lateral control for uavs using sliding mode technique

Lateral Control for UAVs using Sliding Mode Technique

M. Zamurad Shah*, R. Samar**, A.I. Bhatti**

* Mohammad Ali Jinnah University, Islamabad, Pakistan, (Ph.D. Student, email: [email protected])

** Mohammad Ali Jinnah University, Islamabad, Pakistan

Abstract: This paper presents sliding mode based lateral control for UAVs using a nonlinear sliding

approach. The control is shown to perform well in different flight conditions including straight and turning

flight and can recover gracefully from large track errors. Saturation constraints on the control input are met

through the nonlinear sliding surface, while maintaining high performance for small track errors. Stability

of the nonlinear sliding surface is proved using an appropriate Lyapunov function. The main contribution

of this work is to develop a robust lateral control scheme that uses readily available sensor information and

keeps the track error as small as possible without violating control constraints. In the proposed scheme the

only information used in the control law is the lateral track error and the heading error angle. No

information is required about the desired path/mission, which therefore can be changed online during run-

time. This scheme is implemented on a high fidelity nonlinear 6-degrees-of-freedom (6-dof) simulation

and different scenarios are simulated with large and small track errors in windy and calm conditions.

Simulation results illustrate the robustness of the proposed scheme for straight and turning flight, in the

presence of disturbances, both for large and small track errors. Furthermore it is shown that the saturation

limits of the control input are not exceeded in all cases.

1. INTRODUCTION

Two approaches are used for trajectory tracking of UAVs. In

first approach, UAV guidance and control problem is

separated into an outer guidance and an inner control loop.

Based on lateral track error and heading angle, outer

guidance loop generates a desired reference bank (roll) angle

and inner control law generates command to control surface

to follow the desired reference bank angle. In the second

approach, guidance and control laws are designed together in

a single framework.

In most applications, the separate inner and outer loop

approach is commonly used since it is simpler and well-

established design methods are available for inner loop

vehicle control. For outer guidance loop, different approaches

have been used in the past. Linear proportional and derivative

(PD) control (Siouris, 2004; Pappoullias, 1994) has been used

in many UAV applications. But during curved path following

in the present of a persistent disturbance (e.g. wind),

performance of PD control degrades. A nonlinear scheme has

been given by (Park, 2004; Park, 2007) showing improved

performance than PD scheme. But in the case of large track

error, control output of this nonlinear scheme saturates and

there is no stability proof during control saturation. A

conventional linear proportional and derivative lateral control

with some non-linear modifications has been given by

(Samar, 2007) which enhance the tracking performance. But

the control proposed by (Samar, 2007) is an ad-hoc solution

and as such no stability proof exists. In literature, other

different approaches have been proposed by (Nelson, 2006;

Regina 2009; Jia, 2010; Shtessel, 2009).

Sliding mode control is a technique derived from variable

structure control. Ideal sliding mode control is insensitive to

parameter variations and external disturbances, regardless of

nonlinearity and uncertainty (Bandyopadhyay, 2009). For a

certain class of systems, sliding mode controller design

provides a systematic approach to the problem and guarantee

system insensitivity with respect to the matched disturbance

and model uncertainty. The controller so designed is unique

since the performance of the controller depends on the design

of sliding surface and not the states tracking directly [Slotine,

1991]. Idea is to force the trajectory states towards the sliding

surface and once achieved, the states are constrained to

remain on the surface. Although the technique has good

robustness properties, pure sliding mode control presents

drawbacks that include large control requirements and

chattering. Chattering may be settled by smoothing the

control input using boundary layer or bandwidth limited

sliding mode control. Saturations and sigmoid functions are

used, for example as “filters” for the output of a

discontinuous signal in order to obtain a continuous one that

is realizable by mechanical hardware.

In this work, the approach used is to divide the UAV

guidance and control problem into an outer guidance and an

inner control loop (Fig. 1). Here, it is assumed that a fast

inner control law is already designed that can track the

desired (reference) roll angle command with minimal

overshoot. A Sliding mode based outer guidance loop is

proposed here in this paper for robust tracking. Main concern

here is the robustness and performance with minimum

information. The only inputs of guidance loop are lateral

track error & current heading error angle, and output is

desired/reference roll angle. In order to meet the constraints

of high performance in small track error and a bounded

Preprints of the 18th IFAC World CongressMilano (Italy) August 28 - September 2, 2011

Copyright by theInternational Federation of Automatic Control (IFAC)

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heading error angle in the case of large track error, a non-

linear sliding surface is proposed here.

Fig. 1: Guidance & Control approach used in this work

The stability of this non-linear sliding surface is proved with

the help of a Lyapunov function. Also the proposed scheme is

implemented in non-linear 6-dof simulation and simulation

results of different cases are shown here, that validate the

robustness of this proposed scheme.

2. PROBLEM DEFINITION

Notation of different variables used in this paper is same as

used in (Siouris 2004; Samar, 2007). Here, „𝑦‟ is the cross

track error, „𝜓𝐺 ‟ is the velocity heading, „𝜓𝑅‟ is the reference

heading and 𝜓𝐸 = 𝜓𝐺 − 𝜓𝑅 (Fig. 2). Note that magnitude of

𝜓𝐸 should be less than 90 degrees.

Fig. 2: Definition of cross track (𝑦), „𝜓𝐺 ‟, „𝜓𝑅‟ and „𝜓𝐸‟

2.1 Assumptions

For way point tracking during straight path 𝜓 𝑅 = 0, while

during circular path 𝜓 𝑅 is non-zero but may be small

depending on the mission. In this paper, 𝜓 𝑅 is assumed small

and neglected here.

While designing this outer guidance loop it is assumed that a

fast inner control loop is already designed. As the inner

control loop is fast enough (at least 5 times), so we assumed

here that the actual roll angle (𝜙) is approximately equal to

the desired/reference roll angle (𝜙𝑅).

2.2 Model of System Dynamics

During steady turn when vehicle is banked at an angle 𝜙, Lift

(L) is resolved into two components 𝐿 cos 𝜙 and 𝐿 sin 𝜙 that

balance the weight and centrifugal force respectively (Fig. 3)

𝐿 cos 𝜙 = 𝑚𝑔, 𝐿 sin 𝜙 =𝑚 𝑉2

𝑅 (1)

where 𝑚 is the mass, 𝑔 is the gravitational acceleration, V is

the velocity of Aircraft/UAV and R is the radius of turn.

From (1) we have

tan 𝜙 =𝑉2

𝑅 𝑔 (2)

Moreover we also know that during steady turn 𝑉 = 𝑅 𝜓𝐺 , so

(2) becomes

tan 𝜙 =𝑉 𝜓𝐺

𝑔 (3)

Resolving the components of velocity vector in Fig. 2, we

have

𝑦 = 𝑉 sin 𝜓𝐸 (4)

Using 𝜓𝐸 = 𝜓𝐺 − 𝜓𝑅 and assumption 𝜓 𝑅 = 0, (3) becomes

tan 𝜙 =𝑉 𝜓𝐸

𝑔 (5)

Fig. 3: Components of Lift (L) during steady turn

Using assumption 𝜙 ≅ 𝜙𝑅, (5) becomes

tan 𝜙𝑅 =𝑉 𝜓𝐸

𝑔 (6)

In summary, the overall dynamics of the outer guidance loop

is governed by (4) and (6) which can be also written in state-

space form as

𝑦 = 𝑉 sin 𝜓𝐸 (7)

𝜓𝐸 =

𝑔 tan 𝜙𝑅

𝑉 (8)

where cross track error „y‟ and „𝜓𝐸‟ are the state variables

and reference bank/roll angle „𝜙𝑅‟ is the output of outer

guidance loop. Also note that in case of small 𝜓𝐸 , 𝑦 ≅ 𝑉𝜓𝐸

that means that state „𝜓𝐸‟ is approximately equal to „𝑦

𝑉‟.

Yaw rate

roll angle

Ref Roll angle

Current Position & Heading angle

Desired Position &

Heading angle

Aileron

Rudder

Mission

Guidance

Control

Lateral Dynamcis


11122

2.3 Control Task

Using readily available sensor information „y‟ and „𝜓𝐸‟,

control task is to keeps the track error as small as possible by

generating an appropriate „𝜙𝑅‟. Also the output „𝜙𝑅‟ of

control system should be a bounded number (magnitude <30

degrees).

3. PROPOSED LATERAL CONTROL SCHEME

3.1 Why Non-Linear Sliding Surface?

In case of bank-to-turn, the UAV roll angle should be

bounded by some finite number (say approximately 𝜋

6). Hence

the output of guidance block (reference roll angle) should

also be bounded ( 𝜙𝑅 < 𝜙𝑚𝑎𝑥 =𝜋

6). During flight, large

track error is possible due to loss of GPS for long time so the

possibility of large track error can‟t be ignored during flight.

Suppose we choose a linear sliding surface 𝑆 = 𝜓𝐸 + 𝜆 𝑦 for

some positive scalar 𝜆. For performance in case of small

track error, we need a large 𝜆 so that we can reach the origin

𝑦, 𝜓𝐸 = (0,0) in minimum time. But on the other hand

choosing a large 𝜆 implies that in case of large track error we

should have a large 𝜓𝐸 (> 90 degrees) which is not possible

as magnitude of 𝜓𝐸 could not exceed 𝜋

6. So choice of 𝜆 is

trade-off between performance in the case of small track error

and a realizable magnitude of 𝜓𝐸 in the case of large track

error, and both constraints cannot be satisfied at the same

time with a linear sliding surface.

Fig. 4: Linear sliding surface (S) for some positive 𝜆

3.2 Non-Linear Sliding Surface

For performance in case of small track error and also

magnitude of 𝜓𝐸 less than 90 degrees, we choose a non-linear

sliding surface

𝑆 = 𝜓𝐸 + 𝛼 tan−1(𝛽𝑦) (9)

where 𝛼 and 𝛽 are scalars and later we will show that for

stability of sliding surface 𝛼𝛽 > 0. Performance in case of

small track error can be changed by changing parameter 𝛽

while another parameter 𝛼 can be used to keep the magnitude

of 𝜓𝐸 less than 90 degrees ( 𝛼 ≤ 1). A graph of above

sliding surface is shown in Fig. 5 for some particular 𝛼 and 𝛽.

Now an important question arises, is the motion on sliding

surface stable? Motion on sliding surface is described

by 𝑆 = 0, that is

ψE + 𝛼 tan−1(𝛽𝑦) = 0 (10)

The stability of (10) can be proved using Lyapunov theory,

Lyapunov candidate function (𝐕) for stability of (10) is

𝐕 =1

2 y2 + ψE

2 = 1

2 y2 + 𝛼 tan−1 𝛽𝑦 2 (11)

And the derivative of Lyapunov function 𝐕 is

𝐕 = −𝑉𝑦 sin 𝛼 tan−1 𝛽𝑦

−𝛼2𝛽 𝑉

1 + 𝛽2𝑦2tan−1 𝛽𝑦 sin 𝛼 tan−1 𝛽𝑦 12

as −𝜋

2≤ tan−1 𝛽𝑦 ≤

𝜋

2 and 𝛼 ≤ 1, sin 𝛼 tan−1 𝛽𝑦 will

have a same sign as 𝑦 if 𝛼𝛽 > 0. So the derivative of

Lyapunov function 𝐕 in (12) is negative definite if 𝛼𝛽 > 0

and 𝛼 ≤ 1.

In summary, the proposed non-linear sliding surface (9) is a

stable sliding surface subject to the above stated conditions

𝛼𝛽 > 0 and 𝛼 ≤ 1.

3.3 Equivalent Lateral Control

Equivalent control is interpreted (Slotine, 1991) as a

continuous control law that would maintain 𝑆 = 0 if the

dynamics were exactly known. In our case

𝑆 = 𝜓 𝐸 +

αβ

1 + β2y2y (13)

Fig. 5: Nonlinear Sliding Surface 𝑆 = 𝜓𝐸 + 𝛼 tan−1(𝛽𝑦)

𝑆 = 0 implies

0 =𝑔 tan(𝜙𝑅_𝑒𝑞𝑣 )

𝑉+

𝛼𝛽

1 + β2y2𝑉 sin 𝜓𝐸 (14)

From (14), equivalent control can be derived as

𝜙𝑅_𝑒𝑞𝑣 = tan−1 −𝑉2

𝑔

𝛼𝛽

1 + 𝛽2𝑦2sin 𝜓𝐸 (15)

-2000 -1500 -1000 -500 0 500 1000 1500 2000-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

Cross track error (y) in meters

E i

n r

ad

ian

s

-2000 -1500 -1000 -500 0 500 1000 1500 2000-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

Cross track error (y) in meters

E i

n r

ad

ian

s


11123

3.4 Existence of Sliding Mode

A control law should be chosen in such a way that from any

initial condition, the system trajectory is attracted towards the

sliding surface and then slides along the sliding surface

(Bandyopadhyay, 2009). To see the reachability condition,

Lyapunov candidate function 𝐕 =1

2S2 is selected. And the

derivative of Lyapunov function 𝐕 is

𝐕 = 𝑆𝑆

= 𝑆 𝑔

𝑉tan 𝜙𝑅 +

𝛼𝛽

1 + β2y2𝑉 sin 𝜓𝐸 (16)

Let us assume

tan 𝜙𝑅 = −𝐾 𝑠𝑖𝑔𝑛 𝑆 (17)

and replace it in (16), derivative of Lyapunov function 𝐕

becomes

𝐕 = 𝑆 −𝑔

𝑉𝐾 𝑠𝑖𝑔𝑛(𝑆) +

𝛼𝛽

1 + β2y2𝑉 sin 𝜓𝐸 (18)

𝐕 is negative definite if 𝑔

𝑉𝐾 >

𝛼𝛽

1+β2y2 𝑉 sin 𝜓𝐸

or

𝐾 >𝑉2

𝑔

𝛼𝛽

1 + β2y2 sin 𝜓𝐸

In extreme case, the maximum value of right side occurs in

the situation when sin 𝜓𝐸 = 1 and 𝑦 = 0. So (18) is

negative definite in whole flight envelope if

𝐾 >𝑉2

𝑔αβ (19)

3.5 Total Lateral Control

From equations (17) and (19), we have

𝜙𝑅 = tan−1 −𝐾 𝑠𝑖𝑔𝑛 𝑆 where 𝐾 >𝑉2

𝑔αβ

or

𝜙𝑅 = − tan−1 𝐾 𝑠𝑖𝑔𝑛 𝑆 (20)

Total lateral control is the sum of (15) and (20)

𝜙𝑅 = − tan−1 𝑉2

𝑔

𝛼𝛽

1 + 𝛽2𝑦2sin 𝜓𝐸

− tan−1 𝐾 𝑠𝑖𝑔𝑛 𝑆 (21)

where first part is continuous one and the second one is

discontinuous. From practical implementation point of view,

second discontinuous part can be approximated by a

continuous one.

4. CONTROL EFFORT BOUNDEDNESS

As discussed earlier that output of guidance block should be a

bounded number, in this section we have derived conditions

to keep 𝜙𝑅 ≤𝜋

6 during motion on sliding surface. Motion on

sliding surface is described by (10), that is

ψE + 𝛼 tan−1(𝛽𝑦) = 0 (22)

Differentiating (22), we get an expression (15) for equivalent

control that would maintain 𝑆 = 0 if the dynamics were

exactly known. So the equivalent control should always be

less 30 degrees ( 𝜙𝑅_𝑒𝑞𝑣 ≤π

6).

𝑉2

𝑔

𝛼𝛽

1 + 𝛽2𝑦2sin 𝜓𝐸 ≤ tan

π

6 (23)

In above inequality (23), replacing value of 𝜓𝐸 from (22)

𝑉2

𝑔

𝛼𝛽

1 + 𝛽2𝑦2sin(−𝛼 tan−1(𝛽𝑦)) ≤ tan

π

6

That can be written as

𝛼𝛽

1 + 𝛽2𝑦2sin(−𝛼 tan−1(𝛽𝑦)) ≤ tan

π

6

𝑔

𝑉2 (24)

In summary, the choice of 𝛼 and 𝛽 should be such that it

satisfies the inequality (24) in order to avoid the control

saturation during motion on sliding surface.

5. SIMULATION RESULTS

The proposed scheme is implemented in non-linear 6-dof

simulation of a UAV. Prior to implementation of this outer

guidance loop, a robust linear 𝐻∞ based linear control law

was implemented for inner control loop. To implement the

lateral control law (21), signum function is approximated

by 𝑆𝑖𝑔𝑛(𝑆) ≈𝑆

𝑆 +𝜖, where 𝜖 is a small scalar positive

number, in our case its value is 0.1. After a number of

simulation runs, 𝛼 and 𝛽 values are kept at 0.3333 and 0.003

respectively for better performance and robustness.

Fig. 6: Validation of inequality (21) for particular 𝛼 and 𝛽

0 500 1000 1500 20000

0.5

1

1.5x 10

-4

Lateral Track Error [m]

LHS

RHS


11124

Fig. 7: Performance for 2000m cross track error (No wind)

The above selected values of 𝛼 and 𝛽 also satisfy the

inequality (21) and maximum control effort during motion on

sliding surface is expected at cross track error of ~ 250

meters (Fig. 6). For the above selected 𝛼 and𝛽, the minimum

value of K in (21) should be greater than 4 (K=10 used in all

simulation results).

Straight Path Following

Simulation results for 2000m track error in the absence of

wind and in the presence of 30m/s side wind are shown in

Fig. 7 and Fig. 8 respectively. In the absence of wind, it takes

~26.72 seconds to reach the sliding surface and the

subsequent motion takes place on the sliding surface. On the

other hand in the case of side wind of 30m/s, it takes ~0.78

seconds more to reach the sliding surface. In both cases (with

& without wind), there is no overshoot in cross track error

and magnitude of steady state track error is less than 2.5

meters. Fig. 7 and Fig. 8 also validate that during motion on

sliding surface maximum control effort is at ~250 meters

cross track error.

Circular Path Following

In Fig. 9, the simulation results are plotted for a curved

(~circular) path in the presence of a north wind of 30m/s.

Even in this worst case scenario, the maximum magnitude of

cross track error is less than 45 meters.

6. CONCLUSIONS

A new scheme of lateral control for UAVs is proposed here

in this paper. This new scheme is based on a sliding mode

technique with a non-linear sliding surface. Due to limitations

of performance in the case of small track error and keep

𝜓𝐸 ≤ 𝜋

2 for whole flight envelope, a single linear sliding

surface is not a feasible solution. So a non-linear sliding

surface that can meet both these limitations is proposed here

in this paper and stability of that non-linear sliding surface is

proved with the help of a Lyapunov function. After the

selection of non-linear sliding surface, a sliding mode based

lateral control law is proposed and condition for reachability

to sliding surface is derived using a Lyapunov function.

Another limitation of guidance block output saturation

(reference roll angle saturation) is also addressed here in this

paper and conditions for saturation avoidance are derived

here.

Above proposed law is implemented in simulation (with

approximation of signum function), and simulation results are

shown here for straight & curved path. To see the robustness

of this proposed algorithm, simulations are performed in the

presence of disturbance (wind). Maximum steady state error

in the case of straight path following and circular path

following is 2.5 meters and 45 meters respectively even in the

presence of a lateral wind of 30m/s. Simulation results also

validate that there is no control effort saturation during

motion on sliding surface. In reaching phase (initial phase

when the system trajectory has not reached the sliding

surface), there is control saturation but reaching phase can be

eliminated using different techniques like a method discussed

in (Bandyopadhyay, 2009).

Problem of chattering in standard sliding mode technique is

settled by smoothing the control input using boundary layer

during implementation of this proposed algorithm. But in a

future work, this problem will be settled by using higher

order sliding mode (HOSM).

40 60 80 100 120 140 1600

1000

2000

3000

Time [sec]

Cro

ss T

rack E

rror

[m]

40 60 80 100 120 140 160-40

-20

0

20

Time [sec]

E [

deg]

-1000 -500 0 500 1000 1500 2000 2500-0.5

0

0.5

X: 1544

Y: -0.435

Z: 76.72

Cross Track Error [m]

E [

rad]

Sliding Surface

UAV Phase Portrait

40 60 80 100 120 140 160-40

-20

0

20

40

X: 76.72

Y: -8.212

Time [sec]

Roll

Refe

rence [

deg]


11125

Fig. 8: Performance for 2000m cross track error (30m/s wind)

REFERENCES

B. Bandyopadhyay, F. Deepak, and K.S. Kim (2009). Sliding

Mode Control Using Novel Sliding Surfaces. Springer-

Verlag, Berlin Heidelberg.

D. Nelson, B. Barber, T. McLein, R. Beard. (2006). Vector

field path following for small unmanned air vehicles.

Proceedings of the IEEE American control conference,

IEEE, Piscatawey, NJ.

George M. Siouris (2004). Missile Guidance and Control

Systems. Springer-verlag, USA.

Lei jia and Meng Xiuyun (2010). Application of Sliding

Mode Control Based on DMC in Guidance Control

System. 2nd

International Conference on Advanced

Computer Control, Shenyang, China

Marius Niculescu (2001). Lateral Track Control Law for

Aerosonde UAV. 39th AIAA Aerospace Sciences

Meeting and Exhibit. Reno, NV, USA.

N. Regina and M. Zanzi (2009). 2D Tracking and Over-

Flight of Target by Means of a Non-Linear Guidance

Law for UAV. IEEE Aerospace Conference, Big Sky,

MT.

R. Samar, S. Ahmed and F. Aftab (2007). Lateral Control

with Improved Performance for UAVs. 17th

IFAC World

Congress. Seoul, Korea.

Sanghyuk Park (2004). Avionics and Control System

Development for Mid-Air Rendezvous of Two Unmanned

Aerial Vehicles. Ph.d. Thesis, Chapter 3. Massachusetts

Institute of Technology. USA.

Sanghyuk Park, John Deyst, Jonathan P. How (2007),

Performance and Lyapunov stability of a nonlinear

pathfollowing guidance method, Journal of guidance,

control and dynamics, Vol. 30, No. 6, 1718–1728.

Slotine, J.-J. E. and Li, W., Applied Nonlinear Control,

chapter- 7, Prentice Hall, 1991.

Y.B. Shtessel, I.A. Shkolnikov and A. Levant (2009).

Guidance and Control of Missile Interceptor using

Second-Order Sliding Modes. IEEE Transaction on

Aerospace and Electronic Systems, Vol. 45, No. 1.

Fig. 9: Performance for curved path (30m/s North wind)

40 60 80 100 120 140 1600

500

1000

1500

2000

2500

Time [sec]

Cro

ss T

rack E

rror

[m]

40 60 80 100 120 140 160-40

-20

0

20

Time [sec]

E [

deg]

-1000 -500 0 500 1000 1500 2000 2500-0.5

0

0.5

X: 1494

Y: -0.4341

Z: 77.5

Cross Track Error [m]

E [

rad]

Sliding Surface

UAV Phase Portrait

40 60 80 100 120 140 160-40

-20

0

20

40

Time [sec]

Roll

Refe

rence [

deg]

X: 77.5

Y: -6.472

1300 1400 1500 1600 1700 1800-20

0

20

40

60

Time [sec]

Cro

ss T

rack E

rror

[m]

1300 1400 1500 1600 1700 1800-2

-1

0

1

2

Time [sec]

E [

deg]

1300 1400 1500 1600 1700 1800-30

-20

-10

0

10

Time [sec]

Roll

Refe

rence [

deg]

63.4 63.5 63.6 63.7 63.8

25.4

25.45

25.5

25.55

25.6

25.65

Track( Lat/Long )

long (deg)

lat

(deg)

0

1

2

3

4

8

9


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lateral control for uavs using sliding mode technique

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