Summary of doctoral thesis Electrical, electronic and Telecommuication engineering: Adaptive dynamic surface trajectory tracking control for the four-wheeled omnidirectional mobile robot

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Research and propose a novel adaptive trajectory tracking algorithm for FWOMR ‘s nonlinear uncertain model which is influenced by the change of robot parameters and the effect of noise when operating on another plane; construct the physical model of FWOMR and the controller based on the microchip and embedded programming technique to experiment with the proposed algorithms.

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Nội dung Text: Summary of doctoral thesis Electrical, electronic and Telecommuication engineering: Adaptive dynamic surface trajectory tracking control for the four-wheeled omnidirectional mobile robot

MINISTRY OF EDUCATION VIETNAM ACADEMY OF SCIENCE AND TRAINING AND TECHNOLOGY GRADUATE UNIVERSITY OF SCIENCE AND TECHNOLOGY ------------------------------- HA THI KIM DUYEN ADAPTIVE DYNAMIC SURFACE TRAJECTORY TRACKING CONTROL FOR THE FOUR-WHEELED OMNIDIRECTIONAL MOBILE ROBOT Majors: Control and Automation Engineering Code: 9 52 02 16 ELECTRICAL, ELECTRONIC AND TELECOMMUICATION ENGINEERING Hanoi – 2020
Publications at: Graduate University of Science and Technology – Vietnam Academy of Science anf Technology Advisor 1: Prof. Phan Xuan Minh Advisor 2: Dr. Pham Van Bach Ngoc Reviewer 1: Reviewer 2: Reviewer 3: The thesis is defended at the PhD dissertation committee, which meets at Graduate University of Science and Technology – Vietnam Academy of Science and Technology at…………,2020 Thesis can be found at: - Library of Graduate University of Science and Technology - National Library of Vietnam
LIST OF AUTHOR’S PUBLICATIONS SCIENTIFIC JOURNAL 1. Duyen Ha Thi Kim, Tien Ngo Manh, Cuong Nguyen Manh, Nhan Duc Nguyen, Manh Tran Van, Dung Pham Tien, Minh Phan Xuan. “Adaptive Control for Uncertain Model of Omni-directional Mobile Robot Based on Radial Basis Function Neural Network”. International Journal of Control, Automation, and Systems (SCI-E Q2, Impact Factor: 2.7) (Accepted 2020) 2. Ha Thi Kim Duyen, Ngo Manh Tien, Pham Ngoc Minh, Quang Vinh Thai, Phan Xuan Minh, Pham Tien Dung, Nguyen Duc Dinh, Hiep Do Quang, “Fuzzy Adaptive Dynamic Surface Control for Omnidirectional Robot”, the Springer-Verlag book series “Computational Intelligence” indexed in Scopus and Compendex (Ei). ISSN 1860-9503 (electronic), ISBN 978-3-030-49536-7 (eBook). https://doi.org/10.1007/978-3-030- 49536-7. (2020) 3. Hà Thị Kim Duyên, Phạm Thị Thanh Huyền, Trương Bích Liên, Ngô Mạnh Tiến, Lê Việt Anh, Nguyễn Mạnh Cường, “Điều khiển bám quỹ đạo đối thượng Robot tự hành bằng thuật toán điều khiển trượt theo hàm mũ”, Tạp chí Nghiên cứu KH&CN quân sự, Số Đặc san ACMEC, 07 – 2017. ISSN 1859 - 1043 4. H T ị D , Ngô Mạnh Tiến, Phan Xuân, Minh Lê Xuân Hải, Vũ Đức Thuận, Nguyễn Minh Huy, “Điều khiển bám quỹ đạo Omni robot bốn bánh bằng phương pháp thích nghi mờ trượt”. Tạp chí Nghiên cứu Khoa học và Công nghệ quân sự. Số đặc san ACMEC 07-2017. ISSN 1859 - 1043 5. Ngo Manh Tien, Nguyen Nhu Chien, Do Hoang Viet, Ha Thi Kim Duyen “Research And Development Artificial Intelligence To Track Trajectory And Automatically Path Planning For Auto Car”. Journal of Military Science and Technology; ISSN 1859 – 1043. 11/2018 6. Duyen – Ha Thi Kim, Tien – Ngo Manh, Chien – Nguyen Nhu, Viet – Do Hoang, Huong-Nguyen Thi Thu Kien-Phung Chi, “Tracking Control For Electro-Optical System In Vibration Enviroment Based On Self-Tuning Fuzzy Sliding Mode Control”, Journal of Computer Science and Cybernetics, Vol 02, 6.2019. SCIENTIFIC CONFERENCE 7. Ngô Mạnh Tiến, Nguyễn Như Chiến, Đỗ Hoàng Việt, H T ị D , Nguyễn Tuấn Nghĩa, “Trajectory Tracking Control for Four Wheeled Omnidirectional Mobile Robots using Adaptive Fuzzy Dynamic Surface Control Algorithm”, Proceedings the 4th Vietnam International Conference and Exhibition on Control and Automation VCCA- 2017; ISBN 978-604-73-5569-3 8. Duyen Ha Thi Kim, Tien Ngo Manh, Tuan Pham Duc and Ngoc Pham Van Bach, “Trajectory Tracking Control for Omnidirectional Mobile Robots Using Direct Adaptive Neural Network Dynamic Surface Controller”. The 2019 First International Symposium on Instrumentation, Control, Artificial Intelligence, and Robotics. 1/2019. NSPEC Accession Number: 18473513, DOI: 10.1109/ICA-SYMP.2019.8646146. 9. Ha Thi Kim Duyen, Cuong Nguyen Manh, Hoang Thuat Vo, Manh Tran Van, Dinh Nguyen Duc, Anh Dung Bui, “Trajectory tracking control for four wheeled Omni- directional mobile Robot using backstepping technique aggregated with sliding mode control”, The 2019 First International Symposium on Instrumentation, Control, Artificial Intelligence, and Robotics. 1/2019. INSPEC Accession Number: 18473501, DOI: 10.1109/ICA-SYMP.2019.8646041. 10. Ngô Mạnh Tiến, Nguyễn Mạnh Cường, H T ị D , Phan Sỹ Thuần, Nguyễn Ngọc Hải, Trần Văn Hoàng, Nguyễn Văn Dũng, “Giám sát định vị, bản đồ hóa và điều hướng cho robot tự hành đa hướng sử dụng hệ điều hành lập trình ROS”, Hội nghị Quốc gia lần thứ XXII về Điện tử, Truyền thông và Công nghệ Thông tin lần thứ 22, 2019.
1 INTRODUCTION 1. Rasionale of the thesis Omni-directional mobile robot (OMR) is a holonomic robot using Omni or Mecanum wheel, which can move in any direction without changing the robot’s position and rotation angle. Due to the outstandingly moving in narrow environmental conditions when using the particular wheel structure and wheel layout, OMR is being widely applied and developed not only in research but has quickly become widely used in production and life fields. In robot control, the problem of trajectory tracking control, dealing with external disturbance, and the uncertain part existing in the robot model, including mass, moment, and friction,… is the content which is significantly focused in research. The task which is to guarantee the high accuracy in the robot motion is commonly difficult because of the nonlinear and uncertain parts typically existing in the robot model. 2. Research Objective - Research and propose a novel adaptive trajectory tracking algorithm for FWOMR ‘s nonlinear uncertain model which is influenced by the change of robot parameters and the effect of noise when operating on another plane. - Construct the physical model of FWOMR and the controller based on the microchip and embedded programming technique to experiment with the proposed algorithms. 3. Object and scope The object of the study: a four-wheeled omnidirectional mobile robot. The thesis puts emphasis on the construction of the mathematical robot model and the adaptive trajectory tracking control algorithms for FWOMR. The study scope: Design an adaptive controller for FWOMR model which is influenced by the uncertain elements in the flat environment affected the surface friction and bounded abitrary noise. 4. Scientific significance and new contributions of the thesis 1. Propose an adaptive fuzzy dynamic surface control (AFDSC) for the four-wheeled omnidirectional mobile robot. The algorithm is constructed based on the DSC technique. To take advantage of the DSC, AFDSC uses a fuzzy rule tuning the DSC parameters to guarantee the trajectory tracking quality when the FWOMR’s parameters are variable and are affected by the unknown disturbance. Until now, the DSC tuned by the fuzzy rule has not ever been installed in any mobile robot platforms in domestic and overseas. 2. Propose an adaptive fuzzy neural network dynamic surface control (AFNNDSC) for FWOMR which contains uncertain parameters and its model is affected by the disturbance. The algorithm is developed based on the DSC technique, its adaptive characteristic is created by the combination of the radial basis function neural network (RBFNN) and the fuzzy law. The RBFNN is to approximate uncertain parameters of the FWOMR, while the fuzzy law simultaneously tunes the control parameter. The system stability is verified by the Lyapunov standard. Simulation results and experiments show the accuracy and effectiveness of the proposed method and its practical application ability. The AFNNDSC has not ever been installed in any robot platforms in domestic and overseas. Moreover, the AFNNDSC has highly flexible and adaptive characteristics when the robot is
2 adversely influenced by the disturbance, and the model’s parameters are variable. That leads to the expansion of the robot’s operation scope. The algorithms are installed and successfully experimented in the four-wheeled omnidirectional mobile robot. The constructed robot has the high-performance processor control circuit and the software built on ROS. 5. Thesis layout Chapter 1: “An overview of the four-wheeled omnidirectional mobile robot”. An overview study of FWOMR, domestic and international research, detailed analysis of advantages and disadvantages of previously researched works according to the content of the subjects, the research scope of the thesis, from there draw appropriate research directions for the thesis. Chương 2: “Modeling and trajectory control algorithms for the four-wheeled omnidirectional mobile robot”. This chapter shows the construction of the dynamic model for FWOMR and a presentation of some typical trajectory control algorithms for FWOMR. Moreover, the simulation, evaluation, and analysis the results of these algorithms are to evaluate and draw lessons learned in the proposed study of the new adaptive orbital control algorithm. Chapter 3: “Design an adaptive trajectory tracking controller for the four-wheeled omnidirectional robot”. The chapter is the main contribution of the thesis. In this chapter, an adaptive trajectory tracking controller for the four-wheeled omnidirectional robot using a dynamic surface control is presented. The DSC algorithm is the platform for the control quality improvement using the proposed adaptive trajectory tracking controller. The DSC controller is combined with a radial basis function neural network (RBFNN) and fuzzy logic system (FLS) to design a novel adaptive DSC controller which is proposed in the thesis. Chương 4: “Manufacture the four-wheeled omnidirectional mobile robot and experiment with the proposed control algorithm”. This chapter shows the design and manufactures the four-wheeled omnidirectional mobile robot model, and programming and experiment of the proposed algorithms to verify and evaluate the practical applicability. “Conclusion” presents the abstract of the main contributions and future works. AN OVERVIEW OF THE FOUR-WHEELED OMNIDIRECTIONAL MOBILE ROBOT Omnidirectional mobile robots (OMR) are capable to move in any direction without the changing its direction and position. With a particular wheel structure and the advantage of flexible movement in the environment which is narrow and difficult to change the position. Nowadays, OMR is widespreadly applied not only in research but in the various production fields due to its ability to flexible and omnidirectional move. 1.1. Autonomous mobile robot built with omnidirectional wheel. In the thesis scope, a holonomic mobile robot is constructed using four Omni wheels with a consistent structure, which ensures that the robot can promptly move in horizontal direction. 1.2. Trajectory tracking control problem The structure of the motion control system for OMR can be devided into three phases:
3 - Set up the motion plan - Plan the reference trajectory - Control robot to track the reference trajectory 1.3. Overview of domestic and international research 1.3.1. Domestic research situation Institute of Information Technology and Institute of Mechanics, Vietnam Academy of Science and Technology Institute of Information Technology and Institute of Mechanics, Vietnam Academy of Science and Technology have a number of published research on autonomous robot research, such as [1] presenting new control methods to compensate the impact of the slip phenomena for mobile robots in the existence of wheel slip, uncertainties, and external noise for 3-wheel mobile robotics, [2] presenting the design and control of a nonholonomic mobile robot for warehouse applications. Institute of Physics, Vietnam Academy of Science and Technology is also a research group with many published works on autonomous robots, including [3] presenting research orientation for trajectory tracking control for nonholonomic mobile robots with the integration of the image processing technology to identify some parameters and track the target. Moreover, [4] conducted research on applying the trajectory tracking control algorithm for the nonholonomic mobile robot using the ideal function adaptation. Nowadays, there are a few domestic studies on OMR, in which, [5] is an obstacle avoidance control for OMR using Kinect image processing technology. In addition, this work focuses more on image processing than the field of robot control for OMR. There are not many publications about the trajectory tracking control algorithm for the four-wheeled robots using Ommi wheels in Vietnam. 1.3.2. International research situation The robot was modeled using kinematic and dynamic models. Robot modelling was researched based on Euler-Lagrange equations, which used theoretical and experimental methods [31], [32], [33], [34], [35], [36], [37], [38], [39], [40]. A mount of research focused on the navigation system and motion control based on the robot kinematic model of OMR [43], [44], [45], and [46]. The PID algorithm was applied for trajectory tracking control of the four-wheeled OMR in [43] and [44]. However, recent research has considered both kinematic and dynamic models to enhance the accuracy in robot movement [42] and [43]. The design of the trajectory tracking control algorithm for OMR taking into account all the kinematic and dynamic models has been considered in [39]. The dynamic model is constructed in [47] and [48], followed by a number of trajectory tracking control algorithms for this full model in [49] and [50]. Studies have used PI controllers to optimize orbital adhesion [52], [57], and [79]. On the other hand, the algorithm using the predictive model has also been mentioned in [51]. More and more research is focused on feedback control methods for the nonlinear model 52], [53], [54], [55], [56], and [57]. 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 0  q  Hv   sinθ cosθ 0  v (2.1)    0 0 1 cosθ -sinθ 0  where: H =  sinθ cosθ 0  is a transition matrix.  0 0 1 From (2.1), we calculate an equation presenting a relationship between the robot’s position and the velocity of wheels:  1   x     y   g ( ) 2  với g ( )  HH   2 (2.4) 3      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 M (q) v  Cv  Gsgn( v)  τ d  Bτ (2.8) with: v  [ vx vy  ]T is a velocity vector  2 2 2 2     2r 2r 2r 2r   2 2 2 2 B    is a control parameter matrix.  2r 2r 2r 2r   d d d d   r r r r    m 0 0  M (q)   0 m 0  is a matrix with m is the robot mass and J is the inertia moment.  0 0 J   Bx 0 0 C x 0 0 C   0 By 0  and G   0 C y 0  are viscous friction parameter matrix and     0 0 B   0 0 C  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   x (t )   xd     xd  e1   y (t )    y d   g ( )  2    y d      (2.11) 3  (t )  d    d  4  We need to find the angular velocity vector of the wheels for the closed-loop controller to be stable.  t     xe d    1   0      xe   t   2   g T ( )( g ( ) g T ( ))1    K P  ye   K I  ye d   3     0   (2.12)    e    4   t   d      0 e   with K P and K I 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. x  q From (2.1) and (2.8), let  1 , we have state equations: x 2  v x 1  Hx 2  (2.19) Mx 2  Cx 2  Gsgn( x 2 )  τ d  Bτ with τ d 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.  e  x1  x1d Define the errors  1 with x1d is a reference trajectory and x 2d  H 1x 1d is e  2  x 2  x 2d reference velocity. Choose the sliding surface S  e1  e1 (2.20) with  >0 is a sliding surface coefficient. Take the sliding surface’s derivative S  He   He  e  H (M 1 (Bτ  Cx  Gsgn(x ))  x  (  H 1H 2 2 1 2 2 2d  )e ) 2 (2.21) Choose a Lyapunov candidate function
7 1 V  S2 (2.22) 2 Take its derivative, we obtain V  SS  SH (M 1 (Bτ  Cx2  Gsgn(x 2 ))  x 2 d  (  H -1H  )e ) 2 (2.23) With the control signal is chosen as follows τ  BT (BBT )1 (M ((  H 1H )e  x )  Cx  Gsgn(x )  K sgn(S)) (2.24) 2 2d 2 2 1 V  SK sgn(S)  0 , which satisfies the Lyapunov standard. 1 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 x 1  Hx 2  (2.36) Mx 2  Cx 2  Gsgn(x 2 )  τ d  Bτ x  vx  with x1   y  và x 2  v y           - Consider the sliding surface  S11  S1   S12   x1  x1d (2.37)    S13  - Take derivative of S1 and use (2.37), we obtain S  x  x  Hx  x 1 1 1d 2 1d (2.38) Choose a virtual control signal x 2 d  H 1 ( K1S1  x 1d ) (2.39) - Choose the first Lyapunov candidate function 1 (2.40) V1  S1T S1 2 - Take derivative of V1 , and use (2.38) and (2.39) V  ST S  ST K S 1 1 2 1 1 1 (2.41) - Consider S 2 as the second sliding surface S 2  H(x 2  x 2 d ) (2.42) Taking derivative of S 2 S 2  H(x 2  x 2 d )  H  (x  x ) 2 2d (2.43)  (x  x )  H (M 1 (Bτ  Cx 2  Gsgn(x 2 ))  x 2 d )  H 2 2d Combine (2.39), (2.40), (2.43), and (2.44), we obtain:
8 S 1  HS 2  K1S1 (2.44) - Choose th control signal as follows: τ  BT (BBT )1 (M (H 1H (x  x ) )  x )  Cx  Gsgn(x )  K S ) (2.45) 2 2d 2d 2 2 2 2 We have: S 2   K 2S 2 (2.46) Choose the second Lyapunov candidate function 1 1 (2.47) V2  S1T S1  ST2 S 2 2 2 Take derivative of V2 and combine with (2.45), (2.46), (2.47), and (2.48) V  S S  S S   K ST S  K ST S  ST S 2 1 1 2 2 1 1 1 2 2 2 1 2 (2.48) We have V2   K1 S1  K 2 S 2  S1 S 2 2 2 (2.49) 1 - Choose K1  K 2  K  with K  0 , and we obtain: 2 1 1 V2   K S1  K S 2  S1  S 2  S1 S 2 2 2 2 2 2 2 2 2 1   K S1  K S 2  ( S1  S 2 )2 2 - Thus, V2 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 x 2d 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 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: x1  q  [ x y  ]T  (3.1) x 2  v  [ vx v y  ] T and we obtain the system state equations as follows x 1  Hx 2  (3.2) Mx 2  Cx 2  Gsgn  x 2   τ d  Bτ With the assumption that an accuracy model is identified and τ d is considered as the unknown external noises, the FWOMR model with the existence of disturbances has the formula as follows x 1  Hx 2 (3.3)  Mx 2  Cx 2  Gsgn  x 2   Bτ T First, define e1  x1  x1d as a tracking error vector, where x1d   xd yd  d  is the desired trajectory vector. The control target is to ensure that x1 approach x1d or e1 tends to 0. Take derivative of e1 e1  x 1  x 1d  Hx2  x 1d (3.4) Assuming that αf is a virtual control signal in the design of DSC controller. α is an input of the first-order lowpass filter α  H 1  c1e1  x 1d  (3.5)  c1x 0 0  with c1   0 c1 y 0  is a appropriate diagnonal matrix containing positive  0 0 c1   elements. The first-order lowpass filter has a formula Tα f  αf  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 V1  e1T e1 (3.7) 2 Take derivative of V1 V1  e1T e1  e1T  Hx 2  x 1d   e1T c1e1  e1T  c1e1  Hx 2  x 1d  (3.8) It can be seen that from (3.8) with the virtual control signal (3.5), V1  e1T c1e1  0 and that leads to the condition V  e T c e  0 is satisfied. 1 1 1 1 Define the virtual signal error e2  x2  α f (3.9) Choose the sliding surface S  e1  He 2 (3.10) where  is a coefficient. Take derivative of S S  e1  He 2  He   e  He 2 1   H M 1  Cx  Gsgn  x   Bτ   α 2 2 2 f  (3.11) The second Lyapunove candidate function is chosen as 1 V2  ST S (3.12) 2 The control signal includes the two elements τ eq và τ sw τ eq keeps the system states on the sliding surface. τ eq is calculated from solving S  0 .      x τ eq  BT (BBT ) 1 M H 1  e1  He 2  2 d  Cx 2  Gsgn  x 2   (3.13) The equation of τ sw is chosen as follows: τ sw  BT (BBT ) 1 MH 1  c2sgn  S   c3S  (3.14)  c2 x 0 0   c3 x 0 0  with c2   0 c2 y 0  and c3   0 c3 y   0  are the diagonal positive definite  0 0 c2   0 0 c3    matrixes. Finally, the control signal is the sum of τ eq and τ sw : τ  τ eq  τ sw (3.15) Theorem 3.1: Consider the FWOMR model is described by (2.3), the controller (3.15) with τ eq in (3.13) and τ sw in (3.14) guarantees that the close-loop system is stable and the tracking error tends to 0. Proof Taking derivative of V2 V2  ST S (3.16) From (3.11), V2 becomes
11 V2  ST  e1  He    H M 1  Cx  Gsgn  x   Bτ   α 2 2  2 f  (3.17) With the control signal (3.13), the derivative of V2 can be rewritten as V  ST c sgn  S   ST c S 2 2 3 (3.18) By choosing appropriate values for c2 and c3 , we obtain V  ST c sgn  S   ST c S  0 2 2 3 (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 (c1 , c2 , c3 ) . c1 is a parameter directly affecting the tracking quality of the robot, while c2 and c3 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 e1 and its derivative e1 . Fuzzy sets for linguistic variables are described in Figure 3.8 and Figure 3.9. NB NS Z PS PB NB NS Z PS PB -10 -5 -0.01 0 0.01 5 10 -25 -12 -0.06 0 0.06 12 25 Figure 3.8. Fuzzy set for e1 Figure 3.9. Fuzzy set for e1
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 e1 và e1 are triangle forms, while c1 , c2 , c3 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 Bảng 3.2. Output values variable Output Meanin Output Output Linguistic Linguistic Meaning variabl g value of values of c2 e1 e1 e c1 and c 3 NB NB Negative big VS Very 1.5 20 NS NS Negative small small Z Z Zero S Small 4.25 25 PS PS Positive small M Medium 6.5 30 PB PB Positive big B Big 8 35 VB Very 10 40 big Bảng 3.3. Fuzzy rule of c1 Bảng 3.4. Fuzzy rule of c2 ( c3 ) e1 e1 e1 e1 NB NS Z PS PB NB NS Z PS PB N M S VS S M NB M B VB B M NS B M S M B NS S M B M S Z VS B M B VS Z VB S M S VB PS B M S M B PS S M B M S PB M S VS S M 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: xr  r0 cos(t ) yr  r0 sin(t ) r   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 m  10kg ; J=0.56 kgm 2 ; d  0.3m; r  0.06m Trajectory parameters 0  t  15, r0  10m Control parameters   diag (10,10,10); b  25 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 (c1 , c2 , c3 ) of the AFDSC are in the Figures 3.14, 3.15, 3.16. Figure 3.14. c1 Figure 3.15. c2 Figure 3.16. c3 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
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 τ d 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: Θ  M 1  Cx 2  Gsgn  x 2   τ d  (3.20) which is a (3x1) vector containing the uncertainties of FWOMR. The equations describing FWOMR is rewritten as: x 1  Hx 2  (3.21) x 2  Θ  M Bτ 1 Conduct the calculation steps which are the same as previous chapter for control design, the sliding surface’s derivative has the form S  e  He  He 1 2 2   e  He  H Θ  M 1Bτ  x 1 2  2d  (3.22) The system control signal is τ  τ eq  τ sw (3.23) with     x  Θ τ eq  BT (BBT )1 M H 1  e1  He ˆ  (3.24) 2 2d
15 τ sw  BT (BBT ) 1 MH 1  c2sgn  S   c3S  (3.25) where Θ ˆ 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 ˆ Lựa chọn các giá trị để tính toán luật thích nghi cho Θ Θ  RT γ  ε (3.26 ˆ R Θ ˆ Tγ (3.27) with Θ is idea value of the uncertainty. While, Θ ˆ is the neural network output and it is also the value used for the controller. Next, R   R R ˆ is defined as the weight error. The hidden layer output is calculated by a radial basis function  x1  1i 2  x 2   2i 2  γ i  exp    (3.28)   i2   where x1 and x 2 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 R  ˆ   γST H   S 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 R F N   (3.30) S  4 c3min is satisfied, then the system stability is validated according to the Lyapunov standard. Proof Consider a Lyapunov candidate function: 1 1  T  1R  V2  ST S  tr R 2 2   (3.31) Take derivative of V2
16 ˆ  V2  ST S  tr R  T  1R )  (3.32) Combine (3.22) with the control signal (3.23), the derivative of V2 becomes ˆ V2  ST c2sgn  S   ST c3S  ST H(Θ  Θ  T  1R ˆ )  tr R   (3.33) Use (3.22), (3.24), and (3.25), we obtain ˆ  T γ  tr R V2  ST c2sgn  S   ST c3S  ST Hε  ST HΘ  T  1R   (3.34) After some calculation steps, the derivative of V2 becomes ˆ   T  1R V2  ST c2sgn  S   ST c3S  ST H  tr R  γST HM 1  (3.35) With the adaptive law (3.29), V2 is rewritten as  T R R V2  ST c2sgn  S   ST c3S  ST Hε   S tr R     (3.36) Apply the Cauchy-Schwarz inequality  T R R tr R     R   R   R 2 (3.37) F F F We obtain V2  ST c2sgn  S   ST c3S  S  N   S   R F R F   R 2 F  (3.38) With the bouned condition (3.30), V2 becomes 2   1  V2  ST c2sgn  S    S  R  R F  (3.39)  F 2  V2  0 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 e1x , e1 y , e1 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 c1i  i  x, y,  . The other is the value for c2i  i  x, y,  và c3i  i  x, y,  . To simplify the paramters choosing for AFNNDSC, c2i is chosen so that it is equal c3i . 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 c1i (c2i ) Figure 3.7. Output values of c1i (c2i ) e1 e1 VS 3.0 (5) NB NS Z PS PB S 4.15 (10) NB M(M) S(B) VS(VB) S(B) M(M) M (20) NS B(S) M(M) S(B) M(M) B(S) B 7.5 (25) Z VS(VB) B(S) M(M) B(S) VS(VB) VB 12 (30) PS B(S) M(M) S(B) M(M) B(S) PB M(M) S(B) VS(VB) S(B) M(M) 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 r0  5m ; 0  t  10 . 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 ( x; y )  (2;2) 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:   diag 10,10,10  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 controllers, including DSC, adaptive neural network dynamic surface control (ANNDSC), and adaptive fuzzy neural network dynamic surface control (AFNNDSC). Figure 3.22. x-axis error Figure 3.23. y-axis error Figure 3.24. angular error