Fp minus Fs

m

[umax 0]

[umax minusumin]


J

[0 minusumin]


[minusumin 0]

[minusumin umax]

[0 umax]

[0 0]



119903as the


119903


119865

119903cos (120595

119903) = V119909119903+ 119889

119906V119909119903

119865

119903sin (120595

119903) = V119910119903+ 119889

119906V119910119903

(8)



120595

119903= atan2 ( 119910

119903+ 119889

119906119910

119903

119903+ 119889

119906

119903) + 119899120587

119865

119903= (minus1)

119899radic

(

119903+ 119889

119906

119903)

2

+ ( 119910

119903+ 119889

119906119910

119903)

2

(9)



119903and 119903



119903= 119903

119903+119889

119903119903



119903(119905) 119910

119903(119905)]



follows


119903and 120591

119903are such that

119865



1 120591

1)

(119865

2 120591

2) (1198653 120591

3) and (119865

4 120591

4) such that

1003816

1003816

1003816

1003816

119865

119903(119905)

1003816

1003816

1003816

1003816

lt min (1198651 119865

2 minus119865

3 minus119865

4)

1003816

1003816

1003816

1003816

120591

119903(119905)

1003816

1003816

1003816

1003816

lt min (minus1205911 120591

2 minus120591

3 120591

4)

(10)


119904and 119865



119904and 119865






119903(119905) and 120591




F =Fp + Fs

m

[umax umax]

[umax 0]

[umax minusumin]


J

[0 minusumin]


[minusumin 0]

[minusumin umax]

[0 umax]

1

2

3

4

5

6

7

8

9

(Fr 120591r)

F0

minus1205910 1205910

minusF0

[0 0]

A


119903(119905)) lies on the




119904and 119865

119901to solve the






119901=

[119909 minus 119909

119903 119910 minus 119910

119903 V119909minus V119909119903 V119910minus V119910119903]





if the noise 119899


119898are small





119888to be tracked


119904and 119865


Table 2


119890

119909= 119909 minus 119909

119903

119890

119910= 119910 minus 119910

119903

119890V119909 = V119909minus V119909119903

119890V119910 = V119910minus V119910119903

(11)


119890

119909= 119890V119909

119890

119910= 119890V119910

119890V119909 = 119865 cos (120595) minus 119889

119906119890V119909 minus 119865

119909119903+ 119901V119909

119890V119910 = 119865 sin (120595) minus 119889

119906119890V119910 minus 119865

119910119903+ 119901V119910

(12)

where 119865119909119903

= 119865

119903cos(120595

119903) and 119865

119910119903= 119865




References

x

Positioncontrol

120595 r

120595c rc


Selector


25

2

15

1

05

0

minus05

minus1

minus15

F=(F

b+Fe)m

Fe

Fb

Hovercraft

x

Sign(F)

[umax umax]

[umax 0]

[umax minusumin]

[0 minusumin]


[minusumin 0]

[minusumin umax]

[0 umax]

[0 0]





119904

119909= 119890

119909+ 119896

1119890V119909

119904

119910= 119890

119910+ 119896

1119890V119910

(13)


1if 119904119909

and 119904



119909and 119904



119904

119909= 119896

1119865 cos (120595) + (1 minus 119889

119906119896

1) 119890V119909 minus 119896

1119865

119909119903+ 119896

1119901V119909

119904

119910= 119896

1119865 sin (120595) + (1 minus 119889

119906119896

1) 119890V119910 minus 119896

1119865

119910119903+ 119896

1119901V119910

(14)


119911

1= 119904

119909cos (120595) + 119904

119910sin (120595)

119911

2= minus119904

119909sin (120595) + 119904

119910cos (120595)

(15)


1= 119896

1119865 minus 119896

1119865

119903cos (120595 minus 120595

119903) + 119911

2119903

+ (1 minus 119889

119906119896

1) (119890V119909 cos (120595) + 119890V119910 sin (120595)) + 119896

1119901

1

(16)

2= 119896

1119865

119903sin (120595 minus 120595

119903) minus 119911

1119903

+ (1 minus 119889

119906119896

1) (minus119890V119909 sin (120595) + 119890V119910 cos (120595)) + 119896

1119901

2

(17)

where 1199011and 119901


119901

1= 119901V119909 cos (120595) + 119901V119910 sin (120595)

119901

2= minus119901V119909 sin (120595) + 119901V119910 cos (120595)

p =1003817

1003817

1003817

1003817

1003817

1003817

[119901

1 119901

2]

1198791003817

1003817

1003817

1003817

1003817

1003817

le 119901

119898

(18)


2 given by (16) and




sign (119865) =

minus sign (1199111) if 1003816

1003816

1003816

1003816

119911

1

1003816

1003816

1003816

1003816

ge 120576

sign (119865minus) if 1003816

1003816

1003816

1003816

119911

1

1003816

1003816

1003816

1003816

lt 120576

(19)


1|




1| lt 120576


119903sin(120595 minus 120595

119903)must be



120595

119888= 120595

119903minus 119896

2tanh (119911

2) sign (119865

119903) (20)

119903

119888=

120595

119888=

120595

119903minus

2119896


2)

2

) sign (119865119903) (21)



119888and 0 lt 119896

2lt 1205872

then 119865

119903sin(120595119888minus 120595

119903) = minus|119865

119903| sin(119896

2tanh(119911

2)) that is opposed




119890

120595= 120595 minus 120595

119888 (22)

119890

119903= 119903 minus 119903

119888 (23)


1= 119896

1119865 minus 119896

1119865

119903cos (119890

120595+ 120595

119888minus 120595

119903) + 119911

2119903 + 119896

1119901

1

+ (1 minus 119889

119906119896

1) (119890V119909 cos (119890120595 + 120595

119888) + 119890V119910 sin (119890120595 + 120595

119888))

(24)

2= minus 119896

1

1003816

1003816

1003816

1003816

119865

119903

1003816

1003816

1003816

1003816

sin (1198962tanh (119911

2)) minus 119911

1119903 + 119896

1119865

119903119908 + 119896

1119901

2

+ (1 minus 119889

119906119896

1) (119890V119910 cos (119890120595 + 120595

119888) minus 119890V119909 sin (119890120595 + 120595

119888))

(25)



119903) minus sin(120595

119888minus 120595

119903) bounded by

|119908| le

1003816

1003816

1003816

1003816

120595 minus 120595

119888

1003816

1003816

1003816

1003816

=

1003816

1003816

1003816

1003816

1003816

119890

120595

1003816

1003816

1003816

1003816

1003816

(26)



2



119890

120595=

120595 minus

120595

119888= 119903 minus

120595

119888= 119890

119903

119890

119903= 120591 minus 120591

119888minus 119889

119903119890

119903+ 119901

119903

(27)

where 120591119888= 119889

119903

120595

119888minus

120595



gain 119896


120595as

follows

119904

120595= 119890

120595+ 119896

3119890

119903 (28)




119904

120595= 119896

3(120591 minus 120591

119888+ 119901

119903) + (1 minus 119896

3119889

119903) 119890

119903 (29)




sign (120591) =

minus sign (119904120595) if 1003816

1003816

1003816

1003816

1003816

119904

120595

1003816

1003816

1003816

1003816

1003816

ge 120576

sign (120591minus) if 1003816

1003816

1003816

1003816

1003816

119904

120595

1003816

1003816

1003816

1003816

1003816

lt 120576

(30)


120595|






119909and 119904

119910are


1= 119911

1+ 119899

1and

2= 119911

2+ 119899

2 where

1003816

1003816

1003816

1003816

119899

1

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119899

2

1003816

1003816

1003816

1003816

le (z + radic

2 (1 + 119896

1)) 119899

119898 (31)



119909


the estimate 120595





derivative

120595



120595+ 119891

3 where

1003816

1003816

1003816

1003816

119891

3

1003816

1003816

1003816

1003816

le 119896

3119896

2119896

1119901

119898+ 119899

119898(1 + 119896

3+ 119899

119898+ 119896

1

1003816

1003816

1003816

1003816

119865

119903

1003816

1003816

1003816

1003816

+ |119903|)

+ 119896

2119899

119898(1 + 119896

3

1003816

1003816

1003816

1003816

2

1003816

1003816

1003816

1003816

) (z + radic

2 (1 + 119896

1))

+ 2

radic

2119899

119898

1003816

1003816

1003816

1003816

1 minus 119889

119906119896

1

1003816

1003816

1003816

1003816

radic119890

2

V119909 + 119890

2

V119910

(32)


sign (119865) =


1003816

1003816

1003816

1

1003816

1003816

1003816

1003816

ge 120576

sign (119865minus) if 1003816

1003816

1003816

1003816

1

1003816

1003816

1003816

1003816

lt 120576

(33)

sign (120591) =

minus sign (119904120595) if 1003816

1003816

1003816

1003816

1003816

119904

120595

1003816

1003816

1003816

1003816

1003816

ge 120576

sign (120591minus) if 1003816

1003816

1003816

1003816

1003816

119904

120595

1003816

1003816

1003816

1003816

1003816

lt 120576

(34)


119890and 119865


Table 2










1 1198962

and 119908



119898 then

z (119905) le 120573 (z (0) 119905) + 120574 (119908

119898+ 119899

119898+ 119901

119898+ 120576) (35)

Proof on Section A2



that for |119908| le 119908

119898

1003817

1003817

1003817

1003817

1003817

e119901(119905)

1003817

1003817

1003817

1003817

1003817

le 120573

2(

1003817

1003817

1003817

1003817

1003817

e119901(0)

1003817

1003817

1003817

1003817

1003817

119905) + 120574

2(119908

119898+ 119899

119898+ 119901

119898+ 120576) (36)

Proof in Section A3







119898 1198963 119886 119887 and 119888 and a K class

function 120574


119903and any


2 if z le 119911

119898 then

1003816

1003816

1003816

1003816

1003816

119904

120595(119905)

1003816

1003816

1003816

1003816

1003816

le max (1003816100381610038161003816

1003816

119904

120595(0)

1003816

1003816

1003816

1003816

1003816

minus 119886119905 120574

3(119887 + 119911

119898+ 119901

119898) 119899

119898+ 119888119901

119898120576)

(37)

Proof in Section A4


119898

1003816

1003816

1003816

1003816

1003816

119890

120595(119905)

1003816

1003816

1003816

1003816

1003816

le 119889 sdot 119890

minus1199051198963+ 120574

3(119887 + 119911

119898+ 119901

119898) 119899

119898+ 119888119901

119898+ 120576 (38)

Proof in Section A5














that1003817

1003817

1003817

1003817

1003817

e119901(119905)

1003817

1003817

1003817

1003817

1003817

le 120573

3(

1003817

1003817

1003817

1003817

1003817

e119901(0)

1003817

1003817

1003817

1003817

1003817

119905) + 120574

4(119899

119898+ 119901

119898+ 120576) (39)



Proof in Section A6






7 Results



1= 6 s 119896

2= 1 s 119896

3= 1 s and 120576 = 001


119903= 02179ms2 and 120591

119903= 04339 rads2) so the





120595 is





references 120595119903and 119903

119903 too



minus2

minus15

minus1

minus05

0

05

1

15

2

x(m

)

y (m)


0 10 20 30 40 50 60minus3

minus2

minus1

0123

x(m

)

t (s)

(a)

0 10 20 30 40 50 60minus3

minus2

minus1

0123

y(m

)

t (s)

(b)



119909and V119910 rad



0 10 20 30 40 50 60

minus04minus02

0020406

x(m

s)

t (s)

(a)

0 10 20 30 40 50 60

minus04minus02

0020406

t (s)

y(m

s)

(b)


0 10 20 30 40 50 60minus4

minus2

0

2

4

6120595

(rad

)

t (s)

(a)


0020406

r(r

ads

)

t (s)

(b)






minus3

minus2

minus1

0

1

2

3

4

x(m

)

y (m)


0 10 20 30 40 50 600

05

1

15

2

25

3

35

4

e p(m

)

t (s)





119903(119905) = 2 sin(38119905) 119910

119903(119905) = 3 cos(28119905)



minus06

minus04

minus02

0

02

04

06

08

1

12

120591 (rads2)

F(m

s2)



minus2

minus15

minus1

minus05

0

05

1

15

2x

(m)

y (m)






value than 120591








minus06

minus04

minus02

0

02

04

06

08

x(m

)

y (m)







0 5 10 15 20 250

01

02

03

04

05

06

07

08

09

1

Dist

ance

(m)

t





119865


radio controller





0020406

x(m

)

t (s)

(a)


002040608

y(m

)

t (s)

(b)


t (s)0 5 10 15 20 25

minus1

minus05

0

05

1

x(m

s)

(a)

0 5 10 15 20 25minus1

minus05

0

05

1

y(m

s)

t (s)

(b)


8 Conclusions



0 5 10 15 20 25minus04

minus02

0020406

Fs

(N)

t (s)

(a)

0 5 10 15 20 25minus04

minus02

0020406

t (s)

F p(N

)

(b)






Appendix

A Proofs



119871

2= V21199102 and 119871

3= 119903

2


119871

1= minusV119909(119889

119906V119909minus119865 cos(120595)+119901V119909)



119871

1can be bounded by

119871

1lt minus119889

119906V2119909(1 minus

119865max + 119901

119898

119889

119906

1003816

1003816

1003816

1003816

V119909

1003816

1003816

1003816

1003816

) (A1)


119871


119889

119906|V119909| gt (119865max + 119901


119909is ultimately



119898)119889


with 119871

2and 119871


Ω = x isin R6

1003816

1003816

1003816

1003816

V119909

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

V119910

1003816

1003816

1003816

1003816

1003816

le

119865max + 119901

119898

119889

119906

|119903| le

120591max + 119901

119898

119889

119903

(A2)



119903| minus 119901




radic

2

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

(1 minus 119889

119906119896

1)

119889

119906119896

1

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

(2119865max + 119901

119898) le 120575120576

1 (A3)


119898so that

radic

2 (119901

119898+ 120575120576

1+

1003816

1003816

1003816

1003816

119865

119903

1003816

1003816

1003816

1003816

119908

119898) lt min (120575 100381610038161003816

1003816

119865

119903

1003816

1003816

1003816

1003816

sin (1198962)) (A4)


(119911

2

1+ 119911

2


119881 = 119896

1119911

1(119865 minus 119865

119903cos (120595 minus 120595

119903) + 119901

1)

minus 119896

1119911

2(

1003816

1003816

1003816

1003816

119865

119903

1003816

1003816

1003816

1003816

sin (1198962tanh (119911

2)) minus 119865

119903119908 minus 119901

2)

+ (1 minus 119889

119906119896

1) (119904

1119890V119909 + 119904

2119890V119910)

(A5)


Case A (|1199111| gt 119899



119903cos(120595 minus 120595


by |119865119903| and |119911


following function

119892 (119909) = min( 120575

radic2

119865

119903

radic2

sin(1198962tanh( 119909

radic2

))) (A6)


1| minus |119865

119903119911

2| sin(119896

2tanh (|119911


119881 le minus119896

1z (119892 (z) minus (120575120576

1+

1003816

1003816

1003816

1003816

119865

119903

1003816

1003816

1003816

1003816

|119908| + 119901

119898)) (A7)


1+ |119865

119903|119908

119898+ 119901

119898


119911max 1 = radic

2 tanhminus1 ( 1

119896

2

sinminus1 (radic2(

119901max + 120575120576

1003816

1003816

1003816

1003816

119865

119903

1003816

1003816

1003816

1003816

+ 119908

119898)))

(A8)



Case B (|1199111| le 119899



2| and

with (31) we find

z le radic

2

1003816

1003816

1003816

1003816

119911

1

1003816

1003816

1003816

1003816

le

radic

2119899

1+ 120576 le

radic

2 (z + radic

2 (1 + 119896

1)) 119899

119898+ 120576

(A9)


z leradic2 (

radic2 (1 + 119896

1) 119899

119898+ 120576)


119898)

= 120576

2 (A10)


2| then z le radic

2|119911

2| so we


2| + 120576

2and |119911

1| le 2(|119911

2| +

1 minus 119896

1)119899


119881 le 119896

1(2 (

1003816

1003816

1003816

1003816

119911

2

1003816

1003816

1003816

1003816

+ 1 minus 119896

1) 119899

119898+ 120576) (

1003816

1003816

1003816

1003816

119865max1003816

1003816

1003816

1003816

+

1003816

1003816

1003816

1003816

119865

119903

1003816

1003816

1003816

1003816

+ 119901

119898)

minus 119896

1

1003816

1003816

1003816

1003816

119911

2

1003816

1003816

1003816

1003816

(

1003816

1003816

1003816

1003816

119865

119903

1003816

1003816

1003816

1003816

sin (1198962tanh (100381610038161003816

1003816

119911

2

1003816

1003816

1003816

1003816

)) minus

1003816

1003816

1003816

1003816

119865

119903119908

1003816

1003816

1003816

1003816

minus 119901

119898)

+ 2 (

radic

2

1003816

1003816

1003816

1003816

119911

2

1003816

1003816

1003816

1003816

+ 120576

2) 119889

119906119896

1120575120576

1

(A11)


3and 120576

4 (A11) can be

rewritten as

119881 le minus 119896

1(

1003816

1003816

1003816

1003816

119911

2

1003816

1003816

1003816

1003816

(

1003816

1003816

1003816

1003816

119865

119903

1003816

1003816

1003816

1003816

sin (1198962tanh (100381610038161003816

1003816

119911

2

1003816

1003816

1003816

1003816

))

minus

1003816

1003816

1003816

1003816

119865

119903119908

1003816

1003816

1003816

1003816

minus 119901

119898minus 120576

3) + 120576

4)

(A12)

Now we define


119896

2

sinminus1 (119901

119898

1003816

1003816

1003816

1003816

119865

119903

1003816

1003816

1003816

1003816

+ 120576

3+ 119908

119898+ 120575)) (A13)



119881 le 119896




119898


119898 119899119898 119901119898 and 120576 then

(35) holds


119909and 119904

119910

119896

1119890

119909= 119904

119909minus 119890

119909

119896

1119890

119910= 119904

119910minus 119890

119910

(A14)


119898 119901119898 119908119898 and 120576 then (36) holds



119903| le |119903| + |119903

119903| le |120591

119898|119889

119903+ |119903

119903| The



z le 1003816

1003816

1003816

1003816

1

1003816

1003816

1003816

1003816

+

1003816

1003816

1003816

1003816

2

1003816

1003816

1003816

1003816

le 119888

1+ 119888

2119911

119898+ 119888

3119901

119898

1003816

1003816

1003816

1003816

2

1003816

1003816

1003816

1003816

le 119888

4+ 119888

5119911

119898+ 119888

6119901

119898

(A15)


120591

119888= 120591

119903minus 119896

2sign (119865

119903) (1 minus tanh2 (119911

2)) (119889

119903

2+

2)

+ 2119896

2sign (119865

119903) (1 minus tanh2 (119911

2)) tanh (119911

2)

2

2

(A16)


120595and 120591

119888can be




1003816

1003816

1003816

1003816

120591

119888

1003816

1003816

1003816

1003816

le

1003816

1003816

1003816

1003816

120591

119903

1003816

1003816

1003816

1003816

+ 119896

2120574

4(119911

119898+ 119901

119898) (A17)

1003816

1003816

1003816

1003816

119891

3

1003816

1003816

1003816

1003816

le 120574

5(119888

7+ 119911

119898+ 119901

119898) 119899

119898+ 119896

3119896

2119896

1119901

119898 (A18)



120575 Now we define 119911119898as

119911

119898= minus

1

2

120574

4

minus1

(

120575

4119896

2

) (A19)

When z lt 119911


119896

2120574

4(2119901

119898)+119901

119898lt 1205754 and with (A17) then |120591

119894|minus|120591

119888|minus|119901

119898| ge


1003816

1003816

1003816

1003816

1 minus 119896

3119889

119903

1003816

1003816

1003816

1003816

119896

3119889

119903

(

1003816

1003816

1003816

1003816

120591

119898

1003816

1003816

1003816

1003816

+ 119889

119903

1003816

1003816

1003816

1003816

119903

119903

1003816

1003816

1003816

1003816

) le

120575

4

(A20)


2= 119904

2

1205952 whose time

derivative is

119881

2= 119904

120595(119896

3(120591 minus 120591

119888+ 119901

119903) + (1 minus 119896

3119889

119903)119890

119903) Consider

first that |119904120595| ge 119891



119881

2= 119904

120595119904

120595le minus120575119896

3(

1

2

minus

1

4

)

1003816

1003816

1003816

1003816

1003816

119904

120595

1003816

1003816

1003816

1003816

1003816

997888rarr 119904

120595le minus

120575119896

3

4

sign (119904120595)

(A21)


120595(119905)| le |119904

120595(0)| minus

120575119896

34119905 If 119904


120595| lt 119891

3+120576 (A21) does not




case |119904120595(119905)| le max(|119904

120595(0)| minus 120575119896

34119905 119891





120595= 119904

120595minus 119890

120595 whose

solution is

119890

120595(119905) = 119890

120595(0) 119890

minus1199051198963+

1

119896

3

int

119905

0

119890

minus(119905minus120591)1198963119904

120595(120591) 119889120591 (A22)


1003816

1003816

1003816

1003816

1003816

119890

120595(119905)

1003816

1003816

1003816

1003816

1003816

le

1003816

1003816

1003816

1003816

1003816

119890

120595(0)

1003816

1003816

1003816

1003816

1003816

119890

minus1199051198963+

1003816

1003816

1003816

1003816

1003816

119904

120595(0)

1003816

1003816

1003816

1003816

1003816

(119890

|119904120595(0)|1198861198963minus 1) 119890

minus1199051198963

+ 120574

3(119887 + 119911

119898+ 119901

119898) 119899

119898+ 119888119901

119898+ 120576

(A23)


120595| + |119904

120595|119890

|119904120595|1198861198963






119898Therefore



119898=

2119872 as follows

119896

2=

120575

4120574

4(4119872)

(A24)


119881 le 119896

1

1003816

1003816

1003816

1003816

119911

2

1003816

1003816

1003816

1003816

(

1003816

1003816

1003816

1003816

119865

119903119908

1003816

1003816

1003816

1003816

+ 119901

2) + (1 minus 119889

119906119896

1) (119904

1119890V119909 + 119904

2119890V119910)

le 119887

1|z| + 119887

2

(A25)


2 Select 119872 gt 119887

2119887

1then z gt

119887

2119887

1and thus

119881 = zz le 2119887


1rarr z le

z(0) + 2119887


2119887

1and therefore 119879 gt 1198722119887

2


3(119887 + 119872 + 119901

119898)119899

119898+ 119888119901

119898+ 120576 le 119889119890


120595(119879)| le 2119889 sdot 119890


max(119901

119898+ 120576

5

1003816

1003816

1003816

1003816

119865

119903

1003816

1003816

1003816

1003816

119901

119898+ 120575120576

1003816

1003816

1003816

1003816

119865

119903

1003816

1003816

1003816

1003816

) lt 119889 sdot 119890

minus1198791198963 (A26)


119911max (119879) = radic


4(4119872)

120575


minus119872211988821198963))

(A27)

Function 120574



119898



1for which z(119905



2for which z(119905

2) = 2119872 and 119881

grows in 119905



120595(119905

2)| le 2119889119890


119911max(|119890120595(1199052)|) = 119872 because z lt 119911

119898for all 119879 le 119905

1le 119905

2 This




2



Acknowledgments


References









































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










120595

r

Fp

Fs

o

X

Y

l(x y)

U

u

V = [x t]T




2 Vehicle Model


= V119909 (1)

119910 = V119910 (2)

V119909= 119865 cos (120595) minus 119889

119906V119909+ 119901V119909 (3)

V119910= 119865 sin (120595) minus 119889

119906V119910+ 119901V119910 (4)

120595 = 119903 (5)

119903 = 120591 minus 119889

119903119903 + 119901

119903 (6)



119909and V

119910


119906and 119889


where 119889

119906= 119863

119906119898 119889

119903= 119863

119903119869 119898 is the mass of the


119906and 119863

119903are


119901+119865

119904)119898 and



119897 0075m119906max 0545N119906min 0347N119889

11990603588 sminus1

119889

11990317357 sminus1


120591 gt 0 120591 lt 0

119865 gt 0 119865

119901= 119906max 119865119904 = minus119906min 119865

119901= minus119906min 119865119904 = 119906max

119865 lt 0 119865

119901= 0 119865

119904= 119906min 119865

119901= 119906min 119865119904 = 0

torque 120591 = 119897(119865

119901minus 119865

119904)119869 where 119865

119904and 119865




[119899

119909 119899

119910 119899V119909 119899V119910 119899120595 119899119903]



sup119905ge0

p (119905) le 119901

119898 sup

119905ge0

1003817

1003817

1003817

1003817

1003817

1003817

1003817

1003817

119889p119889119905

1003817

1003817

1003817

1003817

1003817

1003817

1003817

1003817

le 119901

119898

sup119905ge0

n (119905) le 119899

119898

(7)





119904and 119865



119904and 119865




119903(119905) and 119910



[umax umax]

F =Fp minus Fs

m

[umax 0]

[umax minusumin]


J

[0 minusumin]


[minusumin 0]

[minusumin umax]

[0 umax]

[0 0]



119903as the


119903


119865

119903cos (120595

119903) = V119909119903+ 119889

119906V119909119903

119865

119903sin (120595

119903) = V119910119903+ 119889

119906V119910119903

(8)



120595

119903= atan2 ( 119910

119903+ 119889

119906119910

119903

119903+ 119889

119906

119903) + 119899120587

119865

119903= (minus1)

119899radic

(

119903+ 119889

119906

119903)

2

+ ( 119910

119903+ 119889

119906119910

119903)

2

(9)



119903and 119903



119903= 119903

119903+119889

119903119903



119903(119905) 119910

119903(119905)]



follows


119903and 120591

119903are such that

119865



1 120591

1)

(119865

2 120591

2) (1198653 120591

3) and (119865

4 120591

4) such that

1003816

1003816

1003816

1003816

119865

119903(119905)

1003816

1003816

1003816

1003816

lt min (1198651 119865

2 minus119865

3 minus119865

4)

1003816

1003816

1003816

1003816

120591

119903(119905)

1003816

1003816

1003816

1003816

lt min (minus1205911 120591

2 minus120591

3 120591

4)

(10)


119904and 119865



119904and 119865






119903(119905) and 120591




F =Fp + Fs

m

[umax umax]

[umax 0]

[umax minusumin]


J

[0 minusumin]


[minusumin 0]

[minusumin umax]

[0 umax]

1

2

3

4

5

6

7

8

9

(Fr 120591r)

F0

minus1205910 1205910

minusF0

[0 0]

A


119903(119905)) lies on the




119904and 119865

119901to solve the






119901=

[119909 minus 119909

119903 119910 minus 119910

119903 V119909minus V119909119903 V119910minus V119910119903]





if the noise 119899


119898are small





119888to be tracked


119904and 119865


Table 2


119890

119909= 119909 minus 119909

119903

119890

119910= 119910 minus 119910

119903

119890V119909 = V119909minus V119909119903

119890V119910 = V119910minus V119910119903

(11)


119890

119909= 119890V119909

119890

119910= 119890V119910

119890V119909 = 119865 cos (120595) minus 119889

119906119890V119909 minus 119865

119909119903+ 119901V119909

119890V119910 = 119865 sin (120595) minus 119889

119906119890V119910 minus 119865

119910119903+ 119901V119910

(12)

where 119865119909119903

= 119865

119903cos(120595

119903) and 119865

119910119903= 119865




References

x

Positioncontrol

120595 r

120595c rc


Selector


25

2

15

1

05

0

minus05

minus1

minus15

F=(F

b+Fe)m

Fe

Fb

Hovercraft

x

Sign(F)

[umax umax]

[umax 0]

[umax minusumin]

[0 minusumin]


[minusumin 0]

[minusumin umax]

[0 umax]

[0 0]





119904

119909= 119890

119909+ 119896

1119890V119909

119904

119910= 119890

119910+ 119896

1119890V119910

(13)


1if 119904119909

and 119904



119909and 119904



119904

119909= 119896

1119865 cos (120595) + (1 minus 119889

119906119896

1) 119890V119909 minus 119896

1119865

119909119903+ 119896

1119901V119909

119904

119910= 119896

1119865 sin (120595) + (1 minus 119889

119906119896

1) 119890V119910 minus 119896

1119865

119910119903+ 119896

1119901V119910

(14)


119911

1= 119904

119909cos (120595) + 119904

119910sin (120595)

119911

2= minus119904

119909sin (120595) + 119904

119910cos (120595)

(15)


1= 119896

1119865 minus 119896

1119865

119903cos (120595 minus 120595

119903) + 119911

2119903

+ (1 minus 119889

119906119896

1) (119890V119909 cos (120595) + 119890V119910 sin (120595)) + 119896

1119901

1

(16)

2= 119896

1119865

119903sin (120595 minus 120595

119903) minus 119911

1119903

+ (1 minus 119889

119906119896

1) (minus119890V119909 sin (120595) + 119890V119910 cos (120595)) + 119896

1119901

2

(17)

where 1199011and 119901


119901

1= 119901V119909 cos (120595) + 119901V119910 sin (120595)

119901

2= minus119901V119909 sin (120595) + 119901V119910 cos (120595)

p =1003817

1003817

1003817

1003817

1003817

1003817

[119901

1 119901

2]

1198791003817

1003817

1003817

1003817

1003817

1003817

le 119901

119898

(18)


2 given by (16) and




sign (119865) =

minus sign (1199111) if 1003816

1003816

1003816

1003816

119911

1

1003816

1003816

1003816

1003816

ge 120576

sign (119865minus) if 1003816

1003816

1003816

1003816

119911

1

1003816

1003816

1003816

1003816

lt 120576

(19)


1|




1| lt 120576


119903sin(120595 minus 120595

119903)must be



120595

119888= 120595

119903minus 119896

2tanh (119911

2) sign (119865

119903) (20)

119903

119888=

120595

119888=

120595

119903minus

2119896


2)

2

) sign (119865119903) (21)



119888and 0 lt 119896

2lt 1205872

then 119865

119903sin(120595119888minus 120595

119903) = minus|119865

119903| sin(119896

2tanh(119911

2)) that is opposed




119890

120595= 120595 minus 120595

119888 (22)

119890

119903= 119903 minus 119903

119888 (23)


1= 119896

1119865 minus 119896

1119865

119903cos (119890

120595+ 120595

119888minus 120595

119903) + 119911

2119903 + 119896

1119901

1

+ (1 minus 119889

119906119896

1) (119890V119909 cos (119890120595 + 120595

119888) + 119890V119910 sin (119890120595 + 120595

119888))

(24)

2= minus 119896

1

1003816

1003816

1003816

1003816

119865

119903

1003816

1003816

1003816

1003816

sin (1198962tanh (119911

2)) minus 119911

1119903 + 119896

1119865

119903119908 + 119896

1119901

2

+ (1 minus 119889

119906119896

1) (119890V119910 cos (119890120595 + 120595

119888) minus 119890V119909 sin (119890120595 + 120595

119888))

(25)



119903) minus sin(120595

119888minus 120595

119903) bounded by

|119908| le

1003816

1003816

1003816

1003816

120595 minus 120595

119888

1003816

1003816

1003816

1003816

=

1003816

1003816

1003816

1003816

1003816

119890

120595

1003816

1003816

1003816

1003816

1003816

(26)



2



119890

120595=

120595 minus

120595

119888= 119903 minus

120595

119888= 119890

119903

119890

119903= 120591 minus 120591

119888minus 119889

119903119890

119903+ 119901

119903

(27)

where 120591119888= 119889

119903

120595

119888minus

120595



gain 119896


120595as

follows

119904

120595= 119890

120595+ 119896

3119890

119903 (28)




119904

120595= 119896

3(120591 minus 120591

119888+ 119901

119903) + (1 minus 119896

3119889

119903) 119890

119903 (29)




sign (120591) =

minus sign (119904120595) if 1003816

1003816

1003816

1003816

1003816

119904

120595

1003816

1003816

1003816

1003816

1003816

ge 120576

sign (120591minus) if 1003816

1003816

1003816

1003816

1003816

119904

120595

1003816

1003816

1003816

1003816

1003816

lt 120576

(30)


120595|






119909and 119904

119910are


1= 119911

1+ 119899

1and

2= 119911

2+ 119899

2 where

1003816

1003816

1003816

1003816

119899

1

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119899

2

1003816

1003816

1003816

1003816

le (z + radic

2 (1 + 119896

1)) 119899

119898 (31)



119909


the estimate 120595





derivative

120595



120595+ 119891

3 where

1003816

1003816

1003816

1003816

119891

3

1003816

1003816

1003816

1003816

le 119896

3119896

2119896

1119901

119898+ 119899

119898(1 + 119896

3+ 119899

119898+ 119896

1

1003816

1003816

1003816

1003816

119865

119903

1003816

1003816

1003816

1003816

+ |119903|)

+ 119896

2119899

119898(1 + 119896

3

1003816

1003816

1003816

1003816

2

1003816

1003816

1003816

1003816

) (z + radic

2 (1 + 119896

1))

+ 2

radic

2119899

119898

1003816

1003816

1003816

1003816

1 minus 119889

119906119896

1

1003816

1003816

1003816

1003816

radic119890

2

V119909 + 119890

2

V119910

(32)


sign (119865) =


1003816

1003816

1003816

1

1003816

1003816

1003816

1003816

ge 120576

sign (119865minus) if 1003816

1003816

1003816

1003816

1

1003816

1003816

1003816

1003816

lt 120576

(33)

sign (120591) =

minus sign (119904120595) if 1003816

1003816

1003816

1003816

1003816

119904

120595

1003816

1003816

1003816

1003816

1003816

ge 120576

sign (120591minus) if 1003816

1003816

1003816

1003816

1003816

119904

120595

1003816

1003816

1003816

1003816

1003816

lt 120576

(34)


119890and 119865


Table 2










1 1198962

and 119908



119898 then

z (119905) le 120573 (z (0) 119905) + 120574 (119908

119898+ 119899

119898+ 119901

119898+ 120576) (35)

Proof on Section A2



that for |119908| le 119908

119898

1003817

1003817

1003817

1003817

1003817

e119901(119905)

1003817

1003817

1003817

1003817

1003817

le 120573

2(

1003817

1003817

1003817

1003817

1003817

e119901(0)

1003817

1003817

1003817

1003817

1003817

119905) + 120574

2(119908

119898+ 119899

119898+ 119901

119898+ 120576) (36)

Proof in Section A3







119898 1198963 119886 119887 and 119888 and a K class

function 120574


119903and any


2 if z le 119911

119898 then

1003816

1003816

1003816

1003816

1003816

119904

120595(119905)

1003816

1003816

1003816

1003816

1003816

le max (1003816100381610038161003816

1003816

119904

120595(0)

1003816

1003816

1003816

1003816

1003816

minus 119886119905 120574

3(119887 + 119911

119898+ 119901

119898) 119899

119898+ 119888119901

119898120576)

(37)

Proof in Section A4


119898

1003816

1003816

1003816

1003816

1003816

119890

120595(119905)

1003816

1003816

1003816

1003816

1003816

le 119889 sdot 119890

minus1199051198963+ 120574

3(119887 + 119911

119898+ 119901

119898) 119899

119898+ 119888119901

119898+ 120576 (38)

Proof in Section A5














that1003817

1003817

1003817

1003817

1003817

e119901(119905)

1003817

1003817

1003817

1003817

1003817

le 120573

3(

1003817

1003817

1003817

1003817

1003817

e119901(0)

1003817

1003817

1003817

1003817

1003817

119905) + 120574

4(119899

119898+ 119901

119898+ 120576) (39)



Proof in Section A6






7 Results



1= 6 s 119896

2= 1 s 119896

3= 1 s and 120576 = 001


119903= 02179ms2 and 120591

119903= 04339 rads2) so the





120595 is





references 120595119903and 119903

119903 too



minus2

minus15

minus1

minus05

0

05

1

15

2

x(m

)

y (m)


0 10 20 30 40 50 60minus3

minus2

minus1

0123

x(m

)

t (s)

(a)

0 10 20 30 40 50 60minus3

minus2

minus1

0123

y(m

)

t (s)

(b)



119909and V119910 rad



0 10 20 30 40 50 60

minus04minus02

0020406

x(m

s)

t (s)

(a)

0 10 20 30 40 50 60

minus04minus02

0020406

t (s)

y(m

s)

(b)


0 10 20 30 40 50 60minus4

minus2

0

2

4

6120595

(rad

)

t (s)

(a)


0020406

r(r

ads

)

t (s)

(b)






minus3

minus2

minus1

0

1

2

3

4

x(m

)

y (m)


0 10 20 30 40 50 600

05

1

15

2

25

3

35

4

e p(m

)

t (s)





119903(119905) = 2 sin(38119905) 119910

119903(119905) = 3 cos(28119905)



minus06

minus04

minus02

0

02

04

06

08

1

12

120591 (rads2)

F(m

s2)



minus2

minus15

minus1

minus05

0

05

1

15

2x

(m)

y (m)






value than 120591








minus06

minus04

minus02

0

02

04

06

08

x(m

)

y (m)







0 5 10 15 20 250

01

02

03

04

05

06

07

08

09

1

Dist

ance

(m)

t





119865


radio controller





0020406

x(m

)

t (s)

(a)


002040608

y(m

)

t (s)

(b)


t (s)0 5 10 15 20 25

minus1

minus05

0

05

1

x(m

s)

(a)

0 5 10 15 20 25minus1

minus05

0

05

1

y(m

s)

t (s)

(b)


8 Conclusions



0 5 10 15 20 25minus04

minus02

0020406

Fs

(N)

t (s)

(a)

0 5 10 15 20 25minus04

minus02

0020406

t (s)

F p(N

)

(b)






Appendix

A Proofs



119871

2= V21199102 and 119871

3= 119903

2


119871

1= minusV119909(119889

119906V119909minus119865 cos(120595)+119901V119909)



119871

1can be bounded by

119871

1lt minus119889

119906V2119909(1 minus

119865max + 119901

119898

119889

119906

1003816

1003816

1003816

1003816

V119909

1003816

1003816

1003816

1003816

) (A1)


119871


119889

119906|V119909| gt (119865max + 119901


119909is ultimately



119898)119889


with 119871

2and 119871


Ω = x isin R6

1003816

1003816

1003816

1003816

V119909

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

V119910

1003816

1003816

1003816

1003816

1003816

le

119865max + 119901

119898

119889

119906

|119903| le

120591max + 119901

119898

119889

119903

(A2)



119903| minus 119901




radic

2

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

(1 minus 119889

119906119896

1)

119889

119906119896

1

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

(2119865max + 119901

119898) le 120575120576

1 (A3)


119898so that

radic

2 (119901

119898+ 120575120576

1+

1003816

1003816

1003816

1003816

119865

119903

1003816

1003816

1003816

1003816

119908

119898) lt min (120575 100381610038161003816

1003816

119865

119903

1003816

1003816

1003816

1003816

sin (1198962)) (A4)


(119911

2

1+ 119911

2


119881 = 119896

1119911

1(119865 minus 119865

119903cos (120595 minus 120595

119903) + 119901

1)

minus 119896

1119911

2(

1003816

1003816

1003816

1003816

119865

119903

1003816

1003816

1003816

1003816

sin (1198962tanh (119911

2)) minus 119865

119903119908 minus 119901

2)

+ (1 minus 119889

119906119896

1) (119904

1119890V119909 + 119904

2119890V119910)

(A5)


Case A (|1199111| gt 119899



119903cos(120595 minus 120595


by |119865119903| and |119911


following function

119892 (119909) = min( 120575

radic2

119865

119903

radic2

sin(1198962tanh( 119909

radic2

))) (A6)


1| minus |119865

119903119911

2| sin(119896

2tanh (|119911


119881 le minus119896

1z (119892 (z) minus (120575120576

1+

1003816

1003816

1003816

1003816

119865

119903

1003816

1003816

1003816

1003816

|119908| + 119901

119898)) (A7)


1+ |119865

119903|119908

119898+ 119901

119898


119911max 1 = radic

2 tanhminus1 ( 1

119896

2

sinminus1 (radic2(

119901max + 120575120576

1003816

1003816

1003816

1003816

119865

119903

1003816

1003816

1003816

1003816

+ 119908

119898)))

(A8)



Case B (|1199111| le 119899



2| and

with (31) we find

z le radic

2

1003816

1003816

1003816

1003816

119911

1

1003816

1003816

1003816

1003816

le

radic

2119899

1+ 120576 le

radic

2 (z + radic

2 (1 + 119896

1)) 119899

119898+ 120576

(A9)


z leradic2 (

radic2 (1 + 119896

1) 119899

119898+ 120576)


119898)

= 120576

2 (A10)


2| then z le radic

2|119911

2| so we


2| + 120576

2and |119911

1| le 2(|119911

2| +

1 minus 119896

1)119899


119881 le 119896

1(2 (

1003816

1003816

1003816

1003816

119911

2

1003816

1003816

1003816

1003816

+ 1 minus 119896

1) 119899

119898+ 120576) (

1003816

1003816

1003816

1003816

119865max1003816

1003816

1003816

1003816

+

1003816

1003816

1003816

1003816

119865

119903

1003816

1003816

1003816

1003816

+ 119901

119898)

minus 119896

1

1003816

1003816

1003816

1003816

119911

2

1003816

1003816

1003816

1003816

(

1003816

1003816

1003816

1003816

119865

119903

1003816

1003816

1003816

1003816

sin (1198962tanh (100381610038161003816

1003816

119911

2

1003816

1003816

1003816

1003816

)) minus

1003816

1003816

1003816

1003816

119865

119903119908

1003816

1003816

1003816

1003816

minus 119901

119898)

+ 2 (

radic

2

1003816

1003816

1003816

1003816

119911

2

1003816

1003816

1003816

1003816

+ 120576

2) 119889

119906119896

1120575120576

1

(A11)


3and 120576

4 (A11) can be

rewritten as

119881 le minus 119896

1(

1003816

1003816

1003816

1003816

119911

2

1003816

1003816

1003816

1003816

(

1003816

1003816

1003816

1003816

119865

119903

1003816

1003816

1003816

1003816

sin (1198962tanh (100381610038161003816

1003816

119911

2

1003816

1003816

1003816

1003816

))

minus

1003816

1003816

1003816

1003816

119865

119903119908

1003816

1003816

1003816

1003816

minus 119901

119898minus 120576

3) + 120576

4)

(A12)

Now we define


119896

2

sinminus1 (119901

119898

1003816

1003816

1003816

1003816

119865

119903

1003816

1003816

1003816

1003816

+ 120576

3+ 119908

119898+ 120575)) (A13)



119881 le 119896




119898


119898 119899119898 119901119898 and 120576 then

(35) holds


119909and 119904

119910

119896

1119890

119909= 119904

119909minus 119890

119909

119896

1119890

119910= 119904

119910minus 119890

119910

(A14)


119898 119901119898 119908119898 and 120576 then (36) holds



119903| le |119903| + |119903

119903| le |120591

119898|119889

119903+ |119903

119903| The



z le 1003816

1003816

1003816

1003816

1

1003816

1003816

1003816

1003816

+

1003816

1003816

1003816

1003816

2

1003816

1003816

1003816

1003816

le 119888

1+ 119888

2119911

119898+ 119888

3119901

119898

1003816

1003816

1003816

1003816

2

1003816

1003816

1003816

1003816

le 119888

4+ 119888

5119911

119898+ 119888

6119901

119898

(A15)


120591

119888= 120591

119903minus 119896

2sign (119865

119903) (1 minus tanh2 (119911

2)) (119889

119903

2+

2)

+ 2119896

2sign (119865

119903) (1 minus tanh2 (119911

2)) tanh (119911

2)

2

2

(A16)


120595and 120591

119888can be




1003816

1003816

1003816

1003816

120591

119888

1003816

1003816

1003816

1003816

le

1003816

1003816

1003816

1003816

120591

119903

1003816

1003816

1003816

1003816

+ 119896

2120574

4(119911

119898+ 119901

119898) (A17)

1003816

1003816

1003816

1003816

119891

3

1003816

1003816

1003816

1003816

le 120574

5(119888

7+ 119911

119898+ 119901

119898) 119899

119898+ 119896

3119896

2119896

1119901

119898 (A18)



120575 Now we define 119911119898as

119911

119898= minus

1

2

120574

4

minus1

(

120575

4119896

2

) (A19)

When z lt 119911


119896

2120574

4(2119901

119898)+119901

119898lt 1205754 and with (A17) then |120591

119894|minus|120591

119888|minus|119901

119898| ge


1003816

1003816

1003816

1003816

1 minus 119896

3119889

119903

1003816

1003816

1003816

1003816

119896

3119889

119903

(

1003816

1003816

1003816

1003816

120591

119898

1003816

1003816

1003816

1003816

+ 119889

119903

1003816

1003816

1003816

1003816

119903

119903

1003816

1003816

1003816

1003816

) le

120575

4

(A20)


2= 119904

2

1205952 whose time

derivative is

119881

2= 119904

120595(119896

3(120591 minus 120591

119888+ 119901

119903) + (1 minus 119896

3119889

119903)119890

119903) Consider

first that |119904120595| ge 119891



119881

2= 119904

120595119904

120595le minus120575119896

3(

1

2

minus

1

4

)

1003816

1003816

1003816

1003816

1003816

119904

120595

1003816

1003816

1003816

1003816

1003816

997888rarr 119904

120595le minus

120575119896

3

4

sign (119904120595)

(A21)


120595(119905)| le |119904

120595(0)| minus

120575119896

34119905 If 119904


120595| lt 119891

3+120576 (A21) does not




case |119904120595(119905)| le max(|119904

120595(0)| minus 120575119896

34119905 119891





120595= 119904

120595minus 119890

120595 whose

solution is

119890

120595(119905) = 119890

120595(0) 119890

minus1199051198963+

1

119896

3

int

119905

0

119890

minus(119905minus120591)1198963119904

120595(120591) 119889120591 (A22)


1003816

1003816

1003816

1003816

1003816

119890

120595(119905)

1003816

1003816

1003816

1003816

1003816

le

1003816

1003816

1003816

1003816

1003816

119890

120595(0)

1003816

1003816

1003816

1003816

1003816

119890

minus1199051198963+

1003816

1003816

1003816

1003816

1003816

119904

120595(0)

1003816

1003816

1003816

1003816

1003816

(119890

|119904120595(0)|1198861198963minus 1) 119890

minus1199051198963

+ 120574

3(119887 + 119911

119898+ 119901

119898) 119899

119898+ 119888119901

119898+ 120576

(A23)


120595| + |119904

120595|119890

|119904120595|1198861198963






119898Therefore



119898=

2119872 as follows

119896

2=

120575

4120574

4(4119872)

(A24)


119881 le 119896

1

1003816

1003816

1003816

1003816

119911

2

1003816

1003816

1003816

1003816

(

1003816

1003816

1003816

1003816

119865

119903119908

1003816

1003816

1003816

1003816

+ 119901

2) + (1 minus 119889

119906119896

1) (119904

1119890V119909 + 119904

2119890V119910)

le 119887

1|z| + 119887

2

(A25)


2 Select 119872 gt 119887

2119887

1then z gt

119887

2119887

1and thus

119881 = zz le 2119887


1rarr z le

z(0) + 2119887


2119887

1and therefore 119879 gt 1198722119887

2


3(119887 + 119872 + 119901

119898)119899

119898+ 119888119901

119898+ 120576 le 119889119890


120595(119879)| le 2119889 sdot 119890


max(119901

119898+ 120576

5

1003816

1003816

1003816

1003816

119865

119903

1003816

1003816

1003816

1003816

119901

119898+ 120575120576

1003816

1003816

1003816

1003816

119865

119903

1003816

1003816

1003816

1003816

) lt 119889 sdot 119890

minus1198791198963 (A26)


119911max (119879) = radic


4(4119872)

120575


minus119872211988821198963))

(A27)

Function 120574



119898



1for which z(119905



2for which z(119905

2) = 2119872 and 119881

grows in 119905



120595(119905

2)| le 2119889119890


119911max(|119890120595(1199052)|) = 119872 because z lt 119911

119898for all 119879 le 119905

1le 119905

2 This




2



Acknowledgments


References









































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










[umax umax]

F =Fp minus Fs

m

[umax 0]

[umax minusumin]


J

[0 minusumin]


[minusumin 0]

[minusumin umax]

[0 umax]

[0 0]



119903as the


119903


119865

119903cos (120595

119903) = V119909119903+ 119889

119906V119909119903

119865

119903sin (120595

119903) = V119910119903+ 119889

119906V119910119903

(8)



120595

119903= atan2 ( 119910

119903+ 119889

119906119910

119903

119903+ 119889

119906

119903) + 119899120587

119865

119903= (minus1)

119899radic

(

119903+ 119889

119906

119903)

2

+ ( 119910

119903+ 119889

119906119910

119903)

2

(9)



119903and 119903



119903= 119903

119903+119889

119903119903



119903(119905) 119910

119903(119905)]



follows


119903and 120591

119903are such that

119865



1 120591

1)

(119865

2 120591

2) (1198653 120591

3) and (119865

4 120591

4) such that

1003816

1003816

1003816

1003816

119865

119903(119905)

1003816

1003816

1003816

1003816

lt min (1198651 119865

2 minus119865

3 minus119865

4)

1003816

1003816

1003816

1003816

120591

119903(119905)

1003816

1003816

1003816

1003816

lt min (minus1205911 120591

2 minus120591

3 120591

4)

(10)


119904and 119865



119904and 119865






119903(119905) and 120591




F =Fp + Fs

m

[umax umax]

[umax 0]

[umax minusumin]


J

[0 minusumin]


[minusumin 0]

[minusumin umax]

[0 umax]

1

2

3

4

5

6

7

8

9

(Fr 120591r)

F0

minus1205910 1205910

minusF0

[0 0]

A


119903(119905)) lies on the




119904and 119865

119901to solve the






119901=

[119909 minus 119909

119903 119910 minus 119910

119903 V119909minus V119909119903 V119910minus V119910119903]





if the noise 119899


119898are small





119888to be tracked


119904and 119865


Table 2


119890

119909= 119909 minus 119909

119903

119890

119910= 119910 minus 119910

119903

119890V119909 = V119909minus V119909119903

119890V119910 = V119910minus V119910119903

(11)


119890

119909= 119890V119909

119890

119910= 119890V119910

119890V119909 = 119865 cos (120595) minus 119889

119906119890V119909 minus 119865

119909119903+ 119901V119909

119890V119910 = 119865 sin (120595) minus 119889

119906119890V119910 minus 119865

119910119903+ 119901V119910

(12)

where 119865119909119903

= 119865

119903cos(120595

119903) and 119865

119910119903= 119865




References

x

Positioncontrol

120595 r

120595c rc


Selector


25

2

15

1

05

0

minus05

minus1

minus15

F=(F

b+Fe)m

Fe

Fb

Hovercraft

x

Sign(F)

[umax umax]

[umax 0]

[umax minusumin]

[0 minusumin]


[minusumin 0]

[minusumin umax]

[0 umax]

[0 0]





119904

119909= 119890

119909+ 119896

1119890V119909

119904

119910= 119890

119910+ 119896

1119890V119910

(13)


1if 119904119909

and 119904



119909and 119904



119904

119909= 119896

1119865 cos (120595) + (1 minus 119889

119906119896

1) 119890V119909 minus 119896

1119865

119909119903+ 119896

1119901V119909

119904

119910= 119896

1119865 sin (120595) + (1 minus 119889

119906119896

1) 119890V119910 minus 119896

1119865

119910119903+ 119896

1119901V119910

(14)


119911

1= 119904

119909cos (120595) + 119904

119910sin (120595)

119911

2= minus119904

119909sin (120595) + 119904

119910cos (120595)

(15)


1= 119896

1119865 minus 119896

1119865

119903cos (120595 minus 120595

119903) + 119911

2119903

+ (1 minus 119889

119906119896

1) (119890V119909 cos (120595) + 119890V119910 sin (120595)) + 119896

1119901

1

(16)

2= 119896

1119865

119903sin (120595 minus 120595

119903) minus 119911

1119903

+ (1 minus 119889

119906119896

1) (minus119890V119909 sin (120595) + 119890V119910 cos (120595)) + 119896

1119901

2

(17)

where 1199011and 119901


119901

1= 119901V119909 cos (120595) + 119901V119910 sin (120595)

119901

2= minus119901V119909 sin (120595) + 119901V119910 cos (120595)

p =1003817

1003817

1003817

1003817

1003817

1003817

[119901

1 119901

2]

1198791003817

1003817

1003817

1003817

1003817

1003817

le 119901

119898

(18)


2 given by (16) and




sign (119865) =

minus sign (1199111) if 1003816

1003816

1003816

1003816

119911

1

1003816

1003816

1003816

1003816

ge 120576

sign (119865minus) if 1003816

1003816

1003816

1003816

119911

1

1003816

1003816

1003816

1003816

lt 120576

(19)


1|




1| lt 120576


119903sin(120595 minus 120595

119903)must be



120595

119888= 120595

119903minus 119896

2tanh (119911

2) sign (119865

119903) (20)

119903

119888=

120595

119888=

120595

119903minus

2119896


2)

2

) sign (119865119903) (21)



119888and 0 lt 119896

2lt 1205872

then 119865

119903sin(120595119888minus 120595

119903) = minus|119865

119903| sin(119896

2tanh(119911

2)) that is opposed




119890

120595= 120595 minus 120595

119888 (22)

119890

119903= 119903 minus 119903

119888 (23)


1= 119896

1119865 minus 119896

1119865

119903cos (119890

120595+ 120595

119888minus 120595

119903) + 119911

2119903 + 119896

1119901

1

+ (1 minus 119889

119906119896

1) (119890V119909 cos (119890120595 + 120595

119888) + 119890V119910 sin (119890120595 + 120595

119888))

(24)

2= minus 119896

1

1003816

1003816

1003816

1003816

119865

119903

1003816

1003816

1003816

1003816

sin (1198962tanh (119911

2)) minus 119911

1119903 + 119896

1119865

119903119908 + 119896

1119901

2

+ (1 minus 119889

119906119896

1) (119890V119910 cos (119890120595 + 120595

119888) minus 119890V119909 sin (119890120595 + 120595

119888))

(25)



119903) minus sin(120595

119888minus 120595

119903) bounded by

|119908| le

1003816

1003816

1003816

1003816

120595 minus 120595

119888

1003816

1003816

1003816

1003816

=

1003816

1003816

1003816

1003816

1003816

119890

120595

1003816

1003816

1003816

1003816

1003816

(26)



2



119890

120595=

120595 minus

120595

119888= 119903 minus

120595

119888= 119890

119903

119890

119903= 120591 minus 120591

119888minus 119889

119903119890

119903+ 119901

119903

(27)

where 120591119888= 119889

119903

120595

119888minus

120595



gain 119896


120595as

follows

119904

120595= 119890

120595+ 119896

3119890

119903 (28)




119904

120595= 119896

3(120591 minus 120591

119888+ 119901

119903) + (1 minus 119896

3119889

119903) 119890

119903 (29)




sign (120591) =

minus sign (119904120595) if 1003816

1003816

1003816

1003816

1003816

119904

120595

1003816

1003816

1003816

1003816

1003816

ge 120576

sign (120591minus) if 1003816

1003816

1003816

1003816

1003816

119904

120595

1003816

1003816

1003816

1003816

1003816

lt 120576

(30)


120595|






119909and 119904

119910are


1= 119911

1+ 119899

1and

2= 119911

2+ 119899

2 where

1003816

1003816

1003816

1003816

119899

1

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119899

2

1003816

1003816

1003816

1003816

le (z + radic

2 (1 + 119896

1)) 119899

119898 (31)



119909


the estimate 120595





derivative

120595



120595+ 119891

3 where

1003816

1003816

1003816

1003816

119891

3

1003816

1003816

1003816

1003816

le 119896

3119896

2119896

1119901

119898+ 119899

119898(1 + 119896

3+ 119899

119898+ 119896

1

1003816

1003816

1003816

1003816

119865

119903

1003816

1003816

1003816

1003816

+ |119903|)

+ 119896

2119899

119898(1 + 119896

3

1003816

1003816

1003816

1003816

2

1003816

1003816

1003816

1003816

) (z + radic

2 (1 + 119896

1))

+ 2

radic

2119899

119898

1003816

1003816

1003816

1003816

1 minus 119889

119906119896

1

1003816

1003816

1003816

1003816

radic119890

2

V119909 + 119890

2

V119910

(32)


sign (119865) =


1003816

1003816

1003816

1

1003816

1003816

1003816

1003816

ge 120576

sign (119865minus) if 1003816

1003816

1003816

1003816

1

1003816

1003816

1003816

1003816

lt 120576

(33)

sign (120591) =

minus sign (119904120595) if 1003816

1003816

1003816

1003816

1003816

119904

120595

1003816

1003816

1003816

1003816

1003816

ge 120576

sign (120591minus) if 1003816

1003816

1003816

1003816

1003816

119904

120595

1003816

1003816

1003816

1003816

1003816

lt 120576

(34)


119890and 119865


Table 2










1 1198962

and 119908



119898 then

z (119905) le 120573 (z (0) 119905) + 120574 (119908

119898+ 119899

119898+ 119901

119898+ 120576) (35)

Proof on Section A2



that for |119908| le 119908

119898

1003817

1003817

1003817

1003817

1003817

e119901(119905)

1003817

1003817

1003817

1003817

1003817

le 120573

2(

1003817

1003817

1003817

1003817

1003817

e119901(0)

1003817

1003817

1003817

1003817

1003817

119905) + 120574

2(119908

119898+ 119899

119898+ 119901

119898+ 120576) (36)

Proof in Section A3







119898 1198963 119886 119887 and 119888 and a K class

function 120574


119903and any


2 if z le 119911

119898 then

1003816

1003816

1003816

1003816

1003816

119904

120595(119905)

1003816

1003816

1003816

1003816

1003816

le max (1003816100381610038161003816

1003816

119904

120595(0)

1003816

1003816

1003816

1003816

1003816

minus 119886119905 120574

3(119887 + 119911

119898+ 119901

119898) 119899

119898+ 119888119901

119898120576)

(37)

Proof in Section A4


119898

1003816

1003816

1003816

1003816

1003816

119890

120595(119905)

1003816

1003816

1003816

1003816

1003816

le 119889 sdot 119890

minus1199051198963+ 120574

3(119887 + 119911

119898+ 119901

119898) 119899

119898+ 119888119901

119898+ 120576 (38)

Proof in Section A5














that1003817

1003817

1003817

1003817

1003817

e119901(119905)

1003817

1003817

1003817

1003817

1003817

le 120573

3(

1003817

1003817

1003817

1003817

1003817

e119901(0)

1003817

1003817

1003817

1003817

1003817

119905) + 120574

4(119899

119898+ 119901

119898+ 120576) (39)



Proof in Section A6






7 Results



1= 6 s 119896

2= 1 s 119896

3= 1 s and 120576 = 001


119903= 02179ms2 and 120591

119903= 04339 rads2) so the





120595 is





references 120595119903and 119903

119903 too



minus2

minus15

minus1

minus05

0

05

1

15

2

x(m

)

y (m)


0 10 20 30 40 50 60minus3

minus2

minus1

0123

x(m

)

t (s)

(a)

0 10 20 30 40 50 60minus3

minus2

minus1

0123

y(m

)

t (s)

(b)



119909and V119910 rad



0 10 20 30 40 50 60

minus04minus02

0020406

x(m

s)

t (s)

(a)

0 10 20 30 40 50 60

minus04minus02

0020406

t (s)

y(m

s)

(b)


0 10 20 30 40 50 60minus4

minus2

0

2

4

6120595

(rad

)

t (s)

(a)


0020406

r(r

ads

)

t (s)

(b)






minus3

minus2

minus1

0

1

2

3

4

x(m

)

y (m)


0 10 20 30 40 50 600

05

1

15

2

25

3

35

4

e p(m

)

t (s)





119903(119905) = 2 sin(38119905) 119910

119903(119905) = 3 cos(28119905)



minus06

minus04

minus02

0

02

04

06

08

1

12

120591 (rads2)

F(m

s2)



minus2

minus15

minus1

minus05

0

05

1

15

2x

(m)

y (m)






value than 120591








minus06

minus04

minus02

0

02

04

06

08

x(m

)

y (m)







0 5 10 15 20 250

01

02

03

04

05

06

07

08

09

1

Dist

ance

(m)

t





119865


radio controller





0020406

x(m

)

t (s)

(a)


002040608

y(m

)

t (s)

(b)


t (s)0 5 10 15 20 25

minus1

minus05

0

05

1

x(m

s)

(a)

0 5 10 15 20 25minus1

minus05

0

05

1

y(m

s)

t (s)

(b)


8 Conclusions



0 5 10 15 20 25minus04

minus02

0020406

Fs

(N)

t (s)

(a)

0 5 10 15 20 25minus04

minus02

0020406

t (s)

F p(N

)

(b)






Appendix

A Proofs



119871

2= V21199102 and 119871

3= 119903

2


119871

1= minusV119909(119889

119906V119909minus119865 cos(120595)+119901V119909)



119871

1can be bounded by

119871

1lt minus119889

119906V2119909(1 minus

119865max + 119901

119898

119889

119906

1003816

1003816

1003816

1003816

V119909

1003816

1003816

1003816

1003816

) (A1)


119871


119889

119906|V119909| gt (119865max + 119901


119909is ultimately



119898)119889


with 119871

2and 119871


Ω = x isin R6

1003816

1003816

1003816

1003816

V119909

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

V119910

1003816

1003816

1003816

1003816

1003816

le

119865max + 119901

119898

119889

119906

|119903| le

120591max + 119901

119898

119889

119903

(A2)



119903| minus 119901




radic

2

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

(1 minus 119889

119906119896

1)

119889

119906119896

1

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

(2119865max + 119901

119898) le 120575120576

1 (A3)


119898so that

radic

2 (119901

119898+ 120575120576

1+

1003816

1003816

1003816

1003816

119865

119903

1003816

1003816

1003816

1003816

119908

119898) lt min (120575 100381610038161003816

1003816

119865

119903

1003816

1003816

1003816

1003816

sin (1198962)) (A4)


(119911

2

1+ 119911

2


119881 = 119896

1119911

1(119865 minus 119865

119903cos (120595 minus 120595

119903) + 119901

1)

minus 119896

1119911

2(

1003816

1003816

1003816

1003816

119865

119903

1003816

1003816

1003816

1003816

sin (1198962tanh (119911

2)) minus 119865

119903119908 minus 119901

2)

+ (1 minus 119889

119906119896

1) (119904

1119890V119909 + 119904

2119890V119910)

(A5)


Case A (|1199111| gt 119899



119903cos(120595 minus 120595


by |119865119903| and |119911


following function

119892 (119909) = min( 120575

radic2

119865

119903

radic2

sin(1198962tanh( 119909

radic2

))) (A6)


1| minus |119865

119903119911

2| sin(119896

2tanh (|119911


119881 le minus119896

1z (119892 (z) minus (120575120576

1+

1003816

1003816

1003816

1003816

119865

119903

1003816

1003816

1003816

1003816

|119908| + 119901

119898)) (A7)


1+ |119865

119903|119908

119898+ 119901

119898


119911max 1 = radic

2 tanhminus1 ( 1

119896

2

sinminus1 (radic2(

119901max + 120575120576

1003816

1003816

1003816

1003816

119865

119903

1003816

1003816

1003816

1003816

+ 119908

119898)))

(A8)



Case B (|1199111| le 119899



2| and

with (31) we find

z le radic

2

1003816

1003816

1003816

1003816

119911

1

1003816

1003816

1003816

1003816

le

radic

2119899

1+ 120576 le

radic

2 (z + radic

2 (1 + 119896

1)) 119899

119898+ 120576

(A9)


z leradic2 (

radic2 (1 + 119896

1) 119899

119898+ 120576)


119898)

= 120576

2 (A10)


2| then z le radic

2|119911

2| so we


2| + 120576

2and |119911

1| le 2(|119911

2| +

1 minus 119896

1)119899


119881 le 119896

1(2 (

1003816

1003816

1003816

1003816

119911

2

1003816

1003816

1003816

1003816

+ 1 minus 119896

1) 119899

119898+ 120576) (

1003816

1003816

1003816

1003816

119865max1003816

1003816

1003816

1003816

+

1003816

1003816

1003816

1003816

119865

119903

1003816

1003816

1003816

1003816

+ 119901

119898)

minus 119896

1

1003816

1003816

1003816

1003816

119911

2

1003816

1003816

1003816

1003816

(

1003816

1003816

1003816

1003816

119865

119903

1003816

1003816

1003816

1003816

sin (1198962tanh (100381610038161003816

1003816

119911

2

1003816

1003816

1003816

1003816

)) minus

1003816

1003816

1003816

1003816

119865

119903119908

1003816

1003816

1003816

1003816

minus 119901

119898)

+ 2 (

radic

2

1003816

1003816

1003816

1003816

119911

2

1003816

1003816

1003816

1003816

+ 120576

2) 119889

119906119896

1120575120576

1

(A11)


3and 120576

4 (A11) can be

rewritten as

119881 le minus 119896

1(

1003816

1003816

1003816

1003816

119911

2

1003816

1003816

1003816

1003816

(

1003816

1003816

1003816

1003816

119865

119903

1003816

1003816

1003816

1003816

sin (1198962tanh (100381610038161003816

1003816

119911

2

1003816

1003816

1003816

1003816

))

minus

1003816

1003816

1003816

1003816

119865

119903119908

1003816

1003816

1003816

1003816

minus 119901

119898minus 120576

3) + 120576

4)

(A12)

Now we define


119896

2

sinminus1 (119901

119898

1003816

1003816

1003816

1003816

119865

119903

1003816

1003816

1003816

1003816

+ 120576

3+ 119908

119898+ 120575)) (A13)



119881 le 119896




119898


119898 119899119898 119901119898 and 120576 then

(35) holds


119909and 119904

119910

119896

1119890

119909= 119904

119909minus 119890

119909

119896

1119890

119910= 119904

119910minus 119890

119910

(A14)


119898 119901119898 119908119898 and 120576 then (36) holds



119903| le |119903| + |119903

119903| le |120591

119898|119889

119903+ |119903

119903| The



z le 1003816

1003816

1003816

1003816

1

1003816

1003816

1003816

1003816

+

1003816

1003816

1003816

1003816

2

1003816

1003816

1003816

1003816

le 119888

1+ 119888

2119911

119898+ 119888

3119901

119898

1003816

1003816

1003816

1003816

2

1003816

1003816

1003816

1003816

le 119888

4+ 119888

5119911

119898+ 119888

6119901

119898

(A15)


120591

119888= 120591

119903minus 119896

2sign (119865

119903) (1 minus tanh2 (119911

2)) (119889

119903

2+

2)

+ 2119896

2sign (119865

119903) (1 minus tanh2 (119911

2)) tanh (119911

2)

2

2

(A16)


120595and 120591

119888can be




1003816

1003816

1003816

1003816

120591

119888

1003816

1003816

1003816

1003816

le

1003816

1003816

1003816

1003816

120591

119903

1003816

1003816

1003816

1003816

+ 119896

2120574

4(119911

119898+ 119901

119898) (A17)

1003816

1003816

1003816

1003816

119891

3

1003816

1003816

1003816

1003816

le 120574

5(119888

7+ 119911

119898+ 119901

119898) 119899

119898+ 119896

3119896

2119896

1119901

119898 (A18)



120575 Now we define 119911119898as

119911

119898= minus

1

2

120574

4

minus1

(

120575

4119896

2

) (A19)

When z lt 119911


119896

2120574

4(2119901

119898)+119901

119898lt 1205754 and with (A17) then |120591

119894|minus|120591

119888|minus|119901

119898| ge


1003816

1003816

1003816

1003816

1 minus 119896

3119889

119903

1003816

1003816

1003816

1003816

119896

3119889

119903

(

1003816

1003816

1003816

1003816

120591

119898

1003816

1003816

1003816

1003816

+ 119889

119903

1003816

1003816

1003816

1003816

119903

119903

1003816

1003816

1003816

1003816

) le

120575

4

(A20)


2= 119904

2

1205952 whose time

derivative is

119881

2= 119904

120595(119896

3(120591 minus 120591

119888+ 119901

119903) + (1 minus 119896

3119889

119903)119890

119903) Consider

first that |119904120595| ge 119891



119881

2= 119904

120595119904

120595le minus120575119896

3(

1

2

minus

1

4

)

1003816

1003816

1003816

1003816

1003816

119904

120595

1003816

1003816

1003816

1003816

1003816

997888rarr 119904

120595le minus

120575119896

3

4

sign (119904120595)

(A21)


120595(119905)| le |119904

120595(0)| minus

120575119896

34119905 If 119904


120595| lt 119891

3+120576 (A21) does not




case |119904120595(119905)| le max(|119904

120595(0)| minus 120575119896

34119905 119891





120595= 119904

120595minus 119890

120595 whose

solution is

119890

120595(119905) = 119890

120595(0) 119890

minus1199051198963+

1

119896

3

int

119905

0

119890

minus(119905minus120591)1198963119904

120595(120591) 119889120591 (A22)


1003816

1003816

1003816

1003816

1003816

119890

120595(119905)

1003816

1003816

1003816

1003816

1003816

le

1003816

1003816

1003816

1003816

1003816

119890

120595(0)

1003816

1003816

1003816

1003816

1003816

119890

minus1199051198963+

1003816

1003816

1003816

1003816

1003816

119904

120595(0)

1003816

1003816

1003816

1003816

1003816

(119890

|119904120595(0)|1198861198963minus 1) 119890

minus1199051198963

+ 120574

3(119887 + 119911

119898+ 119901

119898) 119899

119898+ 119888119901

119898+ 120576

(A23)


120595| + |119904

120595|119890

|119904120595|1198861198963






119898Therefore



119898=

2119872 as follows

119896

2=

120575

4120574

4(4119872)

(A24)


119881 le 119896

1

1003816

1003816

1003816

1003816

119911

2

1003816

1003816

1003816

1003816

(

1003816

1003816

1003816

1003816

119865

119903119908

1003816

1003816

1003816

1003816

+ 119901

2) + (1 minus 119889

119906119896

1) (119904

1119890V119909 + 119904

2119890V119910)

le 119887

1|z| + 119887

2

(A25)


2 Select 119872 gt 119887

2119887

1then z gt

119887

2119887

1and thus

119881 = zz le 2119887


1rarr z le

z(0) + 2119887


2119887

1and therefore 119879 gt 1198722119887

2


3(119887 + 119872 + 119901

119898)119899

119898+ 119888119901

119898+ 120576 le 119889119890


120595(119879)| le 2119889 sdot 119890


max(119901

119898+ 120576

5

1003816

1003816

1003816

1003816

119865

119903

1003816

1003816

1003816

1003816

119901

119898+ 120575120576

1003816

1003816

1003816

1003816

119865

119903

1003816

1003816

1003816

1003816

) lt 119889 sdot 119890

minus1198791198963 (A26)


119911max (119879) = radic


4(4119872)

120575


minus119872211988821198963))

(A27)

Function 120574



119898



1for which z(119905



2for which z(119905

2) = 2119872 and 119881

grows in 119905



120595(119905

2)| le 2119889119890


119911max(|119890120595(1199052)|) = 119872 because z lt 119911

119898for all 119879 le 119905

1le 119905

2 This




2



Acknowledgments


References









































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










F =Fp + Fs

m

[umax umax]

[umax 0]

[umax minusumin]


J

[0 minusumin]


[minusumin 0]

[minusumin umax]

[0 umax]

1

2

3

4

5

6

7

8

9

(Fr 120591r)

F0

minus1205910 1205910

minusF0

[0 0]

A


119903(119905)) lies on the




119904and 119865

119901to solve the






119901=

[119909 minus 119909

119903 119910 minus 119910

119903 V119909minus V119909119903 V119910minus V119910119903]





if the noise 119899


119898are small





119888to be tracked


119904and 119865


Table 2


119890

119909= 119909 minus 119909

119903

119890

119910= 119910 minus 119910

119903

119890V119909 = V119909minus V119909119903

119890V119910 = V119910minus V119910119903

(11)


119890

119909= 119890V119909

119890

119910= 119890V119910

119890V119909 = 119865 cos (120595) minus 119889

119906119890V119909 minus 119865

119909119903+ 119901V119909

119890V119910 = 119865 sin (120595) minus 119889

119906119890V119910 minus 119865

119910119903+ 119901V119910

(12)

where 119865119909119903

= 119865

119903cos(120595

119903) and 119865

119910119903= 119865




References

x

Positioncontrol

120595 r

120595c rc


Selector


25

2

15

1

05

0

minus05

minus1

minus15

F=(F

b+Fe)m

Fe

Fb

Hovercraft

x

Sign(F)

[umax umax]

[umax 0]

[umax minusumin]

[0 minusumin]


[minusumin 0]

[minusumin umax]

[0 umax]

[0 0]





119904

119909= 119890

119909+ 119896

1119890V119909

119904

119910= 119890

119910+ 119896

1119890V119910

(13)


1if 119904119909

and 119904



119909and 119904



119904

119909= 119896

1119865 cos (120595) + (1 minus 119889

119906119896

1) 119890V119909 minus 119896

1119865

119909119903+ 119896

1119901V119909

119904

119910= 119896

1119865 sin (120595) + (1 minus 119889

119906119896

1) 119890V119910 minus 119896

1119865

119910119903+ 119896

1119901V119910

(14)


119911

1= 119904

119909cos (120595) + 119904

119910sin (120595)

119911

2= minus119904

119909sin (120595) + 119904

119910cos (120595)

(15)


1= 119896

1119865 minus 119896

1119865

119903cos (120595 minus 120595

119903) + 119911

2119903

+ (1 minus 119889

119906119896

1) (119890V119909 cos (120595) + 119890V119910 sin (120595)) + 119896

1119901

1

(16)

2= 119896

1119865

119903sin (120595 minus 120595

119903) minus 119911

1119903

+ (1 minus 119889

119906119896

1) (minus119890V119909 sin (120595) + 119890V119910 cos (120595)) + 119896

1119901

2

(17)

where 1199011and 119901


119901

1= 119901V119909 cos (120595) + 119901V119910 sin (120595)

119901

2= minus119901V119909 sin (120595) + 119901V119910 cos (120595)

p =1003817

1003817

1003817

1003817

1003817

1003817

[119901

1 119901

2]

1198791003817

1003817

1003817

1003817

1003817

1003817

le 119901

119898

(18)


2 given by (16) and




sign (119865) =

minus sign (1199111) if 1003816

1003816

1003816

1003816

119911

1

1003816

1003816

1003816

1003816

ge 120576

sign (119865minus) if 1003816

1003816

1003816

1003816

119911

1

1003816

1003816

1003816

1003816

lt 120576

(19)


1|




1| lt 120576


119903sin(120595 minus 120595

119903)must be



120595

119888= 120595

119903minus 119896

2tanh (119911

2) sign (119865

119903) (20)

119903

119888=

120595

119888=

120595

119903minus

2119896


2)

2

) sign (119865119903) (21)



119888and 0 lt 119896

2lt 1205872

then 119865

119903sin(120595119888minus 120595

119903) = minus|119865

119903| sin(119896

2tanh(119911

2)) that is opposed




119890

120595= 120595 minus 120595

119888 (22)

119890

119903= 119903 minus 119903

119888 (23)


1= 119896

1119865 minus 119896

1119865

119903cos (119890

120595+ 120595

119888minus 120595

119903) + 119911

2119903 + 119896

1119901

1

+ (1 minus 119889

119906119896

1) (119890V119909 cos (119890120595 + 120595

119888) + 119890V119910 sin (119890120595 + 120595

119888))

(24)

2= minus 119896

1

1003816

1003816

1003816

1003816

119865

119903

1003816

1003816

1003816

1003816

sin (1198962tanh (119911

2)) minus 119911

1119903 + 119896

1119865

119903119908 + 119896

1119901

2

+ (1 minus 119889

119906119896

1) (119890V119910 cos (119890120595 + 120595

119888) minus 119890V119909 sin (119890120595 + 120595

119888))

(25)



119903) minus sin(120595

119888minus 120595

119903) bounded by

|119908| le

1003816

1003816

1003816

1003816

120595 minus 120595

119888

1003816

1003816

1003816

1003816

=

1003816

1003816

1003816

1003816

1003816

119890

120595

1003816

1003816

1003816

1003816

1003816

(26)



2



119890

120595=

120595 minus

120595

119888= 119903 minus

120595

119888= 119890

119903

119890

119903= 120591 minus 120591

119888minus 119889

119903119890

119903+ 119901

119903

(27)

where 120591119888= 119889

119903

120595

119888minus

120595



gain 119896


120595as

follows

119904

120595= 119890

120595+ 119896

3119890

119903 (28)




119904

120595= 119896

3(120591 minus 120591

119888+ 119901

119903) + (1 minus 119896

3119889

119903) 119890

119903 (29)




sign (120591) =

minus sign (119904120595) if 1003816

1003816

1003816

1003816

1003816

119904

120595

1003816

1003816

1003816

1003816

1003816

ge 120576

sign (120591minus) if 1003816

1003816

1003816

1003816

1003816

119904

120595

1003816

1003816

1003816

1003816

1003816

lt 120576

(30)


120595|






119909and 119904

119910are


1= 119911

1+ 119899

1and

2= 119911

2+ 119899

2 where

1003816

1003816

1003816

1003816

119899

1

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119899

2

1003816

1003816

1003816

1003816

le (z + radic

2 (1 + 119896

1)) 119899

119898 (31)



119909


the estimate 120595





derivative

120595



120595+ 119891

3 where

1003816

1003816

1003816

1003816

119891

3

1003816

1003816

1003816

1003816

le 119896

3119896

2119896

1119901

119898+ 119899

119898(1 + 119896

3+ 119899

119898+ 119896

1

1003816

1003816

1003816

1003816

119865

119903

1003816

1003816

1003816

1003816

+ |119903|)

+ 119896

2119899

119898(1 + 119896

3

1003816

1003816

1003816

1003816

2

1003816

1003816

1003816

1003816

) (z + radic

2 (1 + 119896

1))

+ 2

radic

2119899

119898

1003816

1003816

1003816

1003816

1 minus 119889

119906119896

1

1003816

1003816

1003816

1003816

radic119890

2

V119909 + 119890

2

V119910

(32)


sign (119865) =


1003816

1003816

1003816

1

1003816

1003816

1003816

1003816

ge 120576

sign (119865minus) if 1003816

1003816

1003816

1003816

1

1003816

1003816

1003816

1003816

lt 120576

(33)

sign (120591) =

minus sign (119904120595) if 1003816

1003816

1003816

1003816

1003816

119904

120595

1003816

1003816

1003816

1003816

1003816

ge 120576

sign (120591minus) if 1003816

1003816

1003816

1003816

1003816

119904

120595

1003816

1003816

1003816

1003816

1003816

lt 120576

(34)


119890and 119865


Table 2










1 1198962

and 119908



119898 then

z (119905) le 120573 (z (0) 119905) + 120574 (119908

119898+ 119899

119898+ 119901

119898+ 120576) (35)

Proof on Section A2



that for |119908| le 119908

119898

1003817

1003817

1003817

1003817

1003817

e119901(119905)

1003817

1003817

1003817

1003817

1003817

le 120573

2(

1003817

1003817

1003817

1003817

1003817

e119901(0)

1003817

1003817

1003817

1003817

1003817

119905) + 120574

2(119908

119898+ 119899

119898+ 119901

119898+ 120576) (36)

Proof in Section A3







119898 1198963 119886 119887 and 119888 and a K class

function 120574


119903and any


2 if z le 119911

119898 then

1003816

1003816

1003816

1003816

1003816

119904

120595(119905)

1003816

1003816

1003816

1003816

1003816

le max (1003816100381610038161003816

1003816

119904

120595(0)

1003816

1003816

1003816

1003816

1003816

minus 119886119905 120574

3(119887 + 119911

119898+ 119901

119898) 119899

119898+ 119888119901

119898120576)

(37)

Proof in Section A4


119898

1003816

1003816

1003816

1003816

1003816

119890

120595(119905)

1003816

1003816

1003816

1003816

1003816

le 119889 sdot 119890

minus1199051198963+ 120574

3(119887 + 119911

119898+ 119901

119898) 119899

119898+ 119888119901

119898+ 120576 (38)

Proof in Section A5














that1003817

1003817

1003817

1003817

1003817

e119901(119905)

1003817

1003817

1003817

1003817

1003817

le 120573

3(

1003817

1003817

1003817

1003817

1003817

e119901(0)

1003817

1003817

1003817

1003817

1003817

119905) + 120574

4(119899

119898+ 119901

119898+ 120576) (39)



Proof in Section A6






7 Results



1= 6 s 119896

2= 1 s 119896

3= 1 s and 120576 = 001


119903= 02179ms2 and 120591

119903= 04339 rads2) so the





120595 is





references 120595119903and 119903

119903 too



minus2

minus15

minus1

minus05

0

05

1

15

2

x(m

)

y (m)


0 10 20 30 40 50 60minus3

minus2

minus1

0123

x(m

)

t (s)

(a)

0 10 20 30 40 50 60minus3

minus2

minus1

0123

y(m

)

t (s)

(b)



119909and V119910 rad



0 10 20 30 40 50 60

minus04minus02

0020406

x(m

s)

t (s)

(a)

0 10 20 30 40 50 60

minus04minus02

0020406

t (s)

y(m

s)

(b)


0 10 20 30 40 50 60minus4

minus2

0

2

4

6120595

(rad

)

t (s)

(a)


0020406

r(r

ads

)

t (s)

(b)






minus3

minus2

minus1

0

1

2

3

4

x(m

)

y (m)


0 10 20 30 40 50 600

05

1

15

2

25

3

35

4

e p(m

)

t (s)





119903(119905) = 2 sin(38119905) 119910

119903(119905) = 3 cos(28119905)



minus06

minus04

minus02

0

02

04

06

08

1

12

120591 (rads2)

F(m

s2)



minus2

minus15

minus1

minus05

0

05

1

15

2x

(m)

y (m)






value than 120591








minus06

minus04

minus02

0

02

04

06

08

x(m

)

y (m)







0 5 10 15 20 250

01

02

03

04

05

06

07

08

09

1

Dist

ance

(m)

t





119865


radio controller





0020406

x(m

)

t (s)

(a)


002040608

y(m

)

t (s)

(b)


t (s)0 5 10 15 20 25

minus1

minus05

0

05

1

x(m

s)

(a)

0 5 10 15 20 25minus1

minus05

0

05

1

y(m

s)

t (s)

(b)


8 Conclusions



0 5 10 15 20 25minus04

minus02

0020406

Fs

(N)

t (s)

(a)

0 5 10 15 20 25minus04

minus02

0020406

t (s)

F p(N

)

(b)






Appendix

A Proofs



119871

2= V21199102 and 119871

3= 119903

2


119871

1= minusV119909(119889

119906V119909minus119865 cos(120595)+119901V119909)



119871

1can be bounded by

119871

1lt minus119889

119906V2119909(1 minus

119865max + 119901

119898

119889

119906

1003816

1003816

1003816

1003816

V119909

1003816

1003816

1003816

1003816

) (A1)


119871


119889

119906|V119909| gt (119865max + 119901


119909is ultimately



119898)119889


with 119871

2and 119871


Ω = x isin R6

1003816

1003816

1003816

1003816

V119909

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

V119910

1003816

1003816

1003816

1003816

1003816

le

119865max + 119901

119898

119889

119906

|119903| le

120591max + 119901

119898

119889

119903

(A2)



119903| minus 119901




radic

2

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

(1 minus 119889

119906119896

1)

119889

119906119896

1

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

(2119865max + 119901

119898) le 120575120576

1 (A3)


119898so that

radic

2 (119901

119898+ 120575120576

1+

1003816

1003816

1003816

1003816

119865

119903

1003816

1003816

1003816

1003816

119908

119898) lt min (120575 100381610038161003816

1003816

119865

119903

1003816

1003816

1003816

1003816

sin (1198962)) (A4)


(119911

2

1+ 119911

2


119881 = 119896

1119911

1(119865 minus 119865

119903cos (120595 minus 120595

119903) + 119901

1)

minus 119896

1119911

2(

1003816

1003816

1003816

1003816

119865

119903

1003816

1003816

1003816

1003816

sin (1198962tanh (119911

2)) minus 119865

119903119908 minus 119901

2)

+ (1 minus 119889

119906119896

1) (119904

1119890V119909 + 119904

2119890V119910)

(A5)


Case A (|1199111| gt 119899



119903cos(120595 minus 120595


by |119865119903| and |119911


following function

119892 (119909) = min( 120575

radic2

119865

119903

radic2

sin(1198962tanh( 119909

radic2

))) (A6)


1| minus |119865

119903119911

2| sin(119896

2tanh (|119911


119881 le minus119896

1z (119892 (z) minus (120575120576

1+

1003816

1003816

1003816

1003816

119865

119903

1003816

1003816

1003816

1003816

|119908| + 119901

119898)) (A7)


1+ |119865

119903|119908

119898+ 119901

119898


119911max 1 = radic

2 tanhminus1 ( 1

119896

2

sinminus1 (radic2(

119901max + 120575120576

1003816

1003816

1003816

1003816

119865

119903

1003816

1003816

1003816

1003816

+ 119908

119898)))

(A8)



Case B (|1199111| le 119899



2| and

with (31) we find

z le radic

2

1003816

1003816

1003816

1003816

119911

1

1003816

1003816

1003816

1003816

le

radic

2119899

1+ 120576 le

radic

2 (z + radic

2 (1 + 119896

1)) 119899

119898+ 120576

(A9)


z leradic2 (

radic2 (1 + 119896

1) 119899

119898+ 120576)


119898)

= 120576

2 (A10)


2| then z le radic

2|119911

2| so we


2| + 120576

2and |119911

1| le 2(|119911

2| +

1 minus 119896

1)119899


119881 le 119896

1(2 (

1003816

1003816

1003816

1003816

119911

2

1003816

1003816

1003816

1003816

+ 1 minus 119896

1) 119899

119898+ 120576) (

1003816

1003816

1003816

1003816

119865max1003816

1003816

1003816

1003816

+

1003816

1003816

1003816

1003816

119865

119903

1003816

1003816

1003816

1003816

+ 119901

119898)

minus 119896

1

1003816

1003816

1003816

1003816

119911

2

1003816

1003816

1003816

1003816

(

1003816

1003816

1003816

1003816

119865

119903

1003816

1003816

1003816

1003816

sin (1198962tanh (100381610038161003816

1003816

119911

2

1003816

1003816

1003816

1003816

)) minus

1003816

1003816

1003816

1003816

119865

119903119908

1003816

1003816

1003816

1003816

minus 119901

119898)

+ 2 (

radic

2

1003816

1003816

1003816

1003816

119911

2

1003816

1003816

1003816

1003816

+ 120576

2) 119889

119906119896

1120575120576

1

(A11)


3and 120576

4 (A11) can be

rewritten as

119881 le minus 119896

1(

1003816

1003816

1003816

1003816

119911

2

1003816

1003816

1003816

1003816

(

1003816

1003816

1003816

1003816

119865

119903

1003816

1003816

1003816

1003816

sin (1198962tanh (100381610038161003816

1003816

119911

2

1003816

1003816

1003816

1003816

))

minus

1003816

1003816

1003816

1003816

119865

119903119908

1003816

1003816

1003816

1003816

minus 119901

119898minus 120576

3) + 120576

4)

(A12)

Now we define


119896

2

sinminus1 (119901

119898

1003816

1003816

1003816

1003816

119865

119903

1003816

1003816

1003816

1003816

+ 120576

3+ 119908

119898+ 120575)) (A13)



119881 le 119896




119898


119898 119899119898 119901119898 and 120576 then

(35) holds


119909and 119904

119910

119896

1119890

119909= 119904

119909minus 119890

119909

119896

1119890

119910= 119904

119910minus 119890

119910

(A14)


119898 119901119898 119908119898 and 120576 then (36) holds



119903| le |119903| + |119903

119903| le |120591

119898|119889

119903+ |119903

119903| The



z le 1003816

1003816

1003816

1003816

1

1003816

1003816

1003816

1003816

+

1003816

1003816

1003816

1003816

2

1003816

1003816

1003816

1003816

le 119888

1+ 119888

2119911

119898+ 119888

3119901

119898

1003816

1003816

1003816

1003816

2

1003816

1003816

1003816

1003816

le 119888

4+ 119888

5119911

119898+ 119888

6119901

119898

(A15)


120591

119888= 120591

119903minus 119896

2sign (119865

119903) (1 minus tanh2 (119911

2)) (119889

119903

2+

2)

+ 2119896

2sign (119865

119903) (1 minus tanh2 (119911

2)) tanh (119911

2)

2

2

(A16)


120595and 120591

119888can be




1003816

1003816

1003816

1003816

120591

119888

1003816

1003816

1003816

1003816

le

1003816

1003816

1003816

1003816

120591

119903

1003816

1003816

1003816

1003816

+ 119896

2120574

4(119911

119898+ 119901

119898) (A17)

1003816

1003816

1003816

1003816

119891

3

1003816

1003816

1003816

1003816

le 120574

5(119888

7+ 119911

119898+ 119901

119898) 119899

119898+ 119896

3119896

2119896

1119901

119898 (A18)



120575 Now we define 119911119898as

119911

119898= minus

1

2

120574

4

minus1

(

120575

4119896

2

) (A19)

When z lt 119911


119896

2120574

4(2119901

119898)+119901

119898lt 1205754 and with (A17) then |120591

119894|minus|120591

119888|minus|119901

119898| ge


1003816

1003816

1003816

1003816

1 minus 119896

3119889

119903

1003816

1003816

1003816

1003816

119896

3119889

119903

(

1003816

1003816

1003816

1003816

120591

119898

1003816

1003816

1003816

1003816

+ 119889

119903

1003816

1003816

1003816

1003816

119903

119903

1003816

1003816

1003816

1003816

) le

120575

4

(A20)


2= 119904

2

1205952 whose time

derivative is

119881

2= 119904

120595(119896

3(120591 minus 120591

119888+ 119901

119903) + (1 minus 119896

3119889

119903)119890

119903) Consider

first that |119904120595| ge 119891



119881

2= 119904

120595119904

120595le minus120575119896

3(

1

2

minus

1

4

)

1003816

1003816

1003816

1003816

1003816

119904

120595

1003816

1003816

1003816

1003816

1003816

997888rarr 119904

120595le minus

120575119896

3

4

sign (119904120595)

(A21)


120595(119905)| le |119904

120595(0)| minus

120575119896

34119905 If 119904


120595| lt 119891

3+120576 (A21) does not




case |119904120595(119905)| le max(|119904

120595(0)| minus 120575119896

34119905 119891





120595= 119904

120595minus 119890

120595 whose

solution is

119890

120595(119905) = 119890

120595(0) 119890

minus1199051198963+

1

119896

3

int

119905

0

119890

minus(119905minus120591)1198963119904

120595(120591) 119889120591 (A22)


1003816

1003816

1003816

1003816

1003816

119890

120595(119905)

1003816

1003816

1003816

1003816

1003816

le

1003816

1003816

1003816

1003816

1003816

119890

120595(0)

1003816

1003816

1003816

1003816

1003816

119890

minus1199051198963+

1003816

1003816

1003816

1003816

1003816

119904

120595(0)

1003816

1003816

1003816

1003816

1003816

(119890

|119904120595(0)|1198861198963minus 1) 119890

minus1199051198963

+ 120574

3(119887 + 119911

119898+ 119901

119898) 119899

119898+ 119888119901

119898+ 120576

(A23)


120595| + |119904

120595|119890

|119904120595|1198861198963






119898Therefore



119898=

2119872 as follows

119896

2=

120575

4120574

4(4119872)

(A24)


119881 le 119896

1

1003816

1003816

1003816

1003816

119911

2

1003816

1003816

1003816

1003816

(

1003816

1003816

1003816

1003816

119865

119903119908

1003816

1003816

1003816

1003816

+ 119901

2) + (1 minus 119889

119906119896

1) (119904

1119890V119909 + 119904

2119890V119910)

le 119887

1|z| + 119887

2

(A25)


2 Select 119872 gt 119887

2119887

1then z gt

119887

2119887

1and thus

119881 = zz le 2119887


1rarr z le

z(0) + 2119887


2119887

1and therefore 119879 gt 1198722119887

2


3(119887 + 119872 + 119901

119898)119899

119898+ 119888119901

119898+ 120576 le 119889119890


120595(119879)| le 2119889 sdot 119890


max(119901

119898+ 120576

5

1003816

1003816

1003816

1003816

119865

119903

1003816

1003816

1003816

1003816

119901

119898+ 120575120576

1003816

1003816

1003816

1003816

119865

119903

1003816

1003816

1003816

1003816

) lt 119889 sdot 119890

minus1198791198963 (A26)


119911max (119879) = radic


4(4119872)

120575


minus119872211988821198963))

(A27)

Function 120574



119898



1for which z(119905



2for which z(119905

2) = 2119872 and 119881

grows in 119905



120595(119905

2)| le 2119889119890


119911max(|119890120595(1199052)|) = 119872 because z lt 119911

119898for all 119879 le 119905

1le 119905

2 This




2



Acknowledgments


References









































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










References

x

Positioncontrol

120595 r

120595c rc


Selector


25

2

15

1

05

0

minus05

minus1

minus15

F=(F

b+Fe)m

Fe

Fb

Hovercraft

x

Sign(F)

[umax umax]

[umax 0]

[umax minusumin]

[0 minusumin]


[minusumin 0]

[minusumin umax]

[0 umax]

[0 0]





119904

119909= 119890

119909+ 119896

1119890V119909

119904

119910= 119890

119910+ 119896

1119890V119910

(13)


1if 119904119909

and 119904



119909and 119904



119904

119909= 119896

1119865 cos (120595) + (1 minus 119889

119906119896

1) 119890V119909 minus 119896

1119865

119909119903+ 119896

1119901V119909

119904

119910= 119896

1119865 sin (120595) + (1 minus 119889

119906119896

1) 119890V119910 minus 119896

1119865

119910119903+ 119896

1119901V119910

(14)


119911

1= 119904

119909cos (120595) + 119904

119910sin (120595)

119911

2= minus119904

119909sin (120595) + 119904

119910cos (120595)

(15)


1= 119896

1119865 minus 119896

1119865

119903cos (120595 minus 120595

119903) + 119911

2119903

+ (1 minus 119889

119906119896

1) (119890V119909 cos (120595) + 119890V119910 sin (120595)) + 119896

1119901

1

(16)

2= 119896

1119865

119903sin (120595 minus 120595

119903) minus 119911

1119903

+ (1 minus 119889

119906119896

1) (minus119890V119909 sin (120595) + 119890V119910 cos (120595)) + 119896

1119901

2

(17)

where 1199011and 119901


119901

1= 119901V119909 cos (120595) + 119901V119910 sin (120595)

119901

2= minus119901V119909 sin (120595) + 119901V119910 cos (120595)

p =1003817

1003817

1003817

1003817

1003817

1003817

[119901

1 119901

2]

1198791003817

1003817

1003817

1003817

1003817

1003817

le 119901

119898

(18)


2 given by (16) and




sign (119865) =

minus sign (1199111) if 1003816

1003816

1003816

1003816

119911

1

1003816

1003816

1003816

1003816

ge 120576

sign (119865minus) if 1003816

1003816

1003816

1003816

119911

1

1003816

1003816

1003816

1003816

lt 120576

(19)


1|




1| lt 120576


119903sin(120595 minus 120595

119903)must be



120595

119888= 120595

119903minus 119896

2tanh (119911

2) sign (119865

119903) (20)

119903

119888=

120595

119888=

120595

119903minus

2119896


2)

2

) sign (119865119903) (21)



119888and 0 lt 119896

2lt 1205872

then 119865

119903sin(120595119888minus 120595

119903) = minus|119865

119903| sin(119896

2tanh(119911

2)) that is opposed




119890

120595= 120595 minus 120595

119888 (22)

119890

119903= 119903 minus 119903

119888 (23)


1= 119896

1119865 minus 119896

1119865

119903cos (119890

120595+ 120595

119888minus 120595

119903) + 119911

2119903 + 119896

1119901

1

+ (1 minus 119889

119906119896

1) (119890V119909 cos (119890120595 + 120595

119888) + 119890V119910 sin (119890120595 + 120595

119888))

(24)

2= minus 119896

1

1003816

1003816

1003816

1003816

119865

119903

1003816

1003816

1003816

1003816

sin (1198962tanh (119911

2)) minus 119911

1119903 + 119896

1119865

119903119908 + 119896

1119901

2

+ (1 minus 119889

119906119896

1) (119890V119910 cos (119890120595 + 120595

119888) minus 119890V119909 sin (119890120595 + 120595

119888))

(25)



119903) minus sin(120595

119888minus 120595

119903) bounded by

|119908| le

1003816

1003816

1003816

1003816

120595 minus 120595

119888

1003816

1003816

1003816

1003816

=

1003816

1003816

1003816

1003816

1003816

119890

120595

1003816

1003816

1003816

1003816

1003816

(26)



2



119890

120595=

120595 minus

120595

119888= 119903 minus

120595

119888= 119890

119903

119890

119903= 120591 minus 120591

119888minus 119889

119903119890

119903+ 119901

119903

(27)

where 120591119888= 119889

119903

120595

119888minus

120595



gain 119896


120595as

follows

119904

120595= 119890

120595+ 119896

3119890

119903 (28)




119904

120595= 119896

3(120591 minus 120591

119888+ 119901

119903) + (1 minus 119896

3119889

119903) 119890

119903 (29)




sign (120591) =

minus sign (119904120595) if 1003816

1003816

1003816

1003816

1003816

119904

120595

1003816

1003816

1003816

1003816

1003816

ge 120576

sign (120591minus) if 1003816

1003816

1003816

1003816

1003816

119904

120595

1003816

1003816

1003816

1003816

1003816

lt 120576

(30)


120595|






119909and 119904

119910are


1= 119911

1+ 119899

1and

2= 119911

2+ 119899

2 where

1003816

1003816

1003816

1003816

119899

1

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119899

2

1003816

1003816

1003816

1003816

le (z + radic

2 (1 + 119896

1)) 119899

119898 (31)



119909


the estimate 120595





derivative

120595



120595+ 119891

3 where

1003816

1003816

1003816

1003816

119891

3

1003816

1003816

1003816

1003816

le 119896

3119896

2119896

1119901

119898+ 119899

119898(1 + 119896

3+ 119899

119898+ 119896

1

1003816

1003816

1003816

1003816

119865

119903

1003816

1003816

1003816

1003816

+ |119903|)

+ 119896

2119899

119898(1 + 119896

3

1003816

1003816

1003816

1003816

2

1003816

1003816

1003816

1003816

) (z + radic

2 (1 + 119896

1))

+ 2

radic

2119899

119898

1003816

1003816

1003816

1003816

1 minus 119889

119906119896

1

1003816

1003816

1003816

1003816

radic119890

2

V119909 + 119890

2

V119910

(32)


sign (119865) =


1003816

1003816

1003816

1

1003816

1003816

1003816

1003816

ge 120576

sign (119865minus) if 1003816

1003816

1003816

1003816

1

1003816

1003816

1003816

1003816

lt 120576

(33)

sign (120591) =

minus sign (119904120595) if 1003816

1003816

1003816

1003816

1003816

119904

120595

1003816

1003816

1003816

1003816

1003816

ge 120576

sign (120591minus) if 1003816

1003816

1003816

1003816

1003816

119904

120595

1003816

1003816

1003816

1003816

1003816

lt 120576

(34)


119890and 119865


Table 2










1 1198962

and 119908



119898 then

z (119905) le 120573 (z (0) 119905) + 120574 (119908

119898+ 119899

119898+ 119901

119898+ 120576) (35)

Proof on Section A2



that for |119908| le 119908

119898

1003817

1003817

1003817

1003817

1003817

e119901(119905)

1003817

1003817

1003817

1003817

1003817

le 120573

2(

1003817

1003817

1003817

1003817

1003817

e119901(0)

1003817

1003817

1003817

1003817

1003817

119905) + 120574

2(119908

119898+ 119899

119898+ 119901

119898+ 120576) (36)

Proof in Section A3







119898 1198963 119886 119887 and 119888 and a K class

function 120574


119903and any


2 if z le 119911

119898 then

1003816

1003816

1003816

1003816

1003816

119904

120595(119905)

1003816

1003816

1003816

1003816

1003816

le max (1003816100381610038161003816

1003816

119904

120595(0)

1003816

1003816

1003816

1003816

1003816

minus 119886119905 120574

3(119887 + 119911

119898+ 119901

119898) 119899

119898+ 119888119901

119898120576)

(37)

Proof in Section A4


119898

1003816

1003816

1003816

1003816

1003816

119890

120595(119905)

1003816

1003816

1003816

1003816

1003816

le 119889 sdot 119890

minus1199051198963+ 120574

3(119887 + 119911

119898+ 119901

119898) 119899

119898+ 119888119901

119898+ 120576 (38)

Proof in Section A5














that1003817

1003817

1003817

1003817

1003817

e119901(119905)

1003817

1003817

1003817

1003817

1003817

le 120573

3(

1003817

1003817

1003817

1003817

1003817

e119901(0)

1003817

1003817

1003817

1003817

1003817

119905) + 120574

4(119899

119898+ 119901

119898+ 120576) (39)



Proof in Section A6






7 Results



1= 6 s 119896

2= 1 s 119896

3= 1 s and 120576 = 001


119903= 02179ms2 and 120591

119903= 04339 rads2) so the





120595 is





references 120595119903and 119903

119903 too



minus2

minus15

minus1

minus05

0

05

1

15

2

x(m

)

y (m)


0 10 20 30 40 50 60minus3

minus2

minus1

0123

x(m

)

t (s)

(a)

0 10 20 30 40 50 60minus3

minus2

minus1

0123

y(m

)

t (s)

(b)



119909and V119910 rad



0 10 20 30 40 50 60

minus04minus02

0020406

x(m

s)

t (s)

(a)

0 10 20 30 40 50 60

minus04minus02

0020406

t (s)

y(m

s)

(b)


0 10 20 30 40 50 60minus4

minus2

0

2

4

6120595

(rad

)

t (s)

(a)


0020406

r(r

ads

)

t (s)

(b)






minus3

minus2

minus1

0

1

2

3

4

x(m

)

y (m)


0 10 20 30 40 50 600

05

1

15

2

25

3

35

4

e p(m

)

t (s)





119903(119905) = 2 sin(38119905) 119910

119903(119905) = 3 cos(28119905)



minus06

minus04

minus02

0

02

04

06

08

1

12

120591 (rads2)

F(m

s2)



minus2

minus15

minus1

minus05

0

05

1

15

2x

(m)

y (m)






value than 120591








minus06

minus04

minus02

0

02

04

06

08

x(m

)

y (m)







0 5 10 15 20 250

01

02

03

04

05

06

07

08

09

1

Dist

ance

(m)

t





119865


radio controller





0020406

x(m

)

t (s)

(a)


002040608

y(m

)

t (s)

(b)


t (s)0 5 10 15 20 25

minus1

minus05

0

05

1

x(m

s)

(a)

0 5 10 15 20 25minus1

minus05

0

05

1

y(m

s)

t (s)

(b)


8 Conclusions



0 5 10 15 20 25minus04

minus02

0020406

Fs

(N)

t (s)

(a)

0 5 10 15 20 25minus04

minus02

0020406

t (s)

F p(N

)

(b)






Appendix

A Proofs



119871

2= V21199102 and 119871

3= 119903

2


119871

1= minusV119909(119889

119906V119909minus119865 cos(120595)+119901V119909)



119871

1can be bounded by

119871

1lt minus119889

119906V2119909(1 minus

119865max + 119901

119898

119889

119906

1003816

1003816

1003816

1003816

V119909

1003816

1003816

1003816

1003816

) (A1)


119871


119889

119906|V119909| gt (119865max + 119901


119909is ultimately



119898)119889


with 119871

2and 119871


Ω = x isin R6

1003816

1003816

1003816

1003816

V119909

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

V119910

1003816

1003816

1003816

1003816

1003816

le

119865max + 119901

119898

119889

119906

|119903| le

120591max + 119901

119898

119889

119903

(A2)



119903| minus 119901




radic

2

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

(1 minus 119889

119906119896

1)

119889

119906119896

1

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

(2119865max + 119901

119898) le 120575120576

1 (A3)


119898so that

radic

2 (119901

119898+ 120575120576

1+

1003816

1003816

1003816

1003816

119865

119903

1003816

1003816

1003816

1003816

119908

119898) lt min (120575 100381610038161003816

1003816

119865

119903

1003816

1003816

1003816

1003816

sin (1198962)) (A4)


(119911

2

1+ 119911

2


119881 = 119896

1119911

1(119865 minus 119865

119903cos (120595 minus 120595

119903) + 119901

1)

minus 119896

1119911

2(

1003816

1003816

1003816

1003816

119865

119903

1003816

1003816

1003816

1003816

sin (1198962tanh (119911

2)) minus 119865

119903119908 minus 119901

2)

+ (1 minus 119889

119906119896

1) (119904

1119890V119909 + 119904

2119890V119910)

(A5)


Case A (|1199111| gt 119899



119903cos(120595 minus 120595


by |119865119903| and |119911


following function

119892 (119909) = min( 120575

radic2

119865

119903

radic2

sin(1198962tanh( 119909

radic2

))) (A6)


1| minus |119865

119903119911

2| sin(119896

2tanh (|119911


119881 le minus119896

1z (119892 (z) minus (120575120576

1+

1003816

1003816

1003816

1003816

119865

119903

1003816

1003816

1003816

1003816

|119908| + 119901

119898)) (A7)


1+ |119865

119903|119908

119898+ 119901

119898


119911max 1 = radic

2 tanhminus1 ( 1

119896

2

sinminus1 (radic2(

119901max + 120575120576

1003816

1003816

1003816

1003816

119865

119903

1003816

1003816

1003816

1003816

+ 119908

119898)))

(A8)



Case B (|1199111| le 119899



2| and

with (31) we find

z le radic

2

1003816

1003816

1003816

1003816

119911

1

1003816

1003816

1003816

1003816

le

radic

2119899

1+ 120576 le

radic

2 (z + radic

2 (1 + 119896

1)) 119899

119898+ 120576

(A9)


z leradic2 (

radic2 (1 + 119896

1) 119899

119898+ 120576)


119898)

= 120576

2 (A10)


2| then z le radic

2|119911

2| so we


2| + 120576

2and |119911

1| le 2(|119911

2| +

1 minus 119896

1)119899


119881 le 119896

1(2 (

1003816

1003816

1003816

1003816

119911

2

1003816

1003816

1003816

1003816

+ 1 minus 119896

1) 119899

119898+ 120576) (

1003816

1003816

1003816

1003816

119865max1003816

1003816

1003816

1003816

+

1003816

1003816

1003816

1003816

119865

119903

1003816

1003816

1003816

1003816

+ 119901

119898)

minus 119896

1

1003816

1003816

1003816

1003816

119911

2

1003816

1003816

1003816

1003816

(

1003816

1003816

1003816

1003816

119865

119903

1003816

1003816

1003816

1003816

sin (1198962tanh (100381610038161003816

1003816

119911

2

1003816

1003816

1003816

1003816

)) minus

1003816

1003816

1003816

1003816

119865

119903119908

1003816

1003816

1003816

1003816

minus 119901

119898)

+ 2 (

radic

2

1003816

1003816

1003816

1003816

119911

2

1003816

1003816

1003816

1003816

+ 120576

2) 119889

119906119896

1120575120576

1

(A11)


3and 120576

4 (A11) can be

rewritten as

119881 le minus 119896

1(

1003816

1003816

1003816

1003816

119911

2

1003816

1003816

1003816

1003816

(

1003816

1003816

1003816

1003816

119865

119903

1003816

1003816

1003816

1003816

sin (1198962tanh (100381610038161003816

1003816

119911

2

1003816

1003816

1003816

1003816

))

minus

1003816

1003816

1003816

1003816

119865

119903119908

1003816

1003816

1003816

1003816

minus 119901

119898minus 120576

3) + 120576

4)

(A12)

Now we define


119896

2

sinminus1 (119901

119898

1003816

1003816

1003816

1003816

119865

119903

1003816

1003816

1003816

1003816

+ 120576

3+ 119908

119898+ 120575)) (A13)



119881 le 119896




119898


119898 119899119898 119901119898 and 120576 then

(35) holds


119909and 119904

119910

119896

1119890

119909= 119904

119909minus 119890

119909

119896

1119890

119910= 119904

119910minus 119890

119910

(A14)


119898 119901119898 119908119898 and 120576 then (36) holds



119903| le |119903| + |119903

119903| le |120591

119898|119889

119903+ |119903

119903| The



z le 1003816

1003816

1003816

1003816

1

1003816

1003816

1003816

1003816

+

1003816

1003816

1003816

1003816

2

1003816

1003816

1003816

1003816

le 119888

1+ 119888

2119911

119898+ 119888

3119901

119898

1003816

1003816

1003816

1003816

2

1003816

1003816

1003816

1003816

le 119888

4+ 119888

5119911

119898+ 119888

6119901

119898

(A15)


120591

119888= 120591

119903minus 119896

2sign (119865

119903) (1 minus tanh2 (119911

2)) (119889

119903

2+

2)

+ 2119896

2sign (119865

119903) (1 minus tanh2 (119911

2)) tanh (119911

2)

2

2

(A16)


120595and 120591

119888can be




1003816

1003816

1003816

1003816

120591

119888

1003816

1003816

1003816

1003816

le

1003816

1003816

1003816

1003816

120591

119903

1003816

1003816

1003816

1003816

+ 119896

2120574

4(119911

119898+ 119901

119898) (A17)

1003816

1003816

1003816

1003816

119891

3

1003816

1003816

1003816

1003816

le 120574

5(119888

7+ 119911

119898+ 119901

119898) 119899

119898+ 119896

3119896

2119896

1119901

119898 (A18)



120575 Now we define 119911119898as

119911

119898= minus

1

2

120574

4

minus1

(

120575

4119896

2

) (A19)

When z lt 119911


119896

2120574

4(2119901

119898)+119901

119898lt 1205754 and with (A17) then |120591

119894|minus|120591

119888|minus|119901

119898| ge


1003816

1003816

1003816

1003816

1 minus 119896

3119889

119903

1003816

1003816

1003816

1003816

119896

3119889

119903

(

1003816

1003816

1003816

1003816

120591

119898

1003816

1003816

1003816

1003816

+ 119889

119903

1003816

1003816

1003816

1003816

119903

119903

1003816

1003816

1003816

1003816

) le

120575

4

(A20)


2= 119904

2

1205952 whose time

derivative is

119881

2= 119904

120595(119896

3(120591 minus 120591

119888+ 119901

119903) + (1 minus 119896

3119889

119903)119890

119903) Consider

first that |119904120595| ge 119891



119881

2= 119904

120595119904

120595le minus120575119896

3(

1

2

minus

1

4

)

1003816

1003816

1003816

1003816

1003816

119904

120595

1003816

1003816

1003816

1003816

1003816

997888rarr 119904

120595le minus

120575119896

3

4

sign (119904120595)

(A21)


120595(119905)| le |119904

120595(0)| minus

120575119896

34119905 If 119904


120595| lt 119891

3+120576 (A21) does not




case |119904120595(119905)| le max(|119904

120595(0)| minus 120575119896

34119905 119891





120595= 119904

120595minus 119890

120595 whose

solution is

119890

120595(119905) = 119890

120595(0) 119890

minus1199051198963+

1

119896

3

int

119905

0

119890

minus(119905minus120591)1198963119904

120595(120591) 119889120591 (A22)


1003816

1003816

1003816

1003816

1003816

119890

120595(119905)

1003816

1003816

1003816

1003816

1003816

le

1003816

1003816

1003816

1003816

1003816

119890

120595(0)

1003816

1003816

1003816

1003816

1003816

119890

minus1199051198963+

1003816

1003816

1003816

1003816

1003816

119904

120595(0)

1003816

1003816

1003816

1003816

1003816

(119890

|119904120595(0)|1198861198963minus 1) 119890

minus1199051198963

+ 120574

3(119887 + 119911

119898+ 119901

119898) 119899

119898+ 119888119901

119898+ 120576

(A23)


120595| + |119904

120595|119890

|119904120595|1198861198963






119898Therefore



119898=

2119872 as follows

119896

2=

120575

4120574

4(4119872)

(A24)


119881 le 119896

1

1003816

1003816

1003816

1003816

119911

2

1003816

1003816

1003816

1003816

(

1003816

1003816

1003816

1003816

119865

119903119908

1003816

1003816

1003816

1003816

+ 119901

2) + (1 minus 119889

119906119896

1) (119904

1119890V119909 + 119904

2119890V119910)

le 119887

1|z| + 119887

2

(A25)


2 Select 119872 gt 119887

2119887

1then z gt

119887

2119887

1and thus

119881 = zz le 2119887


1rarr z le

z(0) + 2119887


2119887

1and therefore 119879 gt 1198722119887

2


3(119887 + 119872 + 119901

119898)119899

119898+ 119888119901

119898+ 120576 le 119889119890


120595(119879)| le 2119889 sdot 119890


max(119901

119898+ 120576

5

1003816

1003816

1003816

1003816

119865

119903

1003816

1003816

1003816

1003816

119901

119898+ 120575120576

1003816

1003816

1003816

1003816

119865

119903

1003816

1003816

1003816

1003816

) lt 119889 sdot 119890

minus1198791198963 (A26)


119911max (119879) = radic


4(4119872)

120575


minus119872211988821198963))

(A27)

Function 120574



119898



1for which z(119905



2for which z(119905

2) = 2119872 and 119881

grows in 119905



120595(119905

2)| le 2119889119890


119911max(|119890120595(1199052)|) = 119872 because z lt 119911

119898for all 119879 le 119905

1le 119905

2 This




2



Acknowledgments


References









































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119903) minus sin(120595

119888minus 120595

119903) bounded by

|119908| le

1003816

1003816

1003816

1003816

120595 minus 120595

119888

1003816

1003816

1003816

1003816

=

1003816

1003816

1003816

1003816

1003816

119890

120595

1003816

1003816

1003816

1003816

1003816

(26)



2



119890

120595=

120595 minus

120595

119888= 119903 minus

120595

119888= 119890

119903

119890

119903= 120591 minus 120591

119888minus 119889

119903119890

119903+ 119901

119903

(27)

where 120591119888= 119889

119903

120595

119888minus

120595



gain 119896


120595as

follows

119904

120595= 119890

120595+ 119896

3119890

119903 (28)




119904

120595= 119896

3(120591 minus 120591

119888+ 119901

119903) + (1 minus 119896

3119889

119903) 119890

119903 (29)




sign (120591) =

minus sign (119904120595) if 1003816

1003816

1003816

1003816

1003816

119904

120595

1003816

1003816

1003816

1003816

1003816

ge 120576

sign (120591minus) if 1003816

1003816

1003816

1003816

1003816

119904

120595

1003816

1003816

1003816

1003816

1003816

lt 120576

(30)


120595|






119909and 119904

119910are


1= 119911

1+ 119899

1and

2= 119911

2+ 119899

2 where

1003816

1003816

1003816

1003816

119899

1

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119899

2

1003816

1003816

1003816

1003816

le (z + radic

2 (1 + 119896

1)) 119899

119898 (31)



119909


the estimate 120595





derivative

120595



120595+ 119891

3 where

1003816

1003816

1003816

1003816

119891

3

1003816

1003816

1003816

1003816

le 119896

3119896

2119896

1119901

119898+ 119899

119898(1 + 119896

3+ 119899

119898+ 119896

1

1003816

1003816

1003816

1003816

119865

119903

1003816

1003816

1003816

1003816

+ |119903|)

+ 119896

2119899

119898(1 + 119896

3

1003816

1003816

1003816

1003816

2

1003816

1003816

1003816

1003816

) (z + radic

2 (1 + 119896

1))

+ 2

radic

2119899

119898

1003816

1003816

1003816

1003816

1 minus 119889

119906119896

1

1003816

1003816

1003816

1003816

radic119890

2

V119909 + 119890

2

V119910

(32)


sign (119865) =


1003816

1003816

1003816

1

1003816

1003816

1003816

1003816

ge 120576

sign (119865minus) if 1003816

1003816

1003816

1003816

1

1003816

1003816

1003816

1003816

lt 120576

(33)

sign (120591) =

minus sign (119904120595) if 1003816

1003816

1003816

1003816

1003816

119904

120595

1003816

1003816

1003816

1003816

1003816

ge 120576

sign (120591minus) if 1003816

1003816

1003816

1003816

1003816

119904

120595

1003816

1003816

1003816

1003816

1003816

lt 120576

(34)


119890and 119865


Table 2










1 1198962

and 119908



119898 then

z (119905) le 120573 (z (0) 119905) + 120574 (119908

119898+ 119899

119898+ 119901

119898+ 120576) (35)

Proof on Section A2



that for |119908| le 119908

119898

1003817

1003817

1003817

1003817

1003817

e119901(119905)

1003817

1003817

1003817

1003817

1003817

le 120573

2(

1003817

1003817

1003817

1003817

1003817

e119901(0)

1003817

1003817

1003817

1003817

1003817

119905) + 120574

2(119908

119898+ 119899

119898+ 119901

119898+ 120576) (36)

Proof in Section A3







119898 1198963 119886 119887 and 119888 and a K class

function 120574


119903and any


2 if z le 119911

119898 then

1003816

1003816

1003816

1003816

1003816

119904

120595(119905)

1003816

1003816

1003816

1003816

1003816

le max (1003816100381610038161003816

1003816

119904

120595(0)

1003816

1003816

1003816

1003816

1003816

minus 119886119905 120574

3(119887 + 119911

119898+ 119901

119898) 119899

119898+ 119888119901

119898120576)

(37)

Proof in Section A4


119898

1003816

1003816

1003816

1003816

1003816

119890

120595(119905)

1003816

1003816

1003816

1003816

1003816

le 119889 sdot 119890

minus1199051198963+ 120574

3(119887 + 119911

119898+ 119901

119898) 119899

119898+ 119888119901

119898+ 120576 (38)

Proof in Section A5














that1003817

1003817

1003817

1003817

1003817

e119901(119905)

1003817

1003817

1003817

1003817

1003817

le 120573

3(

1003817

1003817

1003817

1003817

1003817

e119901(0)

1003817

1003817

1003817

1003817

1003817

119905) + 120574

4(119899

119898+ 119901

119898+ 120576) (39)



Proof in Section A6






7 Results



1= 6 s 119896

2= 1 s 119896

3= 1 s and 120576 = 001


119903= 02179ms2 and 120591

119903= 04339 rads2) so the





120595 is





references 120595119903and 119903

119903 too



minus2

minus15

minus1

minus05

0

05

1

15

2

x(m

)

y (m)


0 10 20 30 40 50 60minus3

minus2

minus1

0123

x(m

)

t (s)

(a)

0 10 20 30 40 50 60minus3

minus2

minus1

0123

y(m

)

t (s)

(b)



119909and V119910 rad



0 10 20 30 40 50 60

minus04minus02

0020406

x(m

s)

t (s)

(a)

0 10 20 30 40 50 60

minus04minus02

0020406

t (s)

y(m

s)

(b)


0 10 20 30 40 50 60minus4

minus2

0

2

4

6120595

(rad

)

t (s)

(a)


0020406

r(r

ads

)

t (s)

(b)






minus3

minus2

minus1

0

1

2

3

4

x(m

)

y (m)


0 10 20 30 40 50 600

05

1

15

2

25

3

35

4

e p(m

)

t (s)





119903(119905) = 2 sin(38119905) 119910

119903(119905) = 3 cos(28119905)



minus06

minus04

minus02

0

02

04

06

08

1

12

120591 (rads2)

F(m

s2)



minus2

minus15

minus1

minus05

0

05

1

15

2x

(m)

y (m)






value than 120591








minus06

minus04

minus02

0

02

04

06

08

x(m

)

y (m)







0 5 10 15 20 250

01

02

03

04

05

06

07

08

09

1

Dist

ance

(m)

t





119865


radio controller





0020406

x(m

)

t (s)

(a)


002040608

y(m

)

t (s)

(b)


t (s)0 5 10 15 20 25

minus1

minus05

0

05

1

x(m

s)

(a)

0 5 10 15 20 25minus1

minus05

0

05

1

y(m

s)

t (s)

(b)


8 Conclusions



0 5 10 15 20 25minus04

minus02

0020406

Fs

(N)

t (s)

(a)

0 5 10 15 20 25minus04

minus02

0020406

t (s)

F p(N

)

(b)






Appendix

A Proofs



119871

2= V21199102 and 119871

3= 119903

2


119871

1= minusV119909(119889

119906V119909minus119865 cos(120595)+119901V119909)



119871

1can be bounded by

119871

1lt minus119889

119906V2119909(1 minus

119865max + 119901

119898

119889

119906

1003816

1003816

1003816

1003816

V119909

1003816

1003816

1003816

1003816

) (A1)


119871


119889

119906|V119909| gt (119865max + 119901


119909is ultimately



119898)119889


with 119871

2and 119871


Ω = x isin R6

1003816

1003816

1003816

1003816

V119909

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

V119910

1003816

1003816

1003816

1003816

1003816

le

119865max + 119901

119898

119889

119906

|119903| le

120591max + 119901

119898

119889

119903

(A2)



119903| minus 119901




radic

2

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

(1 minus 119889

119906119896

1)

119889

119906119896

1

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

(2119865max + 119901

119898) le 120575120576

1 (A3)


119898so that

radic

2 (119901

119898+ 120575120576

1+

1003816

1003816

1003816

1003816

119865

119903

1003816

1003816

1003816

1003816

119908

119898) lt min (120575 100381610038161003816

1003816

119865

119903

1003816

1003816

1003816

1003816

sin (1198962)) (A4)


(119911

2

1+ 119911

2


119881 = 119896

1119911

1(119865 minus 119865

119903cos (120595 minus 120595

119903) + 119901

1)

minus 119896

1119911

2(

1003816

1003816

1003816

1003816

119865

119903

1003816

1003816

1003816

1003816

sin (1198962tanh (119911

2)) minus 119865

119903119908 minus 119901

2)

+ (1 minus 119889

119906119896

1) (119904

1119890V119909 + 119904

2119890V119910)

(A5)


Case A (|1199111| gt 119899



119903cos(120595 minus 120595


by |119865119903| and |119911


following function

119892 (119909) = min( 120575

radic2

119865

119903

radic2

sin(1198962tanh( 119909

radic2

))) (A6)


1| minus |119865

119903119911

2| sin(119896

2tanh (|119911


119881 le minus119896

1z (119892 (z) minus (120575120576

1+

1003816

1003816

1003816

1003816

119865

119903

1003816

1003816

1003816

1003816

|119908| + 119901

119898)) (A7)


1+ |119865

119903|119908

119898+ 119901

119898


119911max 1 = radic

2 tanhminus1 ( 1

119896

2

sinminus1 (radic2(

119901max + 120575120576

1003816

1003816

1003816

1003816

119865

119903

1003816

1003816

1003816

1003816

+ 119908

119898)))

(A8)



Case B (|1199111| le 119899



2| and

with (31) we find

z le radic

2

1003816

1003816

1003816

1003816

119911

1

1003816

1003816

1003816

1003816

le

radic

2119899

1+ 120576 le

radic

2 (z + radic

2 (1 + 119896

1)) 119899

119898+ 120576

(A9)


z leradic2 (

radic2 (1 + 119896

1) 119899

119898+ 120576)


119898)

= 120576

2 (A10)


2| then z le radic

2|119911

2| so we


2| + 120576

2and |119911

1| le 2(|119911

2| +

1 minus 119896

1)119899


119881 le 119896

1(2 (

1003816

1003816

1003816

1003816

119911

2

1003816

1003816

1003816

1003816

+ 1 minus 119896

1) 119899

119898+ 120576) (

1003816

1003816

1003816

1003816

119865max1003816

1003816

1003816

1003816

+

1003816

1003816

1003816

1003816

119865

119903

1003816

1003816

1003816

1003816

+ 119901

119898)

minus 119896

1

1003816

1003816

1003816

1003816

119911

2

1003816

1003816

1003816

1003816

(

1003816

1003816

1003816

1003816

119865

119903

1003816

1003816

1003816

1003816

sin (1198962tanh (100381610038161003816

1003816

119911

2

1003816

1003816

1003816

1003816

)) minus

1003816

1003816

1003816

1003816

119865

119903119908

1003816

1003816

1003816

1003816

minus 119901

119898)

+ 2 (

radic

2

1003816

1003816

1003816

1003816

119911

2

1003816

1003816

1003816

1003816

+ 120576

2) 119889

119906119896

1120575120576

1

(A11)


3and 120576

4 (A11) can be

rewritten as

119881 le minus 119896

1(

1003816

1003816

1003816

1003816

119911

2

1003816

1003816

1003816

1003816

(

1003816

1003816

1003816

1003816

119865

119903

1003816

1003816

1003816

1003816

sin (1198962tanh (100381610038161003816

1003816

119911

2

1003816

1003816

1003816

1003816

))

minus

1003816

1003816

1003816

1003816

119865

119903119908

1003816

1003816

1003816

1003816

minus 119901

119898minus 120576

3) + 120576

4)

(A12)

Now we define


119896

2

sinminus1 (119901

119898

1003816

1003816

1003816

1003816

119865

119903

1003816

1003816

1003816

1003816

+ 120576

3+ 119908

119898+ 120575)) (A13)



119881 le 119896




119898


119898 119899119898 119901119898 and 120576 then

(35) holds


119909and 119904

119910

119896

1119890

119909= 119904

119909minus 119890

119909

119896

1119890

119910= 119904

119910minus 119890

119910

(A14)


119898 119901119898 119908119898 and 120576 then (36) holds



119903| le |119903| + |119903

119903| le |120591

119898|119889

119903+ |119903

119903| The



z le 1003816

1003816

1003816

1003816

1

1003816

1003816

1003816

1003816

+

1003816

1003816

1003816

1003816

2

1003816

1003816

1003816

1003816

le 119888

1+ 119888

2119911

119898+ 119888

3119901

119898

1003816

1003816

1003816

1003816

2

1003816

1003816

1003816

1003816

le 119888

4+ 119888

5119911

119898+ 119888

6119901

119898

(A15)


120591

119888= 120591

119903minus 119896

2sign (119865

119903) (1 minus tanh2 (119911

2)) (119889

119903

2+

2)

+ 2119896

2sign (119865

119903) (1 minus tanh2 (119911

2)) tanh (119911

2)

2

2

(A16)


120595and 120591

119888can be




1003816

1003816

1003816

1003816

120591

119888

1003816

1003816

1003816

1003816

le

1003816

1003816

1003816

1003816

120591

119903

1003816

1003816

1003816

1003816

+ 119896

2120574

4(119911

119898+ 119901

119898) (A17)

1003816

1003816

1003816

1003816

119891

3

1003816

1003816

1003816

1003816

le 120574

5(119888

7+ 119911

119898+ 119901

119898) 119899

119898+ 119896

3119896

2119896

1119901

119898 (A18)



120575 Now we define 119911119898as

119911

119898= minus

1

2

120574

4

minus1

(

120575

4119896

2

) (A19)

When z lt 119911


119896

2120574

4(2119901

119898)+119901

119898lt 1205754 and with (A17) then |120591

119894|minus|120591

119888|minus|119901

119898| ge


1003816

1003816

1003816

1003816

1 minus 119896

3119889

119903

1003816

1003816

1003816

1003816

119896

3119889

119903

(

1003816

1003816

1003816

1003816

120591

119898

1003816

1003816

1003816

1003816

+ 119889

119903

1003816

1003816

1003816

1003816

119903

119903

1003816

1003816

1003816

1003816

) le

120575

4

(A20)


2= 119904

2

1205952 whose time

derivative is

119881

2= 119904

120595(119896

3(120591 minus 120591

119888+ 119901

119903) + (1 minus 119896

3119889

119903)119890

119903) Consider

first that |119904120595| ge 119891



119881

2= 119904

120595119904

120595le minus120575119896

3(

1

2

minus

1

4

)

1003816

1003816

1003816

1003816

1003816

119904

120595

1003816

1003816

1003816

1003816

1003816

997888rarr 119904

120595le minus

120575119896

3

4

sign (119904120595)

(A21)


120595(119905)| le |119904

120595(0)| minus

120575119896

34119905 If 119904


120595| lt 119891

3+120576 (A21) does not




case |119904120595(119905)| le max(|119904

120595(0)| minus 120575119896

34119905 119891





120595= 119904

120595minus 119890

120595 whose

solution is

119890

120595(119905) = 119890

120595(0) 119890

minus1199051198963+

1

119896

3

int

119905

0

119890

minus(119905minus120591)1198963119904

120595(120591) 119889120591 (A22)


1003816

1003816

1003816

1003816

1003816

119890

120595(119905)

1003816

1003816

1003816

1003816

1003816

le

1003816

1003816

1003816

1003816

1003816

119890

120595(0)

1003816

1003816

1003816

1003816

1003816

119890

minus1199051198963+

1003816

1003816

1003816

1003816

1003816

119904

120595(0)

1003816

1003816

1003816

1003816

1003816

(119890

|119904120595(0)|1198861198963minus 1) 119890

minus1199051198963

+ 120574

3(119887 + 119911

119898+ 119901

119898) 119899

119898+ 119888119901

119898+ 120576

(A23)


120595| + |119904

120595|119890

|119904120595|1198861198963






119898Therefore



119898=

2119872 as follows

119896

2=

120575

4120574

4(4119872)

(A24)


119881 le 119896

1

1003816

1003816

1003816

1003816

119911

2

1003816

1003816

1003816

1003816

(

1003816

1003816

1003816

1003816

119865

119903119908

1003816

1003816

1003816

1003816

+ 119901

2) + (1 minus 119889

119906119896

1) (119904

1119890V119909 + 119904

2119890V119910)

le 119887

1|z| + 119887

2

(A25)


2 Select 119872 gt 119887

2119887

1then z gt

119887

2119887

1and thus

119881 = zz le 2119887


1rarr z le

z(0) + 2119887


2119887

1and therefore 119879 gt 1198722119887

2


3(119887 + 119872 + 119901

119898)119899

119898+ 119888119901

119898+ 120576 le 119889119890


120595(119879)| le 2119889 sdot 119890


max(119901

119898+ 120576

5

1003816

1003816

1003816

1003816

119865

119903

1003816

1003816

1003816

1003816

119901

119898+ 120575120576

1003816

1003816

1003816

1003816

119865

119903

1003816

1003816

1003816

1003816

) lt 119889 sdot 119890

minus1198791198963 (A26)


119911max (119879) = radic


4(4119872)

120575


minus119872211988821198963))

(A27)

Function 120574



119898



1for which z(119905



2for which z(119905

2) = 2119872 and 119881

grows in 119905



120595(119905

2)| le 2119889119890


119911max(|119890120595(1199052)|) = 119872 because z lt 119911

119898for all 119879 le 119905

1le 119905

2 This




2



Acknowledgments


References









































Volume 2014




Journal of











Journal of


Function Spaces






Algebra















119898 1198963 119886 119887 and 119888 and a K class

function 120574


119903and any


2 if z le 119911

119898 then

1003816

1003816

1003816

1003816

1003816

119904

120595(119905)

1003816

1003816

1003816

1003816

1003816

le max (1003816100381610038161003816

1003816

119904

120595(0)

1003816

1003816

1003816

1003816

1003816

minus 119886119905 120574

3(119887 + 119911

119898+ 119901

119898) 119899

119898+ 119888119901

119898120576)

(37)

Proof in Section A4


119898

1003816

1003816

1003816

1003816

1003816

119890

120595(119905)

1003816

1003816

1003816

1003816

1003816

le 119889 sdot 119890

minus1199051198963+ 120574

3(119887 + 119911

119898+ 119901

119898) 119899

119898+ 119888119901

119898+ 120576 (38)

Proof in Section A5














that1003817

1003817

1003817

1003817

1003817

e119901(119905)

1003817

1003817

1003817

1003817

1003817

le 120573

3(

1003817

1003817

1003817

1003817

1003817

e119901(0)

1003817

1003817

1003817

1003817

1003817

119905) + 120574

4(119899

119898+ 119901

119898+ 120576) (39)



Proof in Section A6






7 Results



1= 6 s 119896

2= 1 s 119896

3= 1 s and 120576 = 001


119903= 02179ms2 and 120591

119903= 04339 rads2) so the





120595 is





references 120595119903and 119903

119903 too



minus2

minus15

minus1

minus05

0

05

1

15

2

x(m

)

y (m)


0 10 20 30 40 50 60minus3

minus2

minus1

0123

x(m

)

t (s)

(a)

0 10 20 30 40 50 60minus3

minus2

minus1

0123

y(m

)

t (s)

(b)



119909and V119910 rad



0 10 20 30 40 50 60

minus04minus02

0020406

x(m

s)

t (s)

(a)

0 10 20 30 40 50 60

minus04minus02

0020406

t (s)

y(m

s)

(b)


0 10 20 30 40 50 60minus4

minus2

0

2

4

6120595

(rad

)

t (s)

(a)


0020406

r(r

ads

)

t (s)

(b)






minus3

minus2

minus1

0

1

2

3

4

x(m

)

y (m)


0 10 20 30 40 50 600

05

1

15

2

25

3

35

4

e p(m

)

t (s)





119903(119905) = 2 sin(38119905) 119910

119903(119905) = 3 cos(28119905)



minus06

minus04

minus02

0

02

04

06

08

1

12

120591 (rads2)

F(m

s2)



minus2

minus15

minus1

minus05

0

05

1

15

2x

(m)

y (m)






value than 120591








minus06

minus04

minus02

0

02

04

06

08

x(m

)

y (m)







0 5 10 15 20 250

01

02

03

04

05

06

07

08

09

1

Dist

ance

(m)

t





119865


radio controller





0020406

x(m

)

t (s)

(a)


002040608

y(m

)

t (s)

(b)


t (s)0 5 10 15 20 25

minus1

minus05

0

05

1

x(m

s)

(a)

0 5 10 15 20 25minus1

minus05

0

05

1

y(m

s)

t (s)

(b)


8 Conclusions



0 5 10 15 20 25minus04

minus02

0020406

Fs

(N)

t (s)

(a)

0 5 10 15 20 25minus04

minus02

0020406

t (s)

F p(N

)

(b)






Appendix

A Proofs



119871

2= V21199102 and 119871

3= 119903

2


119871

1= minusV119909(119889

119906V119909minus119865 cos(120595)+119901V119909)



119871

1can be bounded by

119871

1lt minus119889

119906V2119909(1 minus

119865max + 119901

119898

119889

119906

1003816

1003816

1003816

1003816

V119909

1003816

1003816

1003816

1003816

) (A1)


119871


119889

119906|V119909| gt (119865max + 119901


119909is ultimately



119898)119889


with 119871

2and 119871


Ω = x isin R6

1003816

1003816

1003816

1003816

V119909

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

V119910

1003816

1003816

1003816

1003816

1003816

le

119865max + 119901

119898

119889

119906

|119903| le

120591max + 119901

119898

119889

119903

(A2)



119903| minus 119901




radic

2

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

(1 minus 119889

119906119896

1)

119889

119906119896

1

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

(2119865max + 119901

119898) le 120575120576

1 (A3)


119898so that

radic

2 (119901

119898+ 120575120576

1+

1003816

1003816

1003816

1003816

119865

119903

1003816

1003816

1003816

1003816

119908

119898) lt min (120575 100381610038161003816

1003816

119865

119903

1003816

1003816

1003816

1003816

sin (1198962)) (A4)


(119911

2

1+ 119911

2


119881 = 119896

1119911

1(119865 minus 119865

119903cos (120595 minus 120595

119903) + 119901

1)

minus 119896

1119911

2(

1003816

1003816

1003816

1003816

119865

119903

1003816

1003816

1003816

1003816

sin (1198962tanh (119911

2)) minus 119865

119903119908 minus 119901

2)

+ (1 minus 119889

119906119896

1) (119904

1119890V119909 + 119904

2119890V119910)

(A5)


Case A (|1199111| gt 119899



119903cos(120595 minus 120595


by |119865119903| and |119911


following function

119892 (119909) = min( 120575

radic2

119865

119903

radic2

sin(1198962tanh( 119909

radic2

))) (A6)


1| minus |119865

119903119911

2| sin(119896

2tanh (|119911


119881 le minus119896

1z (119892 (z) minus (120575120576

1+

1003816

1003816

1003816

1003816

119865

119903

1003816

1003816

1003816

1003816

|119908| + 119901

119898)) (A7)


1+ |119865

119903|119908

119898+ 119901

119898


119911max 1 = radic

2 tanhminus1 ( 1

119896

2

sinminus1 (radic2(

119901max + 120575120576

1003816

1003816

1003816

1003816

119865

119903

1003816

1003816

1003816

1003816

+ 119908

119898)))

(A8)



Case B (|1199111| le 119899



2| and

with (31) we find

z le radic

2

1003816

1003816

1003816

1003816

119911

1

1003816

1003816

1003816

1003816

le

radic

2119899

1+ 120576 le

radic

2 (z + radic

2 (1 + 119896

1)) 119899

119898+ 120576

(A9)


z leradic2 (

radic2 (1 + 119896

1) 119899

119898+ 120576)


119898)

= 120576

2 (A10)


2| then z le radic

2|119911

2| so we


2| + 120576

2and |119911

1| le 2(|119911

2| +

1 minus 119896

1)119899


119881 le 119896

1(2 (

1003816

1003816

1003816

1003816

119911

2

1003816

1003816

1003816

1003816

+ 1 minus 119896

1) 119899

119898+ 120576) (

1003816

1003816

1003816

1003816

119865max1003816

1003816

1003816

1003816

+

1003816

1003816

1003816

1003816

119865

119903

1003816

1003816

1003816

1003816

+ 119901

119898)

minus 119896

1

1003816

1003816

1003816

1003816

119911

2

1003816

1003816

1003816

1003816

(

1003816

1003816

1003816

1003816

119865

119903

1003816

1003816

1003816

1003816

sin (1198962tanh (100381610038161003816

1003816

119911

2

1003816

1003816

1003816

1003816

)) minus

1003816

1003816

1003816

1003816

119865

119903119908

1003816

1003816

1003816

1003816

minus 119901

119898)

+ 2 (

radic

2

1003816

1003816

1003816

1003816

119911

2

1003816

1003816

1003816

1003816

+ 120576

2) 119889

119906119896

1120575120576

1

(A11)


3and 120576

4 (A11) can be

rewritten as

119881 le minus 119896

1(

1003816

1003816

1003816

1003816

119911

2

1003816

1003816

1003816

1003816

(

1003816

1003816

1003816

1003816

119865

119903

1003816

1003816

1003816

1003816

sin (1198962tanh (100381610038161003816

1003816

119911

2

1003816

1003816

1003816

1003816

))

minus

1003816

1003816

1003816

1003816

119865

119903119908

1003816

1003816

1003816

1003816

minus 119901

119898minus 120576

3) + 120576

4)

(A12)

Now we define


119896

2

sinminus1 (119901

119898

1003816

1003816

1003816

1003816

119865

119903

1003816

1003816

1003816

1003816

+ 120576

3+ 119908

119898+ 120575)) (A13)



119881 le 119896




119898


119898 119899119898 119901119898 and 120576 then

(35) holds


119909and 119904

119910

119896

1119890

119909= 119904

119909minus 119890

119909

119896

1119890

119910= 119904

119910minus 119890

119910

(A14)


119898 119901119898 119908119898 and 120576 then (36) holds



119903| le |119903| + |119903

119903| le |120591

119898|119889

119903+ |119903

119903| The



z le 1003816

1003816

1003816

1003816

1

1003816

1003816

1003816

1003816

+

1003816

1003816

1003816

1003816

2

1003816

1003816

1003816

1003816

le 119888

1+ 119888

2119911

119898+ 119888

3119901

119898

1003816

1003816

1003816

1003816

2

1003816

1003816

1003816

1003816

le 119888

4+ 119888

5119911

119898+ 119888

6119901

119898

(A15)


120591

119888= 120591

119903minus 119896

2sign (119865

119903) (1 minus tanh2 (119911

2)) (119889

119903

2+

2)

+ 2119896

2sign (119865

119903) (1 minus tanh2 (119911

2)) tanh (119911

2)

2

2

(A16)


120595and 120591

119888can be




1003816

1003816

1003816

1003816

120591

119888

1003816

1003816

1003816

1003816

le

1003816

1003816

1003816

1003816

120591

119903

1003816

1003816

1003816

1003816

+ 119896

2120574

4(119911

119898+ 119901

119898) (A17)

1003816

1003816

1003816

1003816

119891

3

1003816

1003816

1003816

1003816

le 120574

5(119888

7+ 119911

119898+ 119901

119898) 119899

119898+ 119896

3119896

2119896

1119901

119898 (A18)



120575 Now we define 119911119898as

119911

119898= minus

1

2

120574

4

minus1

(

120575

4119896

2

) (A19)

When z lt 119911


119896

2120574

4(2119901

119898)+119901

119898lt 1205754 and with (A17) then |120591

119894|minus|120591

119888|minus|119901

119898| ge


1003816

1003816

1003816

1003816

1 minus 119896

3119889

119903

1003816

1003816

1003816

1003816

119896

3119889

119903

(

1003816

1003816

1003816

1003816

120591

119898

1003816

1003816

1003816

1003816

+ 119889

119903

1003816

1003816

1003816

1003816

119903

119903

1003816

1003816

1003816

1003816

) le

120575

4

(A20)


2= 119904

2

1205952 whose time

derivative is

119881

2= 119904

120595(119896

3(120591 minus 120591

119888+ 119901

119903) + (1 minus 119896

3119889

119903)119890

119903) Consider

first that |119904120595| ge 119891



119881

2= 119904

120595119904

120595le minus120575119896

3(

1

2

minus

1

4

)

1003816

1003816

1003816

1003816

1003816

119904

120595

1003816

1003816

1003816

1003816

1003816

997888rarr 119904

120595le minus

120575119896

3

4

sign (119904120595)

(A21)


120595(119905)| le |119904

120595(0)| minus

120575119896

34119905 If 119904


120595| lt 119891

3+120576 (A21) does not




case |119904120595(119905)| le max(|119904

120595(0)| minus 120575119896

34119905 119891





120595= 119904

120595minus 119890

120595 whose

solution is

119890

120595(119905) = 119890

120595(0) 119890

minus1199051198963+

1

119896

3

int

119905

0

119890

minus(119905minus120591)1198963119904

120595(120591) 119889120591 (A22)


1003816

1003816

1003816

1003816

1003816

119890

120595(119905)

1003816

1003816

1003816

1003816

1003816

le

1003816

1003816

1003816

1003816

1003816

119890

120595(0)

1003816

1003816

1003816

1003816

1003816

119890

minus1199051198963+

1003816

1003816

1003816

1003816

1003816

119904

120595(0)

1003816

1003816

1003816

1003816

1003816

(119890

|119904120595(0)|1198861198963minus 1) 119890

minus1199051198963

+ 120574

3(119887 + 119911

119898+ 119901

119898) 119899

119898+ 119888119901

119898+ 120576

(A23)


120595| + |119904

120595|119890

|119904120595|1198861198963






119898Therefore



119898=

2119872 as follows

119896

2=

120575

4120574

4(4119872)

(A24)


119881 le 119896

1

1003816

1003816

1003816

1003816

119911

2

1003816

1003816

1003816

1003816

(

1003816

1003816

1003816

1003816

119865

119903119908

1003816

1003816

1003816

1003816

+ 119901

2) + (1 minus 119889

119906119896

1) (119904

1119890V119909 + 119904

2119890V119910)

le 119887

1|z| + 119887

2

(A25)


2 Select 119872 gt 119887

2119887

1then z gt

119887

2119887

1and thus

119881 = zz le 2119887


1rarr z le

z(0) + 2119887


2119887

1and therefore 119879 gt 1198722119887

2


3(119887 + 119872 + 119901

119898)119899

119898+ 119888119901

119898+ 120576 le 119889119890


120595(119879)| le 2119889 sdot 119890


max(119901

119898+ 120576

5

1003816

1003816

1003816

1003816

119865

119903

1003816

1003816

1003816

1003816

119901

119898+ 120575120576

1003816

1003816

1003816

1003816

119865

119903

1003816

1003816

1003816

1003816

) lt 119889 sdot 119890

minus1198791198963 (A26)


119911max (119879) = radic


4(4119872)

120575


minus119872211988821198963))

(A27)

Function 120574



119898



1for which z(119905



2for which z(119905

2) = 2119872 and 119881

grows in 119905



120595(119905

2)| le 2119889119890


119911max(|119890120595(1199052)|) = 119872 because z lt 119911

119898for all 119879 le 119905

1le 119905

2 This




2



Acknowledgments


References









































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











minus2

minus15

minus1

minus05

0

05

1

15

2

x(m

)

y (m)


0 10 20 30 40 50 60minus3

minus2

minus1

0123

x(m

)

t (s)

(a)

0 10 20 30 40 50 60minus3

minus2

minus1

0123

y(m

)

t (s)

(b)



119909and V119910 rad



0 10 20 30 40 50 60

minus04minus02

0020406

x(m

s)

t (s)

(a)

0 10 20 30 40 50 60

minus04minus02

0020406

t (s)

y(m

s)

(b)


0 10 20 30 40 50 60minus4

minus2

0

2

4

6120595

(rad

)

t (s)

(a)


0020406

r(r

ads

)

t (s)

(b)






minus3

minus2

minus1

0

1

2

3

4

x(m

)

y (m)


0 10 20 30 40 50 600

05

1

15

2

25

3

35

4

e p(m

)

t (s)





119903(119905) = 2 sin(38119905) 119910

119903(119905) = 3 cos(28119905)



minus06

minus04

minus02

0

02

04

06

08

1

12

120591 (rads2)

F(m

s2)



minus2

minus15

minus1

minus05

0

05

1

15

2x

(m)

y (m)






value than 120591








minus06

minus04

minus02

0

02

04

06

08

x(m

)

y (m)







0 5 10 15 20 250

01

02

03

04

05

06

07

08

09

1

Dist

ance

(m)

t





119865


radio controller





0020406

x(m

)

t (s)

(a)


002040608

y(m

)

t (s)

(b)


t (s)0 5 10 15 20 25

minus1

minus05

0

05

1

x(m

s)

(a)

0 5 10 15 20 25minus1

minus05

0

05

1

y(m

s)

t (s)

(b)


8 Conclusions



0 5 10 15 20 25minus04

minus02

0020406

Fs

(N)

t (s)

(a)

0 5 10 15 20 25minus04

minus02

0020406

t (s)

F p(N

)

(b)






Appendix

A Proofs



119871

2= V21199102 and 119871

3= 119903

2


119871

1= minusV119909(119889

119906V119909minus119865 cos(120595)+119901V119909)



119871

1can be bounded by

119871

1lt minus119889

119906V2119909(1 minus

119865max + 119901

119898

119889

119906

1003816

1003816

1003816

1003816

V119909

1003816

1003816

1003816

1003816

) (A1)


119871


119889

119906|V119909| gt (119865max + 119901


119909is ultimately



119898)119889


with 119871

2and 119871


Ω = x isin R6

1003816

1003816

1003816

1003816

V119909

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

V119910

1003816

1003816

1003816

1003816

1003816

le

119865max + 119901

119898

119889

119906

|119903| le

120591max + 119901

119898

119889

119903

(A2)



119903| minus 119901




radic

2

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

(1 minus 119889

119906119896

1)

119889

119906119896

1

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

(2119865max + 119901

119898) le 120575120576

1 (A3)


119898so that

radic

2 (119901

119898+ 120575120576

1+

1003816

1003816

1003816

1003816

119865

119903

1003816

1003816

1003816

1003816

119908

119898) lt min (120575 100381610038161003816

1003816

119865

119903

1003816

1003816

1003816

1003816

sin (1198962)) (A4)


(119911

2

1+ 119911

2


119881 = 119896

1119911

1(119865 minus 119865

119903cos (120595 minus 120595

119903) + 119901

1)

minus 119896

1119911

2(

1003816

1003816

1003816

1003816

119865

119903

1003816

1003816

1003816

1003816

sin (1198962tanh (119911

2)) minus 119865

119903119908 minus 119901

2)

+ (1 minus 119889

119906119896

1) (119904

1119890V119909 + 119904

2119890V119910)

(A5)


Case A (|1199111| gt 119899



119903cos(120595 minus 120595


by |119865119903| and |119911


following function

119892 (119909) = min( 120575

radic2

119865

119903

radic2

sin(1198962tanh( 119909

radic2

))) (A6)


1| minus |119865

119903119911

2| sin(119896

2tanh (|119911


119881 le minus119896

1z (119892 (z) minus (120575120576

1+

1003816

1003816

1003816

1003816

119865

119903

1003816

1003816

1003816

1003816

|119908| + 119901

119898)) (A7)


1+ |119865

119903|119908

119898+ 119901

119898


119911max 1 = radic

2 tanhminus1 ( 1

119896

2

sinminus1 (radic2(

119901max + 120575120576

1003816

1003816

1003816

1003816

119865

119903

1003816

1003816

1003816

1003816

+ 119908

119898)))

(A8)



Case B (|1199111| le 119899



2| and

with (31) we find

z le radic

2

1003816

1003816

1003816

1003816

119911

1

1003816

1003816

1003816

1003816

le

radic

2119899

1+ 120576 le

radic

2 (z + radic

2 (1 + 119896

1)) 119899

119898+ 120576

(A9)


z leradic2 (

radic2 (1 + 119896

1) 119899

119898+ 120576)


119898)

= 120576

2 (A10)


2| then z le radic

2|119911

2| so we


2| + 120576

2and |119911

1| le 2(|119911

2| +

1 minus 119896

1)119899


119881 le 119896

1(2 (

1003816

1003816

1003816

1003816

119911

2

1003816

1003816

1003816

1003816

+ 1 minus 119896

1) 119899

119898+ 120576) (

1003816

1003816

1003816

1003816

119865max1003816

1003816

1003816

1003816

+

1003816

1003816

1003816

1003816

119865

119903

1003816

1003816

1003816

1003816

+ 119901

119898)

minus 119896

1

1003816

1003816

1003816

1003816

119911

2

1003816

1003816

1003816

1003816

(

1003816

1003816

1003816

1003816

119865

119903

1003816

1003816

1003816

1003816

sin (1198962tanh (100381610038161003816

1003816

119911

2

1003816

1003816

1003816

1003816

)) minus

1003816

1003816

1003816

1003816

119865

119903119908

1003816

1003816

1003816

1003816

minus 119901

119898)

+ 2 (

radic

2

1003816

1003816

1003816

1003816

119911

2

1003816

1003816

1003816

1003816

+ 120576

2) 119889

119906119896

1120575120576

1

(A11)


3and 120576

4 (A11) can be

rewritten as

119881 le minus 119896

1(

1003816

1003816

1003816

1003816

119911

2

1003816

1003816

1003816

1003816

(

1003816

1003816

1003816

1003816

119865

119903

1003816

1003816

1003816

1003816

sin (1198962tanh (100381610038161003816

1003816

119911

2

1003816

1003816

1003816

1003816

))

minus

1003816

1003816

1003816

1003816

119865

119903119908

1003816

1003816

1003816

1003816

minus 119901

119898minus 120576

3) + 120576

4)

(A12)

Now we define


119896

2

sinminus1 (119901

119898

1003816

1003816

1003816

1003816

119865

119903

1003816

1003816

1003816

1003816

+ 120576

3+ 119908

119898+ 120575)) (A13)



119881 le 119896




119898


119898 119899119898 119901119898 and 120576 then

(35) holds


119909and 119904

119910

119896

1119890

119909= 119904

119909minus 119890

119909

119896

1119890

119910= 119904

119910minus 119890

119910

(A14)


119898 119901119898 119908119898 and 120576 then (36) holds



119903| le |119903| + |119903

119903| le |120591

119898|119889

119903+ |119903

119903| The



z le 1003816

1003816

1003816

1003816

1

1003816

1003816

1003816

1003816

+

1003816

1003816

1003816

1003816

2

1003816

1003816

1003816

1003816

le 119888

1+ 119888

2119911

119898+ 119888

3119901

119898

1003816

1003816

1003816

1003816

2

1003816

1003816

1003816

1003816

le 119888

4+ 119888

5119911

119898+ 119888

6119901

119898

(A15)


120591

119888= 120591

119903minus 119896

2sign (119865

119903) (1 minus tanh2 (119911

2)) (119889

119903

2+

2)

+ 2119896

2sign (119865

119903) (1 minus tanh2 (119911

2)) tanh (119911

2)

2

2

(A16)


120595and 120591

119888can be




1003816

1003816

1003816

1003816

120591

119888

1003816

1003816

1003816

1003816

le

1003816

1003816

1003816

1003816

120591

119903

1003816

1003816

1003816

1003816

+ 119896

2120574

4(119911

119898+ 119901

119898) (A17)

1003816

1003816

1003816

1003816

119891

3

1003816

1003816

1003816

1003816

le 120574

5(119888

7+ 119911

119898+ 119901

119898) 119899

119898+ 119896

3119896

2119896

1119901

119898 (A18)



120575 Now we define 119911119898as

119911

119898= minus

1

2

120574

4

minus1

(

120575

4119896

2

) (A19)

When z lt 119911


119896

2120574

4(2119901

119898)+119901

119898lt 1205754 and with (A17) then |120591

119894|minus|120591

119888|minus|119901

119898| ge


1003816

1003816

1003816

1003816

1 minus 119896

3119889

119903

1003816

1003816

1003816

1003816

119896

3119889

119903

(

1003816

1003816

1003816

1003816

120591

119898

1003816

1003816

1003816

1003816

+ 119889

119903

1003816

1003816

1003816

1003816

119903

119903

1003816

1003816

1003816

1003816

) le

120575

4

(A20)


2= 119904

2

1205952 whose time

derivative is

119881

2= 119904

120595(119896

3(120591 minus 120591

119888+ 119901

119903) + (1 minus 119896

3119889

119903)119890

119903) Consider

first that |119904120595| ge 119891



119881

2= 119904

120595119904

120595le minus120575119896

3(

1

2

minus

1

4

)

1003816

1003816

1003816

1003816

1003816

119904

120595

1003816

1003816

1003816

1003816

1003816

997888rarr 119904

120595le minus

120575119896

3

4

sign (119904120595)

(A21)


120595(119905)| le |119904

120595(0)| minus

120575119896

34119905 If 119904


120595| lt 119891

3+120576 (A21) does not




case |119904120595(119905)| le max(|119904

120595(0)| minus 120575119896

34119905 119891





120595= 119904

120595minus 119890

120595 whose

solution is

119890

120595(119905) = 119890

120595(0) 119890

minus1199051198963+

1

119896

3

int

119905

0

119890

minus(119905minus120591)1198963119904

120595(120591) 119889120591 (A22)


1003816

1003816

1003816

1003816

1003816

119890

120595(119905)

1003816

1003816

1003816

1003816

1003816

le

1003816

1003816

1003816

1003816

1003816

119890

120595(0)

1003816

1003816

1003816

1003816

1003816

119890

minus1199051198963+

1003816

1003816

1003816

1003816

1003816

119904

120595(0)

1003816

1003816

1003816

1003816

1003816

(119890

|119904120595(0)|1198861198963minus 1) 119890

minus1199051198963

+ 120574

3(119887 + 119911

119898+ 119901

119898) 119899

119898+ 119888119901

119898+ 120576

(A23)


120595| + |119904

120595|119890

|119904120595|1198861198963






119898Therefore



119898=

2119872 as follows

119896

2=

120575

4120574

4(4119872)

(A24)


119881 le 119896

1

1003816

1003816

1003816

1003816

119911

2

1003816

1003816

1003816

1003816

(

1003816

1003816

1003816

1003816

119865

119903119908

1003816

1003816

1003816

1003816

+ 119901

2) + (1 minus 119889

119906119896

1) (119904

1119890V119909 + 119904

2119890V119910)

le 119887

1|z| + 119887

2

(A25)


2 Select 119872 gt 119887

2119887

1then z gt

119887

2119887

1and thus

119881 = zz le 2119887


1rarr z le

z(0) + 2119887


2119887

1and therefore 119879 gt 1198722119887

2


3(119887 + 119872 + 119901

119898)119899

119898+ 119888119901

119898+ 120576 le 119889119890


120595(119879)| le 2119889 sdot 119890


max(119901

119898+ 120576

5

1003816

1003816

1003816

1003816

119865

119903

1003816

1003816

1003816

1003816

119901

119898+ 120575120576

1003816

1003816

1003816

1003816

119865

119903

1003816

1003816

1003816

1003816

) lt 119889 sdot 119890

minus1198791198963 (A26)


119911max (119879) = radic


4(4119872)

120575


minus119872211988821198963))

(A27)

Function 120574



119898



1for which z(119905



2for which z(119905

2) = 2119872 and 119881

grows in 119905



120595(119905

2)| le 2119889119890


119911max(|119890120595(1199052)|) = 119872 because z lt 119911

119898for all 119879 le 119905

1le 119905

2 This




2



Acknowledgments


References









































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











minus3

minus2

minus1

0

1

2

3

4

x(m

)

y (m)


0 10 20 30 40 50 600

05

1

15

2

25

3

35

4

e p(m

)

t (s)





119903(119905) = 2 sin(38119905) 119910

119903(119905) = 3 cos(28119905)



minus06

minus04

minus02

0

02

04

06

08

1

12

120591 (rads2)

F(m

s2)



minus2

minus15

minus1

minus05

0

05

1

15

2x

(m)

y (m)






value than 120591








minus06

minus04

minus02

0

02

04

06

08

x(m

)

y (m)







0 5 10 15 20 250

01

02

03

04

05

06

07

08

09

1

Dist

ance

(m)

t





119865


radio controller





0020406

x(m

)

t (s)

(a)


002040608

y(m

)

t (s)

(b)


t (s)0 5 10 15 20 25

minus1

minus05

0

05

1

x(m

s)

(a)

0 5 10 15 20 25minus1

minus05

0

05

1

y(m

s)

t (s)

(b)


8 Conclusions



0 5 10 15 20 25minus04

minus02

0020406

Fs

(N)

t (s)

(a)

0 5 10 15 20 25minus04

minus02

0020406

t (s)

F p(N

)

(b)






Appendix

A Proofs



119871

2= V21199102 and 119871

3= 119903

2


119871

1= minusV119909(119889

119906V119909minus119865 cos(120595)+119901V119909)



119871

1can be bounded by

119871

1lt minus119889

119906V2119909(1 minus

119865max + 119901

119898

119889

119906

1003816

1003816

1003816

1003816

V119909

1003816

1003816

1003816

1003816

) (A1)


119871


119889

119906|V119909| gt (119865max + 119901


119909is ultimately



119898)119889


with 119871

2and 119871


Ω = x isin R6

1003816

1003816

1003816

1003816

V119909

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

V119910

1003816

1003816

1003816

1003816

1003816

le

119865max + 119901

119898

119889

119906

|119903| le

120591max + 119901

119898

119889

119903

(A2)



119903| minus 119901




radic

2

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

(1 minus 119889

119906119896

1)

119889

119906119896

1

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

(2119865max + 119901

119898) le 120575120576

1 (A3)


119898so that

radic

2 (119901

119898+ 120575120576

1+

1003816

1003816

1003816

1003816

119865

119903

1003816

1003816

1003816

1003816

119908

119898) lt min (120575 100381610038161003816

1003816

119865

119903

1003816

1003816

1003816

1003816

sin (1198962)) (A4)


(119911

2

1+ 119911

2


119881 = 119896

1119911

1(119865 minus 119865

119903cos (120595 minus 120595

119903) + 119901

1)

minus 119896

1119911

2(

1003816

1003816

1003816

1003816

119865

119903

1003816

1003816

1003816

1003816

sin (1198962tanh (119911

2)) minus 119865

119903119908 minus 119901

2)

+ (1 minus 119889

119906119896

1) (119904

1119890V119909 + 119904

2119890V119910)

(A5)


Case A (|1199111| gt 119899



119903cos(120595 minus 120595


by |119865119903| and |119911


following function

119892 (119909) = min( 120575

radic2

119865

119903

radic2

sin(1198962tanh( 119909

radic2

))) (A6)


1| minus |119865

119903119911

2| sin(119896

2tanh (|119911


119881 le minus119896

1z (119892 (z) minus (120575120576

1+

1003816

1003816

1003816

1003816

119865

119903

1003816

1003816

1003816

1003816

|119908| + 119901

119898)) (A7)


1+ |119865

119903|119908

119898+ 119901

119898


119911max 1 = radic

2 tanhminus1 ( 1

119896

2

sinminus1 (radic2(

119901max + 120575120576

1003816

1003816

1003816

1003816

119865

119903

1003816

1003816

1003816

1003816

+ 119908

119898)))

(A8)



Case B (|1199111| le 119899



2| and

with (31) we find

z le radic

2

1003816

1003816

1003816

1003816

119911

1

1003816

1003816

1003816

1003816

le

radic

2119899

1+ 120576 le

radic

2 (z + radic

2 (1 + 119896

1)) 119899

119898+ 120576

(A9)


z leradic2 (

radic2 (1 + 119896

1) 119899

119898+ 120576)


119898)

= 120576

2 (A10)


2| then z le radic

2|119911

2| so we


2| + 120576

2and |119911

1| le 2(|119911

2| +

1 minus 119896

1)119899


119881 le 119896

1(2 (

1003816

1003816

1003816

1003816

119911

2

1003816

1003816

1003816

1003816

+ 1 minus 119896

1) 119899

119898+ 120576) (

1003816

1003816

1003816

1003816

119865max1003816

1003816

1003816

1003816

+

1003816

1003816

1003816

1003816

119865

119903

1003816

1003816

1003816

1003816

+ 119901

119898)

minus 119896

1

1003816

1003816

1003816

1003816

119911

2

1003816

1003816

1003816

1003816

(

1003816

1003816

1003816

1003816

119865

119903

1003816

1003816

1003816

1003816

sin (1198962tanh (100381610038161003816

1003816

119911

2

1003816

1003816

1003816

1003816

)) minus

1003816

1003816

1003816

1003816

119865

119903119908

1003816

1003816

1003816

1003816

minus 119901

119898)

+ 2 (

radic

2

1003816

1003816

1003816

1003816

119911

2

1003816

1003816

1003816

1003816

+ 120576

2) 119889

119906119896

1120575120576

1

(A11)


3and 120576

4 (A11) can be

rewritten as

119881 le minus 119896

1(

1003816

1003816

1003816

1003816

119911

2

1003816

1003816

1003816

1003816

(

1003816

1003816

1003816

1003816

119865

119903

1003816

1003816

1003816

1003816

sin (1198962tanh (100381610038161003816

1003816

119911

2

1003816

1003816

1003816

1003816

))

minus

1003816

1003816

1003816

1003816

119865

119903119908

1003816

1003816

1003816

1003816

minus 119901

119898minus 120576

3) + 120576

4)

(A12)

Now we define


119896

2

sinminus1 (119901

119898

1003816

1003816

1003816

1003816

119865

119903

1003816

1003816

1003816

1003816

+ 120576

3+ 119908

119898+ 120575)) (A13)



119881 le 119896




119898


119898 119899119898 119901119898 and 120576 then

(35) holds


119909and 119904

119910

119896

1119890

119909= 119904

119909minus 119890

119909

119896

1119890

119910= 119904

119910minus 119890

119910

(A14)


119898 119901119898 119908119898 and 120576 then (36) holds



119903| le |119903| + |119903

119903| le |120591

119898|119889

119903+ |119903

119903| The



z le 1003816

1003816

1003816

1003816

1

1003816

1003816

1003816

1003816

+

1003816

1003816

1003816

1003816

2

1003816

1003816

1003816

1003816

le 119888

1+ 119888

2119911

119898+ 119888

3119901

119898

1003816

1003816

1003816

1003816

2

1003816

1003816

1003816

1003816

le 119888

4+ 119888

5119911

119898+ 119888

6119901

119898

(A15)


120591

119888= 120591

119903minus 119896

2sign (119865

119903) (1 minus tanh2 (119911

2)) (119889

119903

2+

2)

+ 2119896

2sign (119865

119903) (1 minus tanh2 (119911

2)) tanh (119911

2)

2

2

(A16)


120595and 120591

119888can be




1003816

1003816

1003816

1003816

120591

119888

1003816

1003816

1003816

1003816

le

1003816

1003816

1003816

1003816

120591

119903

1003816

1003816

1003816

1003816

+ 119896

2120574

4(119911

119898+ 119901

119898) (A17)

1003816

1003816

1003816

1003816

119891

3

1003816

1003816

1003816

1003816

le 120574

5(119888

7+ 119911

119898+ 119901

119898) 119899

119898+ 119896

3119896

2119896

1119901

119898 (A18)



120575 Now we define 119911119898as

119911

119898= minus

1

2

120574

4

minus1

(

120575

4119896

2

) (A19)

When z lt 119911


119896

2120574

4(2119901

119898)+119901

119898lt 1205754 and with (A17) then |120591

119894|minus|120591

119888|minus|119901

119898| ge


1003816

1003816

1003816

1003816

1 minus 119896

3119889

119903

1003816

1003816

1003816

1003816

119896

3119889

119903

(

1003816

1003816

1003816

1003816

120591

119898

1003816

1003816

1003816

1003816

+ 119889

119903

1003816

1003816

1003816

1003816

119903

119903

1003816

1003816

1003816

1003816

) le

120575

4

(A20)


2= 119904

2

1205952 whose time

derivative is

119881

2= 119904

120595(119896

3(120591 minus 120591

119888+ 119901

119903) + (1 minus 119896

3119889

119903)119890

119903) Consider

first that |119904120595| ge 119891



119881

2= 119904

120595119904

120595le minus120575119896

3(

1

2

minus

1

4

)

1003816

1003816

1003816

1003816

1003816

119904

120595

1003816

1003816

1003816

1003816

1003816

997888rarr 119904

120595le minus

120575119896

3

4

sign (119904120595)

(A21)


120595(119905)| le |119904

120595(0)| minus

120575119896

34119905 If 119904


120595| lt 119891

3+120576 (A21) does not




case |119904120595(119905)| le max(|119904

120595(0)| minus 120575119896

34119905 119891





120595= 119904

120595minus 119890

120595 whose

solution is

119890

120595(119905) = 119890

120595(0) 119890

minus1199051198963+

1

119896

3

int

119905

0

119890

minus(119905minus120591)1198963119904

120595(120591) 119889120591 (A22)


1003816

1003816

1003816

1003816

1003816

119890

120595(119905)

1003816

1003816

1003816

1003816

1003816

le

1003816

1003816

1003816

1003816

1003816

119890

120595(0)

1003816

1003816

1003816

1003816

1003816

119890

minus1199051198963+

1003816

1003816

1003816

1003816

1003816

119904

120595(0)

1003816

1003816

1003816

1003816

1003816

(119890

|119904120595(0)|1198861198963minus 1) 119890

minus1199051198963

+ 120574

3(119887 + 119911

119898+ 119901

119898) 119899

119898+ 119888119901

119898+ 120576

(A23)


120595| + |119904

120595|119890

|119904120595|1198861198963






119898Therefore



119898=

2119872 as follows

119896

2=

120575

4120574

4(4119872)

(A24)


119881 le 119896

1

1003816

1003816

1003816

1003816

119911

2

1003816

1003816

1003816

1003816

(

1003816

1003816

1003816

1003816

119865

119903119908

1003816

1003816

1003816

1003816

+ 119901

2) + (1 minus 119889

119906119896

1) (119904

1119890V119909 + 119904

2119890V119910)

le 119887

1|z| + 119887

2

(A25)


2 Select 119872 gt 119887

2119887

1then z gt

119887

2119887

1and thus

119881 = zz le 2119887


1rarr z le

z(0) + 2119887


2119887

1and therefore 119879 gt 1198722119887

2


3(119887 + 119872 + 119901

119898)119899

119898+ 119888119901

119898+ 120576 le 119889119890


120595(119879)| le 2119889 sdot 119890


max(119901

119898+ 120576

5

1003816

1003816

1003816

1003816

119865

119903

1003816

1003816

1003816

1003816

119901

119898+ 120575120576

1003816

1003816

1003816

1003816

119865

119903

1003816

1003816

1003816

1003816

) lt 119889 sdot 119890

minus1198791198963 (A26)


119911max (119879) = radic


4(4119872)

120575


minus119872211988821198963))

(A27)

Function 120574



119898



1for which z(119905



2for which z(119905

2) = 2119872 and 119881

grows in 119905



120595(119905

2)| le 2119889119890


119911max(|119890120595(1199052)|) = 119872 because z lt 119911

119898for all 119879 le 119905

1le 119905

2 This




2



Acknowledgments


References









































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












minus06

minus04

minus02

0

02

04

06

08

x(m

)

y (m)







0 5 10 15 20 250

01

02

03

04

05

06

07

08

09

1

Dist

ance

(m)

t





119865


radio controller





0020406

x(m

)

t (s)

(a)


002040608

y(m

)

t (s)

(b)


t (s)0 5 10 15 20 25

minus1

minus05

0

05

1

x(m

s)

(a)

0 5 10 15 20 25minus1

minus05

0

05

1

y(m

s)

t (s)

(b)


8 Conclusions



0 5 10 15 20 25minus04

minus02

0020406

Fs

(N)

t (s)

(a)

0 5 10 15 20 25minus04

minus02

0020406

t (s)

F p(N

)

(b)






Appendix

A Proofs



119871

2= V21199102 and 119871

3= 119903

2


119871

1= minusV119909(119889

119906V119909minus119865 cos(120595)+119901V119909)



119871

1can be bounded by

119871

1lt minus119889

119906V2119909(1 minus

119865max + 119901

119898

119889

119906

1003816

1003816

1003816

1003816

V119909

1003816

1003816

1003816

1003816

) (A1)


119871


119889

119906|V119909| gt (119865max + 119901


119909is ultimately



119898)119889


with 119871

2and 119871


Ω = x isin R6

1003816

1003816

1003816

1003816

V119909

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

V119910

1003816

1003816

1003816

1003816

1003816

le

119865max + 119901

119898

119889

119906

|119903| le

120591max + 119901

119898

119889

119903

(A2)



119903| minus 119901




radic

2

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

(1 minus 119889

119906119896

1)

119889

119906119896

1

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

(2119865max + 119901

119898) le 120575120576

1 (A3)


119898so that

radic

2 (119901

119898+ 120575120576

1+

1003816

1003816

1003816

1003816

119865

119903

1003816

1003816

1003816

1003816

119908

119898) lt min (120575 100381610038161003816

1003816

119865

119903

1003816

1003816

1003816

1003816

sin (1198962)) (A4)


(119911

2

1+ 119911

2


119881 = 119896

1119911

1(119865 minus 119865

119903cos (120595 minus 120595

119903) + 119901

1)

minus 119896

1119911

2(

1003816

1003816

1003816

1003816

119865

119903

1003816

1003816

1003816

1003816

sin (1198962tanh (119911

2)) minus 119865

119903119908 minus 119901

2)

+ (1 minus 119889

119906119896

1) (119904

1119890V119909 + 119904

2119890V119910)

(A5)


Case A (|1199111| gt 119899



119903cos(120595 minus 120595


by |119865119903| and |119911


following function

119892 (119909) = min( 120575

radic2

119865

119903

radic2

sin(1198962tanh( 119909

radic2

))) (A6)


1| minus |119865

119903119911

2| sin(119896

2tanh (|119911


119881 le minus119896

1z (119892 (z) minus (120575120576

1+

1003816

1003816

1003816

1003816

119865

119903

1003816

1003816

1003816

1003816

|119908| + 119901

119898)) (A7)


1+ |119865

119903|119908

119898+ 119901

119898


119911max 1 = radic

2 tanhminus1 ( 1

119896

2

sinminus1 (radic2(

119901max + 120575120576

1003816

1003816

1003816

1003816

119865

119903

1003816

1003816

1003816

1003816

+ 119908

119898)))

(A8)



Case B (|1199111| le 119899



2| and

with (31) we find

z le radic

2

1003816

1003816

1003816

1003816

119911

1

1003816

1003816

1003816

1003816

le

radic

2119899

1+ 120576 le

radic

2 (z + radic

2 (1 + 119896

1)) 119899

119898+ 120576

(A9)


z leradic2 (

radic2 (1 + 119896

1) 119899

119898+ 120576)


119898)

= 120576

2 (A10)


2| then z le radic

2|119911

2| so we


2| + 120576

2and |119911

1| le 2(|119911

2| +

1 minus 119896

1)119899


119881 le 119896

1(2 (

1003816

1003816

1003816

1003816

119911

2

1003816

1003816

1003816

1003816

+ 1 minus 119896

1) 119899

119898+ 120576) (

1003816

1003816

1003816

1003816

119865max1003816

1003816

1003816

1003816

+

1003816

1003816

1003816

1003816

119865

119903

1003816

1003816

1003816

1003816

+ 119901

119898)

minus 119896

1

1003816

1003816

1003816

1003816

119911

2

1003816

1003816

1003816

1003816

(

1003816

1003816

1003816

1003816

119865

119903

1003816

1003816

1003816

1003816

sin (1198962tanh (100381610038161003816

1003816

119911

2

1003816

1003816

1003816

1003816

)) minus

1003816

1003816

1003816

1003816

119865

119903119908

1003816

1003816

1003816

1003816

minus 119901

119898)

+ 2 (

radic

2

1003816

1003816

1003816

1003816

119911

2

1003816

1003816

1003816

1003816

+ 120576

2) 119889

119906119896

1120575120576

1

(A11)


3and 120576

4 (A11) can be

rewritten as

119881 le minus 119896

1(

1003816

1003816

1003816

1003816

119911

2

1003816

1003816

1003816

1003816

(

1003816

1003816

1003816

1003816

119865

119903

1003816

1003816

1003816

1003816

sin (1198962tanh (100381610038161003816

1003816

119911

2

1003816

1003816

1003816

1003816

))

minus

1003816

1003816

1003816

1003816

119865

119903119908

1003816

1003816

1003816

1003816

minus 119901

119898minus 120576

3) + 120576

4)

(A12)

Now we define


119896

2

sinminus1 (119901

119898

1003816

1003816

1003816

1003816

119865

119903

1003816

1003816

1003816

1003816

+ 120576

3+ 119908

119898+ 120575)) (A13)



119881 le 119896




119898


119898 119899119898 119901119898 and 120576 then

(35) holds


119909and 119904

119910

119896

1119890

119909= 119904

119909minus 119890

119909

119896

1119890

119910= 119904

119910minus 119890

119910

(A14)


119898 119901119898 119908119898 and 120576 then (36) holds



119903| le |119903| + |119903

119903| le |120591

119898|119889

119903+ |119903

119903| The



z le 1003816

1003816

1003816

1003816

1

1003816

1003816

1003816

1003816

+

1003816

1003816

1003816

1003816

2

1003816

1003816

1003816

1003816

le 119888

1+ 119888

2119911

119898+ 119888

3119901

119898

1003816

1003816

1003816

1003816

2

1003816

1003816

1003816

1003816

le 119888

4+ 119888

5119911

119898+ 119888

6119901

119898

(A15)


120591

119888= 120591

119903minus 119896

2sign (119865

119903) (1 minus tanh2 (119911

2)) (119889

119903

2+

2)

+ 2119896

2sign (119865

119903) (1 minus tanh2 (119911

2)) tanh (119911

2)

2

2

(A16)


120595and 120591

119888can be




1003816

1003816

1003816

1003816

120591

119888

1003816

1003816

1003816

1003816

le

1003816

1003816

1003816

1003816

120591

119903

1003816

1003816

1003816

1003816

+ 119896

2120574

4(119911

119898+ 119901

119898) (A17)

1003816

1003816

1003816

1003816

119891

3

1003816

1003816

1003816

1003816

le 120574

5(119888

7+ 119911

119898+ 119901

119898) 119899

119898+ 119896

3119896

2119896

1119901

119898 (A18)



120575 Now we define 119911119898as

119911

119898= minus

1

2

120574

4

minus1

(

120575

4119896

2

) (A19)

When z lt 119911


119896

2120574

4(2119901

119898)+119901

119898lt 1205754 and with (A17) then |120591

119894|minus|120591

119888|minus|119901

119898| ge


1003816

1003816

1003816

1003816

1 minus 119896

3119889

119903

1003816

1003816

1003816

1003816

119896

3119889

119903

(

1003816

1003816

1003816

1003816

120591

119898

1003816

1003816

1003816

1003816

+ 119889

119903

1003816

1003816

1003816

1003816

119903

119903

1003816

1003816

1003816

1003816

) le

120575

4

(A20)


2= 119904

2

1205952 whose time

derivative is

119881

2= 119904

120595(119896

3(120591 minus 120591

119888+ 119901

119903) + (1 minus 119896

3119889

119903)119890

119903) Consider

first that |119904120595| ge 119891



119881

2= 119904

120595119904

120595le minus120575119896

3(

1

2

minus

1

4

)

1003816

1003816

1003816

1003816

1003816

119904

120595

1003816

1003816

1003816

1003816

1003816

997888rarr 119904

120595le minus

120575119896

3

4

sign (119904120595)

(A21)


120595(119905)| le |119904

120595(0)| minus

120575119896

34119905 If 119904


120595| lt 119891

3+120576 (A21) does not




case |119904120595(119905)| le max(|119904

120595(0)| minus 120575119896

34119905 119891





120595= 119904

120595minus 119890

120595 whose

solution is

119890

120595(119905) = 119890

120595(0) 119890

minus1199051198963+

1

119896

3

int

119905

0

119890

minus(119905minus120591)1198963119904

120595(120591) 119889120591 (A22)


1003816

1003816

1003816

1003816

1003816

119890

120595(119905)

1003816

1003816

1003816

1003816

1003816

le

1003816

1003816

1003816

1003816

1003816

119890

120595(0)

1003816

1003816

1003816

1003816

1003816

119890

minus1199051198963+

1003816

1003816

1003816

1003816

1003816

119904

120595(0)

1003816

1003816

1003816

1003816

1003816

(119890

|119904120595(0)|1198861198963minus 1) 119890

minus1199051198963

+ 120574

3(119887 + 119911

119898+ 119901

119898) 119899

119898+ 119888119901

119898+ 120576

(A23)


120595| + |119904

120595|119890

|119904120595|1198861198963






119898Therefore



119898=

2119872 as follows

119896

2=

120575

4120574

4(4119872)

(A24)


119881 le 119896

1

1003816

1003816

1003816

1003816

119911

2

1003816

1003816

1003816

1003816

(

1003816

1003816

1003816

1003816

119865

119903119908

1003816

1003816

1003816

1003816

+ 119901

2) + (1 minus 119889

119906119896

1) (119904

1119890V119909 + 119904

2119890V119910)

le 119887

1|z| + 119887

2

(A25)


2 Select 119872 gt 119887

2119887

1then z gt

119887

2119887

1and thus

119881 = zz le 2119887


1rarr z le

z(0) + 2119887


2119887

1and therefore 119879 gt 1198722119887

2


3(119887 + 119872 + 119901

119898)119899

119898+ 119888119901

119898+ 120576 le 119889119890


120595(119879)| le 2119889 sdot 119890


max(119901

119898+ 120576

5

1003816

1003816

1003816

1003816

119865

119903

1003816

1003816

1003816

1003816

119901

119898+ 120575120576

1003816

1003816

1003816

1003816

119865

119903

1003816

1003816

1003816

1003816

) lt 119889 sdot 119890

minus1198791198963 (A26)


119911max (119879) = radic


4(4119872)

120575


minus119872211988821198963))

(A27)

Function 120574



119898



1for which z(119905



2for which z(119905

2) = 2119872 and 119881

grows in 119905



120595(119905

2)| le 2119889119890


119911max(|119890120595(1199052)|) = 119872 because z lt 119911

119898for all 119879 le 119905

1le 119905

2 This




2



Acknowledgments


References









































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











0020406

x(m

)

t (s)

(a)


002040608

y(m

)

t (s)

(b)


t (s)0 5 10 15 20 25

minus1

minus05

0

05

1

x(m

s)

(a)

0 5 10 15 20 25minus1

minus05

0

05

1

y(m

s)

t (s)

(b)


8 Conclusions



0 5 10 15 20 25minus04

minus02

0020406

Fs

(N)

t (s)

(a)

0 5 10 15 20 25minus04

minus02

0020406

t (s)

F p(N

)

(b)






Appendix

A Proofs



119871

2= V21199102 and 119871

3= 119903

2


119871

1= minusV119909(119889

119906V119909minus119865 cos(120595)+119901V119909)



119871

1can be bounded by

119871

1lt minus119889

119906V2119909(1 minus

119865max + 119901

119898

119889

119906

1003816

1003816

1003816

1003816

V119909

1003816

1003816

1003816

1003816

) (A1)


119871


119889

119906|V119909| gt (119865max + 119901


119909is ultimately



119898)119889


with 119871

2and 119871


Ω = x isin R6

1003816

1003816

1003816

1003816

V119909

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

V119910

1003816

1003816

1003816

1003816

1003816

le

119865max + 119901

119898

119889

119906

|119903| le

120591max + 119901

119898

119889

119903

(A2)



119903| minus 119901




radic

2

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

(1 minus 119889

119906119896

1)

119889

119906119896

1

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

(2119865max + 119901

119898) le 120575120576

1 (A3)


119898so that

radic

2 (119901

119898+ 120575120576

1+

1003816

1003816

1003816

1003816

119865

119903

1003816

1003816

1003816

1003816

119908

119898) lt min (120575 100381610038161003816

1003816

119865

119903

1003816

1003816

1003816

1003816

sin (1198962)) (A4)


(119911

2

1+ 119911

2


119881 = 119896

1119911

1(119865 minus 119865

119903cos (120595 minus 120595

119903) + 119901

1)

minus 119896

1119911

2(

1003816

1003816

1003816

1003816

119865

119903

1003816

1003816

1003816

1003816

sin (1198962tanh (119911

2)) minus 119865

119903119908 minus 119901

2)

+ (1 minus 119889

119906119896

1) (119904

1119890V119909 + 119904

2119890V119910)

(A5)


Case A (|1199111| gt 119899



119903cos(120595 minus 120595


by |119865119903| and |119911


following function

119892 (119909) = min( 120575

radic2

119865

119903

radic2

sin(1198962tanh( 119909

radic2

))) (A6)


1| minus |119865

119903119911

2| sin(119896

2tanh (|119911


119881 le minus119896

1z (119892 (z) minus (120575120576

1+

1003816

1003816

1003816

1003816

119865

119903

1003816

1003816

1003816

1003816

|119908| + 119901

119898)) (A7)


1+ |119865

119903|119908

119898+ 119901

119898


119911max 1 = radic

2 tanhminus1 ( 1

119896

2

sinminus1 (radic2(

119901max + 120575120576

1003816

1003816

1003816

1003816

119865

119903

1003816

1003816

1003816

1003816

+ 119908

119898)))

(A8)



Case B (|1199111| le 119899



2| and

with (31) we find

z le radic

2

1003816

1003816

1003816

1003816

119911

1

1003816

1003816

1003816

1003816

le

radic

2119899

1+ 120576 le

radic

2 (z + radic

2 (1 + 119896

1)) 119899

119898+ 120576

(A9)


z leradic2 (

radic2 (1 + 119896

1) 119899

119898+ 120576)


119898)

= 120576

2 (A10)


2| then z le radic

2|119911

2| so we


2| + 120576

2and |119911

1| le 2(|119911

2| +

1 minus 119896

1)119899


119881 le 119896

1(2 (

1003816

1003816

1003816

1003816

119911

2

1003816

1003816

1003816

1003816

+ 1 minus 119896

1) 119899

119898+ 120576) (

1003816

1003816

1003816

1003816

119865max1003816

1003816

1003816

1003816

+

1003816

1003816

1003816

1003816

119865

119903

1003816

1003816

1003816

1003816

+ 119901

119898)

minus 119896

1

1003816

1003816

1003816

1003816

119911

2

1003816

1003816

1003816

1003816

(

1003816

1003816

1003816

1003816

119865

119903

1003816

1003816

1003816

1003816

sin (1198962tanh (100381610038161003816

1003816

119911

2

1003816

1003816

1003816

1003816

)) minus

1003816

1003816

1003816

1003816

119865

119903119908

1003816

1003816

1003816

1003816

minus 119901

119898)

+ 2 (

radic

2

1003816

1003816

1003816

1003816

119911

2

1003816

1003816

1003816

1003816

+ 120576

2) 119889

119906119896

1120575120576

1

(A11)


3and 120576

4 (A11) can be

rewritten as

119881 le minus 119896

1(

1003816

1003816

1003816

1003816

119911

2

1003816

1003816

1003816

1003816

(

1003816

1003816

1003816

1003816

119865

119903

1003816

1003816

1003816

1003816

sin (1198962tanh (100381610038161003816

1003816

119911

2

1003816

1003816

1003816

1003816

))

minus

1003816

1003816

1003816

1003816

119865

119903119908

1003816

1003816

1003816

1003816

minus 119901

119898minus 120576

3) + 120576

4)

(A12)

Now we define


119896

2

sinminus1 (119901

119898

1003816

1003816

1003816

1003816

119865

119903

1003816

1003816

1003816

1003816

+ 120576

3+ 119908

119898+ 120575)) (A13)



119881 le 119896




119898


119898 119899119898 119901119898 and 120576 then

(35) holds


119909and 119904

119910

119896

1119890

119909= 119904

119909minus 119890

119909

119896

1119890

119910= 119904

119910minus 119890

119910

(A14)


119898 119901119898 119908119898 and 120576 then (36) holds



119903| le |119903| + |119903

119903| le |120591

119898|119889

119903+ |119903

119903| The



z le 1003816

1003816

1003816

1003816

1

1003816

1003816

1003816

1003816

+

1003816

1003816

1003816

1003816

2

1003816

1003816

1003816

1003816

le 119888

1+ 119888

2119911

119898+ 119888

3119901

119898

1003816

1003816

1003816

1003816

2

1003816

1003816

1003816

1003816

le 119888

4+ 119888

5119911

119898+ 119888

6119901

119898

(A15)


120591

119888= 120591

119903minus 119896

2sign (119865

119903) (1 minus tanh2 (119911

2)) (119889

119903

2+

2)

+ 2119896

2sign (119865

119903) (1 minus tanh2 (119911

2)) tanh (119911

2)

2

2

(A16)


120595and 120591

119888can be




1003816

1003816

1003816

1003816

120591

119888

1003816

1003816

1003816

1003816

le

1003816

1003816

1003816

1003816

120591

119903

1003816

1003816

1003816

1003816

+ 119896

2120574

4(119911

119898+ 119901

119898) (A17)

1003816

1003816

1003816

1003816

119891

3

1003816

1003816

1003816

1003816

le 120574

5(119888

7+ 119911

119898+ 119901

119898) 119899

119898+ 119896

3119896

2119896

1119901

119898 (A18)



120575 Now we define 119911119898as

119911

119898= minus

1

2

120574

4

minus1

(

120575

4119896

2

) (A19)

When z lt 119911


119896

2120574

4(2119901

119898)+119901

119898lt 1205754 and with (A17) then |120591

119894|minus|120591

119888|minus|119901

119898| ge


1003816

1003816

1003816

1003816

1 minus 119896

3119889

119903

1003816

1003816

1003816

1003816

119896

3119889

119903

(

1003816

1003816

1003816

1003816

120591

119898

1003816

1003816

1003816

1003816

+ 119889

119903

1003816

1003816

1003816

1003816

119903

119903

1003816

1003816

1003816

1003816

) le

120575

4

(A20)


2= 119904

2

1205952 whose time

derivative is

119881

2= 119904

120595(119896

3(120591 minus 120591

119888+ 119901

119903) + (1 minus 119896

3119889

119903)119890

119903) Consider

first that |119904120595| ge 119891



119881

2= 119904

120595119904

120595le minus120575119896

3(

1

2

minus

1

4

)

1003816

1003816

1003816

1003816

1003816

119904

120595

1003816

1003816

1003816

1003816

1003816

997888rarr 119904

120595le minus

120575119896

3

4

sign (119904120595)

(A21)


120595(119905)| le |119904

120595(0)| minus

120575119896

34119905 If 119904


120595| lt 119891

3+120576 (A21) does not




case |119904120595(119905)| le max(|119904

120595(0)| minus 120575119896

34119905 119891





120595= 119904

120595minus 119890

120595 whose

solution is

119890

120595(119905) = 119890

120595(0) 119890

minus1199051198963+

1

119896

3

int

119905

0

119890

minus(119905minus120591)1198963119904

120595(120591) 119889120591 (A22)


1003816

1003816

1003816

1003816

1003816

119890

120595(119905)

1003816

1003816

1003816

1003816

1003816

le

1003816

1003816

1003816

1003816

1003816

119890

120595(0)

1003816

1003816

1003816

1003816

1003816

119890

minus1199051198963+

1003816

1003816

1003816

1003816

1003816

119904

120595(0)

1003816

1003816

1003816

1003816

1003816

(119890

|119904120595(0)|1198861198963minus 1) 119890

minus1199051198963

+ 120574

3(119887 + 119911

119898+ 119901

119898) 119899

119898+ 119888119901

119898+ 120576

(A23)


120595| + |119904

120595|119890

|119904120595|1198861198963






119898Therefore



119898=

2119872 as follows

119896

2=

120575

4120574

4(4119872)

(A24)


119881 le 119896

1

1003816

1003816

1003816

1003816

119911

2

1003816

1003816

1003816

1003816

(

1003816

1003816

1003816

1003816

119865

119903119908

1003816

1003816

1003816

1003816

+ 119901

2) + (1 minus 119889

119906119896

1) (119904

1119890V119909 + 119904

2119890V119910)

le 119887

1|z| + 119887

2

(A25)


2 Select 119872 gt 119887

2119887

1then z gt

119887

2119887

1and thus

119881 = zz le 2119887


1rarr z le

z(0) + 2119887


2119887

1and therefore 119879 gt 1198722119887

2


3(119887 + 119872 + 119901

119898)119899

119898+ 119888119901

119898+ 120576 le 119889119890


120595(119879)| le 2119889 sdot 119890


max(119901

119898+ 120576

5

1003816

1003816

1003816

1003816

119865

119903

1003816

1003816

1003816

1003816

119901

119898+ 120575120576

1003816

1003816

1003816

1003816

119865

119903

1003816

1003816

1003816

1003816

) lt 119889 sdot 119890

minus1198791198963 (A26)


119911max (119879) = radic


4(4119872)

120575


minus119872211988821198963))

(A27)

Function 120574



119898



1for which z(119905



2for which z(119905

2) = 2119872 and 119881

grows in 119905



120595(119905

2)| le 2119889119890


119911max(|119890120595(1199052)|) = 119872 because z lt 119911

119898for all 119879 le 119905

1le 119905

2 This




2



Acknowledgments


References









































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119898)119889


with 119871

2and 119871


Ω = x isin R6

1003816

1003816

1003816

1003816

V119909

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

V119910

1003816

1003816

1003816

1003816

1003816

le

119865max + 119901

119898

119889

119906

|119903| le

120591max + 119901

119898

119889

119903

(A2)



119903| minus 119901




radic

2

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

(1 minus 119889

119906119896

1)

119889

119906119896

1

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

(2119865max + 119901

119898) le 120575120576

1 (A3)


119898so that

radic

2 (119901

119898+ 120575120576

1+

1003816

1003816

1003816

1003816

119865

119903

1003816

1003816

1003816

1003816

119908

119898) lt min (120575 100381610038161003816

1003816

119865

119903

1003816

1003816

1003816

1003816

sin (1198962)) (A4)


(119911

2

1+ 119911

2


119881 = 119896

1119911

1(119865 minus 119865

119903cos (120595 minus 120595

119903) + 119901

1)

minus 119896

1119911

2(

1003816

1003816

1003816

1003816

119865

119903

1003816

1003816

1003816

1003816

sin (1198962tanh (119911

2)) minus 119865

119903119908 minus 119901

2)

+ (1 minus 119889

119906119896

1) (119904

1119890V119909 + 119904

2119890V119910)

(A5)


Case A (|1199111| gt 119899



119903cos(120595 minus 120595


by |119865119903| and |119911


following function

119892 (119909) = min( 120575

radic2

119865

119903

radic2

sin(1198962tanh( 119909

radic2

))) (A6)


1| minus |119865

119903119911

2| sin(119896

2tanh (|119911


119881 le minus119896

1z (119892 (z) minus (120575120576

1+

1003816

1003816

1003816

1003816

119865

119903

1003816

1003816

1003816

1003816

|119908| + 119901

119898)) (A7)


1+ |119865

119903|119908

119898+ 119901

119898


119911max 1 = radic

2 tanhminus1 ( 1

119896

2

sinminus1 (radic2(

119901max + 120575120576

1003816

1003816

1003816

1003816

119865

119903

1003816

1003816

1003816

1003816

+ 119908

119898)))

(A8)



Case B (|1199111| le 119899



2| and

with (31) we find

z le radic

2

1003816

1003816

1003816

1003816

119911

1

1003816

1003816

1003816

1003816

le

radic

2119899

1+ 120576 le

radic

2 (z + radic

2 (1 + 119896

1)) 119899

119898+ 120576

(A9)


z leradic2 (

radic2 (1 + 119896

1) 119899

119898+ 120576)


119898)

= 120576

2 (A10)


2| then z le radic

2|119911

2| so we


2| + 120576

2and |119911

1| le 2(|119911

2| +

1 minus 119896

1)119899


119881 le 119896

1(2 (

1003816

1003816

1003816

1003816

119911

2

1003816

1003816

1003816

1003816

+ 1 minus 119896

1) 119899

119898+ 120576) (

1003816

1003816

1003816

1003816

119865max1003816

1003816

1003816

1003816

+

1003816

1003816

1003816

1003816

119865

119903

1003816

1003816

1003816

1003816

+ 119901

119898)

minus 119896

1

1003816

1003816

1003816

1003816

119911

2

1003816

1003816

1003816

1003816

(

1003816

1003816

1003816

1003816

119865

119903

1003816

1003816

1003816

1003816

sin (1198962tanh (100381610038161003816

1003816

119911

2

1003816

1003816

1003816

1003816

)) minus

1003816

1003816

1003816

1003816

119865

119903119908

1003816

1003816

1003816

1003816

minus 119901

119898)

+ 2 (

radic

2

1003816

1003816

1003816

1003816

119911

2

1003816

1003816

1003816

1003816

+ 120576

2) 119889

119906119896

1120575120576

1

(A11)


3and 120576

4 (A11) can be

rewritten as

119881 le minus 119896

1(

1003816

1003816

1003816

1003816

119911

2

1003816

1003816

1003816

1003816

(

1003816

1003816

1003816

1003816

119865

119903

1003816

1003816

1003816

1003816

sin (1198962tanh (100381610038161003816

1003816

119911

2

1003816

1003816

1003816

1003816

))

minus

1003816

1003816

1003816

1003816

119865

119903119908

1003816

1003816

1003816

1003816

minus 119901

119898minus 120576

3) + 120576

4)

(A12)

Now we define


119896

2

sinminus1 (119901

119898

1003816

1003816

1003816

1003816

119865

119903

1003816

1003816

1003816

1003816

+ 120576

3+ 119908

119898+ 120575)) (A13)



119881 le 119896




119898


119898 119899119898 119901119898 and 120576 then

(35) holds


119909and 119904

119910

119896

1119890

119909= 119904

119909minus 119890

119909

119896

1119890

119910= 119904

119910minus 119890

119910

(A14)


119898 119901119898 119908119898 and 120576 then (36) holds



119903| le |119903| + |119903

119903| le |120591

119898|119889

119903+ |119903

119903| The



z le 1003816

1003816

1003816

1003816

1

1003816

1003816

1003816

1003816

+

1003816

1003816

1003816

1003816

2

1003816

1003816

1003816

1003816

le 119888

1+ 119888

2119911

119898+ 119888

3119901

119898

1003816

1003816

1003816

1003816

2

1003816

1003816

1003816

1003816

le 119888

4+ 119888

5119911

119898+ 119888

6119901

119898

(A15)


120591

119888= 120591

119903minus 119896

2sign (119865

119903) (1 minus tanh2 (119911

2)) (119889

119903

2+

2)

+ 2119896

2sign (119865

119903) (1 minus tanh2 (119911

2)) tanh (119911

2)

2

2

(A16)


120595and 120591

119888can be




1003816

1003816

1003816

1003816

120591

119888

1003816

1003816

1003816

1003816

le

1003816

1003816

1003816

1003816

120591

119903

1003816

1003816

1003816

1003816

+ 119896

2120574

4(119911

119898+ 119901

119898) (A17)

1003816

1003816

1003816

1003816

119891

3

1003816

1003816

1003816

1003816

le 120574

5(119888

7+ 119911

119898+ 119901

119898) 119899

119898+ 119896

3119896

2119896

1119901

119898 (A18)



120575 Now we define 119911119898as

119911

119898= minus

1

2

120574

4

minus1

(

120575

4119896

2

) (A19)

When z lt 119911


119896

2120574

4(2119901

119898)+119901

119898lt 1205754 and with (A17) then |120591

119894|minus|120591

119888|minus|119901

119898| ge


1003816

1003816

1003816

1003816

1 minus 119896

3119889

119903

1003816

1003816

1003816

1003816

119896

3119889

119903

(

1003816

1003816

1003816

1003816

120591

119898

1003816

1003816

1003816

1003816

+ 119889

119903

1003816

1003816

1003816

1003816

119903

119903

1003816

1003816

1003816

1003816

) le

120575

4

(A20)


2= 119904

2

1205952 whose time

derivative is

119881

2= 119904

120595(119896

3(120591 minus 120591

119888+ 119901

119903) + (1 minus 119896

3119889

119903)119890

119903) Consider

first that |119904120595| ge 119891



119881

2= 119904

120595119904

120595le minus120575119896

3(

1

2

minus

1

4

)

1003816

1003816

1003816

1003816

1003816

119904

120595

1003816

1003816

1003816

1003816

1003816

997888rarr 119904

120595le minus

120575119896

3

4

sign (119904120595)

(A21)


120595(119905)| le |119904

120595(0)| minus

120575119896

34119905 If 119904


120595| lt 119891

3+120576 (A21) does not




case |119904120595(119905)| le max(|119904

120595(0)| minus 120575119896

34119905 119891





120595= 119904

120595minus 119890

120595 whose

solution is

119890

120595(119905) = 119890

120595(0) 119890

minus1199051198963+

1

119896

3

int

119905

0

119890

minus(119905minus120591)1198963119904

120595(120591) 119889120591 (A22)


1003816

1003816

1003816

1003816

1003816

119890

120595(119905)

1003816

1003816

1003816

1003816

1003816

le

1003816

1003816

1003816

1003816

1003816

119890

120595(0)

1003816

1003816

1003816

1003816

1003816

119890

minus1199051198963+

1003816

1003816

1003816

1003816

1003816

119904

120595(0)

1003816

1003816

1003816

1003816

1003816

(119890

|119904120595(0)|1198861198963minus 1) 119890

minus1199051198963

+ 120574

3(119887 + 119911

119898+ 119901

119898) 119899

119898+ 119888119901

119898+ 120576

(A23)


120595| + |119904

120595|119890

|119904120595|1198861198963






119898Therefore



119898=

2119872 as follows

119896

2=

120575

4120574

4(4119872)

(A24)


119881 le 119896

1

1003816

1003816

1003816

1003816

119911

2

1003816

1003816

1003816

1003816

(

1003816

1003816

1003816

1003816

119865

119903119908

1003816

1003816

1003816

1003816

+ 119901

2) + (1 minus 119889

119906119896

1) (119904

1119890V119909 + 119904

2119890V119910)

le 119887

1|z| + 119887

2

(A25)


2 Select 119872 gt 119887

2119887

1then z gt

119887

2119887

1and thus

119881 = zz le 2119887


1rarr z le

z(0) + 2119887


2119887

1and therefore 119879 gt 1198722119887

2


3(119887 + 119872 + 119901

119898)119899

119898+ 119888119901

119898+ 120576 le 119889119890


120595(119879)| le 2119889 sdot 119890


max(119901

119898+ 120576

5

1003816

1003816

1003816

1003816

119865

119903

1003816

1003816

1003816

1003816

119901

119898+ 120575120576

1003816

1003816

1003816

1003816

119865

119903

1003816

1003816

1003816

1003816

) lt 119889 sdot 119890

minus1198791198963 (A26)


119911max (119879) = radic


4(4119872)

120575


minus119872211988821198963))

(A27)

Function 120574



119898



1for which z(119905



2for which z(119905

2) = 2119872 and 119881

grows in 119905



120595(119905

2)| le 2119889119890


119911max(|119890120595(1199052)|) = 119872 because z lt 119911

119898for all 119879 le 119905

1le 119905

2 This




2



Acknowledgments


References









































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119903| le |119903| + |119903

119903| le |120591

119898|119889

119903+ |119903

119903| The



z le 1003816

1003816

1003816

1003816

1

1003816

1003816

1003816

1003816

+

1003816

1003816

1003816

1003816

2

1003816

1003816

1003816

1003816

le 119888

1+ 119888

2119911

119898+ 119888

3119901

119898

1003816

1003816

1003816

1003816

2

1003816

1003816

1003816

1003816

le 119888

4+ 119888

5119911

119898+ 119888

6119901

119898

(A15)


120591

119888= 120591

119903minus 119896

2sign (119865

119903) (1 minus tanh2 (119911

2)) (119889

119903

2+

2)

+ 2119896

2sign (119865

119903) (1 minus tanh2 (119911

2)) tanh (119911

2)

2

2

(A16)


120595and 120591

119888can be




1003816

1003816

1003816

1003816

120591

119888

1003816

1003816

1003816

1003816

le

1003816

1003816

1003816

1003816

120591

119903

1003816

1003816

1003816

1003816

+ 119896

2120574

4(119911

119898+ 119901

119898) (A17)

1003816

1003816

1003816

1003816

119891

3

1003816

1003816

1003816

1003816

le 120574

5(119888

7+ 119911

119898+ 119901

119898) 119899

119898+ 119896

3119896

2119896

1119901

119898 (A18)



120575 Now we define 119911119898as

119911

119898= minus

1

2

120574

4

minus1

(

120575

4119896

2

) (A19)

When z lt 119911


119896

2120574

4(2119901

119898)+119901

119898lt 1205754 and with (A17) then |120591

119894|minus|120591

119888|minus|119901

119898| ge


1003816

1003816

1003816

1003816

1 minus 119896

3119889

119903

1003816

1003816

1003816

1003816

119896

3119889

119903

(

1003816

1003816

1003816

1003816

120591

119898

1003816

1003816

1003816

1003816

+ 119889

119903

1003816

1003816

1003816

1003816

119903

119903

1003816

1003816

1003816

1003816

) le

120575

4

(A20)


2= 119904

2

1205952 whose time

derivative is

119881

2= 119904

120595(119896

3(120591 minus 120591

119888+ 119901

119903) + (1 minus 119896

3119889

119903)119890

119903) Consider

first that |119904120595| ge 119891



119881

2= 119904

120595119904

120595le minus120575119896

3(

1

2

minus

1

4

)

1003816

1003816

1003816

1003816

1003816

119904

120595

1003816

1003816

1003816

1003816

1003816

997888rarr 119904

120595le minus

120575119896

3

4

sign (119904120595)

(A21)


120595(119905)| le |119904

120595(0)| minus

120575119896

34119905 If 119904


120595| lt 119891

3+120576 (A21) does not




case |119904120595(119905)| le max(|119904

120595(0)| minus 120575119896

34119905 119891





120595= 119904

120595minus 119890

120595 whose

solution is

119890

120595(119905) = 119890

120595(0) 119890

minus1199051198963+

1

119896

3

int

119905

0

119890

minus(119905minus120591)1198963119904

120595(120591) 119889120591 (A22)


1003816

1003816

1003816

1003816

1003816

119890

120595(119905)

1003816

1003816

1003816

1003816

1003816

le

1003816

1003816

1003816

1003816

1003816

119890

120595(0)

1003816

1003816

1003816

1003816

1003816

119890

minus1199051198963+

1003816

1003816

1003816

1003816

1003816

119904

120595(0)

1003816

1003816

1003816

1003816

1003816

(119890

|119904120595(0)|1198861198963minus 1) 119890

minus1199051198963

+ 120574

3(119887 + 119911

119898+ 119901

119898) 119899

119898+ 119888119901

119898+ 120576

(A23)


120595| + |119904

120595|119890

|119904120595|1198861198963






119898Therefore



119898=

2119872 as follows

119896

2=

120575

4120574

4(4119872)

(A24)


119881 le 119896

1

1003816

1003816

1003816

1003816

119911

2

1003816

1003816

1003816

1003816

(

1003816

1003816

1003816

1003816

119865

119903119908

1003816

1003816

1003816

1003816

+ 119901

2) + (1 minus 119889

119906119896

1) (119904

1119890V119909 + 119904

2119890V119910)

le 119887

1|z| + 119887

2

(A25)


2 Select 119872 gt 119887

2119887

1then z gt

119887

2119887

1and thus

119881 = zz le 2119887


1rarr z le

z(0) + 2119887


2119887

1and therefore 119879 gt 1198722119887

2


3(119887 + 119872 + 119901

119898)119899

119898+ 119888119901

119898+ 120576 le 119889119890


120595(119879)| le 2119889 sdot 119890


max(119901

119898+ 120576

5

1003816

1003816

1003816

1003816

119865

119903

1003816

1003816

1003816

1003816

119901

119898+ 120575120576

1003816

1003816

1003816

1003816

119865

119903

1003816

1003816

1003816

1003816

) lt 119889 sdot 119890

minus1198791198963 (A26)


119911max (119879) = radic


4(4119872)

120575


minus119872211988821198963))

(A27)

Function 120574



119898



1for which z(119905



2for which z(119905

2) = 2119872 and 119881

grows in 119905



120595(119905

2)| le 2119889119890


119911max(|119890120595(1199052)|) = 119872 because z lt 119911

119898for all 119879 le 119905

1le 119905

2 This




2



Acknowledgments


References









































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











1for which z(119905



2for which z(119905

2) = 2119872 and 119881

grows in 119905



120595(119905

2)| le 2119889119890


119911max(|119890120595(1199052)|) = 119872 because z lt 119911

119898for all 119879 le 119905

1le 119905

2 This




2



Acknowledgments


References









































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





















Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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Research Article Nonlinear Control for Trajectory Tracking ...downloads.hindawi.com/journals/mpe/2013/589267.pdf · Research Article Nonlinear Control for Trajectory Tracking of a