Trajectory Tracking Control of the Nonholonomic Mobile Robot using Torque Method and Neural Network T T Hoang, D T Hiep, B G Duong and T Q Vinh Department of Electronics and Computer Engineering University of Engineering and Technology Vietnam National University, Hanoi thuanhoang@donga.edu.vn Abstract— This paper deals with the problem of tracking control of the mobile robot with non-holonomic constraint A controller with two control loops is designed The inner loop generates control laws for the tangent and angular velocities to control the robot to follow the target trajectory It is derived based on the robot kinematics and the Lyapunov theory The outer loop employs the torque method to control the robot dynamics A neural network is implemented to compensate the uncertainties caused by the dynamics model The asymptotic stabilization of the whole system is proven by direct Lyapunov stabilization theory Simulations in MATLAB confirmed the validity of the proposed method 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 978-1-4673-6322-8/13/$31.00 c 2013 IEEE noises, computstion roundoff error, and sampling delay also need be considered Thus, the problem of kinematics and dynamics control of nonholonomic system is challenging 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 However, we find it difficult to implement closely the dynamics model of the mobile robot due to the non-quantified parameters of dynamics as well as the usually-variable robot’s load These errors are mainly caused by the uncertainties of such a model and the nonparameters, which is composed of: 1-high frequency unmodeled dynamics, such as actuator dynamics or structral vibrations; 2- measurement noise; 3-computation roundoff error and sampling delay Thus, the problem of kinematics and dynamics control of non-holonomic system is absolutely challenging The typical method is to solve this is considered as “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 implemented neural network to compensate the uncertainties such as the combination between the backstepping method and the neural network in [9,11] In [21], a combination between the RBFNN controller and the sliding-mode techniques for the path following task of an omnidirectional wheeled mobile manipulators was applied The asymptotically stabilization were proven theoretically and experimentally 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 model If the dynamics model contains non-parameter uncertainties, the asymptotically stabilization is then not assured In this paper, the tracking control algorithm based on the torque method is presented The controller is of feedforward being in combination with proportional type The nonparameter uncertainties and dynamics model errors are compensated by using the RBFNN The asymptotic 1798 stabilization of the whole system is proven by the Lyapunov theory Our main job is that 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 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, This controller is designed by the combination between the feedforward, the scale techniques, and compensatednonparameter-RBFNN type The paper is organized as follows Section briefly is introduced the kinematics and dynamics of the mobile robot Section is described the process to design the controller Section is presented the simulation results and finally is section II THE KINEMATICS AND DYNAMICS OF NONHOLONOMIC MOBILE ROBOT Yp Xp 2R P 2r O X Figure A noholonomic mobile robot platform Prom [9-10], we have: + CȦ = IJ MȦ (1) where ( ) ª r mR2 + I « + Iw « R2 M=« 2 ô r mR I ô 4R ( ) and IJ = [ τr Ȧ = [ ωr ( r mR2 − I ) º ª r mc d θ º » « » » 4R R2 » ;C = « » « r mc d θ » r mR2 + I » » «− + Iw » 2 ¬ ¼ R ¼ 4R ( l ω r ω ( º θω l » −θω rẳ ) êĐ r mR2 + I ã + Iw p = ôă ôă 4R ơâ ( Đ r mR2 I ă ă 4R â ) áã Đ r mc d ã ă áằ âă 4R2 ạáằ ẳ T (4) THE CONTROLLER DESIGN A Outer control loop Let the tangent and angular velocities of the robot be v and ω respectively We have: ª1 R º ª x êcos êr ô r r ằ ª v º « » « » ªvº Ȧ=« »=« (5) » « » ; q = « y » = ôsin 0ằ ô ằ ơẳ l ẳ ô R ằ ơẳ ôơ ằẳ ôơ0 ằ 1ẳ ơô r r ẳằ The target of the control problem is to design an adaptive control law in which the position vecctor q and the velocity q to hold the position vector qr (t) , and the demanded velocity q r (t) in terms of unknown dynamics parameters of the mobile robot The desired position and velocity vectors are represented: d ê =ô r Y(,) l III A typical example of a nonholonomic mobile robot is shown in Fig [9-11] Y axis through P, the wheel with the actuator about the wheel axis respectively We can rewrite the system dynamic Eq.(1) into a linear form, [9]: =IJ (2) Y(Ȧ,Ȧ)p (3) Y(Ȧ,Ȧ)p = MȦ + CȦ where p is a 3x1 vector consisting of the known and unknown robot dynamics, such as mass and moment of is a 2x3 coefficient matrix consisting of the inertia; Y(Ȧ,Ȧ) known functions of the robot velocity Ȧ and acceleration which is referred as the robot regressor For the mobile Ȧ robot shown in fig.1, we could easily compute: ) τl ] is the torque applied on the wheels; T ωl ] is the angular velocity of the right and left T wheels in order; m = mc + 2mw , in which mc is the mass of the mobile robot platform, mw is the mass of one driving wheel with the actuator; I = mc d + 2mw R + I c , in which I c , I w are moments of inertia of platform about the vertical qr = [ xr yr θr ] T xr = vr cos θ r (6) with vr > for all t y r = vr sin θ r θr = ωr The position tracking error between the reference and the actual robot could be expressed in the mobile robot’s coordinator as follows [9-10]: ª e1 º ª cos θ sin θ º ª xr − x º e p = ««e2 »» = «« − sin θ cos θ »» «« yr y ằằ (7) ôơ e3 ằẳ ôơ 0 ẳằ ơô r ẳằ In this paper, we choose the control law for ν , ω like: 2013 IEEE 8th Conference on Industrial Electronics and Applications (ICIEA) 1799 ª vr cos e3 + k1e1 º ªvº « » (8) «ω» = « ω + k e + v e sin e3 » 3 r ẳ ô r ằ e ẳ where k1 , k3 > In this law, when e3 → then sin e3 → , and ω always be bounded e3 ª e1 º ªω e2 − v + vr cos e3 º e p = «e2 » = « −ω e1 + vr sin e3 » (9) ô ằ ô ằ ôơ e3 ẳằ ôơ ằẳ ωr − ω With the control law in equation (8), it is easy to prove asymptotical stability system due to ep → when t → ∞ Choosing a positive definite function V p as follows: ( 1 V p = epT ep = e12 + e22 + e32 2 The derivation of Vp with respect to time Vp is: ) (10) Vp = eTpe p = e1e1 + e2e2 + e3e3 = e1 ( ωe2 −v + vr cos e3 ) + e2 ( −ωe1 + vr sin e3 ) + e3 ( ωr −ω) (11a) = e1 ( −v + vr cos e3 ) + e2vr sin e3 + e3 ( ωr −ω) Replacing (8) into (11a), we have Vp = e1 ( −v +vr cose3 ) +e2vr sine3 +e3 ( ωr −ω) § sine · = e1 ( −vr cose3 −k1e1 +vr cose3 ) +e2vr sine3 +e3 ăr r k3e3 vre2 e3 ¹ © (11b) =−k1e12 −k3e32 It is straightforward to see that Vp is continuous and bounded according to the Barbalat theorem, which means that Vp → when t → ∞ , and e1 → 0, e3 → when t → ∞ According to Barbalat theorem, we have: e1 → 0, e3 → (12) and the equation (8) becomes v → vr (13) ω → ωr (14) Combining (7), (12), (13), (14) infers: e2 → 0, e2 → Thus the control law (8) assures the proximity control system e p → when t → ∞ B Inner control loop The deviation of stick angular velocity of driven wheels is: angular velocity using computed-torque method which is satisfied ec → and ep → when t → ∞ Derivating and multiplying both sides of equation (15) with the matrix M, we obtain: -Ȧ c ) = IJ - Cec - Yc p Me c = M(Ȧ (16-a) In case of non-parametric uncertainty component d in the mobile robot of dynamics model, equation (16-a) is rewritten as below: -Ȧ c ) = IJ - Cec - Yc p + d (16-b) Me c = M(Ȧ c + CȦ c Yc p = MȦ (17) Where Torque output of the controller is: IJ = IJ NN - K Dec + Yc pˆ (18) where K D is the positive definite matrix and pˆ is the matrix esimated by the matrix p Substituting (17) into (16), we have: Me c = IJ NN - (K D + C)ec + d + Yc p (19) with p = pˆ - p Because d is unknown parameter, it then could be considerd as uncertainty component We are able to approximate this by a finite neural network [3,4,6] as follows: d = Wı + İ = dˆ + İ (20) where, where W is the weight matrix of an online updated network; İ is the approximate error and is bounded İ ≤ ε0 by The neural network Wı is approximated by Gaussian RBF network consisting of three layers: input layer, hidden layer with n nodes that contains the Gaussian function, and the output layer with linear function of n neurons (Fig 2) The RBF network structure satisfies the conditions of the Stone-Weierstrass theorem Hidden layer neuron is the Gaussian function with the form: σj (s = exp− j − cj ) λ j2 ; j = 1, 2, n where, c j , λ j are the expectation and variance of the Gaussian function chosen in [17] And we must determine the parameters c j , λ j differently , which cover the uncertainty d in terms of amplitude and frequecy The output dˆ is an approximation of d e Choosing IJ NN = (η + 1) Wı − δ c to satisfy our control ec law ªω − ωcr º ec = Ȧ - Ȧ c = « r (15) ằ l cl ẳ where is the desired angular velocity of the robot calculated by (5), Ȧc is the output of the angular velocity control wheel torque We must find the control law of the 1800 (21) 2013 IEEE 8th Conference on Industrial Electronics and Applications (ICIEA) Because of IJ NN = (Ș + 1)Wı - į ec and the matrix C is ec symetric, ecT Cec = So that: ª e º V = −eTc K D ec + ecT «-δ c - İ ằ ằẳ ơô ec ecT K D ec - δ ec + ec İ ≤ −ecT K D ec - δ ec + ec ε If we choose δ = ε + γ with γ > then V ≤ −ecT K D ec - Ȗ ec Figure The approximate neural network of W σ function Theorem: The dynamics of the mobile robot (1) using neural network (20) will be tracked the desired trajectory qd with the error ep → , if we choose the control algorithm i as follows: IJ and learning neural network w IJ = -K Dec + (η +1) Wı - δ ec + Yc pˆ ec i = −η ecσ i w σ j = exp− (22) ( ecj − c j ) IV λ j2 where the optional parameter K D is a symmetric positive definite matrix, η , δ > Figure Mobile robot control based on torque method with online learning neural network Proof: Selecting a positive function V as follows: ã 1Đ V = ă ecT Mec + p T -1p + Ư w Ti w i 2â i=1 ¹ (23) Derivating V with respect to time, we obtain: V = ecT Me c + pˆ T ī-1p + Ư w Ti w i i=1 Đ ã = ă Mec - Yc p - Ư w i i=1 â From (19), we achieve: V = ecT [ IJ NN - (K D + C)ec - d - ȘWı ] (25) From equation (24), (25), we can see that V → so that ec → , p → and ep → , p → p The control system in Fig is the asymptotic stability q → qd Otherwise, the error between the tracking trajectory and the desired one is closely to zero Additionally, the proposed controller exactly estimate the required parameters of mobile robot dynamics ( pˆ → p ) Both theorem and the global asymptotic stability of the system with torque control using neural network depicted in Fig have proved SIMULATION RESULTS In our scenario, we simulates the mobile robot model with the following paramters: 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: K D = diag(5,5); k1=k3=2; δ = 10 Suppose that we only estimate pˆ = 0.6 p and non-parametric uncertainty components are: d = êơsin(0.25t ) cos ( 0.25t ) ẳ t · § ≤ t < : vr = 0.25 ă cos , r = â t < 25 : vr = 0.5, ωr = T vr 2π t · § 25 ≤ t < 30 : vr = 0.15π ¨ − cos ¸ , ωr = − 1.5 â vr t ã Đ 30 t < 35 : vr = 0.15 ă cos , r = 1.5 â t § · 35 ≤ t < 40 : vr = 0.25 ă1 + cos , r = â 35 t < 40 : vr = 0.5, ωr = (24) eTc = −eTc K D ec - ecT Cec + ecT [ IJ NN - Wı - İ - ȘWı ] 2013 IEEE 8th Conference on Industrial Electronics and Applications (ICIEA) 1801 The simulation results are presented in below figures: (m, rad) A Using the neural network with the parameter estimation algorithm of dynamics model: 0.5 -0.5 3.5 Actually Trajectory Desired Trajectory -1 2.5 -1.5 10 15 Y (m ) 20 25 30 Time (s) 35 X direct Error Y direct Error Orient Error 40 45 50 Figure The position error without τ MN 1.5 0.5 0 X (m) 10 12 Comparing Fig.5 and Fig.8, we can verify the efficiency of components IJ NN ( created by RBFNN) in compensating uncertainties in the model dynamics Figure The desired trajectory and the actual trajectory with τ MN -0.5 X direct Error Y direct Error Orient Error -1.5 10 15 20 25 Time (s) 30 35 40 45 CONCLUSION This paper proposes a control method for trajectory tracking of nonholonomic mobile robot The main contribution is the proposal of a controller with two control loops One is for the kinematics The other is for the dynamics In addition, a neural network is introduced to deal with uncertainties of the dynamics model The global stability of the system is proven The simulation results confirmed the effectiveness of the method 0.5 -1 V ACKNOWLEDGMENT 50 This work was supported by Vietnam National Foundation for Science and Technology Development (NAFOSTED) Figure The position error with τ MN Neural Network Weights w11 w12 w21 w22 0.8 0.6 REFERENCES [1] 0.4 0.2 [2] -0.2 -0.4 10 15 20 25 Time (s) 30 35 40 45 50 Figure The neural network weights with τ MN [3] B Without the neural network with the paramter estimation algorithm of mobile robot: [4] 3.5 Actually Trajectory Desired Trajectory [5] Y (m) 2.5 [6] 1.5 [7] 0.5 0 [8] X (m) 10 12 Figure The desired trajectory and the actual trajectory without τ MN 1802 [9] Augie Widyotriatmo, Keum-Shik Hong, and Lafin H Prayudhi, “Robust stabilization of a wheeled vehicle: Hybrid feedback control design and experimental validation,” Journal of Mechanical Science and Technology 24 (2) (2010) 513~520 Thuan Hoang TRAN, Manh Duong PHUNG, Van Tinh NGUYEN and Quang Vinh TRAN, “A Path Following Algorithm for Wheeled Mobile Robot Using Extended Kalman Filter ,” 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Applications, 6, 662-668, 2012 [20] Ngo, Q H., and Hong K.-S (2012), “Sliding-Mode Anti-Sway Control of an Offshore Container Crane”, IEEE/ASME Transactions on Mechatronics, 17, 201-209 [21] Dong Xu et al., “Trajectory Tracking Control of Omnidirectinal Wheeled Mobile Manipulators: Robust Neural Network-Based Sliding Mode Approach”, IEEE Transactions on Systems, Man, and Cybernetics-B,39, 788-799 2009 2013 IEEE 8th Conference on Industrial Electronics and Applications (ICIEA) 1803 ... a control method for trajectory tracking of nonholonomic mobile robot The main contribution is the proposal of a controller with two control loops One is for the kinematics The other is for the. .. γ > then V ≤ −ecT K D ec - Ȗ ec Figure The approximate neural network of W σ function Theorem: The dynamics of the mobile robot (1) using neural network (20) will be tracked the desired trajectory. .. mc is the mass of the mobile robot platform, mw is the mass of one driving wheel with the actuator; I = mc d + 2mw R + I c , in which I c , I w are moments of inertia of platform about the vertical