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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 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. =2;A?=2;=2;A?;. The transformation matrix from ,0to p,q,r0is given by p q r 2 6 4 3 7 5 10? sin 0cossincos 0? sincoscos 2 6 4 3 7 5 _ _ _ 2 6 6 4 3 7 7 5 1 The dynamical model of the quadrotor can be described by the following equations 5,24,29: x 1 mcos sincos sinsinu1?K1_ x m y 1 mcos sinsin? sincosu1?K2_ y m z 1 mcos cosu1?g?K3_ z m _ _Iy ?Iz Ix Jr Ix _ r l Ixu2? K4l Ix _ _ _ Iz?Ix Iy ?Jr Iy _ r l Iyu3? K5l Iy _ _ _Ix ?Iy Iz 1 Izu4? K6 Iz _ 8 : 2 where Ki denote the drag coeffi cients and positive constants; r1?23?4;i; stand for the angular speed of the propeller i Ix,Iy,Izrepresent the inertias of the quadrotor;Jrdenotes the inertia of the propeller;u1denotes the total thrust on the body in the z-axis;u2and u3represent the roll and pitch inputs, respectively;u4denotes a yawing moment.u1 F1F2F3F4; u2 ?F2F4; u3 ?F1F3; u4 d?F1F2F3F4=b;, where Fi b2 i denote the thrust generated by four rotors and are considered as the real control inputs to the dynamical system, b denotes the lift coeffi 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: z # u1coscos m ?g 1 Izu4 2 4 3 5 ?K3 m_ z _ _Ix ?Iy Iz ?K6 Iz _ 2 4 3 5 3 x y # u1 m cossin sin? cos # cossin sin # ?K1 m_ x ?K2 m_ y 2 4 3 5 # l=Ix0 0l=Iy # u2 u3 # _ _Iy ?Iz Ix ?K4l Ix _ _Iz?Ix Iy ?K5l Iy _ 2 4 3 5 8 : 4 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 simplifi ed. For the UAS, a sliding mode control approach is utilized. The special synthesis control scheme is introduced in the following sections. Fig. 1. Quadrotor UAV. J.-J. Xiong, E.-H. Zheng / ISA Transactions () 2 Please cite this article as: Xiong J-J, Zheng E-H. Position and attitude tracking control for a quadrotor UAV. ISA Transactions (2014), http: //10.1016/j.isatra.2014.01.004i 3.1. A novel robust TSMC for FAS Considering the symmetry of a rigid-body quadrotor, therefore, we get IxIyLet x1 z?0and x2 _ z _ ?0. The fully actuated subsystem is written by _x1 x2; _x2 f1g1u1d1 ( 5 where f1 ?g 0?0;g1 coscos=m 0;0 1=Iz?;u1 u1u4?0 and d1 ?K3_z=m?K6_=Iz?0: To develop the tracking control, the sliding manifolds are defi ned as 18,30 s2_s11s11s1m 0 1=n 0 1 6a s4_s32s32s3m 0 2=n 0 2 6b where s1 zd?z; s3d?, Zdanddare the desired values of state variables Z and , respectively. In addition, the coeffi cients 1;2;1;2 are positive, m01;m02;n01;n02are positive odd integers with m01on01and m02on02. Let s20 and s40. The convergence time is calculated as follows: ts1 n01 1n01?m01 ln 1s10?n 0 1?m 0 1=n 0 11 1 ! 7a ts3 n02 2n02?m02 ln 2s30?n 0 2?m 0 2=n 0 22 2 ! 7b In accordance with Eq. (5) and the time derivative of s2and s4, we have _s2zd?u1 m coscosgK3 m _z1_s11 d dts m01=n01 1 8a _s4 d? 1 Izu4 K6 Iz _2_s32 d dts m02=n02 3 8b The controllers are designed by u1 m coscos zdg1_s11m 0 1 n01s m01?n01=n01 1 _s11s21sm1=n1 2 ? 9a u4 Izd2_s32m 0 2 n02 sm 0 2?n 0 2=n 0 2 3 _s32s42sm2=n2 4 ? 9b where1,2,1, and2are positive,m1, n1, m2, and n2are positive odd integers with m1on1and m2on2: Under the controllers, the state trajectories reach the areas (1,2) of the sliding surfaces s20 and s40 along_s2 ?1s2? 01sm1=n1 2 and_s4 ?2s4?02sm2=n2 4 in fi nite-time, respectively. The time is defi ned as t01r n1 1n1?m1 ln 1s10?n1?m1=n11 1 10a t02r n2 2n2?m2 ln 2s30?n2?m2=n22 2 10b where 011?K3_z=m=jsm1=n1 2 j;1 L1=jsm1=n1 2 j1; L1 jK3_z=mjmax;140;1 fjs2jrL1=1m1=n1g 022?K6_=Iz=jsm2=n2 4 j;2 L2=jsm2=n2 4 j2 L2 jK6_=Izjmax;240;2 fjs4jrL2=2m2=n2g Proof 1. In order to illustrate the subsystem is stable, here, we only choose the state variable z as an example and Lyapunov theory is applied. Considering the Lyapunov function candidate V1 s2 2=2 Invoking Eqs. (8a) and (9a) the time derivative of V1is derived _ V1 s2_s2 s2?1s2?1sm1=n1 2 K3_z=m ?1s22?01sm1 n1=n1 2 Considering that (m1n1) is positive even integer, thats, _ V1r0: The state trajectories of the subsystem converge to the desired equilibrium points in fi nite-time. Therefore, the subsystem is asymptotically stable. 3.2. A SMC approach for UAS In this section, the details about sliding mode control of a classofunderactuatedsystemsarefoundin29.Let Q u1 m cossin sin? cos # ,andy1 Q ?1x y?0;y 2 Q ?1_ x_y?0; y3 ?0;y4 _ _ ?0. The underactuated subsystem is written in a cascaded form _y1 y2; _y2 f2d2; _y3 y4; _y4 f3g2u2d3: 11 According to Eqs. (9a) and (9b) we can select the appropriate parameters to guarantee the control law u1and yaw angle convergetothedesired/referencevaluesinshorttime. Thats,_u1 0,becomes time invariant, then_ 0, Q is time invariant matrix and non-singular because u1is the total thrust and nonzero to overcome the gravity. As a result f2 cossinsin?0;d2 Q ?1diagK1=m K2=m?Qy 2;f3 0;g2 diagl=Ixl=Iy?;u2 u2u3?0;d3 diag?lK4=Ix?lK5=Iy?y4 Defi ne the tracking error equations e1 yd 1?y1; e2_e1_yd 1?y2; e3_e2yd 1?f2; e4_e3 : yd 1? f2 y1y2 f2 y2f2 f2 y3y4 ? 8 : 12 where the vector yd 1 denotes the desired value vector. The sliding manifolds are designed as s c1e1c2e2c3e3e413 where the constants ci40. By making_s ?Msgns?s, we get u2 f2 y3g2 ?1 c1e2c2e3c3e4:y01d? d dt f2 y1y2 hi ? d dt f2 y2f2 hi ? d dt f2 y3 hi y4 ?f2 y3f3d3Msgnss 8 : 9 = ; 14 where M c2d2c32d2jjE1jj23d4jjyjj2; 1Zf2=y1;2Zf2=y2;3Zf2=y3; E1 e1e2e3?0;y y1y2y3y4?0and40; 40;jjd2jjod2jjE1jj2;d2 maxK1=mK2=m jjd3jjod4jjyjj2;d4 maxlK4=IxlK5=Iy: Accordingtof2 y3 ? sinsin ? coscoscos0?; and0 of2=y3cos2coso2 ? ? ? ? ? ? ? ? ? ? , and , therefore, f2= y3is invertible. J.-J. Xiong, E.-H. Zheng / ISA Transactions () 3 Please cite this article as: Xiong J-J, Zheng E-H. Position and attitude tracking control for a quadrotor UAV. ISA Transactions (2014), http: //10.1016/j.isatra.2014.01.004i Proof 2. The stability of the subsystem is illustrated by Lyapunov theory as follows. Consider the Lyapunov function candidate: V 1 2s Ts Invoking Eqs. (13) and (14), the time derivative of V is _ V sT_s sTc1_e1c2_e2c3_e3_e4? sT?Msgnsc2d2c3f2 y2d2 f2 y3d3 ? o ?sTs?M?c2d2c32d2jjE1jj2?3d4jjxjj2jjsjj1 ?sTs?jjsjj1r0: Therefore, under the controllers, the subsystem state trajec- tories can reach, and, thereafter, stay on the manifold S0 in fi nite-time. 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 effi ciency of the proposed synthesis control scheme when faced with external disturbances. The simulations of typical position and attitude tracking are performed on Matlab 46/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 bounded PID controller and a sliding mode controller for the fully actuated subsystem, and, a SMC approach for the underactuated subsystem. 4.1. PID control and SMC In this section, more details of the PID control and SMC method for a quadrotor UAV has been introduced, meanwhile, the simula- tion results and analysis, which verify the effectiveness of the synthesis control scheme, can be found in 29. The approximate 05101520253035404550 0 3 6 Time ( s ) z ( m ) 05101520253035404550 -2 0 2 x ( m ) reference real 05101520253035404550 -1 0 1 y ( m ) Fig. 2. The positions (x,y,z), PID control and SMC. 05101520253035404550 -0.025 0 0.025 ( rad ) reference real 05101520253035404550 -0.06 0 0.06 ( rad ) 05101520253035404550 0 0.5 1 Time ( s ) ( rad ) X: 25.48 Y: 0.5003 Fig. 3. The angles (,), PID control and SMC. 05101520253035404550 7 8 9 10 11 12 13 X: 39.54 Y: 9.801 Time ( s ) u1 ( m/s2 ) Fig. 4. The controller u1, PID control and SMC. Table 1 Quadrotor model parameters. VariableValueUnits m2.0kg IxIy1.25Ns2/rad Iz2.2Ns2/rad K1K2K30.01Ns/m K4K5K60.012Ns/m l0.20m Jr1Ns2/rad b2Ns2 d5N ms2 g9.8m/s2 Table 2 Controller parameters. VariableValueVariableValue 1123 1121 m015m025 n017n027 m11m21 n13n23 110210 1 L1=jsm1=n1 2 j1 2 L2=jsm2=n2 4 j2 10.120.1 c120c222 c381 0.110 2032 J.-J. Xiong, E.-H. Zheng / ISA Transactions () 4 Please cite this article as: Xiong J-J, Zheng E-H. Position and attitude tracking control for a quadrotor UAV. ISA Transactions (2014), http: //10.1016/j.isatra.2014.01.004i simulation tests are shown in Figs. 24, however, the research objects are slightly changed to make obvious comparisons with the following simulation tests. 4.2. Novel robust TSMC and SMC In this section, in order to justify the effectiveness of the pro- posed synthesis control method, the position and attitude tracking of the quadrotor have been performed. The initial position and angle values of the quadrotor for the simulation tests are 0, 0, 0m and 0, 0, 0rad. In addition, the quadrotors model variables are listed in Table 1. The desired/reference position and angle values are used in simulation tests: xd1 m, yd1 m, Zd1 m,dd0 rad,d/ 6.28 rad. Besides, The controller parameters are listed in Table 2. The simulation results are shown in Figs. 510. The overall control scheme managed to effectively hold the quadrotors horizontal position and attitude in fi nite-time, as shown in Figs. 5 and 6. The fi nite-time convergence of the state variables z andis clearly faster than the other state variables, therefore, it is safe to consider the pitch angleas time invariant after 1.163 s. In addition, the altitude z reaches its reference value under the controller u1after 1.779 s, thus, it is reliable to consider the controller u1as time invariant after 1.779 s. These verify the matrix Q is time invariant in short fi nite-time. The other state variables reach their desired values after about 5 s. Even though the smooth curves of the state variablesandshow that they have certain oscillation amplitudes, the amplitudes are varying from ?0.05 rad and 0.05 rad. According to the initial conditions, parameters and desired/reference values, the convergence time of the state variables z andis essentially consistent with the values calculated by invoking Eqs. (7a) and (7b) and . This demonstrates the effectiveness of the proposed synthesis control scheme. The linear and angular velocities, displayed in Figs. 7 and 8, respectively, exhibit the same behavior as the corresponding positions and angles. Indeed, these state variables are driven to their steady states as expected. This, once again, demonstrates the effectiveness of the synthesis control scheme. The behavior of the sliding variables (s2,s4and s), shown in Fig. 9, follows the expectations as all the variables converge to their sliding surfaces. Furthermore, as desired, the fi nite-time convergence of s2and s4 is obviously faster than the fi nite-time convergence of s. Similarly, this exhibits the same behavior as shown in Figs. 5 and 6. Seen from Fig. 10, it can be found that the four control input variables converge to their steady state values (19.66,0,0,0) after several seconds, respectively. Besides, u1 a0, this also verifi es the matrix Q is time invariant in short fi nite-time. In spite of the high initial values, u1and u4are almost no oscillation amplitudes. This also denotes that the time derivative of u1trends to zero. Thus, compared with the equations in 29, Eq. (11) is greatly simplifi ed. Finally, the robustness of the proposed overall control method is demonstrated by considering the aerodynamic forces and moments and air drag taken as the external disturbances into the dynamical model of the quadrotor. Furthermore, these dis- turbance terms are also applied to the controller design. As a result, the effects of these disturbance terms are invisible on all the state variables, sliding variables, and controllers. 5. Discussion The extensive simulation tests have been performed to evaluate the different synthesis control schemes which are based on position and attitude tracking of the quadrotor UAV. It can be 0510152025303540 0 1 2 x ( m ) X: 4.977 Y: 0.9909 reference real 0510152025303540 0 1 2 y ( m ) X: 4.977 Y: 0.9946 0510152025303540 0 1 2 Time ( s ) z ( m ) X: 1.779 Y: 0.9932 Fig. 5. The positions (x,y,z), novel robust TSMC and SMC. 0510152025303540 -0.05 0 0.05 ( rad ) reference real 0510152025303540 -0.05 0 0.05 ( rad ) 0510152025303540 0 0.5 1 X: 1.163 Y: 0.4992 Time ( s ) ( rad ) X: 21.01 Y: 0.5002 Fig. 6. The angles (,), novel robust TSMC and SMC. 0510152025303540 -0.45 0 0.45 u ( m/s ) 0510152025303540 -0.45 0 0.45 v ( m/s ) 0510152025303540 -1.4 0 1.4 Time ( s ) w ( m/s ) Fig. 7. The linear velocities (u,v,w), novel robust TSMC and SMC. J.-J. Xiong, E.-H. Zheng / ISA Transactions () 5 Please cite this article as: Xiong J-J, Zheng E-H. Position and attitude tracking control for a quadrotor UAV. ISA Transactions (2014), http: //10.1016/j.isatra.2014.01.004i clearly seen that while all the state variables converge to their reference values as desired in fi nite-time, the convergence time is obviously different. It was shown that the synthesis control method based on the novel robust TSMC and SMC is a more reliable and effective approach to perform the tracking control for the quadrotor UAV. 6. Conclusion This work studies the position and attitude tracking control of a small quadrotor UAV using the proposed control method based on the aforementioned control algorithms. In order to further test the performances of the designed controllers, the dynamical model of the quadrotor along with the controllers is simulated on Matlab/ Simulink. The main conclusions are summarized as follows. (a) The six degrees of freedom converge to their desired/reference values in fi nite-time, respectively. (b) The convergence time of the state variables z andis essentially consistent with the theoretically calculated values. (c) Compared to other variables, the pitch angle and the controller u1become time invariants in short time. (d) The four control input variables converge to their steady values in fi nite-time,u1and u4are almost no oscillation amplitudes. All above, the effectiveness and robustness of the proposed synthesis control method have been demonstrated, and, the presented simulation results are promising at the position and attitude tracking control for the aircraft. Acknowledgment This work was partially supported by the National Natural Science Foundation of China (60905034). References 1 Raffo GV, Ortega MG, Rubio FR. An integral predictive/nonlinear H1 control structure for a quadrotor helicopter. Automatica 2010;46:2939. 2 Derafa L, Benallegue A, Fridman L. Super twisting control algorithm for the attitude tracking of a four rotors UAV. J Frankl Inst 2012;349:68599. 3 Luque-Vega L, Castillo-Toledo B, Loukianov AG. Robust block second order sliding mode control for a quadrotor. J Frankl Inst 2012;349:71939. 4 Garcia-Delgado L, Dzul A, Santibez V, Llama M. Quadrotors formation bansed on potential functions with obstacle avoidance. IET Control Theory Appl 2012;6(12):1787802. 5 Alexis K, Nikolakopoulos G, Tzes A. Model predictive quadrotor control: attitude, altitude and position experimental studies. IET Co