International Journal of Robotics and Automation, Vol. 30, No. 2, 2015
ADAPTIVE SLIDING MODE CONTROL OF UNMANNED FOUR ROTOR FLYING VEHICLE Shafiqul Islam,∗ Xiaoping P. Liu,∗∗ and Abdulmotaleb El Saddik∗∗∗
proportional-integral-derivative (PID) and linear quadratic regulator (LQR) control algorithms for quadrotor UAV. It is well known that the model-based controller cannot ensure robustness in the presence of uncertainties and disturbances. Authors in [2]–[5] used backstepping control techniques to deal with the problem of coupling in the pitch-yaw-roll and the problem of coupling in kinematics and dynamics of the underactuated flying vehicle. Later, the integral action was included with the backstepping technique in [6]. The idea of including PID term with classical backstepping design was to reduce the steady tracking errors while maintaining asymptotic stability of the whole closed-loop system. Authors in [7] proposed model-based dynamic inversion mechanism for hovering control for the quadrotor system. In [8], authors developed robust H∞ tracking controller by using backstepping controller to stabilize uncertain quadrotor UAV system. Authors in [9] proposed sliding mode design which can ensure the stability of the roll and pitch angles. Using visual feedback signal, classical control technique for quadrotor UAV system was presented in [10]. In [11]–[13], authors obtained perfect tracking accuracy of the quadrotor in indoor environment by using visual motion tracking system. However, the design can only be applied for a priori known tasks and indoor environment [14]. Adaptive backstepping control mechanism was proposed for quadrotor UAV system in the presence of model parameter uncertainty in [15], [16]. In our view, it can be seen from the existing designs that most reported results requires a priori known upper bound of the modeling error and disturbance uncertainty to establish stability of altitude and attitude dynamics in uncertain indoor and outdoor flying environment. In practice, it may be unrealistic to know the exact values of the uncertainty associated with the flying environment, payload mass, moment of inertia, aerodynamic friction and gyroscopic effect on the closed-loop systems. On the other hand, unpredictable changes in indoor and outdoor environment may also increase the modeling errors uncertainty significantly making the flight control system design even more complicated. Under these circumstances, available autonomous tracking system design may be unable to adapt with the change of the flight/plant dynamics during different flight mission. In this paper, we propose adaptive sliding mode control technique for stability and trajectory tracking control
Abstract This paper addresses the stability and tracking control problem of an underactuated four rotor unmanned flying robot vehicle. Algorithm design combines adaptive law with the sliding mode control term to deal with uncertainties associated with flying environment, mass, inertia, aerodynamic force and moment of the vehicle.
Using
Lyapunov analysis, we show that the position and orientation tracking errors and their derivatives are bounded by bounds that can be made close to the origin. Simulation examples on a quadrotor vehicle are given to demonstrate the effectiveness of theoretical development for real-world application.
Key Words Quadrotor, adaptive control, Lyapunov method
1. Introduction Quadrotors are unmanned aerial vehicles (UAVs) that have been setting the waves of the growing interest in the scientific and industrial communities because of their various applications, such as inspection and surveillance, search and rescue mission, first responders, police and military services. The interest in quadrotor UAVs has been recently pushing the limits for technology by sparking the new ideas and practical applications amongst the researchers and industrialists alike. The control system design for micro-scale quadrotor UAV is very difficult as the dynamics of the quadrotor UAV associated with inherent nonlinearity and underactuated property, nonlinear aerodynamical force and moment, strong nonlinear coupling between angular and linear dynamics and disturbances associated with the flying environments. Over the last decade, different types of autonomous tracking systems for quadrotor UAV have proposed in the literature to deal with the modeling errors and disturbance uncertainty. In [1], authors developed model-based ∗
University of Ottawa, Ottawa, Canada, and Carleton University, Ottawa, Canada; e-mail:
[email protected] ∗∗ Carleton University, Ottawa, Canada; e-mail: xpliu@sce. carleton.ca ∗∗∗ University of Ottawa, Ottawa, Canada; e-mail: elsaddik@ uottawa.ca Recommended by Prof. J. Gu (DOI: 10.2316/Journal.206.2015.2.206-3960)
140
problem of small size quadrotor flying vehicle in the presence of uncertainty. The overall design comprises adaptive term with the sliding mode control term. Adaptive control law is employed to learn and compensate uncertainties associated with mass, inertia matrix, external disturbances, and aerodynamic force and moment affecting the system. Algorithms for altitude, position and attitude tracking design are developed through Lyapunov-like energy functional. It is shown in our analysis that tracking errors of the position, orientation and their derivatives are bounded and asymptotically converge to zero. In contrast with the existing design, the proposed method does not rely on the upper bound of the modeling error and disturbance. The bound is obtained by using an adaptation law. To demonstrate the effectiveness of this theoretical arguments, evaluation results on a commercial quadrotor is presented. This evaluation shows that the design can be applied for quadrotor UAV with large parametric uncertainty associated with the payload mass, uncertain environment, moment of inertia, nonlinear aerodynamic friction and gyroscopic effect on the closed-loop systems. This paper is organized as follows. In Section 2, kinematics and dynamics model of the four rotor flying vehicle are given. Adaptive sliding mode control designs are introduced in Section 3. A detail stability analysis is also presented in Section 3. Simulation example is given in Section 4. Finally, conclusion and future work is given in Section 5.
skew-symmetric property. The skew-symmetric matrix S(Ωs ) can be defined as follows: ⎡ ⎢ ⎢ S(Ωs ) = ⎢ Ωs3 ⎣ −Ωs2
(2)
(6)
where m ∈ 1 and I ∈ 3×3 = diag[Ix , Iy , Iz ] denotes the mass and symmetric positive definite constant inertia matrix of the vehicle, respectively. Using the Euler angles, the angular velocity transformation matrix can be used to relate the rate of change of the angular velocities in the body fixed frame as: ⎛
⎞
⎛ 1
⎞⎛ ⎞ φ˙ ⎟⎜ ⎟ ⎟⎜ ⎟ cos θ sin φ ⎟ ⎜ θ˙ ⎟ ⎠⎝ ⎠ cos φ cos θ ϕ˙ − sin θ
0
⎜ ⎟ ⎜ ⎜ ⎟ ⎜ ⎜ Ωs2 ⎟ = ⎜ 0 cos φ ⎝ ⎠ ⎝ Ωs3 0 − sin φ
(7)
It is assumed that nonlinear aerodynamic drag forces Fd varies linearly with the velocities as: ⎞
⎛ Fdx
⎡
⎤ Δx
⎟ ⎜ ⎢ ⎟ ⎜ ⎢ ⎜ Fdy ⎟ = ⎢ 0 ⎠ ⎝ ⎣ Fdz 0
0 Δy 0
0
⎥ ⎥ 0 ⎥ ⎦ Δz
(8)
where Δx = τ1 |vs1 |, Δy = τ2 |vs2 |, Δz = τ3 |vs3 | with positive constants τi and i = 1, 2, 3. Now, the thrust force generated by four rotors can be derived as: ⎛ ⎞ 0 ⎜ ⎟ ⎜ ⎟ Ft = Rs · dΣ4i=1 ωi · ⎜ 0 ⎟ ⎝ ⎠ 1
where the transformation of vectors from rigid body frame to the inertial frame can be derived from the following homogenous translational rotational matrix Rs ∈ 3×3 ⎡
(4)
(5)
˙ s = − (Ωs × IΩs ) + ut + ug + ua IΩ
We first present the model dynamics of unmanned four rotor flying vehicle [1], [5]. To derive the motion dynamics of the UAV system, two main reference frames are considered as earth fixed inertial reference ξ and body fixed frame δ attached to the UAV system. The quadrotor has three translational positions with respect to ξ as defined as xs = [xs , ys , zs ]T ∈ 3 and three orientations with respect to δ represented by three Euler angles as defined as η = [φ, θ, ϕ]T . Then, we consider that the vehicle has three translational velocities as vs = [vs1 , vs2 , vs3 ]T and three rotational velocities as Ωs = [Ω1s , Ωs2 , Ωs3 ]T with respect to the body fixed frame. The kinematic model of the UAV can then be written as:
R˙ s = Rs S(Ωs )
Ωs1
⎥ ⎥ −Ωs1 ⎥ ⎦ 0
⎛ ⎞ 0 ⎜ ⎟ ⎜ ⎟ mv˙ s = −mg ⎜ 0 ⎟ + Ft + Fd ⎝ ⎠ 1
2. Dynamical Model
(1)
0
⎤ Ωs2
Applying Newton and Euler laws in the body-fixed reference frame, the dynamic equation for the quadrotor helicopter subjected to translational forces and control torques developed in the centre of the mass can be derived as:
Ωs1
x˙ s = vs
−Ωs3
0
(9)
with the trust factor d > 0 and ωi is the speed of the rotors with i = 1, 2, 3, 4. The torques developed in the epicenter of a quadrotor helicopter by the propellers can be defined as:
⎤
C C S S C − Cφ Sϕ Cφ Sθ Cϕ + Sφ Sϕ ⎥ ⎢ φ ϕ φ θ ϕ ⎥ ⎢ Rs = ⎢ Cθ Sϕ Sφ Sθ Sϕ + Cφ Cϕ Cφ Sθ Cϕ − Sφ Sϕ ⎥ (3) ⎦ ⎣ −Sφ Sφ Cθ Cφ Cθ
⎡ ⎢ ⎢ ut = ⎢ ⎢ ⎣
where Cθ and Sθ denotes cos θ and sin θ and S(Ωs ) denotes the skew-symmetric matrix S(Ωs ) which satisfies 141
dl(ω32 − ω12 ) dl(ω42
−
ω22 )
αr (ω12 − ω22 + ω32 − ω42 )
⎤ ⎥ ⎥ ⎥ ⎥ ⎦
(10)
where l is the distance between the centre of the mass and the rotor axes and αr is the drag factor for the rotation. The nonlinear aerodynamic torques ua are assumed to be varying with angular velocity of flying vehicles as follows: ⎛ ⎞ ⎡ ⎤ uax 0 Πx 0 ⎜ ⎟ ⎢ ⎥ ⎜ ⎟ ⎢ ⎥ (11) ⎜ uay ⎟ = ⎢ 0 Πy 0 ⎥ ⎝ ⎠ ⎣ ⎦ uaz 0 0 Πy
3. Algorithm Design and Stability Analysis In this section, we design sliding mode control system for autonomous tracking of four rotor flying vehicle. We assumed that the vehicle model parameters, such as flying environment, mass, inertia, damping and moment, are uncertain causing modeling errors uncertainty. In our design, we combine sliding mode control algorithm with adaptive control terms for altitude, attitude and position input such that the position and rotation angle of the flying vehicles track a time varying reference angle and position trajectory in the presence of external disturbance and uncertain model dynamics. Let us first design the altitude controller to generate the desired lifting force in order to maintain the desired distance between the ground and the vehicle. We consider that altitude between the ground and the vehicle are available for measurement. Then, we define the altitude tracking errors as:
where Πx = φ1 Ωs1 , Πy = φ2 Ωs2 , πz = φ3 Ωs3 with the positive constants of aerodynamic coefficients φi and i = 1, 2, 3. Finally, the gyroscopic torques are generated by the rotors as they move along the rotor mast with the body-fixed frame defined as: ⎛ ⎛ ⎞⎞ 0 ⎜ ⎜ ⎟⎟ ⎜ ⎜ ⎟⎟ ug = Ir ⎜Ωs × ⎜ 0 ⎟⎟ · (ω1 − ω2 + ω3 − ω4 ) (12) ⎝ ⎝ ⎠⎠ 1
ez = (zd − z)
(16)
where Ir is the inertia of the rotor blade. In flying robot vehicle, the rotational velocities of the four rotors ωi are usually used as an input variable for designing an autonomous system. Therefore, we consider new input variables for four rotors given as follows:
where zd is the altitude reference. Then, we define the auxiliary tracking error signals by combining ez and e˙ z as follows:
u1 = d(ω12 + ω22 + ω32 + ω42 ), u2 = d(ω32 − ω12 ),
˙ and αz 0. The altitude error dynamics with e˙ z = (z˙d − z) has the following form:
Sz = (e˙ z + αz ez )
(13)
u3 = d(ω42 − ω22 ), u4 = αr (ω12 + ω32 − ω22 − ω42 )
e¨z = z¨d −
Using (7)–(13), one can derive overall dynamical model of the quadrotor vehicle as follows: x ¨ =
Δax 1 (cos φ sin θ cos ϕ + sin ϕ sin φ) u1 + , βa βa
y¨ =
Δay 1 (cos φ sin θ sin ϕ − sin φ cos ϕ) u1 + βa βa
z¨ =
Δaz 1 (cos φ cos θ) u1 − g + βa βa
Δaz 1 (cos φ cos θ)u1 − g + βa βa
(17)
(18)
We then introduce the following altitude control algorithm with the presence of the uncertain model dynamics: βa (¨ zd + ueqm + τm + αz1 e˙ z + ks Sz ) cos φ cos θ Δaz , τm = −kˆz sign(Sz ), = g− βa
u1 = ueqm
(14)
˙ k˜z = Γz signT(Sz )Sz
with βa = m, Δax = −Δx , Δay = −Δy , Δaz = −Δz and the orientation dynamics has the following form
where ks > 0, Γz > 0, αz1 0 and k˜z = kz − kˆz . The control term τm is used to compensate uncertainties appearing from external disturbances and modeling errors. We now choose the following Lyapunov function:
Πax ˙ + pb1 u2 − pb2 F θ˙ + , φ¨ = pb (θ˙ϕ) Ix Πay θ¨ = pc (φ˙ ϕ) ˙ + pc1 u3 + pc2 F φ˙ + , Iy
(19)
(15) Vz =
˙ + pd1 u4 + Πaz ϕ¨ = pd (θ˙φ) Iz
1 T 1 Sz Sz + Γ−1 k˜z k˜z 2 2 z
(20)
We then take the derivative (20) along the closed-loop system constructed by (16)–(19). Then, V˙ becomes:
with Πax = −Πx , Πay = −Πy , Πaz = −Πz , Pb = pIx1 , pc = pIy2 , pd = pIz3 , p1 = (Iy − Iz ), p2 = (Iz − Ix ), p3 = (Ix − Iy ), pb1 = Ilx , pc1 = Ily , pd1 = I1z , pb2 = IIxr , pc2 = IIyr and F = (ω1 − ω2 + ω3 − ω4 ).
V˙ z ≤ −ks SzT Sz ≤ 0
(21)
This implies that Vz ∈ L2 and Vz ∈ L∞ ensuring that the signals ez , e˙ z and Sz are bounded. Then, using Barbalat’s 142
Lemma, we can conclude that the signals ez , e˙ z and Sz converges to zero as the time goes to infinity. We now design the position algorithm to generate the desired horizontal motion for the given time varying desired trajectories xd and yd . The horizontal motion of the vehicle is usually obtained by rolling or pitching via rotating the thrust values to the given desired motion. To design a position algorithm to keep the vehicle over the desired point, we derive the position tracking errors model: e¨x = x ¨d −
Δax 1 τ x u1 + βa βa
(22)
e¨y = y¨d −
1 Δay τ y u1 + βa βa
(23)
and their derivatives are available from inertial measurement unit. The tracking errors for the given desired rolling angles can be defined as: Πax ¨ ˙ ˙ (30) e¨φ = φd − pb (θϕ) ˙ + pb1 u2 − pb2 F θ + Ix where φd is the reference rolling angle. By knowing the values of ϕd , τx and τy , φd can be calculated from the relationship φd = arc sin(τx sin(ϕd ) − τy cos(ϕd )). Then, we introduce the following control torque for generating desired rolling moment as:
where τx and τy are the virtual input and xd and yd are the reference position trajectories in x and y direction. Let us define auxiliary error signals Mx and Ny as sliding surface as: Mx = (e˙ x + αx ex )
(24)
Ny = (e˙ y + αy ey )
(25)
veqv
˙ τφ = −kˆφ sign(Bφ ), k˜φ = Γφ signT (Bφ ) Bφ
(26)
βa (¨ yd + αy1 e˙ y + τyn + ky Ny ) u1 ˙ = −kˆyn sign(Ny ) , k˜yn = Γy signT (Ny ) Ny
(27)
Vφ =
τy = τyn
Πay e¨θ = θ¨d − pc (φ˙ ϕ) ˙ + pc1 u3 + pc2 F φ˙ + Iy
where kx > 0, ky > 0, Γx > 0, u1 > 0, αx1 0, αy1 0, k˜xm = (kxm − kˆxm ), k˜yn = (kyn − kˆyn ) and Γy 0. The closed-loop stability of the longitudinal and lateral motion dynamics can be shown by using the following Lyapunovlike energy functional: Vx =
1 T 1 Mx Mx + Γ−1 k˜xm k˜xm 2 2 x
(28)
Vy =
1 T 1 N Ny + Γ−1 k˜yn k˜yn 2 y 2 y
(29)
1 T 1 B Bφ + Γ−1 k˜φ k˜φ 2 φ 2 φ
(32)
Taking the time derivative of (32) along the trajectories formulated by (30) and (31), we can obtain the bound on Vφ as follows V˙ φ ≤ −kφ1 BφT Bφ ≤ 0. Then, using Barbalat’s Lemma, we can state that the tracking errors of the rolling angles, rolling speeds and Bφ are bounded and their bounds converges to zero as the time goes to infinity. We now design algorithm to generate the desired pitching moment to track the desired pitching angle θd . The tracking errors of the pitching angles can be written as:
τx = τxm
(31)
where Γφ > 0, kφ1 > 0, αφ1 0, k˜φ = (kφ − kˆφ ) and Bφ = (e˙ φ + αφ eφ ). For the closed-loop stability analysis under the proposed rolling moment (31), we consider the following Lyapunov energy function:
where αx 0, αy 0, ex = (xd − x) and ey = (yd − y). We then design the following algorithms for generating motion in x and y direction corresponding to the given reference motion in xd and yd : βa (¨ xd + αx1 e˙ x + τxm + kx Mx ) u1 ˙ = −kˆxm sign(Mx ) , k˜xm = Γx signT (Mx ) Mx
1 ¨ φd + veqv + τφ + αφ1 e˙ φ + kφ1 Bφ , pb1 Πax = −pb (θ˙ϕ) ˙ + pb2 F θ˙ − Ix
u2 =
(33)
Using ϕd , φd , τx and τy , θd can be obtained by using τ sin(ϕd ) + τy cos(ϕd ) the relationship θd = arc sin x . Then, cos(φd ) we introduce the following pitching moment for generating desired pitching motion trajectory: u3 =
1 ¨ θd + weqv + τθ + αθ1 e˙ θ + kθ1 Cθ , pc1
Πay weqv = −pc (φ˙ ϕ) ˙ − pc2 F φ˙ − Iy
Taking the derivative Vx and Vy along the closed-loop trajectory designed by using (22)–(27), one can show that the errors ex , ey , Mx and Ny are bounded and their bounds converges to zero in the Lyapunov sense as V˙ x ≤ −kx MTx Mx ≤ 0, V˙ y ≤ −ky NyT Ny ≤ 0 with Vx ∈ L2 , Vy ∈ L2 and Vx ∈ L∞ , Vy ∈ L∞ . Let us now focus our attention on algorithm design for attitude dynamics of the flying vehicle. The main objective of this algorithm is to generate desired control torques for roll, pitch and yaw orientation in the presence of external disturbance and modeling error uncertainties. It is assumed that the orientation angles
˙ τθ = −kˆθ sign(Cθ ), k˜θ = Γθ signT (Cφ )Cφ
(34)
where kθ1 0, Γθ > 0, αθ1 0, k˜θ = (kθ − kˆθ ) and Cθ = (e˙ θ + αθ eφ ). Using the following Lyapunov-like energy function: Vθ = 143
1 T 1 Cθ Cθ + Γ−1 k˜θ k˜θ 2 2 θ
(35)
Figure 1. The time history of uncertainty along x, y and z in metres, φ, θ and ϕ in radians.
Figure 2. The time history of position tracking x and xd in metres.
Figure 3. The time history of position tracking in y and yd in metres. and Barbalat’s Lemma, we can show that all the signals in (33) under pitching moment (34) asymptotically converges to zero as V˙ θ ≤ −kθ1 CθT Cθ ≤ 0 with Vθ ∈ L2 and Vθ ∈ L∞ . Finally, we design the following error dynamics for the yaw moment: Πaz ˙ ˙ (36) e¨ϕ = ϕ¨d − pd (θφ) + pd1 u4 + Iz
with kϕ1 0, Γϕ > 0, αϕ1 0, k˜ϕ = (kϕ − kˆϕ ) and Dϕ = (e˙ ϕ + αϕ eϕ ). To guarantee the closed-loop stability with the yaw moment provided by input (37), we consider the following positive definite Lyapunov function:
The desired yaw moment is generated by the following control torque:
Take the time derivative (38) along the closed-loop system formulated by tracking error model (36) and control law (37), we can obtain the time derivative of Vϕ as V˙ ϕ ≤ −kϕ1 DϕT Dϕ ≤ 0 with Vϕ ∈ L2 and Vϕ ∈ L∞ . In view of above equation and Barbalat’s Lemma, we can conclude that the pitching error signals eϕ , e˙ ϕ and Dϕ are bounded and their bounds asymptotically converges to zero in Lyapunov sense. Based on our above Lyapunov analysis, we can state our main results in Theorem 1.
Vϕ =
1 [ϕ¨d + χeqv + αϕ1 e˙ ϕ + τϕ + kϕ1 Dϕ ] , pd1 ˙ − Πaz = −pd (θ˙φ) Iz
u4 = χeqv
˙ τϕ = −kˆϕ sign(Dϕ ), k˜ϕ = Γϕ signT (Dφ )Dφ
(37) 144
1 T 1 D Dϕ + Γ−1 k˜ϕ k˜ϕ 2 ϕ 2 ϕ
(38)
Figure 4. The time history of position tracking in z and zd in metres.
Figure 5. The time history of yaw tracking in ϕ and ϕd in metres.
Figure 6. The time history of position tracking x and xd in metres under large model parameters. Theorem 1. Let us assume that the linear and angular velocities of the quadrotor flying vehicle are bounded. Then, all the error signals in the closed-loop systems formulated by error models (18), (22), (23), (30), (33) and (36) under input algorithms (19), (26), (27), (31), (34) and (37) are bounded and their bounds asymptotically converges to zero. Proof: For the closed-loop stability analysis, we consider the following composite Lyapunov-like energy functional: Vc = Vx + Vy + Vz + Vφ + Vθ + Vϕ
performed on a commercial quadrotor UAV system [17]. Our aim in this simulation is to verify the flight control stability and tracking property developed in Theorem 1 with respect to varying mass, aerodynamic damping, inertia matrix and uncertain outdoor environment. In our simulation, the desired motion trajectories in x and 3 y directions are selected as xd (t) = (1 − e−5t ) sin(t) and 3 yd (t) = (1 − e−2t ) cos(0.88t). The desired trajectory for zd (take-off, free flight and landing) is defined by us4 ing the following transfer function H(s) = s2 + 4s + 4 . It is assumed that the external forces are acting on the translational and orientational motion dynamics due to the variation of uncertain flying environment, aerodynamic force and moment, mass and inertia. For our evaluation, we consider the following state independent uncertain forces that are acting along the three translational and orientational motion dynamics Ex = cos(10πt), Ey = cos(4πt), Ez = sin(4πt), Eφ = sin(2πt), Eθ = sin(4πt), Eϕ = −cos(4πt). The physical parameters for the given commercial Pelican quadrotor vehicle [17] are chosen as m = 0.5 kg, l = 0.2 m, d=2.9842×10−5 , s2 s2 g = 9.81 m·s−2 , Ix = 0.00235 Nm· rad , Iy = 0.002535 Nm· rad , 2 2 s s , Ir = 0.0032 Nm· rad , τ1 = 0.002 N· ms , Iz = 0.05263 Nm· rad
(39)
In view of our above Lyapunov analysis, we can obtain the time derivative of Vc as V˙ c ≤ 0. Then, using V˙ c ≤ 0 and Barbalat’s Lemma, we can state that the signals ez , e˙ z , ex , e˙ x , ey , e˙ y , eφ , e˙ φ , eθ , e˙ θ , eϕ , e˙ ϕ , Sz , Mx , Ny , Bφ , Cθ and Dψ are bounded and their bounds converges to zero asymptotically. 4. Simulation Results To examine the stability and tracking property of the proposed algorithm, various simulation studies have been 145
Figure 7. The time history of position tracking in y and yd in metres under large model parameters.
Figure 8. The time history of position tracking in z and zd in metres under large model parameters.
Figure 9. The time history of yaw tracking in ϕ and ϕd in metres under large model parameters. as m = 5 kg, l = 0.2 m, d = 2.9842 × 10−5 , g = 9.81 m·s−2 , s2 s2 s2 Ix = 0.235 Nm· rad , Iy = 0.2535 Nm· rad , Iz = 0.5263 Nm· rad , s2 , τ1 =0.2 N· ms , τ2 =0.5 N· ms , τ3 =0.6 N· ms , Ir =0.0032 Nm· rad s s s φ1 = 0.11 Nm· rad , φ2 = 0.3 Nm· rad and φ3 = 0.6 Nm· rad . Using with these new model parameters, we then implement the proposed design with the same set up and same control design parameters as used in our previous evaluation. The evaluation results are depicted in Figs. 6–9. Notice from these results that the tracking errors remain closed to zero even with the increase of the modeling errors uncertainty.
s , φ2 = τ2 = 0.005 N· ms , τ3 = 0.006 N· ms , φ1 = 0.0011 Nm· rad s s 0.003 Nm· rad and φ3 = 0.006 Nm· rad . Then, we choose the control design parameters arbitrarily as αx = 2, αy = 2, αz = 2, αx1 = 20, αy1 = 20, αz1 = 2, αφ = 5, αθ = 5, αϕ = 5, αφ1 = 150, αθ1 = 150, αϕ1 = 150, kx = 100, ky = 150, ks = 100, kφ1 = 300, kθ1 = 300, kϕ1 = 300, Γx = 1, Γy = 1, Γz = 2, Γφ = 2, Γθ = 2 and Γϕ = 2. Using these design parameters, we then apply the proposed design on the given flying vehicle with and without uncertainty. At first, from 0 to 5 s, the vehicle operates under normal condition. Then, from 5 to 10 s, the vehicle flies in the presence of uncertainty that entering into translational and rotational axes. Finally, from 10 to 20 s, the system returns to normal operating condition without using uncertainty as depicted in Fig. 1. The evaluation results are presented in Figs. 2–5. Let us now increase uncertain model parameters of the vehicle. For this evaluation, the physical parameters of the commercial Pelican quadrotor vehicle [17] are chosen
5. Conclusion and Future Work In this work, we have presented adaptive sliding mode control technique for quadrotor UAV system. The proposed design can be used to ensure stability and tracking of 146
the vehicle in the presence of modeling error and disturbance uncertainty associated with aerodynamic damping and moment, mass, inertia and uncertain flying environment. Adaptive control laws have been used to learn and compensate uncertainty affecting the vehicle dynamics. Simulations studies have been carried out on a commercial Pelican quadrotor UAV to demonstrate theoretical development of this paper. In our future work, the proposed design will be implemented and evaluated on a commercial quadrotor helicopter provided by Asctec Inc. [17].
[14]
[15]
[16]
Acknowledgement [17]
Authors thank editor-in-chief, associate editor and anonymous five reviewers for their constructive comments and suggestion that definitely improves the quality and presentation of this paper. This work is partially supported in part by Natural Sciences and Engineering Research Council of Canada (NSERC).
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Biographies Shafiqul Islam earned his Ph.D. degree in Electrical and Computer Engineering from OttawaCarleton Institute for Electrical and Computer Engineering at the University of Ottawa and Carleton University, Canada. He has a master in Control Engineering and B.Sc. in Electrical and Electronic Engineering. His research was funded by many organizations including Natural Sciences and Engineering Research Council of Canada (NSERC), CMC Electronics, Carleton University, University of Ottawa, Lakehead University, etc. He was awarded Research Excellence in Science and Engineering from Carleton University, Canada, for his outstanding contribution to research and development. He currently holds prestigious NSERC Canada Postdoctoral Research Fellowship award for visiting national and international research laboratory. His research interests are robotics and control-unmanned ground and aerial vehicles, industrial manipulators; haptics and virtual reality and interactive networked and multiagent systems.
References [1] S. Bouabdallah and R. Siegwart, Design and control of an indoor micro quadrotor, Proc. 2003 IEEE Int. Conf. on Robotics and Automation, New Orleans, USA, 53–58, 2004. [2] T. Madani and A. Benallegue, Control of a quadrotor via full state backstepping technique, Proc. 45th IEEE Conf. on Decision and Control, San Diego, CA, December 13–15, 1515– 1520, 2006. [3] T. Madani and A. Benallegue, Backstepping control for a quadrotor helicopter, Proc. IEEE/RSJ Int. Conf. on Intelligent Robots and Systems, 9–15 October, Beijing, China, 2006, 3255–3260. [4] T. Madani and A. Benallegue, Backstepping sliding mode control applied to a miniature quadrotor flying robot, Proc. 32nd Annual Conf. of the IEEE Industrial Electronics Society IECON, 6–10 Nov. Paris, France, 2006, 700–705. [5] S. Bouabdallah and R. Siegwart, Backstepping and slidingmode techniques applied to an indoor micro quadrotor, Proc. 2005 IEEE Int. Conf. on Robotics and Automation, 2259–2264, 2005. [6] S. Bouabdallah and R. Siegwart, Full control of a quadrotor, Proc. 2007 IEEE/RSJ Int. Conf. on Intelligent Robots and Systems, October 29–November 2, San Diego, CA, USA, 2007, 153–158. [7] A. Das, F. Lewis and S. Subbarao, Dynamic inversion of the quadrotor with zero-dynamic stabilization, Proc. of the 17th IEEE Int. Conf. on Control Applications, San Diego, TX, 1189–1194, 2008. [8] G.V. Raffo, M.G. Ortega, and F.R. Rubio, Backstepping/nonlinear H∞ control for path tracking of a quadrotor unmanned aerial vehicle, Proc. 2008 American Control Conf., Seattle, Washington, DC, June, 3356–3361. [9] R. Xu and U. Ozguner, Sliding mode control of a quadrotor helicopter, Proc. 45th IEEE Conf. on Decision and Control, San Diego, CA, December 13–15, 4957–4962, 2006. [10] E. Altug, J.P. Ostrowski, and C.J. Taylor, Quadrotor control using dual camera visual feedback, Proc. 2003 IEEE Int. Conf. on Robotics and Automation, 3, 4294–4299, 2003. [11] S. Lupashin, A. Schollig, and M. Sherback, A simple learning strategy for high-speed quadrocopter multi-flips, IEEE Int. Conf. on Robotics and Automation, 2010, 1642–1648. [12] D. Mellinger and V. Kumar, Minimum snap trajectory generation and control for quadrotors, Proc. IEEE Int. Conf. on Robotics and Automation, Shanghai International Conference Center, May 2011, 2520–2525. [13] Schoellig et al., Synchronizing the motion of a quadrocopter to music, IEEE Int. Conf. on Robotics and Automation,
Xiaoping P. Liu received his Ph.D. degree from the University of Alberta in 2002. He has been with the Department of Systems and Computer Engineering, Carleton University, Canada, since July 2002 and is currently a professor and Canada Research Chair. He has published more than 200 research articles and serves as an associate editor for several journals including IEEE/ASME Transactions on Mechatronics and IEEE Transactions on Automation Science and Engineering. He received a 2007 Carleton Research Achievement Award, a 2006 Province of Ontario Early Researcher Award, a 2006 Carty Research Fellowship, the Best Conference Paper Award of the 2006 147
IEEE International Conference on Mechatronics and Automation and a 2003 Province of Ontario Distinguished Researcher Award. He is a licensed member of the Professional Engineers of Ontario (P.Eng.) and a senior member of IEEE. His research interests are interactive networked systems: teleoperation, telerobotics and telehaptics with applications to telemedicine; haptics with applications to medical simulations; robotics, control and intelligent Systems; context-aware smart networks and wireless sensor networks. Abdulmotaleb El Saddik a University Research Chair and Professor in the School of Electrical Engineering and Computer Science at the University of Ottawa, is an internationally recognized scholar who has made strong contributions to the knowledge and understanding of multimedia computing, communications and applications, particularly in the digitization, communication and security of the sense of touch, or haptics, which is a new medium that is significantly changing the way in which human-to-human and human – computer interactions are performed. He has authored and co-authored four books and more than 400 publications. He has received research grants and contracts totaling more than 18 Mio. and has supervised more than 100 researchers. He received several international awards and is ACM Distinguished Scientist, Fellow of the Engineering Institute of Canada, and Fellow of the Canadian Academy of Engineers and Fellow of IEEE. His research interests are multimedia computing, communications and applications, particularly in the digitization, communication and security of the sense of touch, or haptics.
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