robust stabilization of a quadrotor aerial vehicle in presence of actuator faults

7/31/2019 Robust Stabilization of A Quadrotor Aerial Vehicle in Presence of Actuator Faults

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International Journal of Information Technology, Control and Automation (IJITCA) Vol.2, No.2, April 2012

DOI:10.5121/ijitca.2012.2201 1

ROBUST STABILIZATION OF AQUADROTORAERIAL

VEHICLE IN PRESENCE OFACTUATORFAULTS

Hicham Khebbache 1, Belkacem Sait 2, Fouad Yacef3 and Yassine Soukkou 4

1,2Automatic Laboratory of Setif (LAS), Electrical Engineering Department,

Setif University, [email protected] , [email protected]

3,4Automatic Laboratory of Jijel (LAJ), Automatic Control Department,

Jijel University, [email protected] , [email protected]

ABSTRACT

This paper considers the stabilization problem of an underactuated quadrotor UAV system in presence of

actuator faults. The dynamic model of quadrotor while taking into account various physical phenomena,

which can influence the dynamics of a flying structure is presented. Subsequently, a new control strategybased on backstepping approach and taking into account the actuator faults is developed. Lyapunov

based stability analysis shows that the proposed control strategy design keeps the stability of the closed

loop dynamics of quadrotor UAV even after the presence of actuator faults. Numerical simulation results

are provided to show the good tracking performance of proposed control laws.

KEYWORDS

Backstepping control, Dynamic model of quadrotor, Fault tolerant control (FTC), Robust control, Sliding

mode control (SMC), Unmanned aerial vehicles (UAV).

1. INTRODUCTION

The automatic control of a quadrotor UAV is not a straight on mainly due to its underactuatedproperties. The dynamic model of quadrotor UAV has six degree-of-freedom (DOF) with only

four independent thrust forces generated by four rotors. It is difficult to control all these sixoutputs with only four control inputs. Moreover, uncertainties associate with dynamic modelalso bring more challenge for control design.

Despite the real progress made, researchers must still deal with serious difficulties, related to the

control of such systems, particularly, in the presence of atmospheric turbulences. In addition,

the navigation problem is complex and requires the perception of an often constrained andevolutionary environment, especially in the case of low-altitude flights.

In contrast to terrestrial mobile robots, for which it is often possible to limit the model to

kinematics, the control of aerial robots (quadrotors) requires dynamics in order to account for

gravity effects and aerodynamic forces [9].

In [5-10], authors propose a control-law based on the choice of a stabilizing Lyapunovfunction ensuring the desired tracking trajectories along (X,Z) axis and roll angle only. The

authors in [1], do not take into account frictions due to the aerodynamic torques nor drag forces.They propose a control-laws based on backstepping, and on sliding mode control in order tostabilize the complete system (i.e. translation and orientation). In [12-13], authors take into

account the gyroscopic effects and show that the classical model-independent PD controller can

stabilize asymptotically the attitude of the quadrotor aircraft. Moreover, they used a new
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Lyapunov function, which leads to an exponentially stabilizing controller based upon the PD2

and the compensation of coriolis and gyroscopic torques. In [8] the authors develop a PID

controller in order to stabilize altitude. The authors in [3] propose a control algorithm basedupon sliding mode using backstepping approach allowed the tracking of the various desiredtrajectories expressed in term of the center of mass coordinates along (X,Y,Z) axis and yaw

angle. In [7], authors used a controller design based on backstepping approach. Moreover, theyintroduced two neural nets to estimate the aerodynamic components, one for aerodynamic

forces and one for aerodynamic moments. While in [6] the authors propose a hybridbackstepping control technique and the Frenet-Serret Theory (Backstepping+FST) for attitude

stabilization, that includes estimation of the desired angular acceleration as a function of the

aircraft velocity. The authors in [4] develop a direct adaptive sliding mode control for thequadrotor attitude stabilization and altitude trajectory tracking, with consideration of

disturbances and parameter uncertainties. However, all authors mentioned above do not takeinto account the faults affecting the actuators of quadrotor, which makes these control-laws verylimited and induce undesired behavior of quadrotor, or even to instability of the latter after

occurrence of actuator faults.

In this paper, the stabilization problem of the quadrotor aircraft in presence of actuator faults is

considered. The dynamical model describing the quadrotor aircraft motions and taking into aaccount for various parameters which affect the dynamics of a flying structure such as frictionsdue to the aerodynamic torques, drag forces along (X,Y,Z) axis and gyroscopic effects is

presented. Subsequently, a new control strategy based on backstepping approach and taking into

account the actuator faults is developed. In order to compensate the effects of actuator faults,

this control strategy uses a "sign" function as compensation technique. Finally all synthesizedcontrol laws are highlighted by simulations which gave fairly satisfactory results despite the

occurrence of actuator faults.

2. DYNAMICAL MODEL

2.1. Quadrotor dynamic model

The quadrotor have four propellers in cross configuration. The two pairs of propellers {1,3} and{2,4} as described in Fig. 1, turn in opposite directions. By varying the rotor speed, one can

change the lift force and create motion. Thus, increasing or decreasing the four propellersspeeds together generates vertical motion. Changing the 2 and 4 propellers speed conversely

produces roll rotation coupled with lateral motion. Pitch rotation and the corresponding lateralmotion; result from 1 and 3 propellers speed conversely modified. Yaw rotation is more subtle,

as it results from the difference in the counter-torque between each pair of propellers.

Figure 1. Quadrotor configuration


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3

The quadrotor model (position and orientation dynamic) obtained is given by [2], [3], [4]:

( )

( )

( )

2

2

23

2

4

1

1

1

1

1

1

cos( ) cos( )

y z faxrr

x x x x

fayz x rr

y y y y

x y faz

z z z

ftx

x

ft y

y

ft z

I I KJ lu

I I I I

KI I J l u

I I I I

I I Ku

I I I

Kx x u u

m m

Ky y u u

m m

Kz z g u

m m

= +

= + +

= + = + = +

= +

(1)

with

( )

( )

1 2 3 4

cos cos sin sin sin

cos sin sin sin cos

r

x

y

u

u

= + = + =

(2)

u1, u2, u3 and u4 are the control inputs of the system which are written according to the angularvelocities of the four rotors as follows:

21 1

2

2 2

23 3

2

4 4

0 0

0 0

u b b b b

u lb lb

u lb lb

u d d d d

=

(3)

From (2) it easy to show that :

( ) ( )( )

( ) ( )( )( )

arcsin sin cos

cos sinarcsin

cos

d x d y d

x d y d

d

d

u u

u u

= +

=

(4)

2.2. Rotor dynamic model

The dynamics of a DC motor is given by the following differential equations:

{ }, 1,2,3,4r i i iJ Q i = (5)

With Qi =d i2 is the reactive torque generated, in free air, by the rotor i due to rotor drag, and i is the input torque.

A control law for the input torque i is developed in [12-13], it is given by:

,i i r d i i iQ J k = + (6)where ki, i {1,,4} are four positive parameters, d,i, i {1,,4} are the desired speed of

each rotor and ,i i d i = .


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4

In fact, applying (6) to (5) leads to

ii i

r

k

J = (7)

which shows the exponential convergence of i to d,i and hence the convergence of the

airframe torques to the desired values leading to the attitude stabilization of the quadrotor

aircraft.

In our application, the dc motors are voltage controlled, so we need to obtain the voltage inputto each motor. Assuming that the motor inductance is small and taking into consideration thegear ratio, one can obtain the voltage to be applied to each motor as follows [12-13]:

{ }, 1,2,3,4ai i m g im g

Rv k k i

k k = + (8)

whereRa is the motor resistance, km is the motor torque constant, and kg is the gear ratio.

3. CONTROL STRATEGY OF QUADROTOR WITH ACTUATOR FAULTS

The complete model resulting by adding the actuator faults in the model (1) can be written inthe state-space form:

( ) ( ) ( ) ( ) ( )( ), , ax t x t g x t u t f t= + + (9)

With ( ) nx t is the state vector of the system, ( ) mu t is the input control vector, and

( )q

af t is the resultant vector of actuator faults related to the quadrotor motions, such as:

[ ]1 12,..., , , , , , , , , , , ,TT

X x x x x y y z z = = (10)

From (1), (10) and considering the actuator faults, we obtain the following state representation:

( )

( )

( )

( )

1 2

2

2 1 4 6 2 2 3 4 1 2 1

3 4

2

4 4 2 6 5 4 6 2 2 3

5 6

2

6 7 2 4 8 6 3 4 3

7 8

8 9 8 1

9 10

10 10 10 1

11 12

12 11 12 1 4

1

1

cos( ) cos( )

r a

r a

a

x

y

a

x x

x a x x a x a x b u f

x x

x a x x a x a x b u f

x x

x a x x a x b u f

x x

x a x u u

mx x

x a x u um

x x

x a x g u fm

=

= + + + + =

= + + + +

=

= + + +

=

= +

=

= +

=

= + +

(11)

with


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5

1 2 3 4 5

6 7 8 9 10

11 1 2 3

1

y z fayfax r z x

x x x y y

x y ftyfaz ftxr

y z z

ftz

x y z

I I KK J I I a a a a a

I I I I I

I I KK KJa a a a a

I I I m m

K l la b b b

m I I I

= = = = =

= = = = =

= = = =

(12)

The following assumptions is needed for the analysis,

Assumption 1: The resultant of actuator faults related to the quadrotor motions are bounded,

( ) [ ], i 1, 2,3, 4ai aif t f+ (13)

where {fa1+,fa2

+,fa3+,fa4

+} are positive constants.

Assumption 2: The unknowns parts ai(x,fai, t) including the resultants of actuator faults relatedto the quadrotor motions are also bounded,

( ) ( ) [ ], , , , i 1,2,3,4ai ai i ai aix f t g x t f k+ < (14)

where {ka1, ka2, ka3, ka4} are also positive constants.

The adopted control strategy is based on two loops (internal loop and external loop). Theinternal loop contains four control laws: control of roll, control of pitch, control of yaw and

control of altitude. The external loop includes two control laws of positions x and y. Theexternal control loop generates a desired of roll ( d) and pitch ( d) through the correction block

(illustrated by equation (4)). This block corrects the rotation of the roll and pitch depending on

the desired yaw ( d). The synoptic scheme below shows the control strategy:

Figure 2. Synoptic scheme of the proposed control strategy

Control of

law

Control of

pitch

Control of

roll

Control of

altitudeZ

Correction

blockControl of

y position

Control of

x position

uz

u

u

u

ux

uy

d

d

d

yd

xd

Zd

d

Desired

trajectory of

x position

Desired

trajectory of

y position

Desired

trajectory of

Zaltitude

Desired

trajectory of

law

Internal loop

External loop

, , ,Z

x,y


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Basing on backstepping approach, a recursive algorithm is used to synthesize the control lawsforcing the system to follow the desired trajectory in presence of actuator faults, we simplify all

stages of calculation concerning the tracking errors and Lyapunov functions in the followingway:

[ ][ ]( 1) ( 1) ( 1)

i 1,3,5,7,9,11i 2,4,6,8,10,12

i idi

i i d i i

x xex x c e

= + (15)

such as

[ ]0 i 1, ...,12ic > (16)and

[ ] [ ]( ), i 2, 4, 6,12 and j 1,..., 4j a j ik sign e = (17)

The corresponding Lyapunov functions are given by:

[ ]

[ ]

2

2

1

1i 1,3,5,7,9,11

2

1 i 2,4,6,8,10,122

i

i

i i

e

V

V e

= +

(18)

The synthesized stabilizing control laws are as follows:

( )

( )

( )

2

2 1 1 1 2 1 2 2 1 4 6 2 2 3 4 1 2

1

2

3 3 3 3 4 3 4 4 4 2 6 5 4 6 2 2 4

2

2

4 5 5 5 6 5 6 6 7 2 4 8 6 3 6

3

7

1

1( ) ( )

1( ) ( )

1( ) ( )

(

d r a

d r a

d a

x d

u c c e e e c e a x x a x a x k sign eb

u c c e e e c e a x x a x a x k sign eb

u c c e e e c e a x x a x k sign eb

mu x c cu

= +

= +

= +

=

( )

( )

(

7 7 8 7 8 8 9 8 1

9 9 9 10 9 10 10 10 10 1

1

1 11 11 11 12 11 12 12 11 1

1 3

) / 0

( ) / 0

( )cos( ) cos( )

y d

d

e e e c e a x u

mu y c c e e e c e a x u

u

mu z c c e e e c e a x

x x

+

= +

= +

)2 4 12( )ag k sign e

+

(19)

Proof

For i =1:

1 1 1

2

1 1

1

2

d

e x x

V e= =

(20)

and

1 1 1 1 2 1( )dV e e e x x= = (21)

By application of Lyapunov theorem, the stabilization ofe1 can be obtained by introducing a

new virtual controlx2


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7

( )2 1 1 1 1 1 0ddx x c e c= = > (22)

It results that:

2

1 1 1 0V c e= (23)

For i =2:

2 2 1 1 1

2

2 1 2

1

2

de x x c e

V V e

= +

= +

(24)

and

( ) ( )22 1 1 1 2 2 1 4 6 2 2 3 4 1 1 1 1 2 1 1 2( )r d aV e c e e e a x x a x a x x c c e e b u= + + + + + + + + (25)

We know a priori from (14) that:

1 1 1 1 1 1a a a ab f b f k += < (26)

The stabilization of(e1, e2) can be obtained by introducing the input control u2

( )22 1 1 1 1 2 1 2 2 1 4 6 2 2 3 4 1 21

1( )

d ru x c c e e e c e a x x a x a x c

b= + > (27)

It result that

2 2

2 1 1 2 2 2 1 1( )

aV c e c e e = (28)

In order to compensate the resultant of actuator faults related to the roll motion ( a1), a sign

function is introduced. We take:

1 1 2( )ak sign e = (29)

The equation (28) is then

2 2

2 1 1 2 2 2 1 1( ) 0

a aV c e c e e k (30)

The same steps are followed to extract u3, u4, ux, uy and u1.

4. SIMULATION RESULTS

Two cases are treated to evaluate the performances of the controller developed in this paper.

1- Results without faults are shown in Figure. 3, Figure. 5, Figure. 7 and Figure .9.a.

2- Results with four resultants of actuator faults {fa1,fa2,fa3,fa4} related to roll, pitch, yaw and

altitude motions with 25% of maximum values of inputs control u2, u3, u4 and u1 at 15s,20s, 25s and 30s respectively are shown in Figure. 4, Figure. 6, Figure. 8 and Figure .9.b.

The simulation results are obtained based on the real parameters in Table. 1 (see the appendix).


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0 10 20 30 40 50 60-6

-4

-2

0

2

4

6x 10

-3

Time[sec]

Rollangle[rad]

d

0 10 20 30 40 50 60

-6

-4

-2

0

2

4

6

8x 10

-3

Time[sec]

Pitchangle[rad]

d

0 10 20 30 40 50 600

0.2

0.4

0.6

0.8

1

1.2

1.4

Time[sec]

Yaw

angle[rad]

d

(a) evolution of roll angle (b) evolution of pitch angle (c) evolution of yaw angle

0 10 20 30 40 50 600

1

2

3

4

5

6

Time[sec]

X

position[m]

Xd

X

0 10 20 30 40 50 60-3

-2

-1

0

1

2

3

Time[sec]

Y

position[m]

Yd

Y

0 10 20 30 40 50 60-1

0

1

2

3

4

5

6

Time[sec]

Zattitude[m]

Zd

Z

(d) evolution ofx position (e) evolution of y position (f) evolution of z position

Figure. 3. Tracking simulation results of trajectories along roll ( ), pitch ( ), yaw angle ( ) and

(X,Y,Z) axis, Test 1.

0 10 20 30 40 50 60-8

-6

-4

-2

0

2

4

6

8x 10

-3

Time[sec]

Rollangle[rad]

d

0 10 20 30 40 50 60-6

-4

-2

0

2

4

6

8x 10

-3

Time[sec]

Pitchangle[rad]

d

0 10 20 30 40 50 60

0

0.2

0.4

0.6

0.8

1

1.2

1.4

Time[sec]

Yawangle[rad]

d

(a) evolution of roll angle (b) evolution of pitch angle (c) evolution of yaw angle

0 10 20 30 40 50 600

1

2

3

4

5

6

Time[sec]

X

position[m]

X d

X

0 10 20 30 40 50 60-3

-2

-1

0

1

2

3

Time[sec]

Yposition[m]

Yd

Y

0 10 20 30 40 50 60-1

0

1

2

3

4

5

6

Time[sec]

Zattitude[m]

Zd

Z

(d) evolution ofx position (e) evolution of y position (f) evolution of z position

Figure. 4. Tracking simulation results of trajectories along roll ( ), pitch ( ), yaw angle ( ) and(X,Y,Z) axis, Test 2.

Figure. 3 and Figure. 4 shows a good tracking of the desired trajectories for both Tests, while

seeing a low transient deviations due after occurrence of the resultant of actuator faults relatedto altitude motion at 30s (illustrated respectively by (a), (b) and (f) in Test 2), which explains

well the under-actuating of our system.

According to Figure. 5 and Figure. 6, there is also a good tracking of the desired velocities forboth Tests. Moreover, Figure. 6 clearly shows a small transient peaks (illustrated by (a) and (b)),

2 7 .5 2 8 2 8 .5 2 9 2 9 .5 3 0 3 0 .5 3 1 3 1 .5 3 2 3 2 .5

-6

-5.5

-5

-4.5

-4

-3.5

-3

x 10-3

Time[sec]

Rollangle[rad]

29.2 29.4 29.6 29.8 30 30.2 30.4 30.6 30.8 31 31.2

-2.2

-2

-1.8

-1.6

-1.4

-1.2

x 10-3

Time[sec]

Pitchangle[rad]

3 0. 5 3 1 3 1 . 5 3 2 3 2. 5 3 3

4.2

4.4

4.6

4.8

5

5.2

Time[sec]

Zattitude[m]


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9

and low deviations (illustrated by (c) and (f)) disappears quickly after appearance of theresultants of actuator faults related respectively to yaw and altitude motions.

0 10 20 30 40 50 60-4

-3

-2

-1

0

1

2

3x 10

-3

Time[sec]

Angularvelocityofroll[

rad/s]

dp

p

0 10 20 30 40 50 60-2

-1

0

1

2

3

4x 10

-3

Time[sec]

Angularvelocityofpitch

[rad/s]

dp

p

0 10 20 30 40 50 600

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Time[sec]

Angularvelocityofyaw[rad/s]

dp

p

(a) evolution of angular velocity of roll (b) evolution of angular velocity of pitch (c) evolution of angular velocityof yaw

0 10 20 30 40 50 60-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

Time[sec]

LinearvelocityalongXaxis

[m/s]

xdp

x p

0 10 20 30 40 50 60-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

Time[sec]

LinearvelocityalongYaxis[m/s]

ydp

y p

0 10 20 30 40 50 60-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

Time[sec]

linearvelocityofZattitude[m/s]

zdp

z p

(d) evolution of linear velocity alongX

axis

(e) evolution of linear velocity along Y

axis

(f) evolution of linear velocity alongZ

axis

Figure. 5. Tracking simulation results of the angular and linear velocities, Test 1.

0 10 20 30 40 50 60-4

-3

-2

-1

0

1

2

3x 10

-3

Time[sec]

Angularvelocityofroll[rad/s]

dp

p

0 10 20 30 40 50 60-2

-1

0

1

2

3

4x 10

-3

Time[sec]

Angularvelocityofpitch[rad/s]

dp

p

0 10 20 30 40 50 60-0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

Time[sec]


dp

p

(a) evolution of angular velocity of roll (b) evolution of angular velocity of pitch (c) evolution of angular velocity ofyaw

0 10 20 30 40 50 60-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

Time[sec]

LinearvelocityalongXaxis

[m/s]

xdp

x p

0 10 20 30 40 50 60-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

Time[sec]

LinearvelocityalongYaxis[m/s]

ydp

y p

0 10 20 30 40 50 60-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

Time[sec]


zdp

z p

(d) evolution of linear velocity alongX

axis

(e) evolution of linear velocity along Y

axis

(f) evolution of linear velocity alongZ

axis

Figure. 6. Tracking simulation results of the angular and linear velocities, Test 2.

From Figures. 7 and 8, it is clear to see the appearance of chattering phenomenon in inputscontrol u2, u3 and u4 for both tests, resulting from the use of discontinuous term "sign". Further-

more, Figure. 8 clearly shows a variation of the evolution of inputs control u2 and u3 (illustratedby (b) and (c)) after 15s and 20s. Consequently, the chattering phenomenon in these inputscontrol become more intensive. Otherwise, we see well a considerable deviation in evolution of

inputs control u4 and u1 (illustrated by (d) and (a)) after 25s and 30s.

3 0 3 1 3 2 3 3 3 4 3 5 3 6 3 7 3 8

-0.38

-0.36

-0.34

-0.32

-0.3

-0.28

-0.26

-0.24

-0.22

Time[sec]


25 2 6 2 7 2 8 2 9 3 0 3 1 3 2 3 3 3 4 3 5

-2

-1

0

1

2

3

4x 10

-4

Time[sec]


2 2 2 4 2 6 2 8 3 0 3 2 3 4 3 6

-10

-5

0

5

x 10-4

Time[sec]

Angularvelocityofroll[rad/s]

2 7 2 8 2 9 3 0 3 1 32 331

2

3

4

5

6

7

8

9

10

x 10-4

Time[sec]

Angularvelocityofpitch[rad/s]


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0 10 20 30 40 50 604.6

4.65

4.7

4.75

4.8

4.85

4.9

4.95

5

5.05

5.1

Time[sec]

Inputcontrolu1

0 10 20 30 40 50 60-2

-1.5

-1

-0.5

0

0.5

1

1.5

2x 10

-4

Time[sec]

Inputcontrolu2

(a) evolution of input control u1 (b) evolution of input control u2

0 10 20 30 40 50 60-2

-1.5

-1

-0.5

0

0.5

1

1.5

2x 10

-4

Time[sec]

Inputcontrolu3

0 10 20 30 40 50 60-4

-2

0

2

4

6

8

10

12x 10

-4

Time[sec]

Inputcontrolu4

(c) evolution of input control u3 (d) evolution of input control u4

Figure. 7. Simulation results of all inputs control, Test 1.

0 10 20 30 40 50 603.6

3.8

4

4.2

4.4

4.6

4.8

5

5.2

Time[sec]

Inputcontrolu1

0 10 20 30 40 50 60-2

-1.5

-1

-0.5

0

0.5

1

1.5

2x 10

-4

Time[sec]

Inputcontrolu2

(a) evolution of input control u1 (b) evolution of input control u2

0 10 20 30 40 50 60-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5x 10

-4

Time[sec]

Inputcontrolu3

0 10 20 30 40 50 60-4

-2

0

2

4

6

8

10

12x 10

-4

Time[sec]

Inputcontrolu4

(c) evolution of input control u3 (d) evolution of input control u4

Figure. 8. Simulation results of all inputs control, Test 2.

All these changes in the evolution of inputs control u1, u2, u3 and u4 are resulted from thecompensation of the resultants of actuator faults related to Roll, pitch, yaw, and altitude

motions, whose objective is to ensure the stability of the closed loop dynamics of quadrotorafter the occurrence of these actuator faults.

The simulation results given by Figure. 9 clearly shows a good performances and robustness

towards stability and tracking even after the occurrence of the resultants of actuator faultsrelated to Roll, pitch, yaw, and altitude motions, which explains the efficiency of control

strategy developed in this paper.


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0

2

4

6

-4

-2

0

2

4-2

0

2

4

6

X coordinate [m]

3D position

Y coordinate [m]

Zcoordinate[m]

Reference trajectory

Real trajectory

0

2

4

6

-4

-2

0

2

4-2

0

2

4

6

X coordinate [m]

3D position

Y coordinate [m]

Zcoordinate[m]

Reference trajectory

Real trajectory

(a) evolution of position along (X,Y,Z) axis, Test 1 (b) evolution of position along (X,Y,Z) axis, Test 2

Figure. 9. Global trajectory of quadrotor in 3D.

5. CONCLUSIONS AND FUTUR WORKS

In this paper, we proposed a new control strategy based on backstepping approach and includingthe actuator faults. Firstly, we presented the dynamical model of quadrotor taking into accountthe different physics phenomena which can influence the evolution of our system in the space.

Secondly, we synthesized a stabilizing control laws in the presence of actuator faults. The

simulation results have shown high efficiency of this control Strategy, it keeps the stability andthe performances of quadrotor during a malfunction of these actuators. As prospects we hope to

develop other control strategies in order to eliminate the chattering phenomenon in inputscontrol, while maintaining the stability and the performances of this system, with implement-

ation them on a real system.

6. APPENDIX

Table 1. Quadrotor parameters.

Parameter Value

m 0.486 kg

g 9.806 m/s2

l 0.25 m

b 2.9842 105N/rad/sd 3.2320 107N.m/rad/s

Jr 2.8385 105 kg.m2

Ix 3.8278 103 kg.m2

Iy 3.8278 103 kg.m2

Iz 7.1345 103 kg.m2

Kfax 5.5670 104N/rad/sKfay 5.5670 104N/rad/sK

faz 6.3540 104

N/rad/sKftx 0.0320N/m/s

Kfty 0.0320N/m/s

Kftz 0.0480N/m/s

km 4.3 103N.m/Akg 5.6

Ra 0.67


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7. REFERENCES

[1] S. Bouabdellah, & R.Siegwart. (2005) Backstepping and sliding mode techniques applied to an

indoor micro quadrotor, Proceeding of the IEEE, ICRA, Barcelona, Spain, pp. 2259-2264.

[2] H. Bouadi, M. Bouchoucha, & M. Tadjine. (2007) Modelling and Stabilizing Control Laws

Design Based on Backstepping for an UAV Type-Quadrotor, Proceeding of 6th IFAC

Symposium on IAV, Toulouse, France.[3] H. Bouadi, M. Bouchoucha, & M. Tadjine. (2007) Sliding Mode Control Based on Backstepping

Approach for an UAV Type-Quadrotor, International Journal of Applied Mathematics and

Computer Sciences, Vol. 4, No. 1, pp. 12-17.

[4] H .Bouadi, S. Simoes Cunha, A. Drouin, & F. Mora-Camino (2011) Adaptive Sliding Mode

Control for Quadrotor Attitude Stabilization and Altitude Tracking Proceeding of 12th IEEE

International Symposium on Computational Intelligence and Informatics, Budapest, Hungary, pp.

449-455.

[5] P. Castillo, A. Dzul, & R. Lozano. (2004) Real-Time Stabilization and Tracking of a Four-Rotor

Mini Rotorcraft, IEEE Transactions on Control Systems Technology, Vol. 12, No. 4, pp. 510-

516.

[6] J. Colorado, A. Barrientos, , A. Martinez, B. Lafaverges, & J. Valente. (2010) Mini-quadrotor

attitude control based on Hybrid Backstepping & Frenet-Serret Theory, Proceeding of the IEEE,

ICRA, Anchorage, Alaska, USA, pp. 1617-1622.

[7] A. Das, F. Lewis, K. Subbarao (2009) Backstepping Approach for Controlling a QuadrotorUsing Lagrange Form Dynamics. Journal of Intelligent and Robotic Systems, Vol. 56, No. 1-2,

pp. 127-151.

[8] L. Derafa, T. Madani, & A. Benallegue. (2006) Dynamic modelling and experimental

identification of four rotor helicopter parameters, Proceeding of the IEEE-ICIT, Mumbai, India,

pp. 1834-1839.

[9] N. Guenard, T. Hamel, & R. Mahony. (2008) A Practical Visual Servo Control for an Unmanned

Aerial Vehicle,IEEE Transactions on Robotics, Vol. 24, No. 2, pp. 331-340.

[10] R. Lozano, P. Castillo, & A. Dzul. (2004) Global stabilization of the PVTOL: real time

application to a mini aircraft,International Journal of Control, Vol. 77, No. 8, pp. 735-740.

[11] J.J.E. Slotine, & W. Li. (1991) Applied nonlinear control, Prentice Hall, Englewood Cliffs, NJ.

[12] A. Tayebi, & S. Mcgilvray. (2004) Attitude stabilisation of a four rotor aerial robot, Proceeding

of the IEEE Conference on Decision and Control, Atlantis Paradise Island, Bahamas, pp. 1216-

1217.

[13] A. Tayebi, & S. McGilvray (2006) Attitude stabilization of a VTOL Quadrotor Aircaft, IEEETransactions on Control Systems Technology, Vol. 14, No. 3, pp. 562-571.

Authors

Hicham KHEBBACHE is a Graduate student (Magister) of Automatic Control at the

Electrical Engineering Department of Setif University, Algeria. He received the

Engineer degree in Automatic Control from Jijel University, Algeria, in 2009. He is

with Automatic Laboratory of Setif (LAS). His research interests include Aerial

robotics, Linear and Nonlinear control, Robust control, Fault tolerant control (FTC),Diagnosis, Fault detection and isolation (FDI).

Belkacem SAIT is an Associate Professor at Setif University and member at

Automatic Laboratory of Setif (LAS), Algeria. He received the Engineer degree in

Electrical Engineering from National Polytechnic school of Algiers (ENP) in 1987, theMagister degree in Instrumentation and Control from HCR of Algiers in 1992, and the

Ph.D. in Automatic Product from Setif University in 2007. His research interested

include discrete event systems, hybrid systems, Petri nets, Fault tolerant control (FTC),

Diagnosis, Fault detection and isolation (FDI).


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Fouad YACEF is currently a Ph.D. student at the Automatic Control Department of

Jijel University, Algeria. He received the Engineer degree in Automatic Control from

Jijel University, Algeria, in 2009, and the Magister degree in Control and Command

from Military Polytechnic School (EMP), Algiers, in November 2011. He is with

Automatic Laboratory of Jijel (LAJ). His research interests include Aerial robotics,

Linear and Nonlinear control, LMI optimisation, Analysis and design of intelligent

control systems.

Yassine SOUKKOU is currently a Graduate student (Magister) of Automatic Control

and signal processing at the Automatic Control Department of Jijel University, Algeria.

He received the Engineer degree in Automatic Control from Jijel University, Algeria,

in 2009. His research interests include Image and Signal processing, Robotics, Linear

and Nonlinear control, Fault tolerant control (FTC), Diagnosis, Fault detection and

isolation (FDI).

robust stabilization of a quadrotor aerial vehicle in presence of actuator faults

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