DSpace at VNU: Trajectory Tracking Control of a Mobile Robot by computed torque method with on-line learning neural network

2013 10th IEEE International Conference on Control and Automation (ICCA) Hangzhou, China, June 12-14, 2013 Trajectory Tracking Control of a Mobile Robot by computed torque method with on-line learning neural network Thuan Hoang Tran1, Van Tinh Nguyen2 , Minh Tuan Pham2, Thuong Cat Pham2 University of Engineering and Technology, Vietnam National University, Hanoi The Institute of Information Technology, Vietnamese Institute of Science and Technology, Hanoi thuanhoang@donga.edu.vn, 2nvtinh@ioit.ac.vn  Abstract—This paper proposes a novel control algorithm for the mobile robot with nonholonomic constraint The algorithm consists of two control loops: one is based on the kinematics and Lyapunov theory to derive the control laws for the tangent and angular velocities to control the robot to follow a target trajectory, the other controls the robot dynamic based on the moment method in which a neural network namely RBFNN is introduced to compensate the uncertainty of dynamic parameters The convergence of the estimators based on RBFNN of Stone-Weierstrass is proven The asymptotically stabilization of the whole system is confirmed by direct Lyapunov stabilization theory The effectiveness of the method is verified by simulations in Matlab Keywords — Trajectory tracking, mobile robot, torque control, neural networks, Lyapunov theory I INTRODUCTION Controlling a mobile robot to follow a predefined trajectory is a challenging task due to the nonlinear chacteristic and nonholonomic constraint of the robot According to Brocket theory, a nonholonomic system is not able to be asymptotically stable using the smooth and time invariant control laws Some methods to stablize the nonholonomic system through feedback control have been proposed They however often assume ideal conditions Others focus on determining uncertainties in measurements and model parameters and try to fix them by using hybrid feedback control or velocity chart control These methods are usually complex and difficult to implement Our approach is the use of Lyapunov function technique to design a stable controller for nonholonomic systems [1],[2] The goal is the optimization in motion of the robot during the path following process From the robot kinematics, the uncertainties in system parameters are determined and compenstated by the implementation of an extended Kalman filter But this stage only focuses on the kinematics while the dynamics parameters such as the robot’s load which plays an important role in the stable of the robot are not concerned In addition, non-parameter uncertainties such as high-frequency unmodeled dynamics, actuator dynamics, structural vibrations, measurement noises, computstion roundoff error, and sampling delay also need be considered Thus, the problem of kinematics and dynamics control of nonholonomic system is challenging 978-1-4673-4708-2/13/$31.00 ©2013 IEEE A number of approaches to control the system with nonholonomic constraint have been introduced [3],[4],[6] In [9-16], authors were combined the dynamics model of the mobile robot to the kinematics controller with nonholonomic constraint One of the most popular methods to solve this class of control problem is adaptive control For example, the backstepping method of Wang et al [17] and R.Fierro et al [18], the sliding-mode techniques in [19-20], were applied to reduce sway for an offshore container crane These methods also employed neural network to compensate the uncertainties such as the combination between the backstepping method and the neural network reported in [9, 11] In [21], Dong Xu et al were applied a combination between the RBFNN controller and the sliding-mode techniques for the path following task of an omnidirectional wheeled mobile manipulators In [9], the author presented a control method using neural network in which, online learning law of weight factors is used to compensate the uncertainties caused by error in dynamics modelling The asymptotically stabilization were theorically proven and confirmed by experiments Nevertheless, if the dynamics model contains non-parameter uncertainties, the asymptotically stabilization is then not assured The new point in this paper is the splitting of the path following tasks into two independent control loops The outer loop is employed to control the kinematics such as the determination of tangent and angular velocities so that the errors in position and direction go toward zero (globally asymptotically stabilization according to the Lyapunov theory); output of this controller is sent to the inner control loop The inner control loop is used to control the dynamics In this control loop, we control by computed torque method This controller is designed by the combination between the Feedforward and the scale techniques The RBFFN is used to compensate the non-parameter uncertainties and dynamic modeling errors The globally asymptotic stabilization of the system is proven by the Lyapunov theory The paper is organized as follows Section briefly introduces the kinematics and dynamics of the mobile robot, as presented in [9-11] Section describes the process to design the controller Section presents the simulation results and section is the conclusion 1184 II NONHOLONOMIC MOBILE ROBOT KINEMATICS AND DYNAMIC r r   cos cos    r sin  r sin  2  S z    r r     2R 2R  1    0  A typical example of a nonholonomic mobile robot is shown in Fig [9-11] Y Yp Xp 2R Now, we easily have the following pair of equations [910]:  d P 2r where O X The position of the robot in an inertial  Cartesian frame {O,X,Y} can be described by a vector ON , and orientation between mobile robot base frame {P,Xp,Yp} and Cartesian frame.The configuration of the robot can be described by five generalized coordinates l  T y  r (1) where x, y are the coordinates of P, θ is the orientation angle of mobile robot, and r , l are the angles of the right and left driving wheels Therefore the nonholonomic constraint of the mobile robot can be expressed as, y cos   x sin   x cos   y sin   R  r r x cos   y sin   R  r     r mR2  I   Iw  R2 M 2  r mR  I  4R2   Fig A noholonomic mobile robot platform z  x and τ   r  Mv + Cv = τ (8)     r mc d       4R2 R2  ;C     r mc d   r mR2  I    I  w   4R  4R2  r mR2  I   wheels, v   vr vl  represents the angular velocity of the T right and left wheel, m  mc  2mw , I  mc d  2mw R  I c Here, mc is the mass of the mobile robot platform, mw is the mass of one driving wheel with the actuator, I c , I w are moments of inertia of platform about the vertical axis through P, the wheel with the actuator about the wheel axis, respectively We can rewrite the system dynamic Eq (8) into a linear form, Y  v, v  p = τ (9) Y  v, v  p = Mv + Cv (3) (4) Let S  z  be a matrix formed by a set of smooth and linearly independent vector fields spanning the null space of A  z  , i.e., A zS z = (7) T (2) All kinematic equality constraints are considered independent of time and can be expressed as, 0  sin   cos   Where A  z    cos  sin   r  R  cos  sin   R  r  z = S  z  v  t  l  is the torque applied on the right and left l A  z  z  (6) (5) where p is a 3x1 vector consisting of the known and unknown robot dynamics, such as mass and moment of inertia; Y  v, v  is a 2x3 coefficient matrix consisting of the known functions of the robot velocity v and acceleration v which is referred as the robot regressor For the mobile robot shown in fig.1, we can get: v v  vl  Y  v, v    r l   vl vr vr     r mR2  I  p    Iw    4R   It is easy to verify that S  z  is given by 1185   r mR2  I   R2    T  r mc d       4R2    (10) III CONTROLLER DESIGN Vp  A Outer control loop Let the tangent and angular velocities of the robot be v and  respectively We have: 1 v   r v  r   vl    r R   x  cos   r  v      v     ; q   y   sin      R        0     r   e1e1  e2e2  e3e3 (17)  e1  e2  v  vr cos e3   e2  e1  vr sin e3   e3  r  (11)  e1  v  vr cos e3   e2vr sin e3  e3  r  not known exactly Desired position and velocity vectors are represented by: xr  vr cos  r y r  vr sin  r r  r (16) Vp  eTpe p Replacing (15) into (17), we have Vp  e1  v  vr cos e3   e2vr sin e3  e3  r  r  T (18)  sin e3   e1  vr cos e3  k1e1  vr cos e3   e2vr sin e3  e3  r r  k3e3  vr e2  e3   velocity q r  t  , in case the robot dynamics parameters are yr  Derivation of Vp with respect to time Vp is: The objective of the control problem is to design an adaptive control law so that the position vector q and the velocity q to hold the position vector q r  t  and the desired q r   xr  T e p e p  e12  e22  e32 2  k1e12  k3e32 It is easy to see that Vp is continuous and bounded according to the Barbalat theorem It means that V  p with vr > for all t when t   , consequently e1  0, e3  when t   Appling the Barbalat theorem again for the derviation, we get: e1  0, e3  (19) (12) The tracking error position between the reference robot and the actual robot can be expressed in the mobile basis frame as below [9-10]:  e1   cos  sin    xr  x  e p  e2     sin  cos    yr  y  (13)   r     e3   The derivation of the position tracking error can be expressed as:  e1   e2  v  vr cos e3  e p   e2     e1  vr sin e3       e3   r    (14) There are some methods in the literature to select the smooth velocity input In this research, we choose a new control law for v,  as: vr cos e3  k1e1   v         k e  v e sin e3  3 r    r e3   (15) where k1 , k3 > In this control law, when e3  then sin e3  , and  always be bounded e3 With the control law in equation (15), it is easy to prove asymptotical stability system due to e p  when t   and the equation (15) becomes v  vr (20)   r (21) Combining (14), (19), (20), (21) infers: e2  0, e2  Thus the control law (15) assures the proximity control system e p  when t   B Inner control loop The deviation of stick angular velocity of driven wheels is: v  v  ec  v  v c   r cr  (22)  vl  vcl  where v is the desired angular velocity of the robot calculated by (11), vc is the output of the angular velocity control wheel torque We must find the control law of the angular velocity using computed-torque method in order to ec  and e p  when t   Derivating and multiplying both sides of equation (22) with the matrix M obtain: Me c  M  v  v c   τ  Cec  Yc p (23-a) In case of non-parametric uncertainty component d in the mobile robot dynamics model, equation (23-a) is rewritten as below Me c  M  v  v c   τ  Cec  Yc p  d (23-b) Choosing a positive definite function Vp as follows: Where 1186 Yc p  Mv c  Cv c (24) Torque output of the controller is: τ = τ NN  K D ec  Yc pˆ τ = K D ec    1 Wσ   (25)  i   ec i w where KD is the positive definite matrix pˆ is the matrix estimated by matrix p Substituting (24) into (23), we have Me c  τ NN   K D  C  ec  f (26) Because f  Yc p  d  Yc p  Yc pˆ  d is unknown so it can be seen as an uncertain component We can approximate this uncertain component by a finite neural network [3, 4, 6] which has the following structure: f  Yc p  d  Wσ  ε  fˆ  ε ec  Yc pˆ ec  j  exp (29)  ecj  c j   2j where the optional parameter K D is a symmetric positive definite matrix, the coefficients  ,   (27) where W is the weight matrix of an online updated network; ε is the approximate error and is bounded by ε  0 The neural network Wσ is approximated by Gaussian RBF network consisting of three layers: input layer, hidden layer with nodes that contains the Gaussian function, and the output layer with linear function of neurons (Fig 2) [7] ec1 1 Fig Mobile robot control by torque method with online learning neural network Proof: Selecting a positive function V as follows: V fˆ1   w j1 j  1 T T  ec Mec   w i w i  2 i 1  (30) Derivating V with respect to time gets i V  eTc Me c   w iT w i 1 ec fˆ2   w j 2 j 2 Fig Neural network approximation Wσ funcion  eTc K D ec  eTc Cec  eTc  τ NN  Wσ  ε   Wσ  The RBF network structure satisfies the conditions of the Stone-Weierstrass theorem Hidden layer neuron is the Gaussian function with the form: j s  exp  j  cj   j2 ; j  1, Because of τ NN    1 Wσ   ec ec (32) and the matrix C is symmetric, eTc Cec  So (28)   e V  eTc K D ec  eTc   c  ε  ec   where c j ,  j are the expectation and variance of the Gaussian function chosen as follows [17] Here, we choose τ NN (31)    eTc  Me c    w i i  i 1   According to (23), we have: V  eTc  τ NN   K D  C  ec  f   Wσ   eTc K D ec   ec  ec ε e    1 Wσ   c to satisfy our ec control law Following the Stone-Weierstrass theorem, we have the diagram Mobile robot control by torque method with online learning neural network as shown in the figure with  eTc K D ec   ec  ec  If we choose      with   then V  eTc K D ec   ec We see V  when ec  and V  if and only if ec  According to the Lyapunov stability principle [1], [2] ec  means e p  1187 0.4 0.3 follows the desired trajectory with error Theorem as well as the global asymptotic stability of the system with torque control using neural network depicted in Fig has been proved Position Error The control system in Fig is the asymptotic stability q  q d or in other words, the trajectory of the robot X Direct Y Direct Orient 0.2 0.1 IV SIMULATION RESULTS -0.1 The control algorithm developed in section III was implemented in Matlab-Simulink In the course of the simulation test, we use a mobile robot model having the following dynamic parameters r = 0.15m; R= 0.75m; d=0.2m; mc = 30kg; mw = 30kg; Ic = 15.625 kgm2; Iw = 0.0005kgm2; Im = 0.0025 kgm2 The parameters of the controller are: KD = diag(5,5); k1=k3=2;  = 10 Suppose that we only estimate pˆ  0.6p and non-parametric uncertainty components are: d  sin(0.25t ) cos  0.25t   50 Actual Desired Y Axis (m) -1 -3 10 X Axis (m) 12 14 16 18 20 Fig The tracking errors with τ NN 0.4 X direct Y direct Orient Position Error 0.3 0.2 0.1 -0.1 10 20 Time (s) 30 40 Neural Network Weights Y Axis (m) 45 -2 -0.5 -1 -1.5 -2 -2.5 50 16 18 20 w11 w12 w21 w22 0.8 0.6 0.4 0.2 -0.2 -0.4 14 40 Fig Position error with τ NN 12 35 -2.5 Actual Desired 10 Time X Axis (m)(s) 30 -1.5 T 0.5 25 Time (s) -0.5 Below is the graph reflecting the simulation results 20 Fig shows the choosen reference velocity 15 0.5 vr 2 t   25  t  30 : vr  0.15 1  cos  , r   1.5   vr 2 t   30  t  35 : vr  0.15   cos  , r   1.5   t   35  t  40 : vr  0.25 1  cos  , r    35  t  40 : vr  0.5, r  10 Fig Position error without τ NN Process variability of tangent and angular velocity with time is: t    t  : vr  0.25   cos  , r     t  25 : vr  0.5, r  -3 5 10 15 20 25 Time (s) 30 35 40 45 50 Fig Neural network weights with τ NN Fig The tracking errors without τ NN Comparing Fig.6 and Fig.8, we can verify the efficiency of components τ NN ( created by RBFNN) in compensating uncertainties in the model dynamics V CONCLUSIONS This paper proposes a control model using neural networks to compensate the uncertainties of the robot and 1188 assure the global stability of the system Simulation in Matlab-Simulink is consistent with the principles of the proposed control law ACKNOWLEDGMENT This work was supported by Vietnam National Foundation for Science and Technology Development (NAFOSTED) REFERENCES [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] Augie 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