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published online 24 July 2014Transactions of the Institute of Measurement and ControlM Zamurad Shah, Raza Samar and Aamer I Bhatti

Lateral track control of UAVs using the sliding mode approach: from design to flight testing

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Lateral track control of UAVs using thesliding mode approach: from design toflight testing

M Zamurad Shah, Raza Samar and Aamer I Bhatti

AbstractThis paper develops sliding-mode-based nonlinear logic for guidance of unmanned aerial vehicles (UAVs) for curved and straight path following. UAV

trajectories generally consist of straight path segments, curved arcs, circular loiters and other manoeuvres; tight ground track control is desired

throughout the trajectory. This is achieved by controlling the lateral (cross-track) deviation of the vehicle in flight. The main objective of the guidance

algorithm is to keep the lateral track error of the vehicle as small as possible while performing graceful and stable manoeuvres despite the presence of

uncertainties and disturbing winds. Lateral track control is usually achieved by banking the vehicle, that is, by executing roll manoeuvres. The scheme

must perform well without saturating the roll angle of the vehicle, which serves as the control input for the guidance algorithm. The algorithm pro-

posed here is shown to perform well for both straight and circular path tracking while ensuring control boundedness, and hence no saturation.

Crosswinds are a major source of disturbance for the guidance problem. This is incorporated into the design formulation and guidance gains are

selected to provide the desired performance despite the presence of disturbing winds. A sliding-mode-based scheme is developed which includes a

feedforward component related to the rate of change of the desired path heading. Stability of the algorithm is proved using an appropriate Lyapunov

function. The algorithm is implemented in the flight control computer of a scaled YAK-54 research aircraft; flight test results are presented and com-

pared with those from other guidance algorithms. Flight results demonstrate the effectiveness and performance of the proposed guidance scheme. The

algorithm considers guidance in the 2D lateral plane only and minimizes deviations from the desired ground track of the vehicle.

KeywordsUAV guidance, sliding mode control, unmanned aerial vehicles, lateral guidance, track control, control boundedness, guidance of UAVs in the presence

of winds.

Introduction

In recent years, the use of unmanned aerial vehicles (UAVs)

has increased significantly in military, commercial and civil-

ian applications due to their low cost, low human risk and

operational effectiveness. UAVs offer a unique range of fea-

tures, such as ultra-long endurance, demanding trajectory-

following and high-risk mission acceptance, which cannot be

easily performed by manned aircraft. This has been made

possible due to the technological advancements of the last

two decades; UAVs have now obtained a permanent and crit-

ical role in high-tech military arsenals. Beside the military, the

use of UAVs has risen sharply in civilian and commercial

applications as well. Such applications include surveillance

and law enforcement, search and rescue, environmental stud-

ies, mapping and surveying, media and traffic reporting, for-

est fire monitoring and control, gas/oil/water pipeline and

power line monitoring, agriculture growth patterns monitor-

ing and field spraying/crop dusting, radio/communications

relays and disaster surveying and management operations. In

all these applications, the requirement of closely following the

given ground track, despite complex mission trajectories,

remains a fundamental requirement. For example, in the

applications of agricultural field spraying, forest fire control,

rescue operations, border monitoring, etc., the need for tight

lateral track control is obvious.The ‘guidance’ and ‘control’ design problems for

unmanned aircraft are usually treated separately, and imple-

mented in an inner-loop, outer-loop configuration. The gui-

dance law resides in the outer loop, uses ground track

measurements and generates appropriate commands for the

inner loop to follow. These commands can be in the form of

reference roll angle or reference lateral acceleration com-

mands. The inner loop consists of a tracking controller that

accepts the reference commands generated by the guidance

loop, and performs control and stability augmentation tasks.

This approach to design, being simple and intuitive, can be

found in the literature in various papers; see for example

Dadkhah and Mettler (2012), Niculescu (2001), Rysdyk

(2006), Samar et al. (2008) and Shah et al. (2011a). Design of

the outer loop guidance logic has been inspired by techniques

Mohammad Ali Jinnah University, Islamabad, Pakistan

Corresponding author:

M Zamurad Shah, Department of Electronic Engineering, Mohammad Ali

Jinnah University, Near Kakpul, Sihala, Islamabad, Pakistan.

Email: [email protected]

at NATL AEROSPACE LABORATORIES on November 13, 2014tim.sagepub.comDownloaded from


as diverse as proportional navigation, cognitive and intelli-gent methods, vision-based techniques, and others (Cesetti etal., 2009; Ren and Beard, 2004; Siouris, 2004; Yamasaki andBalakrishnan, 2010; Zhang et al., 2013). The objectiveremains to drive the lateral ground track deviation of the

vehicle to a minimum. The inner loop control design problemis beyond the scope of this paper; numerous well-establishedtechniques for flight control design exist such as linear robustdesign and nonlinear and intelligent control design tech-niques. The reader may refer to any text on advanced controlfor further reading.

Some work has also been done on the so-called integratedapproach in which the guidance and control problems aredealt with in an integrated and unified framework (Dadkhahand Mettler, 2012; Park et al., 2004; Yamasaki andBalakrishnan, 2010; Yu et al., 2011). This approach is morecomplicated due to coupling of the different guidance andcontrol variables and needs more research (Kaminer et al.,1998; Yamasaki et al., 2012a,b). Most of the practising aero-space community favours the two-step approach, and that iswhat is considered in this paper.

A tracking guidance law for a pursuer UAV is consideredin Regina and Zanzi (2011). The UAV tracks a moving targetand the guidance law generates lateral acceleration commandswhich the control system follows. A back-stepping-based gui-dance law is proposed in Ahmed and Subbarao (2010); first-order dynamics are assumed for elevation and heading angles.Design of a path-following controller for a small UAV is pre-sented in Sasongko et al. (2011); the cross-track and courseangle errors are used to generate heading angle corrections.Heading change is accomplished using bank-to-turn man-oeuvres; a linear quadratic regulator (LQR)-based controllerprovides inner-loop roll control. Simulation results are pre-sented to demonstrate the working of the system. Guidancebased on the concept of vector fields for curved path follow-ing has also been studied (Griffiths, 2006; Pisano et al., 2007).Vehicle position relative to the desired path is computed, anda vector field of heading commands is generated to drive the

error to zero. This may, in some cases, generate large headingcommands thereby saturating the control system of thevehicle.

Proportional-derivative (PD) logic for outer loop guidancehas been employed in some UAV applications (Pappoullis,1994; Siouris, 2004). Modifications to the PD scheme for dis-turbance rejection and saturation avoidance are proposed inSamar et al. (2007, 2008); however, no formal stability proofsare provided. A nonlinear guidance algorithm is proposed inPark et al. (2004), and its stability discussed in Deyst et al.(2005); the performance is better than the PD scheme for bothstraight and curved path following. When the track errorsbecome large, however, the control input saturates and stabi-lity is not guaranteed. Also, the algorithm needs a ‘referencepoint’ for computation, which is a point from the missionplan ahead of the current position of the vehicle; if the mis-sion gets changed online, it can give rise to a discontinuity.For a more complete survey of different approaches to UAVguidance, the reader is referred to references Dadkhah andMettler (2012) and Goerzen et al. (2009).

Crosswinds are a major source of disturbance for accuratelateral track control. Wind effect is ignored by most authors

in the design of the guidance logic. UAV dynamics in the pres-

ence of winds is discussed in references Ceccarelli et al. (2007),

McGee and Hedrick (2006) and Osborne and Rysdyk (2005);

however, the emphasis here is not on guidance but on how to

modify the planned mission to best cater for wind conditions.

Beard and McLain (2012) discuss the effect of wind on gui-

dance performance of UAVs and aircraft. In Brezoescu et al.

(2013) an adaptive backstepping-based guidance law (consid-

ering the effect of winds) for UAVs is presented using a skid-

to-turn concept.Direct application of sliding mode control in the outer gui-

dance loop is normally not feasible; interested readers are

referred to Shtessel et al. (2003, 2007) for a detailed discus-

sion. In the authors’ previous work (Shah et al., 2011b, 2014),

a novel nonlinear sliding surface is proposed for lateral gui-

dance of UAVs. The main theme of that work was a high-

performance sliding surface and algorithm for simple straight

paths; the effect of wind was not considered. Derivation of the

algorithm was based on the assumption that the reference

course angle will not change rapidly (i.e. no sharp circular or

curved paths). Flight tests for circular and curved paths in the

presence of wind indicated a need for performance improve-

ment, and hence an extension of that work is required from

an applications point of view. Subsequently assumption of ‘no

rapid change in reference course angle’ was relaxed in Shah et

al. (2012) and theoretical aspects along with simulation results

discussed there. In this work, a generalized nonlinear guidance

law is presented for both straight and curved path following

in the presence of winds. Performance is shown to be quite

good both for small and large track errors. Flight test results

with the proposed algorithm are also presented; these demon-

strate excellent performance for both straight and rapidly

turning paths even in windy conditions. The work presented

considers guidance in the 2D horizontal plane only (lateral

guidance): this can be extended to the more general 3D gui-

dance problem which is the subject of ongoing research.This paper is organized as follows. ‘Problem formulation’

describes the problem under consideration in relative detail.

We define the reference coordinate system and different para-

meters relevant for circular and straight path segments, state

assumptions and develop state equations for guidance design.

In ‘Sliding surface for cross-track control’, a brief review of

previous work regarding nonlinear sliding mode guidance is

given; this forms the basis for further development of the gui-

dance algorithm. ‘Guidance law design’ describes design of a

nonlinear guidance law for straight and circular path follow-

ing. Stability and control boundedness proofs for the proposed

law are also discussed. The guidance law is implemented on a

test aircraft and various flight tests carried out. Flight results

show efficacy of the proposed law; these are presented in

‘Flight test results’. Comparison with previous algorithms is

also made. The final section concludes the paper.

Problem formulation

Path planning for UAVs is an active area of research. The

output of a path planning algorithm is in general a sequence

of waypoints joined by a curved path that the UAV should

follow. The curved path can be a combination of straight lines

2 Transactions of the Institute of Measurement and Control



and circular arcs of different radii. The guidance algorithm

generates suitable commands for the vehicle control system so

that the desired path is followed in the presence of distur-

bances. In this paper, we consider the lateral track control (or

guidance) problem of UAVs during straight as well as circular

paths in the presence of winds.Before proceeding to the guidance problem, we first define

some important parameters as shown in Figure 1. The body

heading angle c is the angle which the nose of the aircraft

(usually referred to as the body x -axis) makes with respect to

north. The airspeed vector ~Va denotes the velocity of the air-

craft relative to air, and is not necessary aligned with the body

axis. The difference between the body heading and the relative

air velocity vector angle is called the sideslip angle b. In steady

level flight without crosswind, the airspeed vector ~Va is aligned

with the nose of the aircraft, implying zero sideslip angle.

Another important parameter is the ground velocity vector~Vg, which gives the direction of travel of the vehicle relative to

ground. In the absence of wind, the vector ~Vg is equal to ~Va.

In the presence of wind, ~Vg is the vector sum of ~Va and the

wind velocity vector ~Vw, as shown in Figure 1. The course

angle cG is the angle of the ground velocity ~Vg with respect to

north. Generally in an inner–outer loop design strategy, kine-

matics is considered in the outer guidance loop and dynamics

is considered in the design of the inner control loop. The

dynamical variables respond much more quickly in time scale,

and hence the inner control loop is usually kept much faster

than the outer guidance loop. The sideslip angle is related to

the ‘dynamics’ of the aircraft and must be considered in the

design of the inner control loop. The course angle cG is a

‘kinematic’ variable, and considered in guidance loop design.Different guidance variables and their sign conventions are

shown in Figures 2 and 3 for straight and circular paths,

respectively. Both figures depict the positive sense of all indi-

cated variables. Consider a straight path segment (Figure 2),

let WP1 and WP2 be two successive waypoints, and let cR be

the angle of the line WP1 –WP2 with respect to north (called

the reference or desired course angle). The lateral (or cross-

track) deviation of the vehicle from the desired path is

denoted by y. In the case of a curved path (Figure 3), let WP1

and WP2 be two successive waypoints on a circular arc of

radius R centred at point O. Point P is the nearest point to

the vehicle on the arc. The reference or desired course angle

cR is defined as the angle of the tangent line at P with respect

to north. The cross-track y is defined as in the case of the

straight path. Another parameter of interest is cE =cG 2 cR,

usually referred to as the intercept course. The main task of

the guidance algorithm is to keep the cross-track error y as

small as possible, and also to keep cE ’ 0 when y ’ 0. In

the case of a non-zero y, the guidance algorithm should

manipulate cE by banking the vehicle and bring y to zero.

Generally for very large track errors, a constant cE (� p/2)

is desired, and when the track error reduces, cE is adjusted

accordingly (a criterion known as the ‘good helmsman’ in lit-

erature; Pettersen and Lefeber, 2001; Rysdyk, 2006).The block diagram of the overall guidance and control

system is shown in Figure 4. The outer guidance block gets

mission information from the mission plan in terms of way-

points, and feedback from sensors measuring the instanta-

neous position and course (cG) of the vehicle. The guidance

algorithm generates a roll angle command (fr) for the inner

control loop to track; the inner loop generates commands for

the control surfaces, that is, the ailerons and the rudder. Here

we shall focus on the design of the outer guidance logic that

ensures minimum cross-track error y for both straight and

curved paths by generating appropriate roll angle commands

in the presence of disturbing winds. This will be done using

the sliding mode approach.

V→

V→

V→

Figure 1. Definition of various angles.

WP-1

WP-2

yV→

Figure 2. The guidance problem definition for a straight path.

Centre of turn (O)

R

P

yRad

ius o

f tur

n (R

)

WP-1

WP-1

V→

Figure 3. The guidance problem definition for a circular path.

Shah et al. 3



Assumptions

We shall make the following simplifying assumptions:

1. It is assumed that a control law for the inner loop isavailable. The inner loop controller may be designedusing well-established techniques, such as robust linearcontrol design methods (Skogestad and Postlethwaite,2005).

2. We also assume that the inner loop dynamics from fr

to f are significantly faster (e.g. 5 ; 10 times faster)

than the outer (guidance) loop dynamics.3. The states y and cE (equation (18)) are assumed to be

perfectly measurable.

System dynamics

As discussed earlier, the ground velocity ~Vg is the vector sum

of the airspeed vector ~Va and the wind velocity vector ~Vw, in

other words, ~Vg =~Va +~Vw. Considering a fixed and level

north–east–down inertial coordinate system, the wind triangle

can be expressed as (Beard and McLain, 2012)

Vg

cos cG cos g

sin cG cos g

� sin g

0@

1A=

Vwn

Vwe

Vwd

0@

1A+Va

cos c cos ga

sin c cos ga

� sin ga

0@

1A ð1Þ

where Vg and Va are the magnitudes of the ground and air-

speed vectors, respectively, and g and ga denote the inertial

referenced and air-mass-referenced flight-path angles respec-

tively, in the longitudinal plane. For level flight (g = 0) the

projection of these vectors on the north–east (2D ground)

plane is written as

Vgcos cG

sin cG

� �=

Vwn

Vwe

� �+Va

cos c

sin c

� �ð2Þ

or alternatively

Vgcos cG

sin cG

� �� Vwn

Vwe

� �=Va

cos c

sin c

� �ð3Þ

Taking the square of equation (3), we have

V 2g � 2Vg(Vwn

cos cG +Vwesin cG)+V 2

w � V 2a = 0 ð4Þ

This equation gives the relation between Vg and Va that

depends on the magnitude and direction of wind and the

course angle cG. For UAVs usually a constant airspeed Va is

maintained at a given altitude either through a closed-loop

speed controller, or using a preset open-loop throttle-setting

table. In the presence of winds, a constant Va can still be

approximately maintained, but Vg will vary depending on the

course angle cG and the direction and magnitude of the wind.

As an example consider a constant airspeed of 30 m/s: the

variation of Vg with cG is shown in Figure 5 for a north wind

of 5 m/s (~Vw =(5, 0)T), and also for an east wind of same

magnitude (~Vw =(0, 5)T). The relation between c and cG can

also be derived using (3); multiplying both sides of the equa-

tion on the left-hand side by the row vector ( 2sin cG,cos cG)

yields

Vwnsin cG � Vwe

cos cG =Va � sin cG cos c+ cos cG sin cð Þð5Þ

which can be written in simplified form as

c� cG = sin�1 1

Va

Vwnsin cG � Vwe

cos cGð Þ� �

ð6Þ

The concept of coordinated turns in the presence of winds

is discussed in detail in Beard and McLain (2012). The term

‘coordinated turn’ is used in the sense that there is no side

force in the body frame of the vehicle, thus implying zero

sideslip angle. Aerial vehicles use a component of the aerody-

namic lift to generate lateral accelerations to correct the lat-

eral (cross-track) errors during flight; these are balanced by

centrifugal accelerations in a coordinated turn. Lateral accel-

eration is produced by banking the vehicle so that a compo-

nent of the lift vector is tilted in the required direction (the

bank-to-turn manoeuvre). The guidance loop generates

appropriate bank (or roll) commands fr for the inner control

loop to follow: fr therefore serves as the control input in our

case. Consider a coordinated turn (b = 0) in the presence of

wind: the aerodynamic lift vector L (Figure 6) can be resolved

into two components as shown in Figure 6. The component

Guidance

Mission

Ref. roll angle (φr)

Waypoints,ψR and ψR

Control UAV lateraldynamics

Aileron

Rudder

Current position and heading angle (ψG)

Yaw rate

Roll angle (φ)

+-

Washout filter

Figure 4. The guidance and control scheme.

0 50 100 150 200 250 300 350 40025

26

27

28

29

30

31

32

33

34

35

ψG [˚]

Vg [

m/s

]

North wind (5 m/s)East wind (5 m/s)

Figure 5. Variation of Vg for constant Va in the presence of wind.




(L cos f) balances the weight of the vehicle, and a part of theother component balances the centrifugal force during a turn

(Beard and McLain, 2012). Summing the forces in the vertical

and radial directions:

L cosf=mg, L sinf cos (cG � c)=mV 2

g

Rð7Þ

Here m denotes the mass, g is the gravitational acceleration

and R is the radius of turn of the vehicle. From (7) above, we

get

tan f cos (cG � c)=V 2

g

Rgð8Þ

During a steady turn Vg =R _cG, so (8) becomes

tan f cos (cG � c)=Vg

_cG

gð9Þ

Referring to Assumption 2 of ‘Assumptions’, where we

assume a significantly faster inner loop, we have fr ’ f, and

so (9) can be approximated as

tan fr cos (cG � c)=Vg

_cG

gð10Þ

Now since cE =cG 2 cR we have

tan fr cos (cG � c)=Vg( _cE + _cR)

gð11Þ

Rearranging (11), we get

_cE =g tanfr

Vg

cos (cG � c)� _cR ð12Þ

Using (6), the above state equation becomes

_cE =g tanfr

Vg

cos sin�1 1

Va

�Vwnsin cG +Vwe

cos cGð Þ� ��

� _cR

ð13Þ

where cE is a state variable and _cR is the rate of change of

the desired heading: this information is available from mis-

sion data. During straight flight, _cR is equal to zero, but it

can become appreciable during sharp turns.

The inertial position (Pn, Pe, Pd)T of the UAV in the fixed

north–east–down frame can be derived using

_Pn

_Pe

_Pd

0@

1A=Va

cos c cos ga

sin c cos ga

sin ga

0@

1A+

Vwn

Vwe

Ved

0@

1A ð14Þ

provided the airspeed and wind information is available.

Alternatively, we can derive this from the ground velocity

vector directly:

_Pn

_Pe

_Pd

0@

1A=Vg

cos cG cos g

sin cG cos g

sin g

0@

1A ð15Þ

For level flight (g = 0), the north–east position of the

UAV is

_Pn

_Pe

� �=Vg

cos cG

sin cG

� �ð16Þ

To compute the cross-track deviation (y) at any point (see

Figures 2 and 3) we can use the relation

_y=Vg sin cE ð17Þ

Equations (17) and (13) represent the overall dynamics for the

outer loop guidance design problem. For simplicity, we may

define u = tan fr as the control input. In state-space form we

can then write

Centre of turn

cos

O

(a) Back view during coordinated turn

(b) Top view during coordinated turn

V→

V→ V

→

L co

s

L

R

L

Figure 6. Components of the lift vector L during a steady turn of

radius R.

Shah et al. 5



_y=Vg sin cE

_cE =gu

Vg

cos sin�1 1

Va

�Vwnsin cG +Vwe


� _cR

ð18Þ

Here y and cE are the state variables, and u = tan fr is thecontrol signal that the guidance loop has to generate for

cross-track control.

The guidance problem

The objective of the guidance logic is to provide smooth andstable control of the lateral deviation of the vehicle and to

keep it as small as possible in the presence of disturbances. If

a track error develops (for example in the initial phase aftertake-off when the guidance loop is first activated, or when the

navigation information gets corrected after a long GPS out-

age), it should be brought to zero in a graceful manner with-out saturating the control input to the vehicle. Also, the error

should be minimized quickly with little or no overshoot for

both small and large track errors.

Sliding surface for cross-track control

In our previous work (Shah et al., 2011a,b, 2014), sliding sur-face design for the guidance loop was discussed in detail.

Limitations of linear surfaces were discussed, and thereafter a

piecewise linear sliding surface was proposed in Shah et al.

(2011b) and a nonlinear sliding surface was proposed in Shahet al. (2011a, 2014). It was shown that linear surfaces do not

perform well for both small and large track errors. If a linear

surface is designed to give good performance for small errors,it causes control saturation for large errors, and if on the

other hand the surface is designed to avoid saturation, poor

performance results for small errors. A novel nonlinear slid-

ing surface (Figure 7) was proposed in Shah et al. (2011a,

2014) that overcomes these limitations and performs well for

both small and large error scenarios. The following equation

represents the nonlinear sliding surface:

s=cE +a arctanby= 0 ð19Þ

Here the two adjustable parameters a, b 2 < (the set of real

numbers), and for stability of the sliding surface we require

ab . 0 (Shah et al., 2014). It is clear from (19) that we need

jaj � 1 in order to ensure jcEj � p/2, which is required for

correct direction of approach while guiding the vehicle

towards the desired path. We now derive a guidance law

based on the sliding surface (19) that works for both straight

and circular paths in the presence of winds.

Guidance law design

Sliding-mode-control design is essentially a two-step process:

sliding-surface design, and derivation of a control law to

ensure that the phase trajectory is attracted towards the slid-

ing surface. A suitable, stable sliding surface is presented in

‘Sliding surface for cross-track control’ above. We discuss a

sliding-mode-based guidance law in this section; conditions to

ensure sliding and control boundedness are also derived.

Equivalent lateral control

Equivalent control is the control input which, when applied

to the system, enables the system to continue sliding once it is

on the sliding surface (Bandyopadhyay and Janardhanan,

−1000 −500 0 500 1000−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

Cross track error (y) [m]

Ψ E[r

ad]

Figure 7. The nonlinear sliding surface (19) for a particular a and b.




2006). The equivalent control that maintains the sliding modeis therefore the input ueq satisfying _s= 0. In our case

_s= _cE +ab

1+b2y2_y ð20Þ

Using state equations (18), _s= 0 can be written as

0=~gueq

~Vg

cos sin�1 1

~Va

�~Vwnsin cG + ~Vwe

cos cG

� ��

� _cR +ab

1+b2y2~Vg sin cE ð21Þ

where ~g and ~V are the measured values of gravity and velo-city, respectively. After simplification, the expression forequivalent control ueq becomes

ueq =1

cos sin�1 1~Va�~Vwn

sin cG + ~Vwecos cG

� �� ~Vg

~g_cR

�� ab

1+b2y2

~V 2g

~gsin cE

! ð22Þ

This control has two parts: a feedback part and a feedforwardpart ((~Vg=~g) _cR) that is directly related to the rate of change ofthe reference course angle. It is seen that the lateral accelera-tion generated by the feedforward component is equal to the

centripetal acceleration during a turn, provided the velocityVg is known exactly. Equivalent control can maintain slidingmotion only if the state trajectory is on the sliding manifold,and the dynamics are perfectly known (no uncertainty). A for-mal control law, possibly variable structure, has to be formu-lated to take care of the uncertainty, and to bring the systemstates onto the sliding manifold in the first place.

The complete guidance law

The guidance algorithm must ensure sliding in the presence ofparametric uncertainties and input disturbances and from anarbitrary initial condition, in finite time. For this a discontin-uous control term of the form 2 k sgn (s) is added to theequivalent control term to ensure sliding in the presence ofuncertainties. The total guidance law is then the sum of theequivalent and discontinuous control terms. The gain k is

selected based on Lyapunov theory to ensure stable slidingmotion. The lateral guidance law therefore becomes

u = 1

cos sin�1 1~Va�~Vwn sin cG + ~Vwe cos cGð Þ

� �� ~Vg

~g_cR

�

� ab

1+b2y2

~V 2g

~g sin cE

�� k sgn(s)

fr = tan�1 (u)

ð23Þ

where ab . 0 and jaj � 1. The term

1


sin cG + ~Vwecos cG

� ��

is for compensation of wind; in the absence of wind it is equal

to 1. Generally wind measurement is not available on board

UAVs, and hence this term has to be estimated. The above

guidance law (23) can be written in the form

u=G~Vg

~g_cR �

ab

1+b2y2

~V 2g

~gsin cE

!� k sgn(s)

fr =tan�1 (u)

ð24Þ

where

G=1


sin cG + ~Vwecos cG

� ��

Here we assume that both state variables y and cE are accu-

rately measured and available for feedback control.

Conditions on k are derived in the following sections to cater

for uncertainties in different parameters, including G.From an applications perspective, two approaches are pos-

sible to deal with the wind disturbance. In the first approach

G is taken as 1, but its maximum variation due to extreme

winds is computed beforehand during the design phase; this

variation is treated as an uncertainty and is considered in

selection of the gain k. In the second approach, an estimator

is used to estimate the wind speed and direction, and hence G

is estimated in real time and used in the guidance law (25);

any estimation error is considered while selecting k. Here in

this paper we adopt the first approach and take G equal to 1;

the wind disturbance is considered in design of the gain k.

The simplified guidance law therefore becomes

u=~Vg

~g_cR �

ab

1+b2y2

~V 2g

~gsin cE � k sgn(s)

fr =tan�1 (u)

ð25Þ

Reachability condition

Reachability implies that the state trajectories are attracted

towards the sliding surface, and, once achieved, it maintains

that sliding motion for subsequent time periods

(Bandyopadhyay et al., 2009; Perruquetti and Barbot, 2002).

To check for reachability, let us take V = s2/2 as the

Lyapunov candidate function. The derivative of V is

_V= s_s

= sgu

Vg

cos sin�1 1

Va

�Vwnsin cG +Vwe


� _cR +ab

1+b2y2Vg sin cE

�ð26Þ

Substituting for the control input u from (25), we get

Shah et al. 7



_V= sg

Vg

cos sin�1 1

Va

�Vwnsin cG +Vwe


~Vg

~g_cR

�� ab

1+b2y2

~V 2g


#

+ s � _cR +ab

1+b2y2Vg sin cE

� �ð27Þ

or

_V= sg

Vg

t~Vg

~g_cR �

ab

1+b2y2

~V 2g


" #

+ s � _cR +ab

1+b2y2Vg sin cE

� � ð28Þ

where

t = cos sin�1 1

Va

�Vwnsin cG +Vwe


The uncertain variables ~Vg and ~g (with some measurement

error) appear from substitution of the control input u above.

Neglecting the error in the gravity term (g ’ ~g), we have

_V=� s

Vg

ab

1+b2y2sin cE t ~V 2

g � V 2g

� �� _cR(t ~Vg � Vg):

+ gkt sgn(s)

ð29Þ

_V will be negative definite if

gktj j. ab

1+b2y2sin cE t ~V 2

g � V 2g

� ��+ _cR(t ~Vg � Vg)

�� ð30Þor

k .1

gt

ab

1+b2y2sin cEj j t ~V 2

g � V 2g

�� + 1

gt_cR

�� t ~Vg � Vg

�� ð31ÞTo find the value of ‘k’ which satisfies the above for the entire

flight envelope, that is, to ensure reachability from any initial

condition, we find the maximum of the right-hand side in the

above inequality. The maximum value of ab/(1 +b2y2) is

ab at y = 0, and the maximum value of jsin cEj is 1 at

cE = 6p/2. Hence, the maximum value in the entire phase

portrait occurs at the point: (cE, y) = (6p/2, 0). Let us also

assume a worst-case error of 10% in the measurement of

ground velocity (i.e. ~Vg ’ 1:1 Vg); now _V will be negative defi-

nite, if

k .V 2

g ab

g1:21� 1

t

� �+

Vg

g_cR

�� 1:1� 1

t

� �ð32Þ

Using the minimum value of 1/t = 1 and the relation

Vg = _cRR, we have

k . 0:21V 2

g ab

g+

0:1

g

V 2g

Rð33Þ

With k as above, the phase trajectory will be attracted

towards the sliding surface across the entire flight envelope of

the vehicle. The parameters affecting the value of k are the

sliding surface variables, velocity and the radius of turn.

Keeping the product ab large will give better performance in

terms of cross-track error regulation, but will result in a big-

ger value of k, and hence a larger control input u. Similarly,

keeping the radius of turn R small would demand a larger u.

In the above discussion, a maximum error of 10% is assumed

in the measurement of velocity; if a different error magnitude

is expected for a particular application, expression (33) may

be modified accordingly.

Control boundedness

As discussed earlier, the aerodynamic lift required to balance

the weight of the vehicle decreases by a factor of cos f as the

UAV banks to follow the roll command generated by the gui-

dance law. The output of the guidance block fr = arctan u

must therefore be bounded in magnitude by an upper bound,

say fmax, during both the reaching and sliding phases. This

will ensure the availability of the aerodynamic lift required to

sustain flight. Here we will derive conditions so that jfrj �fmax throughout the reaching and sliding phases. In order to

avoid control saturation, we must have

�~V 2

g

~g

ab

1+b2y2sin cE +

~Vg

~g_cR � k sgn(s)

�� tanfmax

or in the worst case

~V 2g

~g

ab

1+b2y2sin cE

��+

~Vg

~g_cR

��+ k sgn(s)j j � tan fmax ð34Þ

After simplification, we have

k� tan fmax �~Vg

~g_cR �

~V 2g

~g

ab

1+b2y2sin cE

�� ð35Þ

The maximum value of the term

~V 2g

~g

ab

1+b2y2sin cE

��

cannot exceed (~V 2g =~g)ab in the entire phase plane (for all val-

ues of y and cE). So in order to bound the control input for

all conditions, we must have

k� tan fmax �~Vg

~g_cR �

~V 2g

~gab ð36Þ

Constraints on control gain ‘k’

For both reachability and saturation avoidance, we combine

conditions (33) and (36) to get




0:21V 2

g ab

g+

0:1

g

V 2g

R\k� tan fmax �

~Vg

~g_cR �

~V 2g

~gab ð37Þ

or

0:21V 2

g ab

g+

0:1

g

V 2g

R\k� tan fmax �

1:1Vg

g_cR �

1:21V 2g

gab

ð38Þ

The gain k and the sliding surface parameters a and b should

be selected considering the UAV flight conditions (Vg, g), themaximum allowable roll angle (fmax) and the mission require-

ment (R). First the sliding parameters a and b may be chosen,

followed by the selection of the gain k that satisfies (38). If nofeasible value of k is found, the above procedure may be reit-

erated with comparatively smaller values of the sliding para-meters a and b.

Implementation issues

From an implementation point of view, several solutions have

been developed to avoid chattering in the control signal dueto the signum function. One method is to replace the discon-

tinuous switching action sgn (s) with a continuous sigmoid

approximation s/(jsj + e) ; this is known as the boundary

layer approach (Boiko, 2013; Burton and Zinober, 1986). Asa result of this approximation, system trajectories are con-

fined to a small vicinity of the sliding surface (the boundary

layer) and not exactly to s(t) = 0, which is the ideal slidingmode.

Figure 8 shows the sigmoid approximation of sgn (s) for

different values of e. It is seen that the effective gain becomessmaller as s approaches zero, and hence chattering is reduced

in the control signal. Since the gain is reduced, controlboundedness (condition (36)) is not affected in this case. The

reachability condition also needs to be analysed for this

approximation. The reachability condition for the worst-casescenario (33) gives the maximum gain required so that the

states are attracted towards the sliding surface from any pointin the phase plane. The maximum gain corresponds to the

point (cE, y) = (6p/2, 0). In the generalized reachability

condition (31), the dominant term is

1

g

ab

1+b2y2j sin cEj

Figure 9 shows a coloured mesh plot of this term for the

entire phase plane. Thus we can easily see the gain required

to ensure reachability from any point of the phase portrait.

The sliding surface is also plotted in the figure in black line

for a = 0.7 and b = 0.002. It is evident that the maximum

gain required corresponds to (6p/2, 0). It is also seen that

the gain requirement in the close vicinity of the sliding surface

is not too high: in fact, close to the origin the required gain

reduces to one-fifteenth of its maximum value. The effect of

gain reduction (due to the approximation sgn (s) ’ s/(jsj+ e))

on the reachability condition needs to be analysed for each

specific application; it may be acceptable to bound the state

trajectory within a defined vicinity of the sliding surface. For

−10 −8 −6 −4 −2 0 2 4 6 8 10

−1

−0.5

0

0.5

1

s

s

|s|+

εεε

= 0.7

= 0.3 = 0.1

Figure 8. Approximation of sgn (s) function.

Figure 9. Gain k required for reachability over the phase plane.

Shah et al. 9



example let us assume that we want to slide with an accuracyof 0.1, that is, we want to keep jsj \ 0.1 (the boundary layer

width will be 0.2 in this case). Choosing e = 0.3 yields s/(jsj+ e) = 60.25 for s = 60.1, which implies that the actual

gain delivered to the system at the edges of the boundarylayer will be one-fourth of the chosen value. Figure 9 shows

the required gain to be approximately one-fourth of the maxi-

mum in the neighbourhood of the sliding surface: hence forthis case we can have jsj \ 0.1 for e = 0.3.

Flight test results

Experimental setup

The proposed guidance law is programmed in the flight con-

trol computer of a scaled YAK-54 UAV (Figure 10) todemonstrate its effectiveness; comparisons are also made with

other guidance laws. The test vehicle is designed and produced

by EG Aircraft and is powered by a DLE-55 engine. The

structure is suitably modified to accommodate the flight com-

puter and related sensors. The layout of different avionics

modules is shown in Figure 11. An MPC-565 microcontroller-

based generic board forms the heart of the control computer

and generates Pulse Width Modulation (PWM) commands

for the actuators; it communicates with different on-board

sensors and the ground terminal through serial ports. With

the available fuel capacity, the UAV can fly autonomously for

about one hour. Basic data of said vehicle is listed in Table 1.

Parameter selection

For implementation of the guidance law (25), the sliding sur-

face is s =cE+a arctan by = 0, where a and b are selected

Figure 10. A photograph of the test vehicle during take-off.

Figure 11. Main interfaces of the flight control computer.

AHRS: Attitude and heading reference system.




as 0.7 and 0.002 respectively. The minimum turn radius is

taken as 400 m, the speed during flight is approximately

35 m/s and the cruise altitude is 800 m above sea level. With

these parameter values, conditions for reachability and con-

trol boundedness (38) become 0.06 \ k \ 0.212. A bound-

ary layer approximation sgn (s) ’ s/(jsj+ e) is used (see

‘Implementation issues’) to avoid chattering in the computa-

tion of fr.

For flight we want to achieve a sliding accuracy of 0.1 (i.e.

jsj \ 0.1) and this can be achieved with e = 0.3 for k = 0.06

as discussed in ‘Implementation issues’. But here we choose a

larger gain of 0.21 (;3.5 times more than that required) for

implementation. With this larger gain we will have greater dis-

turbance rejection outside the boundary layer, and can

achieve jsj \ 0.1 even with e = 0.7, which gives a much

smoother control signal. Hence for this application we choose

k = 0.21 and e = 0.7.

Flight results

Effectiveness of the proposed algorithm for straight path fol-

lowing is equivalent to that of the algorithm presented in

Shah et al. (2011a, 2014). Here we shall focus on test results

for different cases of curved path following; comparisons will

also be drawn with our previously proposed algorithm (Shah

et al., 2014), and with Park’s algorithm (Park, 2004; Park et

al., 2004). Park’s algorithm generally performs well for

straight flight and mild turns. However, as discussed in

‘Introduction’, if track errors become large, the control input

(roll angle) can saturate. The lateral acceleration command

generated by Park’s algorithm for a circular loiter requires

perfect knowledge of the vehicle velocity for path following

with zero steady-state error. In the case of measurement

errors in velocity, a steady-state error will develop. Also, the

performance of this algorithm is sensitive to uncertainty in

the input channel (the roll channel), for example, a bias or

shift in roll angle following.Flight results for two different circular loiters are discussed

first. The take-off point is taken as the origin, the northward

distance travelled is denoted by x (or Posx), while the east-

ward distance is denoted by y (or Posy). The UAV has an

open loop speed control which is adjustable from the ground.

Initially a circular loiter is performed with a fixed throttle set-

ting; ground speed Vg versus course angle cG is plotted in

Figure 12. A nearly constant airspeed is maintained with the

fixed throttle; variation in ground speed is seen due to wind.

From Figure 12 it is estimated that there was a wind of ;5 m/

s during the flight; all subsequent flight results show the per-

formance of the guidance algorithm in the presence of wind of

approximately 5 m/s.Flight results for a circular loiter of radius 800 m are

shown in Figures 13 to 15, with three different guidance algo-

rithms. In Figure 13, the reference trajectory (the direction of

flight is clockwise) is shown in dashed line and the actual tra-

jectory flown by the vehicle is shown in solid line. The initial

part of the circular loiter is executed with our previous algo-

rithm (presented in Shah et al., 2014): the trajectory-following

is not accurate. After some time, the newly proposed algo-

rithm (25) is activated, the point of transition is marked in

the figure, and thereafter the tracking accuracy improves sub-

stantially. Park’s algorithm (Park et al., 2004) is brought

online in a later part of the flight. Figure 14 shows the cross-

track error y and the heading error angle cE versus time for

the same part of the flight. Initially (1340–1412.5 s), the gui-

dance law of Shah et al. (2014) is online and yields a cross-

track error of ~ 55 m. This improves to around 2 m when the

proposed law (25) is activated (from 1412.5 s to 1494 s). In

the last section (1494–1527 s), Park’s algorithm is activated; it

keeps the error within a 3 m band. Figure 15 shows the refer-

ence roll angle generated by the three guidance algorithms,

and also the actual roll angle of the vehicle along with aileron

deflections. There is no control saturation, the aileron deflec-

tions being nominal.Flight results of the three guidance algorithms for a circular

loiter of radius 400 m (close to the maximum capability of the

UAV) are shown in Figures 16 to 18. For the first 337 s to

413 s of flight our previous algorithm (Shah et al., 2014) is acti-

vated, followed by Park’s algorithm from 413 s to 461.5 s, and

then the algorithm proposed here is brought online in the last

part of the flight (461.5–495 s). It is seen from Figure 17 that

trajectory-following with our previous algorithm is not good,

and a cross-track error of ;80 m develops in the initial part of

the flight. This reduces to ;20 m with Park’s algorithm. With

the newly proposed algorithm (25), trajectory-following

improves and the error reduces to around 5 m. The com-

manded and actual roll angles along with aileron deflections

are shown in Figure 18.Flight results of the proposed algorithm for a straight path

followed by a sharp heading change of approximately 135+are shown in Figures 19 to 21. Figure 19 shows the desired

mission plan along with the actual trajectory flown by the

UAV. The mission consists of three parts: a straight path

WP1 –WP2, a sharp turn (heading change of ;135�), fol-lowed by another straight segment WP2 –WP3. To see

robustness of the proposed algorithm, the speed of the UAV

is varied during the mission and ascend/descend commands

are also given, as shown in Figure 20. Figure 21 shows the

reference roll angle, the actual roll angle and the aileron

deflections. With the proposed guidance algorithm, the air-

craft follows the entire mission successfully. In the last seg-

ment of the mission (straight path WP2 –WP3), the following

remains very good despite ascend/descend manoeuvres and

variations in speed.

Table 1: Geometrical and mass properties of the scaled YAK-54 UAV.

# Parameter Value

1 Wing span 2235 mm

2 Length 2080 mm

3 Wing area 943,869 mm2

4 Wing root/tip airfoil NACA 0016 / 0017

5 Horizontal tail root/tip airfoil NACA 0015 / 0012

6 Vertical tail root/tip airfoil NACA 0009 / 0010

7 Take-off mass 11 kg

8 Moments of inertia: Ixx, Iyy, Izz 1.36,2.848,4.07 kg-m2

Shah et al. 11



−1000 −800 −600 −400 −200 0 200 400 600 800 1000−1000

−800

−600

−400

−200

0

200

400

600

800

1000

Posx (north) [m]

−P

osy (

wes

t) [m

]

Switched to new algorithm

Switched to Park’s algorithm

Start here withprevious algorithm

Figure 13. Trajectory-following for a circular loiter of radius 800 m.

−200 −150 −100 −50 0 50 100 150 20030

32

34

36

38

40

ΨG

[°]

Spe

ed (

Vg

) [m

/s]

−500 0 500

−400

−200

0

200

400

Posx (north) [m]

−P

osy (

wes

t) [m

]

Figure 12. Estimated wind on flight day.




1340 1360 1380 1400 1420 1440 1460 1480 1500 1520−80

−60

−40

−20

0

20

Time [s]

1340 1360 1380 1400 1420 1440 1460 1480 1500 1520−10

−5

0

5

10

15

Time [s]

with previous algorithm

with new algorithm

with Park’salgorithm

Cro

ss-r

ange

(y)

[m

]ψ E

[°]

Figure 14. Cross-track error y and heading error cE vs time for the circular loiter of radius 800 m.

1340 1360 1380 1400 1420 1440 1460 1480 1500 1520−20

−10

0

10

20

30

40

Time [s]

[°]

1340 1360 1380 1400 1420 1440 1460 1480 1500 1520

−6

−4

−2

0

2

4

6

Time [s]

roll ref. roll


with new algorithm with Park’salgorithm

Aile

ron

(δa)

[°]

Figure 15. Roll angle and aileron deflection (da) vs time for the circular loiter of radius 800 m.

Shah et al. 13



−600 −400 −200 0 200 400 600−600

−400

−200

0

200

400

600

Posx (north) [m]

−P

osy (

wes

t) [m

]

Start here with previous algorithm

Switched to Park’s algorithm

Switched to new algorithm

Figure 16. Trajectory-following for a circular loiter of radius 400 m.

340 360 380 400 420 440 460 480 500−80

−60

−40

−20

0

20

Time [s]

Cro

ss-r

ange

(y)

[m

]

340 360 380 400 420 440 460 480 500−15

−10

−5

0

5

10

15

Time [s]

ψ E [°]



with newalgorithm

Figure 17. Cross-track error y and heading error cE vs time for the circular loiter of radius 400 m.




340 360 380 400 420 440 460 480 500−15

−10

−5

0

5

10

15

20

25

30

35

40

Time [s]

[°]

340 360 380 400 420 440 460 480 500

−6

−4

−2

0

2

4

6

Time [s]

Aile

ron

(δa)

[°]

roll ref. roll

with previous algorithmwith newalgorithm


Figure 18. Roll angle and aileron deflection (da) vs time for the circular loiter of radius 400 m.

73.162 73.164 73.166 73.168 73.17 73.172 73.174 73.176 73.178 73.1833.5

33.505

33.51

33.515

33.52

33.525

Longitude [°]

Latit

ude

[°]

WP1

WP2

WP3

Reference pathUAV trajectory

Start here

Figure 19. Trajectory-following for a straight path followed by a sharp turn.

Shah et al. 15



1180 1200 1220 1240 1260 1280 13000

50

100

150

[m]

1180 1200 1220 1240 1260 1280 130010

20

30

40

50

Time [s]

Spe

ed [m

/s]

Reference alt Actual altitude

Figure 20. Altitude and speed for straight path and sharp turn.

1180 1200 1220 1240 1260 1280 1300−20

−10

0

10

20

30

40

[°]

1180 1200 1220 1240 1260 1280 1300−4

−2

0

2

4

6

Time [s]

[°]

δa

roll ref. roll

Figure 21. Roll angle and aileron deflection for straight path and sharp turn.




Conclusion

In this paper a new lateral guidance law for cross-track con-

trol of UAVs is developed. The guidance law is based on the

sliding mode technique. For good performance and accurate

tracking while following curved paths, a feedforward term

which depends on the rate of change of the reference path

heading is included in the guidance law. The algorithm uses

the current mission information (reference path) and sensor

measurements to generate roll commands. Conditions for sta-

bility and control boundedness are derived in terms of upper

and lower bounds on the control gain k. The guidance algo-

rithm is implemented in the flight computer of a scaled YAK-

54 research aircraft, and various flight tests conducted.

Comparison with other algorithms is also made. It is seen

that the performance of the proposed algorithm is better than

the previously suggested algorithm (Shah et al., 2014), espe-

cially during circular loiters. The sliding-mode-based gui-

dance law performs well in the presence of wind, parametric

and input channel uncertainties; it does not saturate the con-

trol input, and meets the criterion of a ‘good helmsman’.

Generalization of this 2D lateral guidance problem to the full

3D trajectory tracking problem is the subject of ongoing

work.

Funding

This research received no specific grant from any fundingagency in the public, commercial, or not-for-profit sectors.

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