Fault Recovery of an Under-Actuated Quadrotor Aerial Vehicle

M. Ranjbaran and K. Khorasani

AbstractMiniature Unmanned Aerial Vehicles (UAVs) withability to vertically take off and land (as in quadrotors)exhibit advantages and features in maneuverability that haverecently gained strong interest in the research community.Reliability of control systems require robustness and faulttolerance capabilities in presence of anomalies and unexpectedfailures in actuators, sensors or subsystems. Development ofan autonomous fault diagnosis and recovery system that cancope with these faults has attracted a lot of interest in thepast several years. Particularly, for small aerial vehicles due tohardware redundancy limitations design of a reliable controlsystem plays an important role in ensuring acceptable andefcient performance.

In this paper, an autonomous fault recovery scheme isproposed in response to actuator faults in an under-actuatedquadrotor aerial vehicle. A self-recovery mechanism, whichextends the capabilities of the quadrotor system to operateunder the presence of faults is proposed. The solution developedtakes into account the management of the control authority byincorporating the post-fault model of the actuator.

I. INTRODUCTION

Unmanned Aerial Vehicles, or UAVs are becoming widelyused as valuable tools in todays society. As their applicationboth in the military and in the industrial sector increases,potential miniature UAVs have steadily gained interest in theresearch community. UAVs have several basic advantagesover manned systems including increased manoeuvrability,low cost, reduced radar signatures and less risk to crews.Vertical take off and landing type UAVs exhibit furtheradvantages in manoeuvrability features. Such vehicles areto require little human intervention from the take-off to thelanding [1]. Quadrotors have become an exciting new area ofunmanned aerial vehicle research in the past few years. It isan aircraft that is lifted and propelled by four rotors in a crossconguration and its basic motions are generated by varyingthe speeds of all the four rotors. The quadrotor rotorcraft isnot a new conguration. It already existed in 1920s [2]. Theuniqueness of this type of UAV is in its vertical landing/take-off capability, hovering ability, great maneuverability andbeing simple to manufacture.

The quadrotor is a 6 Degree of Freedom (DOF) devicewith only four actuators, which make it an under-actuatedvehicle with unstable dynamics and highly coupled states.

This research is supported in part by a Strategic Projects Grant anda Discovery Grant from the Natural Sciences and Engineering ResearchCouncil of Canada (NSERC).

M. Ranjbaran was with the Department of Electrical and ComputerEngineering, Concordia University, Montreal, Quebec, H3G 1M8 Canada.She is now with McGill University.

K. Khorasani is with the Department of Electrical and Computer En-gineering, Concordia University, Montreal, Quebec, H3G 1M8 Canada(Email:kash@ece.concordia.ca).

In order to develop a reliable control to guarantee a stableautonomous ight, development of simple and robust controllaws stabilizing the quadrotor has become an important areaof investigation.

Enhanced reliability and safety of complex and au-tonomous systems due to occurrence of actuator faults areexpected to be achieved by incorporating Fault Diagnosis,Isolation and Recovery (FDIR) mechanisms in the designof the control system. The FDIR module is in charge ofdetecting, identifying, isolating and generating a recoveryprocedure to allow acceptable performance of the systemwhen it is subject to a fault. The main objective of thismodule is to enhance the reliability, performance and sur-vivability of the system.

In general, methods for implementing fault detection andisolation are classied into two categories, namely process-history based methods and process-model based methods.The rst approach depends on the knowledge collectedfrom past experiences and availability of a large amount ofhistorical data, while the latter relies on interactions betweenvarious dynamical system components and variables anda priori knowledge about the process.

The goal of the fault recovery mechanism is to select anoptimal possible conguration of the non-faulty actuators,sensors, and components in the system where a fault hasoccurred and diagnosed, to maintain the quality of the systemperformance despite the presence of faults.

Over the past decade various approaches for fault recoveryhave been proposed in the literature. One of the existingactive approaches for the fault recovery problem is throughadaptive control methods. The following methods fall underthis fault recovery category: Model Reference Adaptive Control (MRAC) or Model

Following method [3], [4], and Adaptive feedback linearization [5], [6].A number of researchers have also developed various

control methods to stabilize a quadrotor. The work done in[7] and [8] have used optimal Linear Quadratic Regulator(LQR) for the controller design. Lyapunov theory is also usedas another design technique [9] and [10]. According to thismethod, it is possible to ensure, under certain conditions, theasymptotic stability of the aerial vehicle. Backstepping andsliding mode control have also been used in [11], [12] and[13]. In these works the convergence of the quadrotor internalstates is guaranteed, however, the computations required arerelatively excessive.

A feedback linearization method was rst used in [14]to make the quadrotor track a reference trajectory. Theydeveloped a nonlinear state space dynamic model and used

49th IEEE Conference on Decision and ControlDecember 15-17, 2010Hilton Atlanta Hotel, Atlanta, GA, USA

978-1-4244-7746-3/10/$26.00 2010 Crown 4385

an exact global feedback linearization and non-interactingcontrol law for controlling the translational motion and yawangle outputs.

The method developed in [14] was also used in [15].In their work a PD controller was designed to controlthe y-axis and the yaw angle, and a feedback linearizationcontroller was implemented to control the x-axis and thez-axis states (translational motions). In [16], a feedbacklinearization scheme with a high-order sliding mode observerwas developed for a quadrotor and in simulations it wasshown to be quite robust against wind disturbances and noise.In [11], feedback linearization and adaptive sliding modecontrol schemes for a quadrotor are also developed and theircapabilities are compared. Given that the quadrotor is anunder-actuated system, the possible set of available solutionsfor control and fault recovery is rather limited. As shownsubsequently, an adaptive feedback linearization strategy isemployed for the purpose of fault recovery that will yield anacceptable behavior and performance in presence of certaintypes of faults in the vehicle actuators.

II. THE QUADROTOR MODEL

The quadrotor simply consists of four dc motors on whichpropellers are mounted in a cross conguration. Each pro-peller is connected to the motor through the reduction gears.All the propellers axes of rotation are xed and parallel. Thefront and the rear propellers rotate counter-clockwise, whilethe left and the right ones turn clockwise. This congurationof opposite pair directions removes the need for a tail rotor(needed instead in the standard helicopter structure).

In Figure 1 the schematic of a simplied quadrotor struc-ture is shown where i(rads1) refers to the propellersrotation speed. While at hovering, all the four propellersrotate at the same speed which implies that i = H fori= 1,2,3,4 to counterbalance the acceleration due to gravity.Although the quadrotor has 6 DOF, it is just equipped withfour propellers, hence it is not possible to reach a desiredset point for all the DOF and the system is under-actuated.However, by selecting four controllable variables properly,it is possible to design a controller that will ensure thevehicle could reach a desired height and attitude. Below, weseparately present the dynamical models corresponding tothe vehicle and the actuators.

1) Dynamic Model: Reference [17] provides a mathemat-ical model of the quadrotor that is derived from the Newton-Euler formulation, that is

x = (cos sin cos+ sin sin)U1my = (cos sin sin sin cos)U1mz =g+(cos cos)U1m = ( IyyIzzIxx )

Jt pIxxT + U2Ixx

= ( IzzIxxIyy )+Jt pIyyT + U3Iyy

= ( IxxIyyIzz )+U4Izz

(1)

where[

x y z]T represents the position of the quadrotor

in the inertial frame, the attitude state variables in the body

Fig. 1. Simplied quadrotor model representation at hovering and thecoordinate systems (the body and the earth reference frames).

frame[

]T represents the roll, the pitch and theyaw angles, respectively, m(Kg) is the overall mass, g refersto the acceleration due to gravity, Ixx, Iyy and Izz(Nms2) arethe inertia moments in the body xed frame and Jt p(Nms2)is the total rotational moment of inertia around the propelleraxis. Furthermore, U1 denotes the normalized total lift force,and U2, U3 and U4 correspond to the control inputs of theroll, the pitch and the yaw moments, respectively, and Trefers to the overall residual propeller angular speed. Theseinput moments are dened according to the equations

U = LUTT (2a)

T =1 +23 +4 (2b)where U =

[U1 U2 U3 U4

]T is the movement vectorand T =

[T1 T2 T3 T4

]T is the thrust vector. Theconstant matrix LUT is dened according to

LUT =

1 1 1 10 l 0 ll 0 l 0 db db db db

(3)

where l(m) is the distance between the center of the quadro-tor and the center of a propeller, b and d denote the thrust andthe drag coefcients, respectively, and Ti is the thrust forcegenerated by each rotor and is proportional to the square ofeach propellerss speed, that is we have

Ti = b2i (4)

2) Actuator Model: The rotors are driven by DC motors.The model consists of three elements in series for the stator,that is, the motor resistance Rmot , the motor inductance L, andthe back-EMF voltage which is given by e= kem. Applyingthe Kirchhoffs current law to the DC motor circuit resultsin

u = Rmot i+Ldidt

+ kem (5)

where u(V ) is the input voltage to the motor, ke (V srad1)is the back-EMF constant, and m(rad s1) is the motorangular speed. The dynamics of the motor is described bythe following representation

Jtmm = Mt Ml (6)

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where Jtm (Nms2) is the total motor moment of inertia,m (rad s2) is the motor angular acceleration, Mt (Nm) isthe motor torque, and Ml (Nm) is the load torque. Since themotor is small and assuming a low inductance, the rotordynamics can be approximated by:

Jtmm = ktu kem

RmotMl

= k2e

RmotmMl + keRmot u

(7)

The actual motor system is composed of the motor itself, thegear box and the propeller. Considering the propeller and thegearbox, the load torque experienced by the motor is givenby the equation

Ml =dr3

2m (8)

where d(Nms2) denotes the aerodynamic drag factor and rand refer to the gearbox reduction ratio and efciencyfactor, respectively. The total inertia seen by the motor canalso be described as

Jtm = Jm +Jpr2

(9)

where Jm (Nms2) is the rotor moment of inertia about themotor axis and Jp (Nms2) is the rotor moment of inertiaabout the propeller axis. Equation (7) can be rewrittenaccording to (8) and (9) as follows

(Jm +Jpr2

)m = k2e

Rmotm dr3

2m +

keRmot

u (10)

Equation (10) is formulated with respect to the motor axis.It is possible to reformulate this equation from the propelleraxis as shown below

(Jp +r2Jm)i = k2e

Rmotr2id2i +

keRmot

ru (11)

From equation (11), the total rotational inertia around thepropeller axis Jt p (Nms2) can be dened as Jt p = (Jp +r2Jm). By setting k

2e

Rmot Jt pr2 = 1t , equation (11) can be

rewritten as:

i = 1tid2i +

1kert

u (12)

The dynamic equation of the propeller angular speedi (rad s1) is dened according to equation (12). On theother hand, the input moments to the quadrotor dened inequation (2a) are related to the propellers speed. Hence, itis possible to derive input moment dynamic equations andcomplete the quadrotor model by considering the effects ofthe motor on the dynamics of the entire system.

From equations (4) and (12) the input voltage to thepropeller i, ui, and to the thrust Ti dynamic equation couldbe obtained as follows:

Ti = 2bii

=2bt2i 2db3i +

2bikert

ui

= 2tTi 2d

bTiTi +

2b

kert

Tiui

(13)

Since the model (13) is nonlinear, a suitable approachwould be to linearize this dynamics around an operatingpoint, T0. The rst order Taylor series approximation yieldsthe linearized model as given by

Ti = AtTi +Btui +Ct (14)

In the above equation, the parameters At , Bt and Ct are thelinearized coefcients and are dened as follows:

At = 2t 3db

T0

Bt =b

kertT0

Ct = dbT32

0

(15)

The set point corresponding to the linearizing conditioncould be calculated from the fact that at hovering the totalthrust should be equal to the gravitational force effective onthe quadrotor. In other words, we have

4

i=1

Ti = mg (16)

Using the linearized dynamic equation for the thrust Tideveloped in (14), it is possible to nd the dynamic equationsfrom the input voltage to the propellers to the movementmoments. For this purpose it is useful to write the dynamicequation in (14) in a matrix form for the thrust vector T =[

T1 T2 T3 T4]T , as follows

T =ATT +BTu+CT (17)where AT = AtI(44) and BT = BtI(44) are constant matricesand CT = Ct

[1 1 1 1

]T . The notation I(44) refersto a 44 identity matrix and u = [ u1 u2 u3 u4 ]T isdened as the vector of the input voltages to the propellers.

By left-multiplying the transfer matrix given by equation(3) between U and T , that is LUT , with (17) the followingequation is obtained

LUT T =(LUTAT )T +(LUTBT )u+(LUTCT ) (18)It should be noted that:{

LUTAT = LUT (AtI(44)) = At(I(44))LTU = ATLUTLUTBT = BTLUT

From equation (2a), it is possible to rewrite equation (18) as

U =ATU +(LUTBT )u+(LUTCT ) (19)Since the quadrotor motion can be assumed close to the

hovering condition, small angular changes occur (especiallyfor the roll and the pitch angles). Since the rates of changein and are small, the terms due to the gyroscopic

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effects appearing in the dynamic equations of and in(1) are also negligible. These assumptions are also veriedthrough simulation results. Moreover, since the structure ofthe quadrotor is symmetric, the body moments of inertia Ixxand Iyy are equal. This fact will also simplify the dynamicequation of in (1). If the altitude z reaches a desired set-point given by zd , then z 0.

As stated earlier, in the hovering condition the total thrustshould be equal to the gravitational force effective on thequadrotor, in other words:

U1 =4

i=1

Ti = mg (20)

Therefore, if z 0 and and are sufciently close tozero, then U1 mg.

By assuming U1 mg and 0, it is possible tosimplify the dynamic equations of the x and y states thatare dened in (1). Considering all the above assumptions,the dynamic equations of the quadrotor system includingthe dynamic equations for the movement vector is speciedaccording to the following model, that is

z =g+(cos cos)U1my =gsinx = gsin = U2Ixx = U3Iyy = U4IzzU =ATU +(LUTBT )u+(LUTCT )

(21)

The control algorithm that is to be designed should providethe appropriate signals to the actuators. Since there areonly four propellers, no more than four variables can becontrolled in the loop. It is possible to dene the positionof the quadrotor in space completely by the linear positionE =

[x y z

]T and the yaw angle (heading angle) .These four variables are indeed selected for control purposesin this work. It can be seen that the system is partitioned intofour semi-decoupled subsystems as the outputs z, y, x and can be controlled by U1, U2, U3 and U4, respectively.

III. LOSS OF EFFECTIVENESS (LOE) FAULT MODELING

In case of a loss of effectiveness (LOE) fault in an actuator,the output speed of the quadrotor becomes different from thecommanded output desired by the controller, that is

i = kici 0 < < ki < 1 (22)

where i refers to the actual output from the ith actuator andci is the commanded output by the controller. Therefore,the resulting thrust force from this actuator varies accordingto the following equation

Ti = b2i= b(kici)2

(23)

The dynamics of Ti dened in equation (14) would alsochange due to the LOE fault, that is

Ti = 2bk2i cici= k2i At + k

2i Btui + k

2i Ct

(24)

or in other words,

Ti = AtiTi +Btiui +Ct i = 1,2,3,4 (25)

where Ati = k2i At and Bti = k2i Bt . It should be noted that we

have assumed that the only coefcients subject to changedue to a fault are the Ati and Bti and the coefcient Ct wouldstay unaffected. The term Ct is proportional to the drag andproportional to the inverse square of the thrust factor, whichmakes it a relatively small constant value.

Equation (25) can now be represented in a matrix form,that is

T = AT0T +BT0u+CT (26)

where

AT0 =

At1 0 0 00 At2 0 00 0 At3 00 0 0 At4

(27a)

BT0 =

Bt1 0 0 00 Bt2 0 00 0 Bt3 00 0 0 Bt4

(27b)

CT =

Ct 0 0 00 Ct 0 00 0 Ct 00 0 0 Ct

(27c)

Now if the thrust dynamics for all the actuators are notidentical, the dynamic equations of the movement vector Uchange since AT = AtI and BT = BtI. Therefore, it is neces-sary to derive the dynamic equations of the movement vectorwhile the actuators do not have the same characteristics, inother words when Ati = At j and Bti = Bt j for i, j = 1, . . . ,4,i = j.

The relationship between the movement vector U and Twas dened in equation (2a). By left-multiplying equation(26) with LUT , we would obtain

LUT T =(LUTAT0)T +(LUTBT0)u+(LUTCT ) (28)From equation (2a), equation (28) can be rewritten as follows

U =(LUTAT0L1UT )U +(LUTBT0)u+(LUTCT ) (29)If the actuators have the same parameters, in other words

Ati = At and Bti = Bt for i = 1, . . . ,4, then (LUTAT0L1UT ) =AT = AtI44 as expected for the healthy system.

IV. ADAPTIVE FEEDBACK LINEARIZATION RECOVERYCONTROL STRATEGY

At the end of Section III, the dynamic equations of themovement vector were derived in equation (29). In thisequation the contribution of each actuator to the resultingmovement vector is specied. As discussed earlier, in caseof a LOE fault the parameters of the actuators also change.A parameter estimation algorithm is now presented in thissection to provide an estimate of the faulty actuator severity

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and to develop a nonlinear adaptive controller to guaranteestability and recovery of the closed-loop system.

The recovery controller is designed by considering thedynamic equation of the movement vector that is obtainedin equation (29). The following equations are derived fordesigning the feedback linearization controller for the (z,U1),(y,U2), (x,U3) and ( ,U4) subsystems, such that the inputappears in the output derivatives equation, namely

z(3) = sin cosU1m cos sinU1

m+ cos cos

U1m

(30)

y(5) =gU2Ixx

cos +gU2Ixx

sin +2g sin +g3 cos (31)

x(5) = gU3Iyy

cos gU3Iyy

sin 2g sin g3 cos (32)

(3) =U4Izz

(33)

It should be noted that the relative degree of the system isequal to the order of the system and no internal dynamicsexists in designing the feedback linearization controller.

Without loss of generality, let us assume that the LOEfault has occurred in the rst actuator and the other threeactuators are healthy, that is{

T1 = At1T1 +Bt1u1 +CtTi = AtTi +Btui +Ct for i = 2,3,4

(34)

where At1 = k2At and Bt1 = k2Bt .The dynamics of the movement vector as dened in (29)

can be written as

U1

U2

U3

U4

=

14 (At1 +3At ) 0

12 (At1 +At ) b4d (At1 +At )

0 At 0 0l4 (At1 +At ) 0 12 (At1 +At ) lb4d (At1At )d4b (At1 +At ) 0 d2lb (At1At ) 14 (At1 +3At )

U1

U2

U3

U4

+

Bt1 Bt Bt Bt

0 lBt 0 BtlBt1 0 lBt 0 db Bt1 db Bt db Bt db Bt

u1

u2

u3

u4

+(LUTCT )

(35)

The above dynamic equation is now substituted in equations(30) to (33) which can be rewritten in a compact matrixform. Note that we are seeking a form to separate the termsthat are related to the unknown variables At1 and Bt1. Thisis achieved in the following equation as shown below

z(3)

y(5)

x(5)

(3)

= F1 +At1F2 +F3

Bt1 0 0 00 Bt 0 00 0 Bt 00 0 0 Bt

u1u2u3u4

(36)where

F1 =

sin cos U1m cos sinU1m +

cos cosm At (

34U1 +

12U3 +

b4dU4)+4Ct

3g U2Ixx sin +g3 cos gIxx cosAtU2

3g U3Iyy sin g 3 cos +gcosIyy

At ( l4U1 +12U3 lb4dU4)

1Izz

At ( d4bU1 d2lbU3 + 34U4)

(37a)

F2 =

cos cosm (

14U1 12U3 b4dU4)

0gcosIyy

( l4U1 + 12U3 + lb4dU4)1IzzAt( d4bU1 + d2lbU3 + 14U4)

(37b)

F3 =

cos cosm

cos cosm

cos cosm

cos cosm

0 gcosIxx l 0 gcosIxx

l gcosIyy l 0

gcosIyy

l 01Izz

( db ) 1Izz ( db ) 1Izz ( db ) 1Izz ( db )

(37c)

The input signal u is dened as

u1u2u3u4

=

F3

Bt1 0 0 00 Bt 0 00 0 Bt 00 0 0 Bt

1

w1w2w3w4

F1 At1F2

(38)

where At1 and Bt1 are the estimates of the unknown pa-rameters At1 and Bt1 and the new control input vectorW =

[w1 w2 w3 w4

]T is to be selected so that thestability and control (LQR) of the feedback linearized systemis achieved.

By applying the control law as dened in (38) to thesystem (36), the closed-loop dynamics can be written as

z(3)

y(5)

x(5)

(3)

= F1 +At1F2+F3

Bt1 0 0 00 Bt 0 00 0 Bt 00 0 0 Bt

Bt1 0 0 00 Bt 0 00 0 Bt 00 0 0 Bt

1

F13

w1w2w3w4

F1 At1F2

(39)

Let us dene the parameter estimation error as the dif-ference between the actual value of the unknown parameterand its estimate, i.e. At1 = At1 At1 and Bt1 = Bt1 Bt1.Therefore, equation (39) can be rewritten as

z(3)

y(5)

x(5)

(3)

= F1 +At1F2

+F3

Bt1Bt1

0 0 0

0 0 0 00 0 0 00 0 0 0

F13

w1w2w3w4

F1 At1F2

+F3F13

w1w2w3w4

F1 At1F2

=

w1w2w3w4

+At1F2

+Bt1F3

1Bt1

0 0 0

0 0 0 00 0 0 00 0 0 0

F13

w1w2w3w4

F1 At1F2

(40)

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Let

F4 = F3

1Bt1

0 0 0

0 0 0 00 0 0 00 0 0 0

F13

w1w2w3w4

F1 At1F2

(41)

Hence, the linearized dynamics of system (40) can be rewrit-ten as

z(3)

y(5)

x(5)

(3)

=

w1w2w3w4

+At1F2 +Bt1F4 (42)

The control inputs w1, w2, w3 and w4 are dened accordingto the following equations

w1 = z(3)d k1z(ez) k2z(ez) k3z(ez) (43a)

w2 = y(5)d k1y(e(4)y ) k2y(e(3)y ) k3y(ey) k4y(e)y k5y(ey)

(43b)w3 = x

(5)d k1x(e(4)x ) k2x(e(3)x ) k3x(e)x k4x(e)x k5x(ex)

(43c)w4 =

(3)d k1(e) k2(e) k3(e) (43d)

In the above equations zd , yd , xd and d denote thedesired output variables and ez = z zd , ey = y yd and ex = x xd and e = d are denedas the tracking error signals. The gain vectors Kz =[

k1z k2z k3z]T

, Ky =[

k1y k2y k3y k4y k5y]T

, Kx =[k1x k2x k3x k4x k5x

]Tand K =

[k1 k2 k3

]Tare

obtained from the LQR design procedure.From the equations (42) and (43a) to (43d), it is possible

to write the dynamic equations of the error signals ez, ey, exand e as follows

e(3)z + k1zez + k2zez + k3zeze(5)y + k1ye

(4)y + k2ye

(3)y + k3yey + k4yey + k5yey

e(5)x + k1xe(4)x + k2xe

(3)x + k3xex + k4xex + k5xex

e(3) + k1 e + k2 e + k3e

= At1F2 +Bt1F4

(44)It is possible to represent the above equation in the state-space form. The selected state variables for this purpose isas follows

X =[ey ex ey ex ez ey ex e ez e

(3)y e

(3)x e ez e

(4)y e

(4)x e

]T(45)

Therefore, one could specify the state-space representationof the system according to

X = A1616X +At1F2 +Bt1F

4 (46)

whereF 2 =

[0 . . . 0 F2

]T116 (47a)

F 4 =[

0 . . . 0 F4]T116 (47b)

All the eigenvalues of the A matrix are negative by properselection of the ki parameters. In other words, A is a Hurwitzmatrix. Now, let us dene the update law for the parameterestimation errors At1 and Bt1 according to the followingrules:

{At1 = At1 =

(F T2 PX +X

TPF 2)

Bt1 = Bt1 =(F T4 PX +X

TPF 4) (48)

where X is the state vector dened in equation (45), and P isa (1616) matrix that is obtained by solving the followingLyapunov equation, that is

ATP+PA =I1616 (49)where the A matrix is dened according to system (46)and the notation I1616 denotes a 16 16 identity matrix.It should be noted that since A is a Hurwitz matrix, P is apositive denite matrix [18].

The following theorem provides a sufcient condition forstability of the resulting closed-loop recovered system.Theorem 1: The state trajectories of the closed-loop system(46) that is subjected to the LOE fault in the rst actuatorand which has employed the update law for the parameterestimation errors At1 and Bt1 given by equation (48) areglobally stable in the sense of Lyapunov.Proof: To carry out the stability analysis, let us choose thefollowing radially unbounded Lyapunov function candidate

V = (XTPX +12 2At1 +

12 2Bt1) (50)

The derivative of this function along the state trajectories ofthe closed-loop system (46) yields

V = XTPX +XTPX +At1 At1 +Bt1 Bt1 (51)

Now, by using (48) the above expression simplies to

V = (AX +At1F2 +Bt1F

4)

TPX +XTP(AX +At1F2 +Bt1F

4)

+At1 (F T2 PXXTPF 2)+Bt1 (F T4 PXXTPF 4)= XTATPX +XTPAX

= XT (ATP+PA)X

(52)

It can be concluded from (49) that

V =XTX (53)which guarantees the negative semi-deniteness of the func-tion V . This implies that the origin is a globally stableequilibrium point of system (46) given that the Lyapunovfunction is radially unbounded.Remark: Provided that the external desired reference tra-jectories are persistently exciting, one can then ensure thatAt1 At1 and Bt1 Bt1, implying that the fault recoverysystem can independently be used to isolate and identify theseverity of the injected fault.

In this section, we have studied the case of a LOE faultrecovery by assuming that the fault has occurred in therst actuator. The same method could be applied to designthree other controllers that accommodate the LOE fault inthe other three actuators. In other words, the fault recoverymodule contains four different controllers and based uponthe information that one can receive from the fault detectionand isolation unit the proper controller can then be selectedand invoked. In each of these controllers, two parameters,namely, Ati and Bti that are related to the faulty ith actuator

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are assumed to be unknown and the parameters related to thehealthy actuators are assumed to be all known. It is clear thatsimilar approach can be employed to design a controller forthe healthy system where all the parameters are consideredto be known.

0 10 20 30 40 50 60 70 80 90 1000

5

10

(a)

time (sec)

x (m

)

0 10 20 30 40 50 60 70 80 90 1000

2

4

6(b)

time (sec)

y (m

)

0 10 20 30 40 50 60 70 80 90 1000

2

4

6(c)

time (sec)

z (m

)

HealthyFaultyRecoveredDelayed recovery

Recovered

Delayed recovery FaultyRecovered

FaultyDelayed recoveryHealthy

Healthy

Fig. 2. Linear position in response to the commanded trajectory corre-sponding to a 25% LOE fault in the rst actuator subject to as well aswithout the fault recovery mechanism invoked: (a) x-axis, (b) y-axis, and(c) z-axis measured in meters.

V. SIMULATION RESULTSIn this section, the behavior of the quadrotor system that

is embedded with the fault recovery mechanism is studiedthrough simulation results. The quadrotor model that isused for simulation is the OS4 that is developed in EcolePolythechnique Federal De Lausanne [17]. The mathematicalmodel that is used for control design is partially nonlinear(the actuator dynamics is linearized for control design),however, we have applied the controller to the fully nonlinearmodel of the quadrotor and have considered the full actuatordynamics. In simulations additive white Gaussian noise isalso added to the input and output channels to simulate amore realistic environment. The noise power is selected sothat the signal to noise ratio is approximately 15 db.

The commanded trajectory starts at the position (x,y,z) =(0,0,0) while the roll, the pitch and the yaw angles areinitially set to zero. The commanded trajectory is to yfrom the initial point to the nal point (5,5,5)(m) in 10seconds and hovering at the nal point. The fault trajectoryconsidered here assumes a 25% partial LOE failure in therst actuator at t = 20sec. We are assuming that a fault detec-tion and isolation mechanism exists to alarm the occurrenceof the fault in the faulty actuator without much delay andthe fault recovery mechanism is initiated after the detectionand isolation of the fault. The performance response of ourproposed fault recovery scheme is also evaluated in case

of a delayed fault detection and isolation by initiating therecovery procedure 8 seconds after the fault occurrence.

Figures 2 and 3 show the linear position and the Eulerangles of the system, respectively, in response to the com-manded trajectory for the healthy system as well as thefaulty and the recovered scenarios by using our proposedfault recovery algorithm with and without the delayed faultdetection and isolation information. Figure 4 depicts theestimated parameters of the faulty actuator correspondingto this case. Table I shows the means and the standarddeviations of the steady state tracking error signals for x,y, z and under four different scenarios where the systemis operating, namely, (I) healthy condition, (II) with a faultyactuator and no fault recovery solution invoked, (III) with afault recovery mechanism invoked, and (IV) with a delayedfault recovery mechanism invoked. It can be seen fromFigures 2 and 3 and Table I that our proposed fault recoverymechanism has a considerable effect on the performance ofthe system in presence of a 25% LOE fault by reducing thesteady state error between the desired and the actual outputs.It should be noted that as intuitively expected, the soonerthe fault recovery is initiated, the improved performance andsmaller error signals are obtained.

0 10 20 30 40 50 60 70 80 90 1000.5

0

0.5(a)

time (sec)

roll

(rad)

0 10 20 30 40 50 60 70 80 90 1000.5

0

0.5(b)

time (sec)

pitc

h (ra

d)

0 10 20 30 40 50 60 70 80 90 100

0

0.5

1

1.5(c)

time (sec)

yaw

(rad)

HealthyFaultyRecoveredDelayed recovery

Faulty

Delayed recovery

RecoveredHealthy

Fig. 3. Euler angles in response to the commanded trajectory correspondingto a 25% LOE fault in the rst actuator subject to as well as without thefault recovery mechanism invoked: (a) Roll angle (rad), (b) Pitch angle (rad)and (c) Yaw angle (rad).

Next, the performance of the quadrotor system is evaluatedin case of a more severe LOE fault. Specically, we considera 50% LOE fault in the rst actuator in a mission similarto the previous case. Table II shows the means and thestandard deviations of the steady state tracking error signalsfor x, y, z and under four different scenarios where thesystem is operating, namely, (I) 50% LOE in the rst actuatorand without fault recovery solution invoked, (II) with animmediate fault recovery mechanism invoked, (III) with a 5

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TABLE IMEAN AND STANDARD DEVIATION (STDV) OF THE TRACKING ERROR

SIGNALS FOR 25% LOE. LEGEND: (I) HEALTHY CONDITION, (II) WITHA FAULTY ACTUATOR AND NO FAULT RECOVERY INVOKED, (III) WITH A

FAULT RECOVERY INVOKED, AND (IV) WITH A DELAYED FAULTRECOVERY INVOKED.

Error (I) (II) (III) (IV)

ex Mean (m) -0.0537 4.2883 0.4226 2.8756ex Stdv (m) 0.2809 2.3790 0.4586 1.8593ey Mean (m) -0.0486 -0.0484 -0.0484 -0.0483ey Stdv (m) 0.1632 0.1634 0.1644 0.1637ez Mean (m) -0.0542 -1.0619 -0.1645 -0.7317ez Stdv (m) 0.1884 0.4445 0.1684 0.3586

e Mean (rad) 1.3461e-004 0.8659 0.0947 0.5818e Stdv (rad) 8.7942e-004 0.4471 0.0566 0.3505

TABLE IIMEAN AND STANDARD DEVIATION (STDV) OF THE TRACKING ERRORSIGNALS FOR 50% LOE. LEGEND: (I) FAULT IN THE FIRST ACTUATORAND WITHOUT FAULT RECOVERY INVOKED, (II) WITH AN IMMEDIATE

FAULT RECOVERY INVOKED, (III) WITH A 5 SECOND DELAY ININITIATING THE FAULT RECOVERY, AND (IV) WITH A 10 SECOND DELAY

IN INITIATING THE FAULT RECOVERY.

Error (I) (II) (III) (IV)

ex Mean (m) 11.8699 0.3915 4.5412 6.5079ex Stdv (m) 5.2980 0.4485 3.5770 4.1577ey Mean (m) -0.0484 -0.0484 -0.0484 -0.0482ey Stdv (m) 0.1637 0.1631 0.1639 0.1637ez Mean (m) -4.3580 -0.1567 -0.8790 -1.5712ez Stdv (m) 1.1027 0.1702 0.6828 0.8786

e Mean (rad) 2.9836 0.0885 0.8147 1.5082e Stdv (rad) 1.0285 0.0548 0.3263 0.8225

second delay in initiating the fault recovery, and (IV) with a10 second delay in initiating the fault recovey. As expected,the fault recovery mechanism performance is considerablybetter when there is smaller delay in the detection andisolation of the fault.

0 20 40 60 80 10015.66

15.64

15.62

15.6

15.58(a)

time (sec)

A t1

0 20 40 60 80 1001.5

2

2.5(b)

time (sec)

B t2

Fig. 4. Estimated actuator parameters in response to the commandedtrajectory corresponding to a 25% LOE fault in the rst actuator with thefault recovery mechanism: (a) At1 and (b) Bt1.

VI. CONCLUSION

In this paper, a fault recovery mechanism is proposedfor reconguring the control system from LOE faults inthe quadrotors actuators. An adaptive feedback linearizationtechnique is employed for the controller design and globalstability of the system with the fault recovery mechanismis shown. This is accomplished by introducing a parameterestimation algorithm and by deriving proper update laws

for the faulty actuator parameters subject to changes inthe quadrotor actuators due to the presence of the LOEfault. Simulation results are also presented to evaluate theperformance of the proposed fault recovery mechanism inpresence of an LOE fault in one of the actuators. It isobserved that by employing the fault recovery algorithm thesteady state tracking errors of the system outputs reduceconsiderably when compared to the responses obtained fromthe faulty system that has not employed our proposed faultrecovery strategy. REFERENCES[1] L. Derafa, T. Madani, and A. Benallegue, Dynamic modelling and

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