Adaptive dynamic surface trajectory tracking control for the four-Wheeled omnidirectional mobile robot

The Backstepping feedback method is a viable solution to solve affined models [58] and [59]. However, with high order nonlinear systems, the computational volume is large, complicated, and takes too many computation time due to the need to calculate the derivative in each iteration step. 4 Sliding mode control (SMC) has also been used [60], [61], [62] and [63 for its superior properties in the case of the system affected by noise. However, the limitation of the SMC algorithm is chattering, and reducing this phenomenon requires an accurate object model. It goes against the properties of the robot model, which is the parameter uncertainty. In order to improve the quality of control as well as to limit some of the disadvantages of the Backstepping and SMC controllers, a dynamic surface control (DSC) technique is introduced in [64] and [65]. The design steps are similar to those of the Backstepping, but to avoid derivative steps for the DSC virtual control signal, a low pass filter is added, just to get information about the lead. medium function to filter the high-frequency internal noises occurring in the control object [65]. For OMR, it is challenging to build an accurate mathematical model because factors such as friction, load change, and environmental conditions are not known. Therefore, the effective modern design methods, in this case, are to use adaptive algorithms to tune controller parameters using Fuzzy logic or approximate the uncertainties using neural networks. This adaptive controller significantly improves the quality of the nonlinear dynamics [60], [61], [62], [66], [68], [69], [70], [71] and [72]. With the above reference and analysis, a new adaptive control structure based on a radial basis function neural network (RBFNN) and fuzzy logic system for the trajectory tracking controller is researched and developed based on the Dynamic Surface Control (DSC) technique. A novel adaptive controller with RBFNN for the approximation of the nonlinear uncertain parameters of the FWOMR and fuzzy logic to tune the controller's parameter is proposed in the thesis. 1.4. Conclusion Chapter 1 presented an overview of mobile robot classification and autonomous mobile robots, which focuses on an autonomous mobile four-wheeled robot (FWOMR) being the main research object of the thesis. Chapter 1 also focused on a research overview of domestic and international research on OMR modeling and trajectory tracking control algorithms for OMR published and analyzed the advantages and disadvantages of these methods from which to draw appropriate research directions for the thesis. 2. MODELING AND TRAJECTORY CONTROL ALGORITHMS FOR THE FOUR-WHEELED OMNIDIRECTIONAL MOBILE ROBOT Building the system of kinematic and dynamic equations for OMR is the very first problem needed for the synthesis of the trajectory tracking control. In the thesis, the research object considered is an autonomous four-wheeled robot using Omni-type wheels, which moves on the plane is affected by friction. 2.1. Building the kinematic and dynamic models of the four-wheeled omnidirectional mobile robot 2.1.1. The Omni wheel Omni wheels are arranged perpendicular to the axis of the motor, the wheels are spaced 3600 / n apart. Omni wheels are widely used in autonomous robots because it allows the robot to move immediately to a position on a plane without having to rotate before. Furthermore, the translational movement along a straight trajectory can be combined with 5 rotational movement that causes the robot to move to the desired position with the accuracy orientational angle. 2.1.2. Kinematic model of the four-wheeled omnidirectional mobile robot [41], [42] An equation presenting a relationship between the two coordinates is also the robot kinematic model. cosθ sinθ sinθ cosθ - 0 0 0 0 1           q Hv v (2.1) where: cosθ sinθ sinθ cosθ - 0 = 0 0 0 1          H is a transition matrix. From (2.1), we calculate an equation presenting a relationship between the robot’s position and the velocity of wheels: 1 2 3 4 ( ) x y g                              với 2( )g   HH (2.4) 2.1.3. Dynamic model of the four-wheeled omnidirectional mobile robot [41], [42] The kinematic and dynamic models of FWOMR are constructed based on a robot model accompanied by the Omni wheels which are positioned 450 apart from the dynamic coordinate and 900 apart from the beside ones. From which, the robot’s dynamic equation has the following formula ( ) sgn( )   M q v Cv G v τ Bτ d (2.8) with: [ ]Tx yv v v is a velocity vector 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 r r r r r r r r d d d d r r r r                      B is a control parameter matrix. 0 0 ( ) 0 0 0 0 m q m J          M is a matrix with m is the robot mass and J is the inertia moment. 0 0 0 0 0 0 x y B B B          C and 0 0 0 0 0 0 x y C C C          G are viscous friction parameter matrix and Coulomb friction matrix, respectively. 6 2.2. Several existing trajectory tracking control algorithms for the four-wheeled omnidirectional mobile robot. 2.2.1. PID controller for FWOMR The PID controller for FWOMR is proposed in [43] and [44]. These studies have designed the PID controller based on the kinematic model of OMR. Hence, the effects of external forces on the system in the robot's dynamic equation were not taken into account. 1 2 1 3 4 ( ) ( ) ( ) ( ) d d d d d d x t x x y t y g y t                                                e           (2.11) We need to find the angular velocity vector of the wheels for the closed-loop controller to be stable. 01 2 1 3 0 4 0 ( )( ( ) ( )) t e e t T T P e I e e t e x d x g g g K y K y d d                                                                 (2.12) with PK and IK are diagonal positive definite matrices. 2.2.2. Sliding mode control for FWOMR SMC in [60], [61], [62] and [63] is commonly used for robot systems in general and FWOMR in particular because of its robust characteristic with external noises. From (2.1) and (2.8), let 1 2    x q x v , we have state equations: 1 2 2 2 2sgn( ) d       x Hx Mx Cx G x τ Bτ   (2.19) with dτ is uncertain and not accurately measured, and thus, this component will not exist during the calculation of SMC, MSSC controllers. Define the sliding surface with conditions and assumptions. Define the errors 1 1 1d 2 2 2d      e x x e x x with 1dx is a reference trajectory and 1 2d 1d x H x is reference velocity. Choose the sliding surface 1 1 S e e (2.20) with  >0 is a sliding surface coefficient. Take the sliding surface’s derivative 1 1 2 2 1 2 2 2 2(M ( sgn( )) ( ) )d           S He He e H Bτ Cx G x x H H e     (2.21) Choose a Lyapunov candidate function 7 21 2 V  S (2.22) Take its derivative, we obtain 1 2 2 2 2( ( sgn( )) ( ) )dV         -1SS SH M Bτ Cx G x x H H e  (2.23) With the control signal is chosen as follows 1 1 2 2 2 2 1( ) ( (( ) ) sgn( ) sgn( )) T T d K        τ B BB M H H e x Cx G x S  (2.24) 1sgn( ) 0V K  S S  , which satisfies the Lyapunov standard. The sliding controller (2.24) is designed for stability and durability when the system exists with model deviation and impact interference. The function V in the formula (2.22) with control law (2.24) for the FWOMR system is the Lyapunov function of the closed system. 2.2.3. Multiple sliding surface control for FWOMR - Consider the robot’s state equations 1 2 2 2 2sgn( ) d       x Hx Mx Cx G x τ Bτ   (2.36) with 1 x y           x và 2 x y v v           x - Consider the sliding surface 11 1 12 1 1 13 d S S S           S x x (2.37) - Take derivative of 1S and use (2.37), we obtain 1 1 1 2 1 d d   S x x Hx x     (2.38) Choose a virtual control signal 1 2 1 1 1( )d dK   x H S x (2.39) - Choose the first Lyapunov candidate function 1 1 1 1 2 TV  S S (2.40) - Take derivative of 1V , and use (2.38) and (2.39) 1 1 2 1 1 1 T TV K S S S S (2.41) - Consider 2S as the second sliding surface 2 2 2 )( d S x xH (2.42) Taking derivative of 2S 2 2 2 2 2 1 2 2 2 2 2 ( ) ( ) ( ( sgn( )) ) ( ) d d d d            S H x x H x x H M Bτ Cx G x x H x x     (2.43) Combine (2.39), (2.40), (2.43), and (2.44), we obtain: 8 1 2 1 1H K S S S  (2.44) - Choose th control signal as follows: 1 1 2 2 2 2 2 2 2( ) ( ( ( ) ) ) sgn( ) ) T T d d K        τ B BB M H H x x x Cx G x S  (2.45) We have: 2 2 2K S S  (2.46) Choose the second Lyapunov candidate function 2 1 1 2 2 1 1 2 2 T TV  S S S S (2.47) Take derivative of 2V and combine with (2.45), (2.46), (2.47), and (2.48) 2 22 1 1 1 1 1 12 2 2 2 T T TV K K    S S S S S SS S S S  (2.48) We have 2 2 2 2 2 2 1 1 1V K K   SS S S  (2.49) - Choose 21 1 2 KK K   with 0K  , and we obtain: 2 2 2 1 1 1 2 1 1 2 2 2 2 2 2 2 2 2 1 1 2 1 ( 2 2 ) V K K K K          S SS S S S S S S S  - Thus, 2V  is the Lyapunov function of the close-loop system. One disadvantage of this approach is that it is necessary to compute the derivative of the virtual control signal 2dx because this input depends on the slip surface and state variables of the system (2.43). That is also the difficulty when using the MSSC method. 2.3. Conclusion In chapter 2, the thesis has obtained the following results: Model a four-wheeled omnidirectional mobile robot with the selected structure, construct the kinetic and dynamics equations and analyze the dynamic of FWOMR based on numerical simulation. Research some typical trajectory tracking control algorithms applied to FWOMR, survey and evaluate the advantages and disadvantages of these control methods by Matlab / Simulink software such as: - PID - Sliding mode control - Multiple sliding surface control Based on theoretical analysis and simulation results, the sliding multi-surface control method (MSSC) will be further researched and developed in the following chapter. 3. DESIGN AN ADAPTIVE TRAJECTORY TRACKING CONTROLLER FOR THE FOUR-WHEELED OMNIDIRECTIONAL ROBOT In Chapter 3, a novel control algorithm is proposed for the FWOMR. The control algorithm is designed based on the basis of the DSC technique which is developed on 9 MSSC combined with the Backstepping technique. An adaptive DSC is constructed using a fuzzy rule and a neural network for FWOMR to overcome the disadvantages of the DSC and expand the application field for FWOMR which has uncertain nonlinear elements and is influenced by noises. The adaptive DSC controller is simulated and evaluated by Matlab- Simulink software. The studies in Chapter 3 propose new adaptive algorithms, including AFDSC and AFNNDSC, to solve the trajectory tracking control problem for FWOMR, in the case of uncertain components in the robot model, as well as the effects of noises. 3.1. Dynamic Surface Control 3.1.1. Design a trajectory tracking controller using the dynamic surface control for FWOMR To simplify the calculation and demonstration of the control system stability, system state variables are set as follows: 1 2 [ ] [ ] T T x y x y v v         x q x v (3.1) and we obtain the system state equations as follows  2 1 2 2 2 sgn d       x Hx Mx Cx G τ Bτx   (3.2) With the assumption that an accuracy model is identified and dτ is considered as the unknown external noises, the FWOMR model with the existence of disturbances has the formula as follows   1 2 22 2 sgn      x Hx Mx Cx G Bτx   (3.3) First, define 1 1 1d e x x as a tracking error vector, where  1 T d d d dx y x is the desired trajectory vector. The control target is to ensure that 1x approach 1dx or 1e tends to 0. Take derivative of 1e 1 1 1 2 1d d   e x x Hx x    (3.4) Assuming that fα is a virtual control signal in the design of DSC controller. α is an input of the first-order lowpass filter  1 1 1 1dc   H e xα  (3.5) with 1 1 1 1 0 0 0 0 0 0 x y c c c c            is a appropriate diagnonal matrix containing positive elements. The first-order lowpass filter has a formula  T f fα α α (3.6) With T is chosen small enough not to increase the calculation time of the DSC. A 10 Lyapunov candidate function is proposed 1 1 1 1 2 TV  e e (3.7) Take derivative of 1V    1 1 1 1 2 1 1 1 1 1 1 1 2 1 T T T T d dV c c       e e e Hx x e e e e Hx x     (3.8) It can be seen that from (3.8) with the virtual control signal (3.5), 1 1 1 1 0 TV c  e e and that leads to the condition 1 1 1 1 0 TV c  e e is satisfied. Define the virtual signal error 2 2 f e x α (3.9) Choose the sliding surface 1 2 S e He (3.10) where  is a coefficient. Take derivative of S    11 2 2 1 2 22 sgn f           S e He He e He H M C G Bτ αxx      (3.11) The second Lyapunove candidate function is chosen as 2 1 2 TV  S S (3.12) The control signal includes the two elements eqτ và swτ eqτ keeps the system states on the sliding surface. eqτ is calculated from solving 0S .      11 2 2 21 2( ) sgndT Teq G      e He x Cτ B BB xM xH   (3.13) The equation of swτ is chosen as follows:   1 3w 1 2( ) sgnT Ts c c   τ B BB M SH S (3.14) with 2 2 2 2 0 0 0 0 0 0 x y c c c c            and 3 3 3 3 0 0 0 0 0 0 x y c c c c            are the diagonal positive definite matrixes. Finally, the control signal is the sum of eqτ and swτ : eq sw τ τ τ (3.15) Theorem 3.1: Consider the FWOMR model is described by (2.3), the controller (3.15) with eqτ in (3.13) and swτ in (3.14) guarantees that the close-loop system is stable and the tracking error tends to 0. Proof Taking derivative of 2V 2 TV  SS  (3.16) From (3.11), 2V  becomes 11     12 1 2 2 2sgn TV         fS e He H M Cx G Bτx α  (3.17) With the control signal (3.13), the derivative of 2V can be rewritten as  2 2 3 T TV c sgn c  S SS S (3.18) By choosing appropriate values for 2c and 3c , we obtain  2 2 3 0 T TV c sgn c   S S SS (3.19) That satisfies the Lyapunov standard, and the Theorem 1 is proven! 3.2. An adaptive fuzzy dynamic surface control for trajectory tracking control for FWOMR 3.2.1. An adaptive fuzzy dynamic surface control. The outstanding point of the DSC controller is its stability with variable system parameters (uncertainties vary in the limited range). However, this strength is only available when the system state is on the sliding surface or the vicinity of the sliding surface. The schematic diagram of a fuzzy DSC system is shown in Figure 3.7. Figure 3.7. The structure of the adaptive fuzzy dynamic surface control system for FWOMR Based on the DSC simulation results for FWOMR, we found that the quality of the system significantly depends the determination of the DSC parameters 1 2 3( , , )c c c . 1c is a parameter directly affecting the tracking quality of the robot, while 2c and 3c take impact on the speed of approaching the sliding surface of the system states, as well as the ability to keep the system states on the sliding surface. In each state, if the right set of parameters is selected, the system will achieve high-quality performance, especially when the system is affected by noise. Thus, in this chapter, an adaptive fuzzy DSC is proposed for FWOMR. The fuzzy inputs are the tracking error 1e and its derivative 1e . Fuzzy sets for linguistic variables are described in Figure 3.8 and Figure 3.9. -10 -5 -0.01 0 0.01 5 10 NB NS Z PS PB -25 -12 -0.06 0 0.06 12 25 NB NS Z PS PB Figure 3.8. Fuzzy set for 1e Figure 3.9. Fuzzy set for 1e 12 With the input and output data obtained when simulating the DSC controller, the fuzzy sets of the input language variable, as well as the output values and the constituent rules for the fuzzy tuner, are built based on the Sugeno fuzzy model. The fuzzy sets for the input linguistic variables 1e và 1e are triangle forms, while 1 2 3, ,c c c are chosen through experiment. Fuzzy linguistic variables and their meanings are shown in Table 3.1. The fuzzy output values are shown in Table 3.2. Table 3.1. Fuzzy sets of the input linguistic variable Linguistic 1e Linguistic 1e Meaning NB NB Negative big NS NS Negative small Z Z Zero PS PS Positive small PB PB Positive big Bảng 3.2. Output values Output variabl e Meanin g Output value of 1c Output values of 2c and 3c VS Very small 1.5 20 S Small 4.25 25 M Medium 6.5 30 B Big 8 35 VB Very big 10 40 Bảng 3.3. Fuzzy rule of 1c 1e 1e NB NS Z PS PB N M S VS S M NS B M S M B Z VS B M B VS PS B M S M B PB M S VS S M Bảng 3.4. Fuzzy rule of 2c ( 3c ) 1e 1e NB NS Z PS PB NB M B VB B M NS S M B M S Z VB S M S VB PS S M B M S PB M B VB M M 3.2.2. Simulation The external disturbance has the form in Figure 3.10. Figure 3.10. The external disturbance The reference trajectory is described by: 0 0 cos( ) sin( ) r r r x r t y r t        The paramters of the FWOMR model and the controller are chosen as in Table 3.5 13 Table 3.5. System parameters and control parameters Dynamic parameters 210 ; J=0.56 kgm ; 0.3 ; 0.06  m kg d m r m Trajectory parameters 00 15, 10t r m   Control parameters (10,10,10); 25diag b   Figure 3.11. x-axis motion Figure 3.12. y-axis motion Figure 3.13. angular motion It can be seen that the controllers ensured the tracking quality, but the AFDSC showed the most considerable performance. The parameters 1 2 3( , , )c c c of the AFDSC are in the Figures 3.14, 3.15, 3.16. Figure 3.17 describes the motion of FWOMR in the Oxy coordinate. It can be seen that the efficiency of the proposed algorithm when the robot’s trajectory tracks remarkably close to the reference. Figure 3.17. Motion of FWOMR Figure 3.15. 2c Figure 3.16. 3c Figure 3.14. 1c 14 3.3. Adaptive fuzzy neural network dynamic surface control for FWOMR AFDSC has been a suitable recommendation to improve the tracking quality for FWOMR in the case of model deviation and noise with a small amplitude. But in the case of large model deviation, the control quality is no longer guaranteed. Therefore, the estimation of model deviation and compensation in controller components will ensure to improve the quality of this controller. Figure 3.18. The structure of AFNNDSC 3.3.1. Approximation of the uncertainty in FWOMR model using the radial basis function neural network. The FWOMR model contains the uncertainties described by dτ in (2.8). Therefore, the calculated control signal τ in the previous chapter may not reach the good quality in many cases. Besides, other uncertainties make the AFDSC difficult to perform. The thesis proposes an estimator using RBF neural network for uncertain components in the AFDSC controller. The uncertain elements are described by an equation:   1 22 sgn d   Θ x G xM C τ (3.20) which is a (3x1) vector containing the uncertainties of FWOMR. The equations describing FWOMR is rewritten as: 1 2 1 2      x Hx x Θ M Bτ   (3.21) Conduct the calculation steps which are the same as previous chapter for control design, the sliding surface’s derivative has the form  11 2 2 1 2 2d         S e He He e He H Θ M Bτ x    (3.22) The system control signal is eq sw τ τ τ (3.23) with   1 1 1 2 2( ) ˆT Teq d     τ B BB M H e He x Θ (3.24)   1 1 2 3( ) sgnT Tsw c c   τ B BB M SH S (3.25) 15 where Θˆ is trained online to approximate the system. The radial basis function neural network contains three layers, including input layer, hidden layer, and output layer. Figure 3.19. Radial basis function neural network Lựa chọn các giá trị để tính toán luật thích nghi cho Θˆ T Θ γ εR (3.26 ˆ ˆ TΘ γR (3.27) with Θ is idea value of the uncertainty. While, Θˆ is the neural network output and it is also the value used for the controller. Next, ˆ R R R is defined as the weight error. The hidden layer output is calculated by a radial basis function 2 2 1 1 2 2 2 exp i ii i              x x γ (3.28) where 1x and 2x are input vectors of the RBFNN. 1i and 2i are the center vectorsn nơ-ron, and i indicates the standard deviation. With the designed neural network structure, the updated law is chosen as  ˆ ˆT   γS H SR R (3.29) where  is a square positive definite matrix with n dimension, in which n is a neural number.  is a learning rate, which is chosen in the range (0,1) . Theorem 3.2: Consider the FWOMR model (3.2), with the controller (3.23) and the adaptive law (3.29), if the bounded condition 2 3 4 F N minc   S R (3.30) is satisfied, then the system stability is validated according to the Lyapunov standard. Proof Consider a Lyapunov candidate function:  12 1 1 2 2 T TV tr   S S  R R (3.31) Take derivative of 2V  12 ˆT TV tr   S S   )R R (3.32) 16 Combine (3.22) with the control signal (3.23), the derivative of 2V becomes    2 2 3 1ˆ ˆs ( )gnT T T TV c c tr      S HS S S S Θ Θ R R (3.33) Use (3.22), (3.24), and (3.25), we obtain    12 2 3 ˆsgn T T TT T TV c c tr      S S Hε HΘ γS S S S  R R (3.34) After some calculation steps, the derivative of 2V becomes     12 2 3 1 ˆsgnT T T T TV c c tr         S S HS S S γS HM R R (3.35) With the adaptive law (3.29), 2V  is rewritten as     2 2 3sgn T T T TV c c tr     S SS S S Hε S  R R R (3.36) Apply the Cauchy-Schwarz inequality    2T FF F tr      R R R R R R (3.37) We obtain    22 2 3sgnT T N FF FV c c       S SS S S S  R R R (3.38) With the bouned condition (3.30), 2V  becomes   2 2 2 1 sgn 2 T FF V c           S SS R R (3.39) 2 0V   and Theorem 2 is proven! 3.3.2. Construct the fuzzy law for AFNNDSC The fuzzy law is described in the section 3.21. The fuzzy inputs are a vector containing 1 1 1, ,x y e e e and its derivative, which are shown in Table 3.6. The fuzzy sets for the input linguistic variables are described in Figure 3.20. In addtion, The fuzzy law outputs are given in Table bẳng 3.7. The first output is the tuned value for  1 , ,ic i x y  . The other is the value for  2 , ,ic i x y  và  3 , ,ic i x y  . To simplify the paramters choosing for AFNNDSC, 2ic is chosen so that it is equal 3ic . The fuzzy rule is given in Table 3.6. Figure 3.20. The fuzzy sets for the input 17 Figure 3.6. Fuzzy rules for 1 2( )i ic c 1e 1e NB NS Z PS PB NB M(M) S(B) VS(VB) S(B) M(M) NS B(S) M(M) S(B) M(M) B(S) Z VS(VB) B(S) M(M) B(S) VS(VB) PS B(S) M(M) S(B) M(M) B(S) PB M(M) S(B) VS(VB) S(B) M(M) Figure 3.7. Output values of 1 2( )i ic c VS 3.0 (5) S 4.15 (10) M (20) B 7.5 (25) VB 12 (30) 3.3.3. Simualtion results. In this section, the simulations are performed in Matlab/Simulink environment. 3.3.3.1. Robot model affected by the external disturbances The reference trajectory for FWOMR is a circular trajectory described by a equation with the radius 0 5r m ; 0 10t  . In addition, the robot parameters are: m=10kg, J=0.56kgm2, d=0.3m, r=0.06m. The initial position of the robot is chosen as ( ; ) (2;2)x y  In this case, the system is simulated and evaluted in the condition, which is directly influenced by the Gaussian-type moment noise affecting the motors. Moreover, the impact of friction force is ignored. The sliding surface coefficient is chosen as:  10,10,10diag  Figure 3.21. Moment noise (Nm) The simulation results are shown as the following figures. The Figure 3.22, 3.23, and 3.24 compare the tracking error of FWOMR when using the three