2012 American Control Conference Fairmont Queen Elizabeth, Montréal, Canada June 27-June 29, 2012 Generalized Fault Recovery of an Under-Actuated Quadrotor Aerial Vehicle M. Ranjbaran and K. Khorasani Abstract— Development of an autonomous fault diagnosis and recovery system for unmanned aerial vehicles has recently attracted a lot of interest in the research community. Design of a reliable control system that can cope with faults and anomalies plays an important role in ensuring efficient performance of small aerial vehicles due to hardware redundancy limitations. In this paper, an autonomous fault recovery scheme is proposed in response to multiple actuator faults in an underactuated quadrotor aerial vehicle. Such a self-recovery mechanism extends the capabilities of the quadrotor system to operate under presence of multiple faults. The developed solution considers a control authority management by incorporating the actuator post-fault model. Simulation results for a quadrotor that is subject to various levels of loss-of-effectiveness faults in multiple control channels are presented to demonstrate the performance of our proposed approach. I. INTRODUCTION Research on autonomous flying robots has recently intensified considerably due to growth of civilian and military interests in Unmanned Aerial Vehicles (UAV). UAVs have several basic advantages over manned systems including increased manoeuvrability, low cost, reduced radar signatures and less risk to crews. Quadrotors have become an exciting new area of UAV research in the past few years. It is an aircraft that is lifted and propelled by four rotors in a cross configuration and its basic motions are generated by varying the speeds of all the four rotors. The uniqueness of this type of UAV is in its vertical landing/take-off capability, hovering ability, great maneuverability and being simple to manufacture. The quadrotor is a 6 Degree of Freedom (DOF) device with only four actuators, which makes it an underactuated vehicle with unstable dynamics and highly coupled states. It is well-known that enhanced reliability and safety of complex and autonomous systems due to occurrence of faults can be accomplished by incorporating Fault Detection, Isolation and Recovery (FDIR) schemes in the design of control systems. The FDIR module is in charge of detecting, identifying, isolating and generating a recovery procedure to ensure acceptable performance specifications of the system when it is subjected to faults. The goal of the fault recovery module is to select an optimal possible configuration of the non-faulty actuators, sensors, and components in the system M. Ranjbaran was with the Department of Electrical and Computer Engineering, Concordia University, Montreal, Quebec, H3G 1M8 Canada. She is now with the McGill University. K. Khorasani is with the Department of Electrical and Computer Engineering, Concordia University, Montreal, Quebec, H3G 1M8 Canada (Email:kash@ece.concordia.ca). 978-1-4577-1096-4/12/$26.00 ©2012 AACC that is subjected to a diagnosed fault and to maintain the overall quality of the system performance requirements. A number of researchers have developed control techniques to stabilize a quadrotor. The works in [1] and [2] have used optimal Linear Quadratic Regulator (LQR) for the controller design. Asymptotic stability of the quadrotor, under certain conditions was also shown by applying Lyapunov theory [3], [4]. The authors in [5] and [6] have used PD2 feedback and PID structures for controller design. Backstepping and sliding mode control techniques have also been used in [7], [8] and [9]. In these work the convergence of the quadrotor internal states is guaranteed, however, the computations required are relatively excessive. The authors in [10] developed a nonlinear dynamic model for a quadrator. They used an exact global feedback linearization and non-interacting control law to control the translational motion and the yaw angle. The method developed in [10] was also used in [11]. In this work, a PD controller was designed to control the y-axis and the yaw angle. Moreover, a feedback linearization controller was implemented to control the x and z-axes states (translational motions). In [12], a feedback linearization scheme with a high-order sliding mode observer was developed for a quadrotor. Simulation studies showed that it is robust against wind disturbances and noise. In [7], feedback linearization and adaptive sliding mode controls for a quadrotor were developed and compared. Given that the quadrotor is an under-actuated system, possible approaches for control and fault recovery are rather limited. For details on a hybrid fault detection and isolation strategy for a network of quadrotor vehicles refer to our earlier work in [13]. As shown subsequently, an adaptive feedback linearization strategy is developed for fault recovery that yields an acceptable performance in presence of certain types of faults in the vehicle actuators. We have recently shown that it is possible to recover from a certain fault in only one of the quadrotor’s actuators [14]. In this paper, a generalization to our previous work on the fault recovery of a quadrotor that is subject to a fault in one or more actuators is presented. The proposed approach for the controller design is shown to be capable of recovering from multiple faults in different actuators. Using this approach there is no need to have a priori knowledge about the actuator’s dynamic parameters and the severity of the occurred faults in the actuators. The remainder of this paper is organized as follows. Section II presents the quadrotor model including the vehicle and the actuator dynamics. Section III discusses the loss-ofeffectiveness fault modeling. An adaptive feedback lineariza- 2515 of inertia around the propeller axis. Furthermore, U1 denotes the normalized total lift force, and U2 , U3 and U4 correspond to the control inputs of the roll, the pitch and the yaw moments, respectively. ΩT refers to the overall residual propeller angular speed. These input moments are defined according to the equations U = LUT T Fig. 1. Simplified quadrotor model representation at hovering and the coordinate systems (the body and the earth reference frames). ΩT = −Ω1 + Ω2 − Ω3 + Ω4 (2b) T is the movement vector where U = U1 U2 U3 U4  T is the thrust vector. The conand T = T1 T2 T3 T4  stant matrix LUT is defined according to tion recovery control strategy is presented in Section IV. Simulation results are presented in Section V followed by concluding remarks in Section VI. II. T HE Q UADROTOR M ODEL The quadrotor consists of four dc motors on which propellers are mounted in a cross configuration. Each propeller is connected to the motor through reduction gears. All the propellers axes of rotation are fixed and parallel. The front and the rear propellers rotate counter-clockwise, while the left and the right ones turn clockwise. In Figure 1, the schematic of a simplified quadrotor structure is shown where Ωi (rads−1 ) refers to the propellers rotational speed. While at hovering, all the four propellers rotate at the same speed to counterbalance the acceleration due to gravity. Basic movements are achieved by the differences between the propellers speeds. Vertical rotation is achieved by creating an angular speed difference between the two pairs of rotors. Increasing or decreasing the speed of the four propellers simultaneously permits climbing and descending. Rotation about the longitudinal and the lateral axis and consequently horizontal motions are achieved by tilting the vehicle. This is possible by conversely changing the propeller speed of one pair of rotors. Below, we present the vehicle and actuators dynamical models. 1) Vehicle Dynamic Model: Reference [15] provides a mathematical model of the quadrotor that is derived from the Newton-Euler formulation, that is   ẍ = (cos φ sin θ cos ψ + sin φ sin ψ) Um1      ÿ = (cos φ sin θ sin ψ − sin φ cos ψ) Um1    z̈ = −g + (cos φ cos θ ) U1 m (1) Jt p I −I θ̇ ΩT + UIxx2 φ̈ = θ̇ φ̇ ( yyIxx zz ) − Ixx    θ̈ = φ̇ ψ̇( Izz −Ixx ) + Jt p φ̇ ΩT + U3   Iyy Iyy Iyy   ψ̈ = φ̇ θ̇ ( Ixx −Iyy ) + U4 Izz  T Izz where x y z represents the position of the quadrotor  T in the inertial frame, φ θ ψ represents the roll, the pitch and the yaw angles, respectively, m (Kg) is the overall mass and g refers to the gravity acceleration. Ixx , Iyy and Izz (N m s2 ) denote the inertia moments in the body fixed frame and Jt p (N m s2 ) denotes the total rotational moment (2a) LUT 1  0 =  −l − db  1 −l 0 d b 1 0 l − db  1 l  0  (3) d b where l(m) is the distance between the center of the quadrotor and the center of a propeller, b and d denote the thrust and the drag coefficients, and Ti is the thrust force generated by each rotor and is proportional to the square of each propellers’s speed, that is Ti = bΩ2i . 2) Actuator Dynamic Model: The rotors are driven by DC motors. The details on the nonlinear and linearized dynamic equations of the propeller angular speed Ωi (rad s−1 ) and the thrust Ti around an operating point T0 are provided in [14] and is summarized below in equation (4) Ṫi = −At Ti + Bt ui +Ct (4) In the above equation, the parameters At , Bt and Ct are the linearized coefficients and are functions of the rotor dynamics [14].The set point corresponding to the linearization is determined from the fact that at hovering the total thrust should be equal to the gravitational force effective on the quadrotor. In other words, we have ∑4i=1 Ti = mg. Using the linearized dynamic equation (4), it is possible to obtain the dynamic equations from the input voltage to the propellers to the movement moments. Towards this end, it is useful to write the dynamic equation (4) in  a matrix form  for the thrust vector T = T1 T2 T3 T4 T , as follows Ṫ = −AT T + BT u +CT (5) where AT = At I(4×4) and BT = Bt I(4×4) are constant matrices,  T CT = Ct 1 1 1 1 , I(4×4) denotes a 4 × 4 identity   matrix, and u = u1 u2 u3 u4 T is defined as the vector of the input voltages to the propellers. Pre-multiplying the transfer matrix given by equation (3) with (5) yields LUT Ṫ = −(LUT AT )T + (LUT BT )u + (LUT CT ) (6) It should be noted that LUT AT = LUT (At I(4×4) ) = At (I(4×4) )LTU = AT LUT , and LUT BT = BT LUT . From equation (2a), it is possible to rewrite equation (6) as U̇ = −AT U + (LUT BT )u + (LUT CT ) (7) Under the condition that the quadrotor motion can be assumed to be close to the hovering condition, small angular changes occur (especially for the roll and the pitch angles). Since the rates of change in θ and φ are small, the terms due to the gyroscopic effects appearing in the dynamic equations 2516 of φ̈ and θ̈ in (1) are also negligible and can be set to zero (note that these assumptions have also been verified and validated through simulation results, but are omitted due to space limitations. Further details can be obtained from [16]). Moreover, since the structure of the quadrotor is symmetric, the body moments of inertia Ixx and Iyy are equal. This also simplifies the dynamic equation of ψ̈ in (1). If the altitude z reaches a desired set-point zd , that is z −→ zd , then z̈ −→ 0. As stated earlier, in the hovering condition (sufficiently small θ and φ ) the total thrust should be equal to the gravitational force effective on the quadrotor, in other words U1 = ∑4i=1 Ti = mg. Therefore, if z̈ −→ 0 and φ and θ are sufficiently close to zero, then U1 −→ mg. By assuming U1 −→ mg and ψ −→ 0, it is possible to simplify the dynamic equations of x and y states in the model (1). Towards this end, the dynamic equations of the quadrotor system including the dynamic equations for the movement vector is now given as follows   z̈ = −g + (cos φ cos θ ) Um1      ÿ = −g sin φ ẍ = g cos φ sin θ (8)  U3 U2 U4   φ̈ = θ̈ = ψ̈ = ; ;  Ixx Iyy Izz    U̇ = −AT U + (LUT BT )u + (LUT CT ) Our objective is now to design appropriate controls for the actuators. Since there are only four propellers, no more than four variables can be controlled in the loop. It is possible to define the position of the quadrotor in space completely by the linear position ΓE = x y z T and the yaw angle (heading angle) ψ. In this work, these four state variables are indeed selected for the purpose of control design. Ṫ = AT 0 T + BT 0 u +CT with AT 0 and BT 0 properly defined and dependent on ki . If the thrust dynamics for all the actuators are not identical, the dynamic equations of the movement vector U change since AT 6= At I and BT 6= Bt I. Therefore, it is necessary to derive the dynamic equations of the movement vector while the actuators do not have the same characteristics, in other words when Ati 6= At j and Bti 6= Bt j for i, j = 1, . . . , 4, i 6= j. The relationship between the movement vector U and T was defined earlier according to equation (2a). By premultiplying equation (10) with LUT and using equation (2a) we get −1 U̇ = −(LUT AT 0 LUT )U + (LUT BT 0 )u + (LUT CT ) Ṫi = Ati Ti + Bti ui +Ct i = 1, . . . , 4 (9) where Ati = ki2 At and Bti = ki2 Bt . Note that we have assumed that the only coefficients that are subject to change due to a fault are the Ati and Bti and the Ct remains unaffected. The term Ct is proportional to the drag as well as the inverse square of the thrust factor, which makes it a relatively small constant. Equation (9) can now be represented in a compact matrix form as (11) If the actuators have the same parameters, Ati = At and −1 Bti = Bt for i = 1, . . . , 4, then LUT AT 0 LUT = AT = At I4×4 as expected for the healthy system. IV. G ENERALIZED A DAPTIVE F EEDBACK L INEARIZATION R ECOVERY C ONTROL S TRATEGY In Section III, the movement vector dynamics were derived in equation (11). It specifies the contribution of each actuator to the resulting movement vector. As discussed earlier, in case of a LOE fault the parameters of the actuators also change. A parameter estimation algorithm is now presented in this section to provide an estimate of the faulty actuator severity gain and to develop a nonlinear adaptive controller to guarantee stability and recovery of the closed-loop system. Considering equation (11), the following equations are derived for designing the feedback linearization controller III. L OSS OF E FFECTIVENESS (LOE) FAULT M ODELING The focus of this work is on developing a recovery control mechanism from the loss of effectiveness (LOE) faults in the actuators of the quadrotor. The LOE fault is characterized by a decrease in the actuator gain from its nominal value. In case of a LOE fault, the speed of the quadrotor deviates from the commanded output that is desired by the controller. In other words, we instead have Ωi = ki Ωci for 0 < ε < ki < 1, where Ωi refers to the actual output from the ith actuator and Ωci is the commanded output by the controller and ki represents the LOE fault gain. Therefore, the resulting thrust force from this actuator changes according to the equation Ti = bΩ2i = b(ki Ωci )2 . The dynamics of Ti as governed by equation (4) would also change due to the LOE fault, that is Ṫi = 2bki2 Ωci Ω̇ci = ki2 At + ki2 Bt ui + ki2Ct , or equivalently (10) U̇1 U1 U1 − θ̇ cos φ sin θ + cos φ cos θ m m m (12) U̇2 U2 cos φ + g φ̇ sin φ + 2gφ̇ φ̈ sin φ + gφ̇ 3 cos φ Ixx Ixx (13) z(3) = −φ̇ sin φ cos θ y(5) = −g x(5) = g U̇2 U̇3 (cos θ cos φ ) + g (− sin θ sin φ ) + f (.) Iyy Ixx ψ (3) = U˙4 Izz (14) (15) where f (.) is defined as a function of the system states and parameters, that is g U3 U2 f (.) = (− + )(3(θ̇ − φ̇ ) sin (θ − φ ) + 3(θ̇ + φ̇ ) sin (θ + φ )) 2 Iyy Ixx g g − (θ̇ − φ̇ )3 cos (θ − φ ) − (θ̇ + φ̇ )3 sin (θ + φ ) 2 2 (16) Note that the relative degree of the system is equal to the order of the system and no internal dynamics exists in designing the feedback linearization controller. In our previous work [14], a fault recovery controller from a LOE fault in a single actuator was developed. In this work, we assume that the thrust dynamic parameters corresponding to all the actuators are unknown and to be estimated and that might have changed due to partial LOE faults. Therefore, 2517 Ṫi = −At Ti + Bt ui +Ct , where Ati = ki2 At and Bti = ki2 Bt for i = 1, . . . , 4 . It is possible to rewrite equation (11) to separate the unknown parameters, namely  U̇1   U̇2    U̇3 U̇4  1 b 1 4 U1 − 2 U3 − 4d U4     0  = −At1    l 1 lb  − 4 U1 + 2 U3 + 4d U4  d d − 4b U1 + 2lb U3 + 14 U4     − At3    1 1 b 4 U1 + 2 U3 − 4d U4 0 1 lb l 4 U1 + 2 U3 − 4d U4 d d − 4b U1 − 2lb U3 + 14 U4  Bt1   0 +   −lBt1 − db Bt1  1 1 b 4 U1 − 2l U2 + 4d U4   1 lb l    − At2  − 4 U1 + 2 U2 − 4d U4   0   d 1 d 4b U1 − 2lb U2 + 4 U4         − At4      Bt2 Bt3 Bt4 −lBt2 0 Bt4 0 lBt3 0 d b Bt2 − db Bt3 d b Bt4 0 d d 1 4b U1 + 2lb U2 + 4 U4 u1    u2     u3 u4    + (LUT CT )   (17) Inserting equation (17) into equations (12) to (15) and rewriting it in a matrix form we get  z(3)  y(5)     x(5)  = H0 − At1 H1 − At2 H2 − At3 H3 − At4 H4 ψ (3)    Bt1 0 0 0 u1 Bt2 0 0   u2   0 + H5  0 0 B 0  u   0 0 Bt4   z(3)  y(5)      x(5)  =  ψ (3) u4  −φ̇ sin φ cos θ Um1 − θ̇ cos φ sin θ Um1 + 4Ct   3 2 3g U  Ixx φ̇ sin φ + gφ̇ cos φ H0 =    f (.) 0    cos φ cos θ  1 1 b U U − U − 4 1 3 m 4 2 4d   0       H1 =  g cos θ cos φ l 1 lb − 4 U1 + 2 U3 + 4d U4  Iyy    d d 1 1 − U U + U + Izz 4 4 4b 1 2lb 3      H2 =       cos φ cos θ 1 1 b m  4 U1 − 2l U2 + 4d U4 −g cos φ lb U4 − 4l U1 + 21 U2 − 4d Ixx   −g sin θ sin φ l 1 lb U − U + U − 4 1 2 Ixx  4 2 4d  d 1 1 d U − U + U 1 2 4 Izz 4b 4 2lb    H3 =        H4 =     cos φ cos θ m  1 1 b 4 U1 + 2 U3 − 4d U4  0   w1 4 w2  4 + ∑ (δAti Hi ) + ∑ (δBti Gi )  w3 i=1 i=1 w4
(22) The control inputs w1 , w2 , w3 and w4 are defined according to the following equations (3) (19b) w1 = zd − k1z (ëz ) − k2z (ėz ) − k3z (ez ) (5) (4) (3) (5) (4) (3) (23a) w2 = yd −k1y (ey )−k2y (ey )−k3y (ëy )−k4y (ė)y −k5y (ey ) (23b) w3 = xd −k1x (ex )−k2x (ex )−k3x (ë)x −k4x (ė)x −k5x (ex ) (23c) (19c) (19d) (23d) where zd , yd , xd and ψd denote the desired output trajectories and ez = z − zd , ey = y − yd and ex = x − xd and eψ = ψ − ψd are defined as the tracking error signals. The gains kiz , k jy , k jx and kiψ for i = 1, 2, 3 and j = 1, · · · , 4 are obtained from the LQR design procedure. Using equations (21) and (23a) to (23d), we can write the error dynamic equations as follows           (21) ti W̄ = W − H0 − Ât1 H1 − Ât2 H2 − Ât3 H3 − Ât4 H4 (3)       cos φ cos θ 1 b U4 U1 + 2l1 U2 + 4d m 4  −g cos φ l 1 lb U U + U + 4 1 2 Ixx 4 2 4d   −g sin θ sin φ l 1 lb U + U + U 1 2 4 Ixx 4 2 4d d d 1 1 U U + U + 1 2 Izz 4b 4 4 2lb (19f) w4 = ψd − k1ψ (ëψ ) − k2ψ (ėψ ) − k3ψ (eψ )            where Gi = H5 Ḡi H5−1 W̄ and Ḡi is a 4 × 4 zero matrix except for the (i, i) element which is equal to B̂1 for i = 1, · · · , 4 and    g cos θ cos φ l 1 lb 4 Iyy 4 U1 + 2 U3 − 4d U   1 d d 1 Izz − 4b U1 − 2lb U3 + 4 U4 (19a) 0 g cos θ cos φ l Iyy 1 d ) (− Izz b  where Âti and B̂ti denote the estimates of the unknown parameters Ati and Bti for i = 1, . . . , 4 and T the new control input vector W = w1 w2 w3 w4 is to be selected so that the stability and control (LQR based) of the feedback linearized system is achieved. Let the parameter estimation error δ be defined as the difference between the actual value of the unknown parameter and its estimate, i.e. δAti = Ati − Âti and δBti = Bti − B̂ti for i = 1, . . . , 4 . By applying the control law (20) to the system (18), the closed-loop dynamics can be written as  where cos φ cos θ m φ − g cos Ixx l g sin θ sin φ − l Ixx 1 d ) ( Izz b −1    B̂t1 0 0 0 u1  0  B̂t2 0 0   u2      u = H5  0 0 B̂t3 0  3 u4 0 0 0 B̂t4    w1  w2    w  − H0 − Ât1 H1 − Ât2 H2 − Ât3 H3 − Ât4 H4  3 w4 (20) 3 t3 0 (18) cos φ cos θ m        cos φ cos θ m g cos φ Ixx l g sin θ sin φ l Ixx 1 d ) ( Izz b The input signal u is selected and designed according to (20) so that the closed-loop system becomes linear, namely  1 b 1 4 U1 + 2l U2 + 4d U4 1 lb l 4 U1 + 2 U2 + 4d U4   0  H5 =  g cos θ cos φ l  − Iyy d 1 ) (− Izz b       cos φ cos θ m  (19e) (3) ez + k1z ëz + k2z ėz + k3z ez  (5) (3)  ey + k1y e(4) y + k2y ey + k3y ëy + k4y ėy + k5y ey  (5) (4) (3)  e + k e + k e + k ë + k ė + k e 1x x 2x x 3x x 4x x  x 5x x (3) eψ + k1ψ ëψ + k2ψ ėψ + k3ψ eψ 4 4    =   ∑ (δAti Hi ) + ∑ (δBti Gi ) i=1 2518 i=1 (24) It is possible to represent the above equation in the statespace form. The selected state variables are as follows h iT (3) (3) (4) (4) X = ey ex ėy ėx ez ëy ëx eψ ėz ey ex ėψ ëz ey ex ëψ (25) Therefore, one could express the state-space representation of the closed-loop system according to 4

4 Ẋ = A16×16 X + ∑ δAti Hi′ + ∑ δBti G′i i=1 (26) i=1 where the vectors Hi′ and G′i are defined as Hi′ = T  T  0 . . . 0 HiT 16×1 and G′i = 0 . . . 0 GTi 16×1 . All the eigenvalues of A are negative by proper selection of ki . In other words, A is a Hurwitz matrix. Now, let us define the update laws for the parameter estimation errors δAti and δBti according to: (
δ̇Ati = − Hi′T PX + X T PHi′
T ′ δ̇Bti = − G′T i PX + X PGi (27) where X is the state vector defined in equation (25), and P is a (16 × 16) matrix that is obtained by solving the Lyapunov equation AT P+PA = −I16×16 where I16×16 denotes a 16×16 identity matrix. It should be noted that since A is a Hurwitz matrix, P is a positive definite matrix [17]. The following theorem provides a sufficient condition for stability of the closed-loop system (26) and (27). Theorem: The state trajectories of the closed-loop system (26) with the update laws for the estimation errors δAti and δBti for i = 1, . . . 4 given by equation (27) are globally stable in the sense of Lyapunov. Proof: To carry out the stability analysis, let us choose the Lyapanov function candidate as V = (X T PX + 1 4 1 4 2 δAti + ∑ δB2ti ) ∑ 2 i=1 2 i=1 (28) The derivative of this Lyapunov function along the trajectories of the system (26) is given by 4 4 i=1 i=1 V̇ = Ẋ T PX + X T PẊ + ∑ δAti δ̇Ati + ∑ δBti δ̇Bti (29) Using equation (27) one arrives at V̇ = −X T X, which guarantees the negative semi-definiteness of V̇ . This implies that the origin is a globally stable equilibrium point of the system (26) and (27). This completes the proof of the theorem.  This theorem implies that the control of the system no longer depends on the prior knowledge of the actuator thrust dynamic parameters. These parameters are estimated from the input and state measurements of the system. Therefore, if a LOE fault occurs in one or more of the actuators, the parameter estimation module is capable of estimating the post fault model of the actuators and in compensating for the resulting error effects in the system by proper commanding the healthy and faulty actuators and guaranteeing the closedloop stability of the system. V. S IMULATION R ESULTS In this section, the performance of the quadrotor system subject to our proposed fault recovery mechanism is studied through simulation scenarios. The quadrotor model used for simulation is the OS4 that is developed in the Ecole Polythechnique Federal De Lausanne [15]. The mathematical model used for control design is partially nonlinear (the actuator dynamics is linearized for control design), however, we have applied the controller to the fully nonlinear model of the quadrotor and have considered the full actuator dynamics. In simulations, additive white Gaussian noise is also added to the input and output channels to simulate a realistic environment. The noise power is selected so that 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 are initially set to zero. The commanded trajectory is to fly from the initial point to the final point (5, 5, 5)(m) in 10 seconds and hovering at the final point. The fault trajectory assumes occurrence of multiple LOE faults in different actuators. The first fault is a 20% LOE in the first actuator at t = 20 sec while the second fault is a 35% LOE in the fourth actuator at t = 35 sec. We are assuming that a fault detection mechanism exists [13] to alert the fault recovery module on the occurrence of the fault without much delay. In other words, the fault recovery mechanism is initiated after the detection of the fault. The performance of our fault recovery scheme is also evaluated in case of a delayed fault detection by initiating the recovery procedure 5 sec after the fault occurrence. Figures 2 and 3 show the linear position and the Euler angles in response to the commanded trajectory for the healthy, faulty and the recovered scenarios in case of multiple fault occurrences by using our proposed fault recovery algorithm with and without the delayed fault detection information. Table I shows the means and the standard deviations (Stdv) of the steady state tracking errors under four different scenarios that the system is operating under, namely, (I) healthy condition, (II) faulty actuator and no fault recovery invoked, (III) fault recovery mechanism invoked, and (IV) delayed fault recovery mechanism invoked. It can be concluded that our proposed fault recovery mechanism has a considerable effect on the performance of the system in presence of multiple LOE faults by reducing the steady state error between the desired and the actual outputs. It should be noted that as intuitively expected, the sooner the fault recovery is initiated, the more improved performance and smaller tracking error signals are obtained. VI. C ONCLUSION In this paper, a fault recovery mechanism is proposed for reconfiguring the control system from multiple loss-ofeffectiveness (LOE) faults in the quadrotor’s actuators. An adaptive feedback linearization technique is employed for the controller design and global stability of the system with the fault recovery mechanism is shown analytically. This is accomplished by developing a parameter estimation algorithm 2519 TABLE I M EAN AND S TDV OF THE TRACKING ERRORS FOR A 20% LOE IN THE FIRST AND A 35% LOE IN THE FOURTH ACTUATORS . (a) 10 8 x (m) 6 4 Faulty 2 Healthy 0 0 10 20 Recovered 30 40 Delayed recovery 50 time (sec) 60 70 80 90 100 90 100 (b) 15 Healthy Faulty Recovered y (m) 10 Delayed recovery 5 Faulty Healthy 0 0 10 20 30 Recovered 40 50 time (sec) 60 Delayed recovery 70 80 Error (I) (II) (III) (IV) ex Mean (m) ex Stdv (m) ey Mean (m) ey Stdv (m) ez Mean (m) ez Stdv (m) eψ Mean (rad) eψ Stdv (rad) -0.0537 0.2809 -0.0486 0.1632 -0.0542 0.1884 1.3461e-004 8.7942e-004 1.6942 2.9598 4.1785 5.1644 0.8837 -1.6082 0.0132 -0.0080 0.3598 0.1870 0.6126 0.3043 0.1939 -0.1190 0.0947 0.0566 1.3164 1.8977 3.2624 3.7113 0.3168 -0.5929 0.1135 -0.0752 (c) 6 z (m) 4 2 Healthy Recovered 0 0 10 20 30 40 50 time (sec) Delayed recovery Faulty 60 70 80 90 100 Fig. 2. Linear position due to the commanded trajectory and a 20% LOE fault in the first actuator at time = 20 sec and a 35% LOE fault in the fourth actuator at time = 35 sec with and without the fault recovery invoked. (a) 1 roll (rad) 0.5 Healthy Faulty Recovered Delayed recovery 0 −0.5 −1 0 10 20 30 40 50 time (sec) 60 70 80 90 100 60 70 80 90 100 70 80 (b) pitch (rad) 1 0.5 0 −0.5 0 10 20 30 40 50 time (sec) (c) 0.2 yaw (rad) 0.1 0 −0.1 Healthy Recevered −0.2 −0.3 Delayed recovery Faulty 0 10 20 30 40 50 time (sec) 60 90 100 Fig. 3. Euler angles due to the commanded trajectory and a 20% LOE fault in the first actuator at time = 20 sec and a 35% LOE fault in the fourth actuator at time = 35 sec with and without the fault recovery invoked. and by deriving proper update laws for the faulty actuator parameters. Simulation results are presented to evaluate the performance of our proposed fault recovery mechanism in presence of LOE fault in one or multiple actuators. 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