無人機(jī)外文翻譯-四旋翼無人機(jī)位置和姿態(tài)跟蹤控制【中文4860字】【PDF+中文WORD】,中文4860字,PDF+中文WORD,無人機(jī),外文,翻譯,四旋翼,位置,姿態(tài),跟蹤,控制,中文,4860,PDF,WORD Contents lists available at ScienceDirect ISA Transactions journal homepage: www.elsevier.com/locate/isatrans ISA Transactions ? (????) ???–??? Research Article Position and attitude tracking control for a quadrotor UAV Jing-Jing Xiong n, En-Hui Zheng College of Mechanical and Electrical Engineering, China Jiliang University, Hangzhou 310018, PR China Please cite this article as: Xiong J-J, Zheng E-H. Position and attitude tracking control for a quadrotor UAV. ISA Transactions (2014), http: //dx.doi.org/10.1016/j.isatra.2014.01.004i a r t i c l e i n f o Article history: Received 3 November 2013 Received in revised form 29 December 2013 Accepted 16 January 2014 This paper was recommended for publication by Jeff Pieper. Keywords: Quadrotor UAV Underactuated Novel robust TSMC SMC Synthesis control a b s t r a c t A synthesis control method is proposed to perform the position and attitude tracking control of the dynamical model of a small quadrotor unmanned aerial vehicle (UAV), where the dynamical model is underactuated, highly-coupled and nonlinear. Firstly, the dynamical model is divided into a fully actuated subsystem and an underactuated subsystem. Secondly, a controller of the fully actuated subsystem is designed through a novel robust terminal sliding mode control (TSMC) algorithm, which is utilized to guarantee all state variables converge to their desired values in short time, the convergence time is so small that the state variables are acted as time invariants in the underactuated subsystem, and, a controller of the underactuated subsystem is designed via sliding mode control (SMC), in addition, the stabilities of the subsystems are demonstrated by Lyapunov theory, respectively. Lastly, in order to demonstrate the robustness of the proposed control method, the aerodynamic forces and moments and air drag taken as external disturbances are taken into account, the obtained simulation results show that the synthesis control method has good performance in terms of position and attitude tracking when faced with external disturbances. & 2014 ISA. Published by Elsevier Ltd. All rights reserved. 1. Introduction The quadrotor unmanned aerial vehicles (UAVs) are being used in several typical missions, such as search and rescue missions, surveillance, inspection, mapping, aerial cinematography and law enforcement [1–5]. Considering that the dynamical model of the quadrotor is an underactuated, highly-coupled and nonlinear system, many con- trol strategies have been developed for a class of similar systems. Among them, sliding mode control, which has drawn researchers' much attention, has been a useful and ef?cient control algorithm for handling systems with large uncertainties, time varying prop- erties, nonlinearities, and bounded external disturbances [6]. The approach is based on de?ning exponentially stable sliding surfaces as a function of tracking errors and using Lyapunov theory to guarantee all state trajectories reach these surfaces in ?nite-time, and, since these surfaces are asymptotically stable, the state trajectories slides along these surfaces till they reach the origin [7]. But, in order to obtain fast tracking error convergence, the desired poles must be chosen far from the origin on the left half of s-plane, simultaneously, this will, in turn, increase the gain of the controller, which is undesirable considering the actuator satura- tion in practical systems [8,9]. n Corresponding author. E-mail addresses: jjxiong357@gmail.com (J.-J. Xiong), ehzheng@cjlu.edu.cn (E.-H. Zheng). Replacing the conventional linear sliding surface with the non- linear terminal sliding surface, the faster tracking error convergence is to obtain through terminal sliding mode control (TSMC). Terminal sliding mode has been shown to be effective for providing faster convergence than the linear hyperplane-based sliding mode around the equilibrium point [8,10,11]. TSMC was proposed for uncertain dynamic systems with pure-feedback form in [12]. In [13], a robust adaptive TSMC technique was developed for n-link rigid robotic manipulators with uncertain dynamics. A global non-singular TSMC for rigid manipulators was presented in [14]. Finite-time control of the robot system was studied through both state feedback and dynamic output feedback control in [15]. A continuous ?nite-time control scheme for rigid robotic manipulators using a new form of terminal sliding modes was proposed in [16]. For the sake of achieving ?nite-time tracking control for the rotor position in the axial direction of a nonlinear thrust active magnetic bearing system, the robust non-singular TSMC was given in [17]. However, the conventional TSMC methods are not the best in the convergence time, the primary reason is that the convergence speed of the nonlinear sliding mode is slower than the linear sliding mode when the state variables are close to the equilibrium points. In [18], a novel TSMC scheme was developed using a function augmented sliding hyperplane for the guarantee that the tracking error converges to zero in ?nite-time, and was proposed for the uncertain single-input and single-output (SISO) nonlinear system with unknown external disturbance. In the most of existing research results, the uncertain external disturbances are not taken into account these nonlinear systems. In order to further demonstrate the robustness of novel 0019-0578/$ - see front matter & 2014 ISA. Published by Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.isatra.2014.01.004 quadrotor is set up in this work by the body-frame B and the earth- frame E as presented in Fig. 1. Let the vector [x,y,z]' denotes the position of the center of the gravity of the quadrotor and the vector [u, v,w]' denotes its linear velocity in the earth-frame. The vector [p,q,r]' represents the quadrotor's angular velocity in the body-frame. m denotes the total mass. g represents the acceleration of gravity. l denotes the distance from the center of each rotor to the center of gravity. The orientation of the quadrotor is given by the rotation matrix R:E-B, where R depends on the three Euler angles [?,θ,ψ]0 , which represent the roll, the pitch and the yaw, respectively. And ?Ae π=2; π=2T; θ Ae π=2; π=2T; ψ Ae π; πT. The transformation matrix from [?,θ,ψ]0 to [p,q,r]0 is given by 2 p 3 2 1 0 sin θ 32 ?_ 3 6 q 7 6 0 cos ? sin ? cos θ 76 θ_ 7 4 5 ? 4 56 7 e1T r 0 sin ? cos ? cos θ 4 ψ_ 5 Fig. 1. Quadrotor UAV. The dynamical model of the quadrotor can be described by the following equations [5,24,29]: 8 x€ ? 1 e cos ? sin θ cos ψ t sin ? sin ψ Tu1 K1 x_ m m > TSMC, the external disturbances are considered into the nonlinear > y€ ? 1  K2 y_ 1 > me cos ? sin θ sin ψ sin ? cos ψ Tu m > systems and are applied to the controller design. > z€ ? 1  K3 z_ <> me cos ? cos θTu1 g m In this work, we combine two components in the proposed ?€ _ Iy Iz Jr θ_ Ω l K4 l?_ e2T control: a novel robust TSMC component for high accuracy > ? θψ_ > Ix t Ix r t Ixu2 Ix tracking performance in the fully actuated subsystem, and a SMC > θ€ ? ψ_ ?_ Iz Ix Jr ?_ Ωr t l u3 K5 lθ_ component that handles the external disturbances in the under- > > > ψ€ Iy Iy _ _ Ix Iy 1 Iy Iy K6 actuated subsystem. Even though many classical, higher order and extended SMC :> ? ?θ Iz t Izu4 Iz ψ_ strategies, have been developed for the ?ight controller design for where Ki denote the drag coef?cients and positive constants; Ωr ? Ω1 Ω2 t Ω3 Ω4; Ωi ; stand for the angular speed of the the quadrotor UAV (see for instance [19–23], and the list is not exhaustive), and, these strategies in the papers [19–23] were propeller i Ix,Iy,Iz represent the inertias of the quadrotor;Jr denotes the inertia of the propeller;u1 denotes the total thrust on the body utilized to dictate a necessity to compensate for the external disturbances, in addition, the other control methods, such as in the z-axis;u2 and u3 represent the roll and pitch inputs, respectively;u4 denotes a yawing moment.u1 ? eF1 t F2 t F3 t F4T; proportional–integral–differential (PID) control [24,25], backstep- ping control [26,27], switching model predictive attitude control u2 ?e F2 tF4T; u3 2 ?e F1 t F3T; u4 ? de F1 t F2 t F3 t F4T=b;, [28], etc., have been proposed for the ?ight controller design, most of the aforementioned control strategies have been proposed in order to make the quadrotor stable in ?nite-time and the stabili- zation time of the aircraft may be too long to re?ect the performance of them. In addition, the stabilization time is essen- tial signi?cance for the quadrotor UAV to quickly recover from some unexpected disturbances. For the sake of decreasing the time, a synthesis control method based on the novel robust TSMC and SMC algorithms is applied to the dynamical model of the quadrotor UAV. The synthesis control method is proposed to guarantee all system state variables converge to their desired where Fi ? bΩi denote the thrust generated by four rotors and are considered as the real control inputs to the dynamical system, b denotes the lift coef?cient;d denotes the force to moment scaling factor. 3. Synthesis control Compared with the brushless motor, the propeller is very light, we ignore the moment of inertia caused by the propeller. Eq. (2) is divided into two parts: values in short time. Furthermore, the convergence time of the state variables are predicted via the equations derived by the novel " z€ # 2 u1 cos ? cos θ 3 2 g m  K3 3 z_ T m 3 robust TSMC, this is demonstrated by the following sections. ψ€ ? 4 1 5t4 ?_ θ_ Ix Iy K6 5 e The organization of this work is arranged as follows. Section 2 presents the dynamical model of a small quadrotor UAV. The 8 " x€ # > Izu4 " cos ψ sin ψ u1 Iz Iz ψ_ #" cos ? sin θ #  x_ 2 K1 3 — m synthesis control method is detailedly introduced in Section 3. In Section 4, simulation results are performed to highlight the overall >> y€ ? m > < sin ψ cos ψ sin ? t4 K2 5 y_ — m validity and the effectiveness of the designed controllers. In > " ?€ # " l=Ix 0 #" u # 2 θ_ ψ_ Iy Iz I  K4 l _ 3 ? I e4T Section 5, a discussion, which is based on different synthesis > 2 x x control schemes, is presented to emphasize the performance of > θ€ ? 0 l=Iy u3 t4 ψ_ ?_ Iz Ix K5 lθ_ 5 >: the proposed synthesis control method in this work, followed by  Iy Iy the concluding remarks in Section 6. 2. Quadrotor model In order to describe the motion situations of the quadrotor model clearly, the position coordinate is to choose. The model of the where Eq. (3) denotes the fully actuated subsystem (FAS), Eq. (4) denotes the underactuated subsystem (UAS). For the FAS, a novel robust TSMC is used to guarantee its state variables converge to their desired values in short time, then the state variables are regarded as time invariants, therefore, the UAS gets simpli?ed. For the UAS, a sliding mode control approach is utilized. The special synthesis control scheme is introduced in the following sections. J.-J. Xiong, E.-H. Zheng / ISA Transactions ? (????) ???–??? 7 3.1. A novel robust TSMC for FAS Considering the symmetry of a rigid-body quadrotor, therefore, we get Ix ? Iy Let x1 ? ?zψ ]0 and x2 ? ? ?z_ψ_ ]0 . The fully actuated Considering the Lyapunov function candidate 2 V 1 ? s2=2 η sm1 =n1 _ Invoking Eqs. (8a) and (9a) the time derivative of V1 is derived subsystem is written by V_ 1 ? s2s_2 ? s2e ε1s2 1 2 t K3z=mT ( x_ 1 ? x2; 2 0 m1 t n1 T=n1 x_ 2 ? f 1 t g1u1 t d1 e5T ? ε1s2  η1se 2 where f 1 ?? g 0]0 ; g1 ?? cos ? cos θ=m 0; 0 1=Iz ]; u1 ? ?u1 u4]0 and d1 ?? K3z_=m K6ψ_ =Iz ]0 : To develop the tracking control, the sliding manifolds are de?ned as [18,30] s2 ? s_1 tω1s1 tξ1s1m1 =n1 e6aT Considering that (m1 tn1) is positive even integer, that’s, V_ 1 r0: The state trajectories of the subsystem converge to the desired equilibrium points in ?nite-time. Therefore, the subsystem is asymptotically stable. 3.2. A SMC approach for UAS 0 0 s4 ? s_3 tω2s3 tξ2s3m2 =n2 e6bT In this section, the details about sliding mode control of a 0 0 class of underactuated systems are found in [29]. Let where s1 ? zd z; s3 ? ψ d ψ, Zd and ψd are the desired values of state variables Z and ψ, respectively. In addition, the coef?cients " cos ψ sin ψ Q ? u1 # , and y1 ? Q ?x y] ; y2 ? Q  ?x_ y_ ] ; eω1; ω2; ξ1; ξ2T are positive, m0 ; m0 ; n0 ; n0 are positive odd integers m sin ψ cos ψ 1 0 1 0 with m0 o n0 and m0 o n0 . 1 2 1 2 y ? ?? θ]0 ; y ? ??_ θ_ ]0 . The underactuated subsystem is written in 1 1 2 2 3 4 Let s2 ? 0 and s4 ? 0. The convergence time is calculated as a cascaded form follows: 1 m1 T=n1 ! y_ 1 ? y2; ξ n0 ω1?s1e0T]en0 0 0 t y_ 2 ? f 2 td2; 1 1 ts1 ? ω n0 0 ln e7aT _ 1e 1 m1T n0 ξ1 ω2?s3e0T]en2 m2 T=n2 tξ ! y3 ? y4; y_ 4 ? f 3 tg2u2 t d3: e11T 0 0 0 2 2 According to Eqs. (9a) and (9b) we can select the appropriate ts3 ? ω ln 2 m0 2en0 2T ξ2 e7bT parameters to guarantee the control law u1 and yaw angle ψ In accordance with Eq. (5) and the time derivative of s2 and s4, we have converge to the desired/reference values in short time. That’s,u_ 1 ? 0,ψ becomes time invariant, then ψ_ ? 0, Q is time invariant matrix and non-singular because u1 is the total thrust u1 K3 d m0 =n0 s_2 ? z€d cos ? cos θ t g t z_ tω1s_1 tξ1 s 1 1 e8aT m m dt 1 and nonzero to overcome the gravity. As a result 2 2 3 2 f ?? cos ? sin θ sin ?]0 ; d2 ? Q 1diag?K1=m K2=m]Qy ; f ? 0; g 1 K6 d m0 =n0 2 2 s_4 ? ψ€ d I u4 t I ψ_ tω2s_3 tξ2dts e8bT ? diag?l=Ix l=Iy]; u ? ?u u ]0 ; d ? diag? lK =I lK =I ]y 3 2 2 3 3 z z 4 x 5 y 4 The controllers are designed by De?ne the tracking error equations m m0 em0 n0 T=n0 m =n 8 e1 ? yd y1; cos ? cos θ 1 1 1 1 u1 ? z€d t g tω1s_1 tξ1  1s 1 n0 1 s_1 tε1s2 tη1s2 e9aT > 1 > < 1 1 > e2 ? e_ 1 ? y_ d d y2; e12T > ／ m0 m0 n0 T=n0 m2 =n2 ＼ e3 ? e_ 2 ? y€ 1 f 2; u4 ? Iz ψ€ d tω2s_3 tξ2 2se 2 2 2 s_3 tε2s4 tη2s e9bT > n0 3 4 > :::d ’ ?f 2  ?f 2  ?f 2 2 > e4 ? e_ 3 ? y1 ?y y2 t?y f 2 t?y y4 : where ε1,ε2,η1, and η2 are positive,m1, n1, m2, and n2 are positive 1 2 3 d odd integers with m1 on1 and m2 on2: Under the controllers, the state trajectories reach the areas (Δ1,Δ2) of the sliding surfaces s2 ? 0 and s4 ? 0 along s_2 ? ε1s2 where the vector y1 denotes the desired value vector. The sliding manifolds are designed as s ? c1e1 tc2e2 tc3e3 te4 e13T η0 m1 =n1 0 m2 =n2 1s2 and s_4 ? ε2s4 η2s4 in ?nite-time, respectively. The time is de?ned as where the constants ci 40. By making s_ ? MsgnesT λs, we get n1 ε1?s1e0T]en1 m1 T=n1 tδ1 ::: h i t0 ln e10aT 8 c1e2 tc2e3 tc3e4 t y0 ?f 2 y 9 1 rε n m δ d d < > 1 y 2 > 1e 1 1T 1 u2 ?  ?f 2 g ?y 2  1>>  d h ?f 2 i f dt ?y2 2 d h?f 2 i y dt ?y3 4 dt ? 1 > > = e14T 3 t0 2 > > n2 ε2?s3e0T]en2 m2 T=n2 tδ2 > ?f > 2 rε 2en2 ln m2T δ2 e10bT >: ?y3 ef 3 td3Tt MsgnesTtλs >; where η0 m1 =n1 m1 =n1 where M ? ec2d2 tc3β d2TjjE1jj2 tβ d4jjξeyTjj2 tρ; 1 ? η1 te K3z_=mT=js2 j; η1 ? L1=js2 jtδ1; 2 3 L1 ? jK3z_=mjmax; δ1 40; Δ1 ? fjs2jreL1=η1T m1 =n1 g β1 Z?f 2=?y1; β2 Z?f 2=?y2; β3 Z?f 2=?y3; η0 m2 =n2 m2 =n2 E1 ? ?e1e2e3] ; ξeyT? ?y1y2y3y4] and λ 40; 2 ? η2 te K6ψ_ =Iz T=js4 j; η2 ? L2=js4 jtδ2 ρ 0 d 0 o d E 0 d max K mK m m2 =n2 4 ; jj 2jj 2jj 1jj2; 2 ? e 1= 2= T L2 ? jK6ψ_ =Iz jmax; δ2 40; Δ2 ? fjs4jreL2=η2T g jjd3jjo d4jjξeyTjj2; d4 ? maxelK4=IxlK5=IyT: Proof 1. In order to illustrate the subsystem is stable, here, we According to?f 2 ? ［ sin ? sin θ cos ? cos θ cos ?0]; only choose the state variable z as an example and Lyapunov and 0 o ｜｜?f 2=?y3｜｜ ? ｜ cos 2? cos θo2, and , therefore, ?f 2= theory is applied. ｜｜ ?y3 ?y3 is invertible. ｜｜ ｜ reference real 2 x ( m ) 0 -2 0 5 10 15 20 25 30 35 40 45 50 1 y ( m ) 0 13 X: 39.54 Y: 9.801 12 u ( m/s 2 ) 11 10 1 9 8 7 0 5 10 15 20 25 30 35 40 45 50 Time ( s ) Fig. 4. The controller u1, PID control and SMC. -1 0 5 10 15 20 25 30 35 40 45 50 6  Table 1 Quadrotor model parameters. z ( m ) Variable Value Units m 2.0 kg Ix ? Iy 1.25 Ns2/rad Iz 2.2 Ns2/rad K1 ? K2 ? K3 0.01 Ns/m K ? K ? K 0.012 Ns/m l 0.20 m Jr 1 Ns2/rad b 2 Ns2 d 5 N ms2 g 9.8 m/s2 3 0 0 5 10 15 20 25 30 35 40 45 50 Time ( s ) Fig. 2. The positions (x,y,z), PID control and SMC.  4 5 6 reference real 0.025 f ( rad ) 0 -0.025 .06 Variable Value Variable Value ω1 ξ1 1 1 ω2 ξ2 3 1 m0 5 0 5 0.06 n1 0 7 n2 7 5 10 15 20 25 30 35 40 45 50 m1 1 m2 1 1 n1 3 n2 3 ε1 10 ε2 10 X: 25.48 Y: 0.5003 η1 L = sm1 =n1 η2 L2=j m2 =n2 0.5 δ1 0.1 δ2 0.1 c1 20 c2 22 c3 8 ρ 1 0 λ 0.1 β1 0 5 10 15 20 25 30 Time ( s ) 35 40 45 50 β2 0 β3 2 0  0 5 10 15 20 25 30 35 40 45 50 Table 2 Controller parameters. q ( rad ) 0 1 m2 0 - 0 y ( rad ) 1 j 2 jtδ1 s4 jtδ2 0 Fig. 3. The angles (?,θ,ψ), PID control and SMC. Proof 2. The stability of the subsystem is illustrated by Lyapunov theory as follows. Consider the Lyapunov function candidate: V 1sT s ? 2 Invoking Eqs. (13) and (14), the time derivative of V is V_ ? sT s_ ? sT ?c1e_ 1 tc2e_ 2 tc3e_ 3 t e_ 4] 4. Simulation results and analysis In this section, the dynamical model of the quadrotor UAV in Eq. (2) is used to test the validity and ef?ciency of the proposed synthesis control scheme when faced with external disturbances. The simulations of typical position and attitude tracking are performed on Matlab 7.1.0.246/Simulink, which is equipped with a computer comprising of a DUO E7200 2.53 GHz CPU with 2 GB of RAM and a 100 GB solid state disk drive. Moreover, the perfor- mance of the synthesis control is demonstrated through the comparison with the control method in [29], which used a rate ／ ? sT MsgnesTtc2d2 tc3  ?f 2 d 2 ?y 2 t  ?f 2 d ＼ 3 ?y3 b