Tài liệu Optimal path planning and adaptive sliding mode control of hexapod model

VIET NAM NATIONAL UNIVERSITY HO CHI MINH CITY HO CHI MINH CITY UNIVERSITY OF TECHNOLOGY --------------- LE HA ANH KHOA OPTIMAL PATH PLANNING AND ADAPTIVE SLIDING MODE CONTROL OF HEXAPOD MODEL HOϩ0ϾNH QUЖ 0ϩO TЁЎ0ϹU KHIϺЎЋT THÍCH NGHI MÔ HÌNH TAY MÁY SONG SONG HEXAPOD MAJOR: Control Engineering and Automation MAJOR CODE: 8520216 MASTER THESIS HO CHI MINH CITY, August 2021 THE RESEARCH IS COMPLETED AT: HO CHI MINH CITY UNIVERSITY OF TECHNOLOGY Ȃ VNU Ȃ HCM Instructor: Dr. Nguyen Vinh Hao Examiner 1: Assoc. Prof. Dr. Nguyen Quoc Chi Examiner 2: Assoc. Prof. Dr. Nguyen Tan Luy The master thesis is defended at Ho Chi Minh city University of Technology (HCMUT), VNU-HCM on Google Meet on August 19th, 2021. Š‡„‘ƒ”†‘ˆ–Š‡ƒ•–‡”ǯ•Thesis Defense Council includes: 1. 2. 3. 4. 5. Prof. Dr. Ho Pham Huy Anh Dr. Pham Viet Cuong Assoc. Prof. Dr. Nguyen Quoc Chi Assoc. Prof. Dr. Nguyen Tan Luy Assoc. Prof. Dr. Nguyወn Minh Tâm Chairman. Secretary. Reviewer 1. Reviewer 2. Member. Verification ‘ˆ–Š‡Šƒ‹”ƒ‘ˆ–Š‡ƒ•–‡”ǯ•Š‡•‹•‡ˆ‡•‡‘—…‹Žand the Dean of faculty of Electrical & Electronics Engineering after the thesis being corrected (If any) CHAIRMAN OF THE COUNCIL DEAN OF FACULTY OF ELECTRICAL & (Full name & signature) ELECTRONICS ENGINEERING (Full name & signature) Prof. Dr. Ho Pham Huy Anh i VIET NAM NATIONAL UNIVERSITY SOCIALIST REPUBLIC OF VIETNAM HO CHI MINH CITY Independence Ȃ Freedom - Happiness UNIVERSITY OF TECHNOLOGY  ǯ Full name: LE HA ANH KHOA MSHV: 1870440 Date of birth: 24 Ȃ 02 Ȃ 1995 Place of birth: Ho Chi Minh City Major: Control Engineering and Automation Major code: 8520216 I. THESIS TOPIC: Optimal Path Planning and Adaptive Sliding Mode Control of Hexapod model. II. NHIϼM VЌ VÀ NЅI DUNG: Design multi-objective optimal path planning based on revolutionary swarm algorithm and then construct adaptive sliding mode control for parallel manipulator Hexapod to track the optimized path. III. TASKS STARTING DATE: February 24th, 2020 IV. TASKS ENDING DATE: June 13th, 2021 V. INSTRUCTOR: Dr. Nguyen Vinh Hao Ho Chi Minh City, August 19th, 2021 INSTRUCTOR HEAD OF DEPARTMENT (Full name & signature) (Full name & signature) Dr. Nguyen Vinh Hao Dr. Nguyen Vinh Hao DEAN OF FACULTY OF ELECTRICAL & ELECTRONICS ENGINEERING (Full name & signature) ii THANK YOU I would like to say thank you to my mom who helps and gives me special care from the beginnings to endings. Besides, I also give my honor to my professor, Dr. Nguyen Vinh Hao who inspires and provides to me a lot of helpful advices and guidances for this project. Šƒ›‘—–‘–Š‡Dz‡™…‡ƒƒ—ˆƒ…–—”‹‰‘Ž—–‹‘•dz…‘’ƒ›™hich donated motors and controllers for building up my mechanical model. Thank you to the DzDelta Technical Supportdz team who trained and instructed me how to use their PLC and Servo system in this project. Thank you to all my friends, my brothers, my colleagues who were beside and encouraged me to pass those most difficult moments. iii ABSTRACT Researchers often think about the spider robot when hearing the term Hexapod. However, in this Master of Science thesis, we talk about a special manipulator which has 6 degrees of freedom in motion invented by Dr. Stewart about a half-century ago. Throughout the thesis, two problems that are examined are Multi-Objective Optimal Path Planning Algorithms and Adaptive Sliding Mode Controllers designing for the high complexity parallel manipulator with 6 degrees of freedom (6-DOFs) in translation and orientation, the Rotary Parallel Manipulator Hexapod. The optimal path planning algorithms are developed base on the formula of B-spline curves with the objectives to a minimum the motion times & path lengths and while satisfying the constraints of the Hexapod mechanical structures such as the limitation of active joint angles, joint angular velocities, joint accelerations, and a maximum of the joint torque outputs of the servo motors. The scheme of solving these objective functions and constraints follows the theories of multi-objective optimization of Particle Swarm Optimization (MOPSO). The optimal solutions are chosen base on the objective weighting method from the distribution of the Pareto sets. Then Adaptive Sliding Mode Controllers (ASMC) are constructed either to overcome the chattering phenomenon of the conventional sliding mode control technic and precisely track the end-effector of the mobile platform to the following optimal path planning process. The Hexapod simulation model is exported from the mechanical CAD design software (SOLIDWORK). All the mathematical models and algorithms are realized using MATLAB/Simulink software with the help of SimScapes Libraries. A lot of simulations have been conducted to proof of the optimal path planning and the asymptotical convergence stability of the control system. iv TÓM TϰT LUϯ$ϩC Các nhà nghiên cዜu –Šዛዕ‰‰ŠÂ˜዆ các rô-bô nh዆n nhዉn Š‹¯ዛዘc nhኽ…¯ዅn thuኼt ngዟ Dz‡šƒ’‘†dzǤ—›Š‹²ǡ–”‘‰Ž—ኼ˜£–Š኶…•Â›ǡ…Šï‰–ƒ¯ƒ‰×‹˜዆ mዒt …ዓ…ኸu ¯዁c biዉt có th዇ chuy዇¯ዒng theo 6 bኼc tዠ do khác nŠƒ—¯ዛዘc phát minh bዖi tiዅ•Â Stewart vào kho኷ng giዟa thዅ kዢ –”ዛዔc. Xuyên suዎt luኼ˜£›ǡhai nዒi dung sዄ ¯ዛዘc kh኷o sát là Ho኶ch 0ዋnh Quዣ 0኶o Tዎi ክu 0a Biዅn ˜0‹዆u Khi዇”ዛዘt Thích Nghi …Š‘…ዓ…ኸu song song …ׯዒ phዜc t኶p cao vዔi 6 bኼc tዠ do (6-DOFs) bao gዏm c኷ –”ዛዘt và quay, cዓ…ኸu song song d኶ng tay quay Hexapod. Gi኷i thuኼt ho኶…Š¯ዋnh tዎ‹ዛ—¯ዛዘc phát tri዇n dዠƒ–”²’Šዛዓ‰–”¿Š¯ዛዕng cong B-spline vዔi các mዙc tiêu cዠc ti዇u thዕi gian dዋch chuy዇n và ¯ዒ dài ¯ዛዕng ¯‹ trong khi vኻn thውa mãn các ràng buዒc v዆ …ዓŠÀŠዛ‰‹ዔi h኶n góc quay cዚa các khዔp, vኼn tዎc góc cዚa các khዔp, các gia tዎc t኶i khዔp, và moment xoኽn cዠ…¯኶i t኶‹¯ኹu ra cዚa các ¯ዒng …ዓ servo. Trình tዠ gi኷‹…ž…’Šዛዓ‰–”¿ŠŠዙc tiêu và các hàm ràng buዒc tuân theo lí thuyዅt tዎ‹ዛ—¯ƒ„iዅn cዚa Gi኷i thuኼt Tዎ‹ዛ— Bኹ›0 (MOPSO). Các kዅt qu኷ ¯ኹu ra cዚa gi኷i thuኼt ¯ዛዘc lዠa chዌn dዠƒ–”²’Šዛዓ‰’Šž’ ¯žŠ trዌng sዎ theo phân bዎ cዚa tኼ’ƒ”‡–‘Ǥƒ—¯×ǡ các bዒ ¯‹዆u khi዇”ዛዘt Thích Nghi ¯ዛዘc xây dዠ‰¯዇ vዝa khኽc phዙc tình tr኶‰”—‰¯ዒng (chattering) vዝa Šዛዔ‰¯‹዇m cuዎi cዚa tኸm ph኿‰†‹¯ዒng bám theo quዣ ¯኶o tዎ‹ዛ—¯ ¯዆ ra. Mô hình mô phው‰ ‡šƒ’‘† ¯ዛዘc xuኸt b኷n tዝ phኹn m዆m thiዅt kዅ …ዓ ŠÀ (SOLIDWORK). Tኸt c኷ các mô hình toán và gi኷i thuኼ–¯ዛዘc thዠc hiዉn trên phኹn m዆m MATLAB/Simulink vዔi sዠ trዘ giúp cዚa bዒ –Šዛ˜‹ዉn SimScapes. Nhi዆u thí nghiዉm mô phው‰¯ ¯ዛዘc tiዅŠŠ¯዇ chዜng minh cho tính tዎ‹ዛ—…ዚa quዣ ¯኶‘¯ Š‘኶…Š¯ዋnh và tính ዐ¯ዋnh tiዉm cኼn cዚa hዉ thዎ‰¯‹዆u khi዇n. v DECLARATION The author confirms that all the contents used in this thesis are unique and are completely quoted and cited. Ho Chi Minh City, August 19th, 2021 (Full name & signature) Le Ha Anh Khoa vi TABLE OF CONTENTS  ǯ ............................................................................. i THANK YOU ................................................................................................................................ii ABSTRACT ................................................................................................................................. iii TÓM TϰT LUϯ$ϩC .............................................................................................. iv DECLARATION ........................................................................................................................... v TABLE OF CONTENTS ............................................................................................................. vi LIST OF FIGURES .................................................................................................................... vii LIST OF TABLES ..................................................................................................................... viii LIST OF ABBREVIATIONS ...................................................................................................... ix Chapter 1. INTRODUCTION ............................................................................................... 1 1.1. Hexapod and The Stewart Platform ................................................................................... 1 1.2. Optimal Path Planning & Trajectory Planning algorithms ...................................... 1 1.3. Adaptive control algorithms.................................................................................................. 2 1.4. The objectives of the thesis .................................................................................................... 3 1.5. Thesis organization ................................................................................................................... 4 Chapter 2. 2.1. LITERATURE REVIEW .................................................................................... 5 Kinematic Analysis..................................................................................................................... 5 2.1.1. Inverse Kinematic .............................................................................................................. 5 2.1.2. Forward Kinematic............................................................................................................ 9 2.2. Dynamic Analysis .................................................................................................................... 10 2.2.1. Velocity And Acceleration Analysis ........................................................................ 10 2.2.2. Dynamic Modelling ......................................................................................................... 14 Chapter 3. MULTI-OBJECTIVE OPTIMAL PATH PLANNING SCHEME ................ 16 3.1. WORKSPACE ANALYSIS ....................................................................................................... 16 3.2. MULTI-OBJECTIVE OPTIMAL PATH PLANNING SCHEME .................................... 18 3.2.1. SingularityȂfree path generation process ............................................................ 18 3.2.2. The path interpolation algorithm using B-spline curve ................................ 18 3.2.3. The Multi-objective path planning using Particle Swarm Optimization 20 vii Chapter 4. ADAPTIVE SLIDING MODE CONTROLLER ............................................ 25 4.1. Conventional Sliding Mode Controller ........................................................................... 25 4.2. Adaptive Sliding Mode Controller .................................................................................... 26 Chapter 5. SIMULATION AND RESULT ........................................................................ 32 5.1. Multi-objective Optimal Path Planning Test Plan ..................................................... 33 5.2. Adaptive Sliding Mode Control Test Plan ..................................................................... 38 5.3. Result of simulation................................................................................................................ 42 5.3.1. Result of Multi-objective Optimal Path planning .............................................. 42 5.3.2. Result of Adaptive Sliding Mode control .............................................................. 47 5.3.3. Discussion ........................................................................................................................... 67 Chapter 6. CONCLUSION AND FUTURE WORK ......................................................... 68 6.1. Achievement .............................................................................................................................. 68 6.2. Limitation.................................................................................................................................... 68 6.3. Future work ............................................................................................................................... 69 LIST OF PUBLICATIONS ....................................................................................................... 70 REFERENCES ............................................................................................................................ 71 viii LIST OF FIGURES Figure 2.1 Model Diagram of Hexapod .............................................................................................. 5 Figure 2.2. The vector loop closure of one active link ................................................................ 6 Figure 2.3. The coordinates of joints Ai & Bi ................................................................................... 8 Figure 2.4. The forward kinematic model using Simscape Simulink................................ 10 Figure 2.5. The 6-DOF body sensor is used for tracking position, velocity, and acceleration of the end-effector of the mobile platform ........................................................ 10 Figure 3.1 The process of workspace analysis ........................................................................... 16 Figure 3.2 The result of the workspace analysis process in 3D View XYZ ..................... 17 Figure 3.3 The result of the workspace analysis process in 2D View XY ........................ 17 Figure 3.4 The Multi-objective optimal Path planning ........................................................... 22 Figure 4.1 The conventional sliding mode controller ............................................................. 26 Figure 4.2 Adaptive Sliding Mode Controller .............................................................................. 26 Figure 5.1. Hexapod CAD model designed on SOLIDWORK software .............................. 32 Figure 5.2. Optimal path planning test plan ................................................................................. 35 ‹‰—”‡ͷǤ͵Ǥ‡–ƒ‹Ž‘ˆ–Š‡Dzƒ–ŠŽƒ‹‰dz„Ž‘… ........................................................................... 35 ‹‰—”‡ͷǤͶǤ‡–ƒ‹Ž‘ˆ–Š‡Dz˜‡”•‡‹‡ƒ–‹…Ƭ›ƒ‹…‘†‡Ždz„Ž‘… ............................ 36 ‹‰—”‡ͷǤͷǤ‡–ƒ‹Ž‘ˆ–Š‡Dz͸- …‘–”‘ŽŽ‡”dz„Ž‘ck ..................................................................... 36 ‹‰—”‡ͷǤ͸Ǥ‡–ƒ‹Ž‘ˆ–Š‡Dzƒ•‡’Žƒ–ˆ‘”ͳ”‹‰‹†dz„Ž‘… ............................................................ 37 ‹‰—”‡ͷǤ͹Ǥ‡–ƒ‹Ž‘ˆ–Š‡Dz’’‡”’Žƒ–ˆ‘”ͳ”‹‰‹†dz„Ž‘… ......................................................... 37 ‹‰—”‡ͷǤͺǤ‡–ƒ‹Ž‘ˆ–Š‡Dz…–—ƒ–‘”͸dz„Ž‘… .................................................................................. 38 Figure 5.9. Adaptive Sliding Mode Control Test Plans ............................................................ 38 ‹‰—”‡ͷǤͳͲǤ‡–ƒ‹Ž‘ˆ–Š‡Dzƒ–Š’Žƒ‹‰dz„Ž‘… ........................................................................ 39 ‹‰—”‡ͷǤͳͳǤ‡–ƒ‹Ž‘ˆ–Š‡Dz‡šƒ’‘†‘†‡Ždz„Ž‘… ..................................................................... 40 ‹‰—”‡ͷǤͳʹǤ‡–ƒ‹Ž‘ˆ–Š‡Dz‘–”‘ŽŽ‡”dz„Ž‘… ................................................................................ 40 ‹‰—”‡ͷǤͳ͵Ǥ‡–ƒ‹Ž‘ˆ–Š‡Dz†ƒ’–‹˜‡ ƒ‹dz„Ž‘… ........................................................................ 41 ‹‰—”‡ͷǤͳͶǤ‡–ƒ‹Ž‘ˆ–Š‡Dzƒ–ƒ’”‘…‡••‹‰ƒ†‡’‘”–‹‰dz„Ž‘… ..................................... 42 Figure 5.15. The Pareto distribution of the three objective function costs ................... 42 ix Figure 5.16. The average best solution in the task space coordinate ............................... 43 Figure 5.17. The average best solution in the joint space coordinate .............................. 44 Figure 5.18. The average best solution in the joint space coordinate with constraint checking........................................................................................................................................................ 45 Figure 5.19. The 3D view of the simulation model of the Initial pose .............................. 46 Figure 5.20. The 3D view of the simulation model of the Final pose................................ 46 Figure 5.21 Space view of the platform position and orientation in the disturbancefree test ......................................................................................................................................................... 47 Figure 5.22 Space view of the platform position and orientation in the disturbance test .................................................................................................................................................................. 47 Figure 5.23. The sensitive response of the system with external disturbance (X-Y-Z axis) .......................................................................................................................................................................... 48 ‹‰—”‡ͷǤʹͶǤŠ‡•‡•‹–‹˜‡”‡•’‘•‡‘ˆ–Š‡•›•–‡™‹–Š‡š–‡”ƒŽ†‹•–—”„ƒ…‡ȋɀ-Ⱦ-Ƚ angles) ........................................................................................................................................................... 49 Figure 5.25. The joint-space coordinate response in the disturbance-free test .......... 50 Figure 5.26. The joint-space coordinate response in the disturbance test .................... 50 Figure 5.27. The performance of Servo 1 in the disturbance-free test ............................ 51 Figure 5.28. The performance of Servo 1 in the disturbance test ...................................... 51 ‹‰—”‡ͷǤʹͻǤŠ‡•‡•‹–‹˜‹–›‘ˆ–Š‡ƒ†ƒ’–‹˜‡‰ƒ‹™‹–Š†‹•–—”„ƒ…‡ɒ† ............................ 52 Figure 5.30 Space view of the platform position and orientation with Mp = 2.2639 kg .......................................................................................................................................................................... 54 Figure 5.31 Space view of the platform position and orientation with Mp =5*2.2639 kg ..................................................................................................................................................................... 54 Figure 5.32. The sensitive response of the system with the change of platform mass (X-Y-Z axis) Ȃ Mp = 2.2639 kg............................................................................................................. 55 Figure 5.33. The sensitive response of the system with the change of platform mass ȋɀ-Ⱦ-Ƚƒ‰Ž‡•ȌȂ Mp = 2.2639 (kg) ................................................................................................. 56 Figure 5.34. The joint-space coordinate response in the Mp test ...................................... 57 x Figure 5.35. The joint-space coordinate response in the 5Mp test ................................... 57 Figure 5.36. The performance of Servo 1 in the Mp test ........................................................ 58 Figure 5.37. The performance of Servo 1 in the 5Mp test ..................................................... 58 Figure 5.38. The comparison of the performance between SMC and ASMC in servo 1 .......................................................................................................................................................................... 60 Figure 5.39 Space view of the platform position and orientation with disturbance-free .......................................................................................................................................................................... 61 Figure 5.40 Space view of the platform position and orientation with disturbance . 61 Figure 5.41. The sensitive response of the system with the disturbance-free (X-Y-Z axis) ................................................................................................................................................................ 62 Figure 5.42. The sensitive response of the system with the disturbance-ˆ”‡‡ȋɀ-Ⱦ-Ƚ angles) ........................................................................................................................................................... 63 Figure 5.43. The joint-space coordinate response in the disturbance-free test .......... 64 Figure 5.44. The joint-space coordinate response in the disturbance test .................... 64 Figure 5.45. The performance of Servo 1 in the disturbance-free test ............................ 65 Figure 5.46. The performance of Servo 1 in the disturbance test ...................................... 65 xi LIST OF TABLES Table 5-1. Hexapod Model Parameter ............................................................................................ 33 Table 5-2. Parameter used for B-spline interpolation ............................................................ 34 Table 5-3. Parameters of MOPSO process..................................................................................... 34 Table 5-4. Simulation Data Of Adaptive Sliding Mode Controllers .................................... 39 Table 5-5. Task space error compare for Case 1 ........................................................................ 53 Table 5-6. Task space error compare for Case 2 ........................................................................ 59 Table 5-7. Task space error compare for Case 4 ........................................................................ 66 xii LIST OF ABBREVIATIONS Abbreviation PD MRAC MOPSO CAD Description Proportional - Derivative Model Reference Adaptive Control Multi Objective Particle Swarm Optimization Computer Aided Design 1 Chapter 1. INTRODUCTION 1.1. Hexapod and The Stewart Platform In this thesis, the Hexapod is designed based on the new structure of the Stewart Platform (SP), invented by Dr. Stewart in 1965 [1]. The Stewart manipulator consists of top-plate and baseplate connected with the help of six variable-length electromechanical actuators with spherical joints (Spherical Ȃ Prismatic Ȃ Spherical SPS type) that are used for rotation and translation of the top-plate concerning the base-plate. In this study, we modify the origin linear actuators mechanism by changing the combination SPS type to Revolute Ȃ Spherical Ȃ Spherical (RSS) type and replacing the current prismatic joints with the new revolute joints. The Hexapod attracts many researchers in the current decade due to its precision, rigidity, and the high force-to-load ratio [2]. The proper coordination of the actuator's angles enables the top plate to follow the desired trajectory with high accuracy. Thus the six inputs to the SP in terms of torque are calculated by the controller and provided by highspeed servo motors. The outputs of the Hexapod are the upper plate's translation and angular positions (in the surge, sway, heave, roll, pitch and yaw) usually sensed by highly precise sensors or vision systems. 1.2. Optimal Path Planning & Trajectory Planning algorithms Although parallel manipulators have strong stiffness and high payload, their workspaces are limit and exit many singular zones, which are the points where the platform becomes uncontrollable [3]. Therefore, to avoid singular poses in the operations, we need to have a singularity-free path planning algorithm. Various projects have been working on these issues. B. Dasgupta et al proposed an algorithm by using via points and Divide and Conquer strategy in [4]. Hao Li et al presented a method to calculate singularity-free task space using Quasi-singularity measures [5]. The paper of H. Abdellatif and B. Heimann used the visibility graph methodology and two heuristic functions to plan singularity-free motions [6]. However, the goals of the path planning process are not only to find singularity-free paths but also to seek ways 2 that let the Hexapod to moving as fast as possible while satisfying the constraints of mechanical structure such as the limits in position, velocity, acceleration, and torque output from actuators. B. Xie et al used Discrete mechanics to generate optimal trajectories [7]. For this kind of multi-objective optimization path planning for the complex parallel manipulators, C. Dong et all have been researched optimal singularity-free trajectory planning based on B-spline curves and evolutionary algorithms for docking systems in [8], [9]. The B-spline curves combined with Evolutionary algorithms are useful for solving multi-objective optimization motion planning because of their flexibility and easy implementation [10]. 1.3. Adaptive control algorithms Last decades, many researchers have worked on sliding mode control and adaptive control technics for the Stewart platform. C. C.Nguyen et al proposed an adaptive control scheme for the Linear SP with the baseplate is above the mobile platform [11]. In [12] Amir Ghobakhloo et al presented an adaptive-robust tracking control design for SP. A sliding mode tracking control of the SP was presented by Chin-I Huang in [13]. S. Iqbal studied the Robust Sliding Ȃ Mode controller design with uncertain dynamics in presence of nonlinearities [14]. In [15], E. Yime et al detailed the robust adaptive control of the SP based on the task space dynamics equations. J. Velasco et al took some experimental validations of a sliding mode control for the SP used in aerospace inspection [16]. Sung-Hua Chen and Li-Chen Fu chose Output Feedback Sliding Mode with a Nonlinear Observer-Based Forward Kinematics Solution to reduce the computational load in solving forward kinematics solutions when controlling for their SP [17]. Ramesh Kumar P. proposes the application of a new algorithm for the position control of SP. In their research, the existing integral sliding mode control law for systems with matched disturbances is modified by replacing the discontinuous part with a continuous modified twisting control [18]. There are various control approaches for the position control problem using sliding mode theories in different perspectives. A sliding mode controller is a variable structure 3 control, which has been successfully implemented in many electromechanical systems for improving their performance and achieving robustness against disturbances [19]. In recent years, a new trend in position tracking control design for robot manipulators in general and also for SP that combines adaptive and sliding mode control is attracting more and more researchers. Ayca Gokhan Ak et al proposed a Fuzzy Sliding Mode Controller with RBF Neural Network for Robotic Manipulator Trajectory Tracking [20]. Wu Dongsu et al provided an adaptive sliding controller in task space on the base of the linear Newton-Euler dynamic equation of motion platform in a six-DOF flight simulator in [21]. Their method uses a nonlinear adaptive controller to identifies constant uncertain parameters and sliding control to eliminate the influences of the time-varying uncertain parameters and external disturbance. Radial Basis Functions and fuzzy neural networks were proposed for adaptive and fixed sliding mode control of robotic manipulators in [22], [23], [24], [25], [26], [27], [28]. Generally, many adaptive sliding mode control designs based on the sliding surface have been conducted for robot manipulators and SP with linear actuators. However, there is a lack of studies on SP with rotary actuators that will be proposed in the next sections. Some previous projects can be addressed such as [29], [30], and [31]. 1.4. The objectives of the thesis In this study, first, we propose a method to calculate the kinematics and dynamics of the Hexapod model. Then from the result of kinematics models, a workspace analysis process is conducted to find non-singularity zones and the limitations of workspace for the path planning processes. Next, we propose optimal path and trajectory generation schemes inspired by B-spline curves and an evolutionary algorithm, the multi-objective Particle Swarm Optimization (MOPSO). An adaptive sliding mode control is developed for tracking the optimal path planning. The adaptive controller is based on the conventional sliding surface combined with a 4 proportional Ȃ derivative (PD) controller whose gains are adjusted online to provide high tracking precision with fast finite-time convergence, less chattering, fast transient response, and high robustness. 1.5. Thesis organization The rest of this thesis is organized as follows. Chapter 2 reviews the kinematic and dynamic models of the platform, respectively. Chapter 3 presents path and trajectory planning with multi-objective optimization methods. Chapter 4 establishes the adaptive sliding mode control using sliding surface combined with adaptive proportional Ȃ derivative gain compensation to get the smooth actuators torques on the planned path. The adaptive laws of the gain compensations are designed to ensure real-time adjustment. Many simulations and evaluations have been conducted in Chapter 5 to prove the optimization results and the fast convergence velocity of tracking errors and the robustness of the adaptive sliding controllers. The final chapter 6 of this thesis will be a summary of the work done and conclusions. Some additional works that can be done in the future are also explained. 5 Chapter 2. LITERATURE REVIEW 2.1. Kinematic Analysis 2.1.1. Inverse Kinematic The inverse kinematics functions are the relations between the positions and orientations of the end-effector of the mobile platform and the lengths/angles of the actuators which are the most important functions in motion planning and control of parallel manipulators [32]. Let define the two coordinate frames: {P} for the end-effector which is attached to the origin of the mobile platform and {O} for the center of the base platform which is the body-fixed frame. In this thesis, we solve the inverse kinematics by finding the functions which describe the relation of the angles of 6 active links (the crank arms) and the translation ܶ ൌ ሾ‫ݔ‬ ‫ݕ‬ ‫ݖ‬ሿ் and orientation ȳ ൌ ሾߛߚߙሿ் which are selected as Euler angles as follows ߛƒ„‘—–ܺ௉ , ߚƒ„‘—–ܻ௉ , and ߙƒ„‘—–ܼ௉ of the mobile platform concerning the base platform: (2-1) ߣ௜ ൌ ݂௜௡௩ ሺܺ௉ ሻܺ௉் ሼ݅ ൌ ͳǡʹǡ ǥ ǡ͸ሽ Where ߣ௜ ሼ݅ ൌ ͳǡʹǡ ǥ ǡ͸ሽ are the angles of active links in the joint-space, ݂௜௡௩ ሺܺ௉ ሻሼܺ௉ ൌ ሾ‫ߙߚߛݖݕݔ‬ሿ் ሽ are non-linear functions that describe the inverse kinematics, and ܺ௉ is the general coordinator of the mobile platform in the task space. Figure 2.1 Model Diagram of Hexapod 6 The inverse kinematics are constructed based on the vector loop closure equations [33]. ‫ܣ‬௜ ‫ܤ‬௜ ൌ ܱܲ ൅ ܴ௉ை ‫ܤ‬௜ െ ‫ܣ‬௜ ሼ݅ ൌ ͳǡʹǡ ǥ ǡ͸ሽ (2-2) ܴ௉ை is the rotation matrix which transforms the frame {P} to base frame {O} ܿߙܿߚ െ‫ ߛܿߙݏ‬൅ ܿߙ‫ ߛݏߙݏ ߛݏߚݏ‬൅ ܿߙ‫ߛܿߚݏ‬ (2-3) ൌ ൥‫ ߛܿߙܿ ߚܿߙݏ‬൅ ‫ ߛݏߚݏߙݏ‬െܿߙ‫ ߛݏ‬൅ ‫ ߛܿߚݏߙݏ‬൩ െ‫ߚݏ‬ ܿߚ‫ߛݏ‬ ܿߚܿߛ Where ܿ ሺήሻ ൌ …‘•ሺήሻ and ‫ݏ‬ሺήሻ ൌ •‹ሺήሻ. However, the Hexapod with the (RSS) type ܴ௉ை contains an extra link per leg: crank arm Ȃ spherical Joint Ȃ rod arm Ȃ spherical joint. ‫ܣ‬௜ ‫ܦ‬௜ ൌ ܱܲ ൅ ܲ‫ܤ‬௜ ൅ ‫ܦ‬௜ ‫ܤ‬௜ െ ܱ‫ܣ‬௜ ሼ݅ ൌ ͳǡʹǡ ǥ ǡ͸ሽ (2-4) ‫ܣ‬௜ ‫ܤ‬௜ ൌ ‫ܣ‬௜ ‫ܦ‬௜ ൅ ‫ܦ‬௜ ‫ܤ‬௜ (2-5) ‫ܣ‬௜ ‫ܦ‬௜ ൌ ‫ݎ‬ሺܿ‫ߣݏ݋‬௜ ܺଵ ൅ ‫ߣ݊݅ݏ‬௜ ܻଵ ሻ (2-6) ‫ܦ‬௜ ‫ܤ‬௜ ൌ ‫ݏ݌‬௜ It is noted that some notations of those equations are as below: x ‫ ݎ‬is the length of the active link (the crank arm length). x ߣ௜ is the rotation angle of the active link ith in the Ai frame x ‫ ݌‬is the length of the vector ‫ܦ‬௜ ‫ܤ‬௜ (the rod arm length). x ‫ݏ‬௜ is the unit vector along the rod arm length. Figure 2.2. The vector loop closure of one active link (2-7)