1st Aviation Engineering Innovations Conference AEIC-2015-000 21-22 March 2015, Luxor, EGYPT
ROBUST MODEL REFERENCE ADAPTIVE CONTROL OF A QUADROTOR UNMANNED AERIAL VEHICLE Ayman El-Badawy
1
Ramy Rashad
2
ABSTRACT Quadrotors are growing rapidly in popularity for use as platforms for robotics research worldwide. This paper describes the development of a robust model reference adaptive controller to a quadrotor unmanned aerial vehicle. Some signi cant characteristics of the system's dynamics are derived for system uncertainties in the case of partial actuator failures. A comparison between the adaptive controller and a linear controller is conducted using simulations. The adaptive controller is found to be more robust to parametric uncertainties. Robust modi cations are made to the adaptive laws to combat against instability phenomenons that occur in the adaptive controller.
INTRODUCTION The rapid advances in sensors, computing and other technology have increased the scope for commercial use of unmanned aerial vehicles. As a consequence, the eld of aerial robotics have evolved in the past decades. One particular rotating wing aircraft have been widely popular for use as a research platform all over the world; the quadrotor. Its high maneuverability and simple mechanical structure make it a suitable candidate for several indoor and outdoor applications when equipped with appropriate sensors and control algorithms. One part of publications related to quadrotor helicopters have focused on developing control algorithms. Typical linear control techniques, such as PID or LQ control posses some robustness properties to such uncertainties. However, these techniques might fail to stabilize the system in the case of severe uncertainties. Adaptive control techniques can be used to handle the uncertainties present in the system. Feedback linearization based approaches have been used by [1, 2] augmented by adaptive laws. Adaptive backstepping was also supplemented by adaptive laws in 1 Prof.
in Mechanical Engineering Department, Al-Azhar University and German University in
Cairo, Email:
[email protected] 2 Teaching Assistant in Mechanical Engineering Department, German University in Cairo, Email:
[email protected]
[3, 4]. To overcome the drawbacks of backstepping approaches, [5] implemented a command ltered backstepping technique with a linear tracking dierentiator to eliminate the time-scale separation assumption usually adopted. A Lyapunov-based adaptive control system was used by [6] to provide robustness against parametric uncertainties and external disturbances. Lee [7] augmented his previously proposed control system expressed on SO(3) with and adaptive law to estimate the quadrotor's inertia matrix. In addition, [8] implemented a MIMO control system based on state feedback output tracking design where adaptation accommodates against parameter uncertainties and varying operating points. Moreover, model reference adaptive control (MRAC) approaches have been used by [9, 10] to provide robustness against partial actuator failures. This paper focuses on the development of an adaptive control system based on the MRAC approach. The control system is tested in simulation only on the linearized quadrotor dynamics about the hovering con guration. The stabilization performance is compared to a linear control system in the case of a partial actuator failure. Moreover, the adaptive laws are modi ed to counteract instability phenomena such as parameter drift. The organization of the paper is as follows. The second section includes the nonlinear and linearized dynamical model of the system in addition to the eect of partial actuator failure on the system dynamics. The third section includes the controller design procedure for the linear and adaptive control systems and the robust modi cations applied to the adaptive laws. In the fourth section, the simulation results are presented and discussed and nally the paper's conclusion is given in section ve.
SYSTEM MODEL Dynamics The equations of motion of quadrotors have been derived and analyzed by several researchers in the literature [11, 12]. The quadrotor's dynamic equations of motion (EoM) are summarized in the following equations (sφ sψ + cφ sθ cψ ) Um1 x¨ y¨ = (−sφ cψ + cφ sθ sψ ) Um1 U1 (cφ cθ ) m − g z¨
(1)
1 p˙ [(I − I )qr − I q Ω + U ] z pz T 2 Ix y 1 q˙ = Iy [(Iz − Ix )pr + Ip z p ΩT + U3 ] , 1 r˙ [(I − I )pq + U ] x y 4 Iz
(2)
where x, y, z represent the quadrotor's Cartesian position, φ, θ, ψ are the roll, pitch and yaw angles respectively, p, q, r are the body angular rates, m, Ix , Iy , Iz are the 2
quadrotor's mass and moments of inertia respectively, Ip is the propeller's moment of inertia about its axis of rotation, ΩT is the algebraic sum of the four rotors, U1 , U2 , U3 , U4 represent the aerodynamic forces and moments generated by the propellers. Finally the terms c(·) and s(·) represent the cosine and sine functions respectively.
System Uncertainties The nonlinear model (1-2) describes most of the dominant dynamics during trivial ight regimes and neglect aerodynamic eects including blade apping, ground eect and induced drag. These neglected aerodynamics are treated as uncertainties in the system model in addition to severe uncertainties that may arise due to actuator failure or structural damage of the propeller blades. The eect of partial actuator failures on the system dynamics will be examined in this section. To model this eect we assume only the rst rotor is damaged without loss of generality. The eective rotor speeds are then related to the rotor speeds calculated by Ω2 1 2 Ω2 2 Ω3 Ω4 2
ef f
λ 0 = 0 0
0 0 0 Ω1 2 1 0 0 Ω2 2 0 1 0 Ω3 2 0 0 1 Ω4 2
(3)
,
calc
where 0 ≤ λ ≤ 1 is a constant representing the eectiveness of the rst rotor where λ = 1 indicates the rotor is undamaged. The relation between the eective forces acting on the quadrotor's rigid body and the calculated ones is given by U 1 U2 U3 U4
ef f
=
3+λ 4
0 − −1+λ 2l
b−bλ 4d
0
1
0
0
(l−lλ) 4
0
bl(−1+λ) 4d
d−dλ 4b
0
1+λ 2 d(−1+λ) 2bl
3+λ 4
U 1 U2 U3 U4
,
(4)
calc
The terms not belonging to the identity matrix, represent a coupling eect between the collective thrust, pitch and yaw axes. In general if any single rotor of was damaged there will be this coupling eect between the collective thrust, yaw and one of the roll or pitch axes depending on which rotor was damaged.
Linearized Dynamics In this section, the quadrotor's linearized EoM will be presented. The quadrotor's hovering con guration is used as the operating point about which the EoM are linearized. 3
Such con guration is de ned by p˜ = [˜ x, y˜, z˜] = [xo , yo , zo ] I ν˜ = [x˜˙ , y˜˙ , z˜˙ ] = [0, 0, 0] I
I
˜ θ, ˜ ψ] ˜ = [0, 0, ψo ] ˜ = [φ, Θ
B
.
(5)
ω ˜ = [˜ p, q˜, r˜] = [0, 0, 0]
In order to achieve such con guration, the total thrust produced by the rotors should be equal to the quadrotor's weight, while the net roll, pitch and yaw moments should ˜ is written as be equal to zero. The nominal control inputs vector U ˜ = [U˜1 , U˜2 , U˜3 , U˜4 ] = [mg, 0, 0, 0]. U
(6)
The translational EoM (1) after linearization about the hovering con guration will be written as x¨ = g(sψo φ + cψo θ) y¨ = g(sψo θ − cψo φ). 8bΩh ∆ΩT z¨ = m
(7)
While the rotational EoM (2) are linearized and written as φ˙ = p θ˙ = q ψ˙ = r
4blΩh ∆Ωφ Ix 4blΩh ∆Ωθ , q˙ = Iy 8dΩh r˙ = ∆Ωψ Iz p˙ =
(8)
where Ωh represents the hovering speed of the propellers, l, b, d are the distance from the rotors to the CoM, thrust and drag factors respectively, ∆ΩT , ∆Ωφ , ∆Ωθ , ∆Ωψ are the deviations in collective thrust, roll, pitch and yaw forces respectively which represent the new control inputs.
MODEL REFERENCE ADAPTIVE CONTROLLER DESIGN In MRAC, the desired system's behavior is described by a reference model which is simply an Linear time invariant (LTI) system driven by the reference input r(t). The control laws are designed such that the closed loop system from the reference input r(t) to the system's states xp (t) is equal to the reference model. In the case of unknown system parameters, the controller gains are directly estimated using adaptive laws such that the tracking error e = xm − xp converges to zero. 4
Problem Formulation The main function of the adaptive controller is to accommodate for any uncertainties present in the linearized system's dynamics (7-8). These linearized EoM can be written in state space form along with the uncertainties as x˙ p = Ap xp + Bp up ,
(9)
where Ap ∈ Rnp ×np is constant, fully populated and unknown which may contain unmodeled cross-coupling terms while Bp ∈ Rnp ×mp is constant and unknown and contains cross-coupling terms that may arise due to the loss of eectiveness of one of the motors as shown previously. The state vector xp contains xp = [x, x, ˙ y, y, ˙ z, z, ˙ φ, p, θ, q, ψ, r]> ,
(10)
and for simplicity we assume all the states are available for measurement. As for the input vector up it contains up = [∆ΩT , ∆Ωφ , ∆Ωθ , ∆Ωψ ]> .
(11)
The system output is de ned to be yp ∈ Rmp which contains x, y, z and ψ . The reference model is described by x˙ m = Am xm + Bm rp , (12) where Am ∈ Rnp ×np is a stable matrix and Bm ∈ Rnp ×mp . The reference model and reference signal r are designed such that xm represent the desired trajectory that xp has to track.
Linear Control System Design In the case of known parameters and no uncertainties, the control law up can be designed using classical design techniques to be up = −K ∗ xp + L∗ r,
(13)
where K ∗ ∈ Rmp ×np is the feedback gains matrix and L∗ ∈ Rmp ×mp is the feedforward gains matrix. The gain matrices can be chosen to satisfy the algebraic matching equations Am = Ap − Bp K ∗ Bm = Bp L∗ , (14) which will cause the closed loop system to be similar to the reference model and thus x(t) → xm (t) exponentially fast for any bounded signal r(t). In the matching equation, the nominal values of the Ap and Bp matrices are then used. In this work we assume a solution of the matching condition (14) exists, i.e., there is su cient structural exibility to meet the control objective. The control law in this paper is also implemented with an integral action on the x, y, z and yaw axis which involves extending the system dynamics to include the integrated output tracking error [13]. 5
Adaptive Control System Design As for the case where the system's parameters are unknown and uncertainties are present, we use the control law (15)
up = −K(t) xp + L(t) rp ,
where K(t), L(t) are the estimates of K ∗ , L∗ respectively which will be generated using an online parameter estimator. First we de ne the tracking error e to be (16)
e = xp − xm ,
which will be used by the gradient-based adaptive laws to estimate the required parameters such that e(t) → 0 as t → 0. The adaptive laws proposed [14] are given by T K˙ = Γ1 Bm P e xTp , T L˙ = −Γ2 Bm P e rpT
(17)
where Γ1 , Γ2 ∈ Rmp ×mp are diagonal, positive de nite matrices of adaptive gains, P = P > ∈ Rnp ×np is the unique solution to the Lyapunov equation (18)
ATm P + P Am = −Q,
where Q = QT is an arbitrary positive de nite matrix chosen in this work to be the identity matrix for simplicity. To account for constant exogenous disturbances that may be applied to the system dynamics, the control input will be modi ed to include an estimate of the disturbance ud (t). The modi ed control law and the corresponding adaptive law for the disturbance's estimate are given by up = −K(t) xp + L(t) rp + ud (t) T u˙ d = −Γ3 Bm P e
.
(19)
The stability of the presented adaptive controller is based on nonlinear theory which has been studied by many researchers throughout the previous decades. For the detailed proof of stability, the reader can refer to [14]
Robust Modi cations It is well known that adaptive control schemes might become unstable in the presence of noise, unmodeled dynamics and other types of uncertainties that are frequently encountered. Robust adaptive control deals with the modi cation of adaptive controllers to make them robust against these uncertainties. In this section we consider the development of robust adaptive laws for the estimation of the controller parameters. One of the instability phenomenons of adaptive laws is what is known as parameter drift, which is mainly due to the integral action of the adaptive laws. The adaptive laws are 6
driven by the tracking error signal e in addition to errors due to unmodeled dynamics. A simple way to eliminate such phenomenon is to adapt only when the tracking error is large relative to the modeling error by introducing a dead-zone. The adaptive laws are modi ed to include a function f (e) = e + g instead of the tracking error e, while g is given by g0 if e < −g0 , (20) g = −g0 if e > g0 −e if − g ≤ e ≤ g 0 0 where g0 is a design parameter that speci es the size of the dead-zone and is chosen in practice to be just greater than the noise present in the system.
SIMULATION RESULTS A simulation run has been carried out with an actuator loss of eectiveness introduced during the run. The results for an actuator eectiveness of 75% in the rst rotor are shown in gures (1-2) which display snapshots of the simulation run in addition to the position and attitude errors. The gures show that after the rst rotor was damaged at t = 8 sec, the control system was able to stabilize the quadrotor due to the integral action. However, the error of the quadrotor's con guration is large and impractical. The MRAC system designed in the previous sections has been simulated with the nonlinear system in the presence of uncertainties due to partial actuator failure. The same simulation run was repeated for the adaptive controller and the results are presented in gures (3-4). The simulation shows that the adaptive controller successfully stabilized the quadrotor after the damage of the rst rotor. The results show a maximum deviation in the quadrotor's position of only 1.5, 0.05 and 3 cm in the x, y and z axes respectively. While the maximum deviations in the quadrotor's attitude are about 0.05 and 6 degrees in the pitch and yaw angles.
CONCLUSION In this paper, a robust model reference adaptive controller was presented. Compared to a control system based on linear theory, the adaptive controller shows superior performance in stabilizing the quadrotor in the case of partial actuator failures. Future work would involve the experimental validation of the presented controllers. In addition, dierent adaptive laws can be compared such as ones based on least squares algorithms.
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(a) t = 0 sec
(b) t = 9 sec
(c) t = 11 sec
(d) t = 13 sec
(e) t = 15 sec
(f ) t = 20 sec
Figure 1: Results - 75 % Actuator Eectiveness Linear Control
Figure 2: Results - 75 % Actuator Eectiveness Position and Attitude Errors 9
(a) t = 0 sec
(b) t = 8 sec
(c) t = 9 sec
(d) t = 10 sec
(e) t = 11 sec
(f ) t = 12 sec
Figure 3: Results - 75 % Actuator Eectiveness Adaptive Control
Figure 4: Results - 75 % Actuator Eectiveness Position and Attitude Errors 10
