2017 IEEE 6th Data Driven Control and Learning Systems Conference May 26-27, 2017, Chongqing, China Fuzzy Neural Network Based Adaptive Iterative Learning Control Scheme for Velocity Tracking of Wheeled Mobile Robots Xiaochun Lu1,2, Juntao Fei1, Jiao Huang1 1. College of IOT Engineering, Hohai University, Changzhou, 213022, China 2. Jiangsu Key Laboratory of Power Transmission & Distribution Equipment Technology, Changzhou, 213022, China E-mail: Luxc@hhu.edu.cn Abstract: The velocity tracking problem of wheeled mobile robots (WMRs) which work with repeatable trajectories and different initial errors is discussed in the paper. Three mathematical models of WMR, namely, kinematic model, dynamic model and DC motor driven model, are deduced and the stratagem of fuzzy neural network based adaptive iterative learning control (FNN-AILC), which includes the components of fuzzy neural network, approximation errors and feedback, is presented. The proposed scheme can deal with MIMO system, which is distinguished from previous research work. The simulation is presented and the result verifies the effectiveness of the controller. Key Words: Fuzzy Neural Network Based Adaptive Iterative Learning Control, Wheeled Mobile Robot, Velocity Tracking Control 1 proposed[15-18]. In [15], Lipschitz condition was relaxed, and Lyapunov-like approach was adopted. An adaptive iterative learning controller, which contains two adaption laws in both time-domain and iteration-domain and does not use any priori knowledge of robot parameters, was designed in [16]. An AILC algorithm including a parametric estimation and an iterative controller is designed to improve the tracking performance of MEMS gyroscope in [17]. Usually, fuzzy technique and neural network are effective approaches for the adaptive control of nonlinear systems, so a fuzzy AILC (FAILC) was proposed in [18]. In some applications, especially in industrial fields, WMRs usually have the repeated tasks and move along with relatively fixed trajectory, which is the typical application area of ILC. Hence, taking advantage of the feature of iterative learn, ILC can be used to solve the problem of WMR trajectory tracking, which means that the promotion of tracking performance is realized by iteration in stationary time intervals. The combination of WMR trajectory tracking and ILC has received attention. In [19], the discrete dynamic kinematical model of WMR was presented, and an ILC strategy was designed to realize trajectory tracking for WMR. An ILC algorithm based on the kinematical model of farm mobile robots was developed in [20], which also analyze the convergence of the controller. An ILC law with predictive, current and past learning items to solve the high-precision trajectory tracking issue of WMR was derived in [21], and the convergence of the algorithm was proofed rigorously. In [22], a FAILC strategy is proposed to implement the velocity tracking of WMRs, and the Lyapunov-like method was used to proof the convergence of the controller. In terms of literature review, few research reports have covered the combination of ILC and trajectory tracking of WMR, especially based on WMR dynamic model. However, dynamic model of WMRs is more suitable to describe a real robots in mathematics than kinetic model. The paper, which introduce a new FNN-AILC algorithm, is an extension of [22].The proposed algorithm can work with the dynamic model of WMRs. The main features of FNN-AILC with Introduction Due to the features of efficiency and flexibility, WMRs have been widely applied in the fields of military, transportation, and industry, and also drawn much attention of researchers. There are three main researching problems in the motion control of WMRs, i.e. path following [1], trajectory tracking [2], and point stabilization [3]. The requirement in common of path following and trajectory tracking is that WMRs must follow a line or a curve. However, the difference between path following and trajectory tracking is specific velocity requirement at specific location, which is needed in trajectory tracking and not needed in path following. Therefore, solving the trajectory tracking with unknown model parameters has profound significance. Over the past decade, many meaningful research results in this field have emerged. A lot of control methods, such as feedback linearization [4], backstepping control [5], sliding mode control [6], adaptive control [7], neural network technique [8], fuzzy system [9] and model predictive method [10, 11], were proposed based on kinematical model and dynamic model. ILC algorithm, which has the advantage of dealing with nonlinear systems, has been made great progress in mathematical description and convergence analyses. As a branch of intelligent control, ILC can work in simple way, especially in low cost of computing [12-14]. The iterative learning stratage, robustness, convergence and application are four essential problems in research of ILC. Initially, D-type, PD-type and PID-type ILC based on contraction map theory were proposed[12]. Traditional ILC schemes are simple and easy to be realized, but these methods need strict Lipschitz condition[13]. In order to remove some strict conditions for technical analysis, adaptive ILC (AILC) which can handle the issue of resetting condition, input noise and iteration-varying desired trajectory, has been * This work is supported by Open Foundation of Jiangsu Key Laboratory of Power Transmission & Distribution Equipment Technology under grant No. 2015JSSPD06. 978-1-5090-5461-9/17/$31.00 ©2017 IEEE 106 DDCLS'17 ª x º ªcos T 0 º « y » «sin T 0 » ªX º (3) « » « » «Z » ¬ ¼ «¬T »¼ «¬0 1 »¼ For the sake of simplicity, Eq. (3) can be rewritten as: (4) q S (q)V application in the velocity tracking of WMRs are highlighted as follows: 1) The algorithm of FNN-AILC which works on the WMR dynamic model, can improve the velocity tracking performance of WMR. 2) The FNN-AILC controller, which contains a FNN approximation component, an approximation error compensator and a feedback controller, has the advantages of deal with MIMO systems and random initial errors. This paper is organized as follows. In section 2, the kinematical model of WMR, the dynamic model and the DC motor driven model are deduced. The FNN-AILC strategy is derived in section 3. Then the simulation results are manifested in section 4. Finally, the section of Conclusion summarizes the whole thesis. 2 where q [ x, y,T ]T is the generalized velocity vector and V By using of the Euler-Lagrangian formulation, the dynamic model of WMR is formulated as: M (q)q C(q, q)q G(q) E(q)W A(q)O where q [ x, y,T ] is generalized acceleration vector; M (q) 3u3 , C (q, q)q 3 and G(q) 3 denote the symmetric positive definite inertia matrix, the centripetal and Coriolis force vector, the gravitational force vector respectively; E (q) 3u2 is input transformation matrix; The diagram of typical two-wheeled mobile robot is shown in Fig. 1. This WMR, which has two independent driving wheels on the same axis, can move in the coordinate system for the inertial frame OXY . The center of mass of the robot locates at point C , which is also the center of driving axle.The location of the robot is at point C with coordinates ( x, y) and the heading angle is T . The length of driving axle and the radius of driving wheels are 2R and r , respectively. The vector of generalized coordinate is defined as q [ x, y,T ]T . W [W L ,W R ]T denotes the input torque vector; A(q)O 3 denotes the constraint forces vector, and O is the Lagrange multiplier. Assume that WMR move on a plane, so its potential energy is constant. Considering that the WMR mass is m and the moment of inertia of WMR is I , and ignoring the moment inertia of wheels, the total kinetic energy is 1 1 m( x 2 y 2 ) IT 2 , and the other parameters are 2 2 ªm 0 0º M (q) «« 0 m 0 »» , C (q, q) 0 , G(q) 0 , «¬ 0 0 I »¼ Y X Y E (q ) C O x X cos T º sin T »» . So, Eq. (5) can be rewritten as: R »¼ q Fig.1 The diagram of WMR S (q)V S (q)V S( (7) T Substitute Eq. (7) into Eq. (6), and multiply by S on both sides of Eq. (6), then we get S T MSV SV S T MS MSV V S T EW S T AO (8) ªm 0º 1 ª 1 1º S 0 , ST E where S T MS « , S T MS , » r «¬ R R »¼ ¬0 I¼ Assuming that the contact of driving wheels and ground satisfies the condition of "pure rolling without sliding", the model of two-wheeled mobile robot can be simplified as unicycle-type mobile robot. The nonholonomic constrains of the WMR, which implies that the robot cannot move sideways, is expressed as: (1) x sin T y cosT 0 Eq. (1) can be written in the form of vector: ª xº T [ i T , cos T , 0] «« y »» A (q)q [sin ¬«T »¼ ªcos T 1« sin T r« «¬ R (6) M (q)q E(q)W A(q)O With the purpose of omitting the Lagrange multiplier O , the derivative of Eq. (4) can be written as 2r 2R (5) T WMR model y [X , w]T is the input vector of WMR kinematical model. 0 ST A 0 . Define M S T MS , B S T E , and substitute the definition into Eq.(8), then we get the simplified formulation: MV V BW (9) The wheels of WMR are driven by DC motors with mechanical gears, and electrical equation of motor armature is formulated as: (2) The linear velocity of WMR at point C is defined as X , and angular velocity at the same point is defined as Z . Then, the kinematical model of WMRs can be formulated as: u 107 L di Ra i KbZm dt (10) DDCLS'17 where u is input voltage, L is armature inductance, Ra is clear that I j (t ) is a monotonic decreasing function in the iterative period [0, T ] . Define the error function as: ª S Ij1(t ) º j i j i (16) ( t ) SI « j » e (t ) S (t ) I (t ) ( t ) ¬S I 2 ¼ j ª º e1 (t ) 0 « Sat ( j ) » ( ) t I 1 » , and sat () is where S j (t ) « j « » ( ) t e 0 Sat ( 2j ) » « I 2(t ) »¼ «¬ saturation function which is defined as j j 1 , e i (t ) !I i (t ) ° j j ° e (t ) e i (t ) j j Sat ( ij ) ® , (i 1, 2) e i (t ) d I i (t ) I i (t ) ° I ij (t ) j j ° 1 , e i (t ) I i (t ) ¯ armature resistance, i is armature current, Zm is angular velocity of motor and K b is back electromotive force constant. With the ignorance of armature inductance, considering relations between angle velocity and torque before gear and after gears ( Zw Zm / n , W n W m )and the relation between armature current and motor torque ( W M KW i ), we get the following expression: where K1 ªW l º ª ul º ªZl º «W » K1 «u » K 2 «Z » ¬ r¼ ¬ r¼ ¬ r¼ nKW / Ra , K2 nKb K1 . (11) According to the relation of velocity vector V and angle velocity of wheels Zw , we have the formulation of 1 º ªZl º rª 1 « » , which can be rewritten as: « 2 ¬ 1/ R 1/ R »¼ ¬Zr ¼ ªZl º 1 ª1 R º ªX º (12) «Z » « »« » ¬ r ¼ r ¬1 R ¼ ¬Z ¼ Substitute Eq. (12) into Eq. (11) , then we have ªW l º ª ul º K 2 ª1 R º ªX º (13) «W » K1 «u » « »« » ¬ r¼ ¬ r ¼ r ¬1 R ¼ ¬Z ¼ ªX º «Z » ¬ ¼ . By use of the definition of saturation function, the error function can be rewritten as e ij (t ) I ij (t ), e ij (t ) !I ij (t ) °° j j . When S Iji (t ) 0 , S I i (t ) ® 0ˈ e ij (t ) d I i (t ) ° j j j j °̄e i (t ) I i (t )ˈ e i (t ) I i (t ) Substitute Eq. (13) into Eq. (9) , and multiply by M -1 at the both sides of Eq. (9), then we get the dynamic model of WMR combining the DC motor model, which is formulated as V 3 t 0, T @ . Because of the feature of the monotonic deceasing function I ij (t ) which will converge to zero rapidly in the 0 º 2 K 2 ª1/ m K ª 1/ m 1/ m º 2K V 1« u » 2 2 « R / I¼ r ¬ R / I R / I »¼ (14) r ¬ 0 f (V ) Bu condition of large value of k , e ij (t ) can converge to zero in the time period 0,T @ . Now, the time derivative of T j j S I (t ) S I (t ) can be deduced as : d ( S Ij (t )T S Ij (t )) 2 S Ij (t )T S Ij (t ) dt 2 S Ij (t )T (e j (t ) I j (t )) ° 0 ® ° j T j j ¯2S I (t ) (e (t ) I (t )) ª sgn(e1j (t )) º j ªsgn 0 2S Ij (t )T (e j (t ) « » I (t )) j sgn(e 2(t )) ¼ 0 ¬ FNN-AILC controller design In the view of ILC system, the dynamic model of WMR can be rewritten as: (15) V j (t ) f (V j (t )) Bu j (t ) where j is the times of iteration, and t [0, T ] . Define the desired velocities as Vd >Xd Zd @ . What T this paper concerns is the tracking problem of WMR. Therefore, the control objective is to design an appropriate control strategy to make WMR tracking the desired velocities Vd asymptotically when the iteration times trends to infinity. The velocity tracking errors can be defined as ª X j Xd º j ªe j º 2u1 , then the e j « 1j » V j Vd « j », e Z Z e d¼ ¬ 2¼ ¬ derivative of velocity tracking errors are e j V j Vd , j 2S Ij (t )T [k e j (t ) sgn(e j (t )T )I (t )] 2S Ij (t )T (V j Vd k e j (t )) j 2k S Ij (t )T S Ij (t ) 2k S Ij (t )T I j (t ) 2 S Ij (t )T I (t ) 2 S Ij (t )T [ Bu (V d f (V j ) k e j (t ))] 2k S Ij (t )T S Ij (t ) 2 S Ij (t )T [ Bu (V d f (V j ) k e j (t ))] where e j 2u1 . In order to handle the problem of random initial errors, design the time-varying boundary layer function as I j (t ) ª¬I 1j (t ) I 2j (t ) º¼ T If Bu =Vd f (V j ) ke j (t ) , d ( S Ij (t )T S Ij (t )) 2k S Ij (t )T S Ij (t ) d 0 , which implies that dt the S Ij (t ) is decreasing function, and S Ij (t ) 0 in the time H j e kt , where T period > 0,T @ since S Ij (0) 0 . However, f (V j ) in dynamic model of WMR is unknown. Usually, fuzzy logic and neural networks can be used to approximate unknown function. Each of them has special ª¬H 1j H 2j º¼ is the absolute value of initial errors in No. j iteration, H 1j ! 0 , H 2j ! 0 , H j 2u1 . It is Hj e j (0) (17) 108 DDCLS'17 properties that make them suited for particular problems and not for others. So, some intelligent hybrid methods which combine two or more techniques, such as fuzzy neural networks, are created to overcome the limitation of individual technique. In this paper, we use FNN technique to approximate f (V j ) . The structure diagram of FNN is illustrated in Fig. 2. The FNN has four layers, including input layer, premise layer, rule layer and output layer. Vd ke j (t ) UC (21) where J ! 0 ˈ T (t ) denotes approximation errorˈ U A is the component of FNN adaptive iterative learning 2u1 j controller, U B is the component of approximation error compensator, and U C is the component of feedback controller. Fig.3 Block diagram of FNN-AILC for WMR For the purpose of ensuring the controller’s convergence along with both time axis and iteration axis, define the parameter errors as Fig. 2 The architecture of FNN j W (t ) T ª¬ x1j x2j º¼ [X j Z j ]T 2u1 as input signal in the input layer of FNN. In premise layer, we choose ( x m )2 Gaussian function exp{ i 2 il } . Define X j V j j j T (t ) ª«T 1 (t ) T 2 (t ) º» ¬ T* il ª¬ L(3)1 ( X j ) L(3)2 ( X j ) L(3) ( ) M ( X j ) º¼ M u1 is the fuzzy basis function vector in rule layer. The FNN has two output signals L(3)( X j ) ª¬ L(4)1 ( X j ,W j ) L(4)2 ( X j ,W j ) º¼ 2u1 , which can be further written as: ªW jT L(3)( X j ) º L(4)( X j ,W j ) « 1 jT W jT L(3)( X j ) (18) j » ¬W2 L(3)( X ) ¼ W j (t )T L(3)( X j (t )) T j (t ) U B j j j ¬ªW1 W2 ¼º ª¬W W º¼ , and then we have f (V j (t )) L(4)( X j (t ),W * ) H ( X j(t )) Ǆ Define virtual input as U Bu Vd ke k j (t ) U A U B UC , U W 1 J (t ) W j (t ) J SIj (t )T L(3)( X j (t )) j 1 (t ) T j (t ) J SIj (t ) W j 1 (t ) proj (W ˈand we (t )) ª proj (T 1 j 1 (t )) º « » « proj (T 2 j 1 (t )) » ¬ ¼ (25) (26) where proj () is mapping function and defined as: c if c(t ) t c °° proj (c(t )) ®c if c(t ) t c ˈ c! 0 . ° c(t ) otherwise °̄ Now, the result is presented as follows. (19) SIi (t )( L(3)( X j )T L(3)( X j )) SIi (t ) 2 2 j 1 ª proj (W 11 (t )) ... proj (W 1n j 1 (t )) º « » « proj (W j 1 (t )) ... proj (W j 1 (t )) » 21 2 n ¬ ¼ B (U Vd ke (t )) . U A , U B J (24) j 1 j W j (t )T L(3)( X j ) T j (t ) (23) and and U C are designed as˖ UA j 1 T j 1 (t ) 2u1 (22) j T * 2 can get the real input u UB U A U B L(4)( X i (t ),W * j (t )) H ( X i (t )) W (t )T L(3)( X j (t )) T (t ) U B Design the adaptive laws of parameters along with iterative axis as: H 2 ( X j (t ) d H , and optional weights are defined as W T j * 2 * 1 T j (t ) T * , where d W (t )T L(3)( X j (t )) T * T j (t ) U B T * ¼ W j (t ) W * , W *T L(3)( X j (t )) H ( X i (t )) w11MM º ª w11 w12 M u2 «w » w w ¬ 21 2M 22 2 M ¼ are the adaptive parameter vectors in output layer. The optional approximation error is defined as ª H ( X j (t ) º H ( X j (t )) « 1 j » , where H1 ( X j (t ) d H1* , ¬H 2 ( X (t ) ¼ where W j T T ª¬H1* H 2* º¼ . U f (V i ) U C T L(4)( X j ,W j ) ªW 1 j (t ) W 2 j (t ) º «¬ »¼ (20) 109 DDCLS'17 Theorem 1. If the WMR dynamic model (Eq. 15) satisfies the assumption of known and invertible input gain matrix B , and the FNN-AILC controllers (Eq. 19, Eq. 20 and Eq. 21) and the parameter adaption laws(Eq. 23-26) are adopted, the following conclusions can be obtained. (1) The parameters of fuzzy system ( W j (t ) , T j (t ) ) are bounded; (2) Error function S Ij (t ) will converge to zero when iteration times trends to infinite, i.e. lim S Ij (t ) j of 0; (3) Tracking errors eij (t ) will converge to zero rapidly after first time if k is large enough. 4 Fig. 4 Linear velocities in 5 times of iterations Simulations For the purpose of demonstrating the effectiveness of the FNN-AILC method for WMR velocity tracking, Matlab/Simulink is utilized for the numerical simulation. Parameters of WMR are listed as: m 15kg , I 10kgm2 , 2R 0.3m , r 0.1m , K1 5.2 , K2 2.3 . In this example of simulation, the desired velocity trajectories can be described as follow: ªX (t ) º ª1.2 1.1sin(t S / 2) º Vd (t ) « d » « » and t >0, 20@ . 0.3sin(0.4t ) ¼ ¬ wd (t ) ¼ ¬ In the FNN system, The Gaussian functions are chosen as ( x m )2 exp{ i 2 il } , which are also called membership V Fig.5 Angular velocity in 5 times of iterations il functions and have five rules for xi . The centers of Gaussian functions are set as [m11 [m21 m25 2 ] m15 1 ] Five times of velocity tracking curves in time interval [0, 20s] are illustrated in Fig. 4 and Fig. 5. It is obviously that the velocity tracking cures converge asymptotically to the desired velocity curves(black) in the presence of different initial states when the iteration times keep increasing. In the No. 5 iterative learning control, actual velocity curves(red) can not converge to the desired curves(black) in the first two seconds because of different initial states, and then the velocity curves almost overlap the desired velocity curves. >0,0.6,1.2,1.8, 2.4@ and >1, 0.5,0,0.5,1@ , and the variances of Gaussian functions are V1l 6 and V 2l 4 . The initial value of FNN-AILC parameters are set as T W 1 (t ) T 1 (t ) ª0.02 ... 0.02 º 1 «0.02 ... 0.02 » and W (t ) ¬ ¼ >0.02 25u2 , 0.02@ . The upper bounder c of project T function is 1. The random initial errors e j (0) step from the 5 random initial states V j (0) . In 5 times of iterations, Conclusions In this paper, the models of two-wheeled mobile robots are deduced in detail, which include the kinematical model, dynamic model and DC motor model. Then, in order to deal with the random initial errors of velocity and the unknown parameters in mathematic model, the FNN-AILC strategy is proposed. The controller for velocity tracking, which contains three components (FNN approximation component, approximation error compensator and feedback control component) and two adaptive laws in iterative direction, can work on MIMO dynamic model. The effectiveness and convergence of the control method is verified by the simulation results. V j (0) are set as ª0.3 0.2 0.15 0.25 0.22 º «0.1 0.1 0.09 0.15 0.05» . ¬ ¼ When k 0.9 and J 0.9 , the velocity tracking curves are shown in Figs.4-5 which show the good performance in velocity tracking. ª¬V 1 (0) V 5 (0) ( )ºº¼ References [1] 110 C. B. Low, and D. W. Wang, Maneuverability and path following control of wheeled mobile robot in the presence of wheel skidding and slipping, Journal of Field Robotics, 27(2): 127–144, 2010. DDCLS'17 [12] D. A. Bristow, M. Tharayil, and A. G. Alleyne, A survey of iterative learning control, IEEE Control Systems Magazine, 26(3), 96–114, 2006. [13] H. S. Ahn, Y. Chen, and K. L. 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