Phương pháp phân tích ưu hóa mô hình động học robot song song

To this end, the numerical simulation and experiment results of parallel Scara robots prove that the proposed method can be applied to solve kinematic problems for a variety of parallel

Optimization Analysis Method of Parallel Manipulator

Kinematic Model

A Dissertation Submitted for the Degree of Doctor

Candidate：Trang Thanh Trung Supervisor：Prof Li Weiguang

South China University of Technology

Guangzhou, China

论文提交日期：2017 年 11 月 11 日 论文答辩日期：2018 年 03 月 12 日 学位授予单位：华南理工大学 学位授予日期： 年 月 日

答辩委员会成员：

主席： 张永俊教授 .

委员： 黄平教授 赵学智教授 姚锡凡教授 李伟光教授 .

摘 要 本文的主要目的是建立一个新的算法，以简化所有类型的并联机器人的运动学问题的解决，而不限制自由度的数量。该算法适用于各种并联机器人结构，具有精度高、可靠性好、执行时间短、比现有方法更易于使用的特点。五连杆并联机器人的数值模拟和实验结果表明，该方法可用于解决各种并联机器人的运动学问题，对于结构复杂和自由度多的并联机器人，该方法也具有计算时间短、精度高、可靠性高、结果收敛快等优点。此外，本文还扩展了该方法在机器人公差设计领域的应用。通过两个仿真实验验证了该方法的可行性；计算和仿真结果也说明了所提出的公差分配方法的准确性和效率。

首先，在研究手臂机器人优化问题的基础上，本论文提供了新的接入方法以寻找运动学参数，即将传统并联机器人运动学问题转换成有约束的非线性最优化问题，其

Rosenbrock-Banana 函数最合适是广义简约算法。从运动学控制试验中直接寻找，将缩短编程开发时间。

了各类机器人的典型并联机器人解决方案。在两个不同的空间（关节空间和工作空间）之间的唯一解决方案的保证已经充分论证。可靠性和精密度试验结果表明，所提出的方法是非常可靠和准确的。通过与其它算法相比较，求解最优运动问题的顺序二次规划和遗传算法，提出的方法的精度更高（约从二个到四个数量级），并且具有较短的执行时间。

第四，逆运动学的结果作为实时控制机器人轨迹的信息，通过 Adams 仿真以及五连杆并联机器人的实验表明，该方法能够实际应用于并联机器人控制。

最后，除了用于并行机器人运动学求解外，本论文提出的方法还可以应用于一个新领域中—机器人制造设计，即确定成品工序容差以保证末端执行器的估计正确度和精度。该技术不仅能应用在并行机器人而且还可以应用给手臂机器人。通过两个实例验证了上述方法的可行性和计算结果, 该方法能准确、有效设计公差构件。

造设计误差。

Abstract The primary objective of this dissertation is to build a new algorithm that simplifies the resolution of the kinematic problems for all types of parallel robots without limiting the number

of degrees of freedom This algorithm applies to various parallel robotic structures in a general order with high accuracy and reliability, shorter execution time, easier to use than current methods To this end, the numerical simulation and experiment results of parallel Scara robots prove that the proposed method can be applied to solve kinematic problems for a variety of parallel robots regardless of its structures and degree of freedom with several advantages such

as shorter computation time, high precision, high reliability and rapid convergence of results

In addition, this dissertation also extends the application of the proposed method in the field of robot tolerance design Two examples are used to verify the feasibility of the proposed method; the accuracy and efficiency of the proposed method for generating tolerance allocations are also illustrated by calculations and simulation results

Firstly, based on optimal problem applied on the robot arm the dissertation proposes a new approach to find kinematic parameters by transforming the kinematic problem of the traditional parallel robot into a nonlinear optimization with the objective function Rosenbrock-Banana Through many tests, the best algorithm for the Rosenbrock-Banana function in the optimal problem is the General Reduced Gradient (GRG) method Direct recovery of the kinematic control resulting from the optimal problem will reduce the preparation time of the programmable data

Secondly, classification of a parallel robot based on texture with or without prismatic joints, the dissertation has been grouped into three types of parallel robots: the non-prismatic parallel robot (type 1) and the prismatic parallel robots including the parallel robot with the active prismatic joints connected to its base (type 2), the parallel robot with the second prismatic joints from its base (type 3) The dissertation presented modeling for all types of parallel robot structures and how to convert the mathematical model of the kinematic problem of the parallel robot to the optimal form The situations that may arise when applying the proposed method on the three types of parallel robots are fully argued With type 1 of parallel robot, the initial mathematical models when transforming into optimal problem, the object function is the quadratic function, so directly apply the GRG to solve the kinematic problem but the initial mathematical models of type 2 and type 3 robots are the quaternary function, which is incompatible with the proposed method Thus, the dissertation proposes to solve this problem

by using the equivalent substitution configuration to downgrade the object function form of the

two types of robots (type 2 and type 3) from quaternary function to quadratic function, which

is compatible with the proposed method

Thirdly, the Microsoft-Excel solver application supports mathematical resolution, to illustrate the example, by solving the kinematic problem of the robot for some typical parallel robots for each type of robot are presented in detail The assurance of a unique solution between two different spaces (joint space and work space) has been fully argued The results of the reliability and precision tests showed that the proposed method was very reliable and accurate

By comparing with other algorithms to solve an optimal problem, which are Sequential Quadratic Programming and the Genetic Algorithm to solve the optimal kinematic problem, the proposed method has exceeded the accuracy (approximately from 2

10 to 10 times) and has 4shorter execution time

Fourthly, the results of the inverse kinematic problem are used as information to control the trajectory of the robot in real time, presented in detail and illustrated by the Adams simulation software as well as experiments in the Scara parallel robot Experimental results demonstrated the capability, accuracy and feasibility of the proposed method when applied to robot control in practice

Finally, in addition to solving the kinematic problem of the parallel robot, the dissertation also developed a new application of the method proposed in the field of the manufacture of robots in order to design the tolerances of the components (links and joints) to ensure the given accuracy of the end effector and vice versa This technique applies not only to parallel robots but also to the robot arm Two examples are used to verify the feasibility of the above method and the calculated result that the method can produce tolerance allocations accurately and efficiently

Key words: Parallel robot; kinematics problem; optimization problem; equivalent structure;

General Reduced Gradient method; tolerance design

目 录

摘 要 I Abstract III

目 录 V Contents VIII 图目录 XI

表目录 XV

第一章 绪论 1

1.1 机器人信息的初始化方法 1

1.2 机器人运动学、模型与解决方法 2

1.2.1 机器人运动学 2

1.2.2 建模的方法 3

1.2.3 解决模型的方法 7

1.2.4 并联机器人运动学问题求解方法综述 9

1.3 研究方向 13

1.4 研究对象和研究方法 14

1.5 本论文的内容 14

第二章 各类机器人运动学问题优化的数学模型 17

2.1 引言 17

2.2 机器人运动学优化形式 17

2.2.1 机器人运动学的最优数学模型 17

2.2.2 手臂机器人优化问题的基础 19

2.2.3 最优运动问题 23

2.2.4 算法图 23

2.2.5 均匀的精密结构 25

2.2.6 差分计算对准确性的影响 27

2.3 并联机器人的关联向量方程类型 30

2.3.1 手臂机器人和并联机器人相关矢量方程的建立方法差异 30

2.3.2 非棱柱并联机器人（类型 1） 31

2.3.3 棱柱并联机器人 36

2.3.4 手臂机器人与并联机器人数学模型的异同点 40

2.4 本章小结 42

第三章 并联机器人运动学问题的广义简化梯度算法研究 43

3.1 引言 43

3.2 广义简化梯度算法 43

3.3 Microsoft-Excel 求解器优化应用介绍 47

3.4 使用广义简化梯度算法解决并行机器人的运动问题 50

3.4.1 并联机器人（类型 1） 50

3.4.2 同等代替结构和变量公式 59

3.4.3 第一棱柱关节连接到固定平台的并联机器人（类型 2） 63

3.4.4 第二棱柱关节连接到固定平台的并联机器人（类型 3） 85

3.4.5 两种不同空间之间的独特的解决方案的保证 104

3.4.6 测试新方法的可靠性 105

3.4.7 测试新方法的精度和准确度与其他方法的比较 108

3.5 本章小结 117

第四章 仿真与实验研究 118

4.1 引言 118

4.2 实验的内容 118

4.3 背景设计实验 118

4.3.1 五连杆并联机器人 118

4.3.2 建立运动学特性的关节的五连杆并联机器人 127

4.4 测试模拟和数值结果的准确性 145

4.4.1 以图形方式检查数学的结果 146

4.4.2 测试结果与数学模拟软件 149

4.5 实验研究 153

4.5.1 实验设置 153

4.5.2 机电-电子-软件基本参数 155

4.5.3 设计控制系统软件 160

4.5.4 经验和讨论结果 168

4.6 本章小结 176

第五章 使用广义简约梯度算法确定机器人运动关节的公差参数 178

5.1 引言 178

5.2 公差设计文献综述 178

5.3 公差最优问题的形成 182

5.4 公差优化问题的求解方法 183

5.5 关节运动公差的确定 183

5.6 通过使用逆运动学确定连杆尺寸和关节自由径向运动的公差 186

5.7 数值模拟实例 188

5.7.1 手臂机器人 188

5.7.2 并联机器人 190

5.8 检查提出的方法的准确性 193

5.9 本章小结 194

第六章 结论和展望 195

6.1 结论 195

6.2 主要创新点 197

6.3 展望 197

参考文献 199

附录 I 211

攻读博士学位期间取得的研究成果 224

致 谢 225

摘 要 I Abstract III

目 录 V Contents VIII List of figures XI List of tables XV

Chapter 1 Introduction 1

1.1 Methods for information initialization of robot 1

1.2 Robot kinematics, models and methods 2

1.2.1 Robot kinematics 2

1.2.2 Modelling phase 3

1.2.3 Model survey phase 7

1.2.4 An overview of methods for solving kinematic problems of parallel robot 9

1.3 Research orientation 13

1.4 Subjects and research methods 14

1.5 Contents of the present thesis 14

Chapter 2 Mathematical Bases for Changing from the Robot Kinematic Problem to the Optimization Problem 17

2.1 Introduction 17

2.2 Robot kinematic under the optimization form 17

2.2.1 The optimal mathematical model of robotic kinematic 17

2.2.2 Bases for optimization problems on the robot arm 19

2.2.3 The optimal movement problem 22

2.2.4 Algorithm diagram 23

2.2.5 The uniform precision structure 25

2.2.6 The effect of the difference calculation on the accuracy of the problem 27

2.3 Types of associated vector equations for parallel robots 30

2.3.1 Difference in the way to build the associated vector equations for robot arms and parallel robots 30

2.3.2 The non-prismatic parallel robot (Type 1) 31

2.3.3 The prismatic parallel robots 36

2.3.4 Identify similarities in the mathematical model of parallel robots and robot arms 40

2.4 Chapter conclusion 42

Chapter 3 Application of Generalized Reduced Gradient Algorithm to Solve the Kinematic Problem of Parallel Robots 43

3.1 Introduction 43

3.2 Generalized Reduced Gradient algorithm 43

3.3 Introduction of optimization application of solver in Microsoft-Excel 47

3.4 Resolution of the Kinematic Problems of Parallel Robots using Generalized Reduced Gradient algorithm 50

3.4.1 Parallel robot of type 1 50

3.4.2 Equivalent substitution configuration and the formulation of variables change 59

3.4.3 Parallel robot of type 2 64

3.4.4 Parallel robot of type 3 86

3.4.5 The assurance of unique solution between two different spaces 105

3.4.6 Testing the reliability of the novel method 107

3.4.7 Testing the precision of the novel method and compare accuracy with other methods 110

3.5 Chapter’s conclusion 119

Chapter 4 Simulation and Experimental Study 120

4.1 Introduction 120

4.2 Content of experiment 120

4.3 Based on experimental design 120

4.3.1 Parallel Scara robot 120

4.3.2 Settings of kinematic characteristics of joints for Parallel Scara robot 129

4.4 Testing simulation and accuracy of numerical results 147

4.4.1 Inspection of results by graphics 148

4.4.2 Inspection of results by simulation software 151

4.5 Experimental study 155

4.5.1 Experimental setup 155

4.5.2 Basic parameters of mechanical-electrical-electronic components 157

4.5.3 Design of control system software 162

4.5.4 Results of experiments and discussion 170

4.6 Chapter conclusions 178

Chapter 5 Application Generalized Reduced Gradient Algorithm to Determine Tolerance Design of Robot Parameters 180

5.1 Introduction 180

5.2 Literature review of tolerance design 180

5.3 The formation of the optimal problem 184

5.4 Solution method for the optimization problem 185

5.5 Determination of the tolerance of joint angle movement 185

5.6 Determination of the deviation of link dimensions and joint free radial movement by using inverse kinematic 187

5.7 The example of numerical simulation 189

5.7.1 Robot arm 189

5.7.2 Parallel Robot 192

5.8 Checking the accuracy of the proposed method 195

5.9 Chapter conclusion 196

Chapter 6 Conclusions and Future Works 197

6.1 Conclusions 197

6.2 The main points of innovation 199

6.3 Future works 199

References 201

Appendix I 211

Achievement of research 226

Acknowledgments 227

List of figures

Figure 1-1 Diagram of closed loop on robot arrm and parallel robot 3

Figure 1-2 Closed loop vector 4

Figure 1-3 Parallel structured Robot 5

Figure 1-4 Kinematics circuit of one limb of parallel robot 6

Figure 1-5 Control Diagram in joint space 8

Figure 1-6 Control Diagram in work space 9

Figure 1-7 The subject of the overall research program 16

Figure 2-1 Algorithm diagram for solving the inverse robot kinematic problem 24

Figure 2-2 The general closed loop scheme for any limb 30

Figure 2-3 Planar parallel robot 3RRR 32

Figure 2-4 Parallel Delta robots (a) and the vector expansion loop for th i limb (b) 33

Figure 2-5 The detailed generalized coordinates for the point C 34

Figure 2-6 Setting up moving reference frames 34

Figure 2-7 The detailed generalized model of the th i limb of the SRS parallel robot 36

Figure 2-8 TPM parallel robot (a) and the generalized model of a one limb vector (b) 37

Figure 2-9 Parallel planar 3RPR Robot 38

Figure 2-10 The detailed generalized model of the th i limb of a 3-RPS parallel robot 39

Figure 2-11 The detailed generalized model of the th i limb of a 6-SPS parallel robot 40

Figure 2-12 Differences and similarities between the mathematical models of Robot Arm and Parallel robot 42

Figure 3-1 Solver parameter dialog box 47

Figure 3-2 Add- Ins of additional setting of Solver 48

Figure 3-3 3RRR planar parallel robot and the moving trajectory across twelve points belong to an ellipse 51

Figure 3-4 Set the objective function and constraints for IKP of 3RRR robot 54

Figure 3-5 Set the objectives function and constraints for FKP of 3RRR robot 57

Figure 3-6 The displacement graph of controlled joint variables i (with i=1,2,3) of 3RRR parallel robot 59

Figure 3-7 TPM robot with prismatic active joint and the substitution configuration for a limb of PRRR robot by using two revolute joints to replace one prismatic joint 60

Figure 3-8 (a) Stewart Gough robot with SPS configuration

(b) The equivalent substitution configuration 61

Figure 3-9 The geometric relation between the original variable  l i and the new one  2i as the actuator type is changed 62

Figure 3-10 Equivalent substitution configuration for some parallel robot structures contains prismatic joints 64

Figure 3-11 The moving trajectory of the PRRR robot is needed to control 64

Figure 3-12 The detailed substitution configuration for a limb of the PRRR robot 66

Figure 3-13 Settings of objective function and constraints of IKP robot PRRR 74

Figure 3-14 Settings of objective functions and constraints of FKP in robot PRRR 80

Figure 3-15 The converted displacement graph of 3 equivalent joint variables between the substitution configuration and the original structure of the PRRR robot 85

Figure 3-16 The moving trajectory of Stewart Gough robot needed to control 86

Figure 3-17 Substitution configuration for one limb of Stewart Gough robot 87

Figure 3-18 Settings of objectives function and constraints of IKP Stewart Gough robot 92

Figure 3-19 Settings of objective functions and constraints of FKP Stewart Gough robot 99

Figure 3-20 The converted displacement graph of 6 equivalent joint variables between the substitution configuration and the original structure of Stewart Gough robot 105

Figure 3-21 The multidirectional relation between joint space and workspace in parallel kinematic robots 106

Figure 3-22 The close relationship of kinematic database of the linking equation 107

Figure 4-1 The parallel scara robot with two translation degrees of freedom and detailed development of right limb 121

Figure 4-2 Equivalent kinematic diagram and detail development of the right limb of Scara parallel robot 122

Figure 4-3 Experimental trajectory need to controlled 123

Figure 4-4 Settings of the objective function and constraints of IKP parallel scara robot 125

Figure 4-5 Settings of the objective function and constraints of FKP parallel scara robot 127

Figure 4-6 The graph of displacement of variable 1 139

Figure 4-7 The displacement graph of variable 1 140

Figure 4-8 The graph of acceleration of variable 1 140

Figure 4-9 The displacement graph of variable 2 146

Figure 4-10 The graph of the velocity of variable 2 147

Figure 4-11 The graph of acceleration of variable 2 147

Figure 4-12 The kinematic problem of parallel scara robot measured by graphics 148

Figure 4-13 The kinematic structure of parallel scara robot in Adams software 152

Figure 4-14 The displacement graph of variable 1 and 2 in the Adams software 152

Figure 4-15 Diagram of experimental robotic structure 155

Figure 4-16 PCMM measurements in the form of robot ….153

Figure 4-17 Drawing of asembly of experimental robotic structure 156

Figure 4-18 Servo motor used in the experiment 158

Figure 4-19 Servo amplifier MR-E-10A E 158

Figure 4-20 Speed reduce gearbox used in the experiment 159

Figure 4-21 Rolling bearings 159

Figure 4-22 Omron encoder 160

Figure 4-23 Electronic caliper 160

Figure 4-24 Interface of database collection in the laboratory 161

Figure 4-25 Rigid couplings 161

Figure 4-26 Arduini Uno R3 microcontrollers and technical parameters 162

Figure 4-27 Diagram of systematic control algorithm 164

Figure 4-28 Interface of the system controlling parallel scara robot 164

Figure 4-29 General Embedded System used in the control system 166

Figure 4-30 Serial communication events 167

Figure 4-31 Motion control method 167

Figure 4-32 The method of moving coordinates of end-effector 168

Figure 4-33 Diagram of systematic hardware principles 169

Figure 4-34 The diagram of installation of robot in practical experiment 170

Figure 4-35 Errors of controlled trajectory between experimental and simulation 176

Figure 4-36 Error of parameter of controlled angles1&2between experimental and simulation 177

Figure 4-37 Error of secondary parameters1&2between experimental and simulation 177 Figure 4-38 Error of translational joint L1&L2 in the orginal configuration between experimental and simulation 177

Figure 5-1 The movement with the smallest step of moving platform between two points in space 186

Figure 5-2 Several types of clearance to be controlled in joints 186

Figure 5-3 The transmission deviation of restricted angle caused by mechanic clearance defined by (5-10) 187

Figure 5-4 Tolerance choice of built-up links 189

Figure 5-5 The use of calculated tolerance results 189

Figure 5-6 The equivalent kinematic diagram of robot Fanuc S900W 190

Figure 5-7 Six allowable moving points of the end-effector in the limited deviation range of a sphere 190

Figure 5-8 The 3-RRR planar parallel robot 193

Figure 5-9 The errors of control trajectory 196

List of tables Table 1-1 Advantages, disadvantages and limitations of some methods solving parallel

kinematic problem 11

Table 2-1 Value of function y in the break points 27

Table 3-1 Solver’s terms in the program interface 49

Table 3-2 Meaning of options in the item Option of Solver 49

Table 3-3 Coordinates of twelve key points of the trajectory and the variation of the angle φ in each key point 51

Table 3-4 Interface of declaration of the kinematic problems of 3RRR parallel robots 53

Table 3-5 IKP result of 3RRR parallel robot 55

Table 3.6 The error of the objective functions, running time (seconds) and the iterations of each key point in IKP of 3RRR parallel robot 55

Table 3-7 FKP results of 3RRR parallel robot 57

Table 3-8 The error of the objective functions, running time (seconds) and iterations of each set of control parameter in FKP of 3RRR parallel robot 57

Table 3-9 The error control of 3RRR parallel robot 58

Table 3-10 Coordinates of 24 key points in the trajectory that needs to controlled 64

Table 3-11 Interface of declaration of IKP of parallel robot PRRR 88

Table 3-12 Results of IKP of parallel robot PRRR, the 1st limb (rad) 74

Table 3-13 Results of IKP of parallel robot PRRR, the 2nd limb (rad) 75

Table 3-14 Results of IKP of parallel robot PRRR, the 3rd limb (rad) 76

Table 3-15 IKP results of parallel robot PRRR in conversion into control variable L in the original configure, error of objective functions, time (second) and iterations of each key point 77

Table 3-16 Interface of declaration of FKP of parallel robot PRRR 79

Table 3-17-3-18 Results of FKP in parallel robot PRRR 81

Table 3-19 Results of FKP in parallel robot PRRR, error of objective function, time (second) and iterations key point 83

Table 3-20 The control error of parallel robot PRRR 84

Table 3-21 Coordinates of twenty-four key point under trajectory that need to be controlled of 6SPS robot 86

Table 3-22 Interface of declaration of IKP with Stewart Gough parallel robot 91

Table 3-23 IKP results of Stewart Gough parallel robot, value of angles 1i 93

Table 3-24 IKP results of Stewart Gough parallel robot, value of angles 2i 94 Table 3-25 IKP results of Stewart Gough parallel robot, value of angles 3i 95 Table 3-26 IKP results of Stewart Gough parallel robot, when exchange into control variable L

in the original configuration The error of objective function, time (seconds) and iterations

in each key point 95 Table 3-27 The interface of declaration of FKP with Stewart Gough parallel robot 96 Table 3-28 FKP results of Stewart Gough parallel robot, value of angles 1i 99 Table 3-29 FKP results of Stewart Gough parallel robot, value of angles 2i 100 Table 3-30 FKP results of Stewart Gough parallel robot, value of angles 3i 101 Table 3-31 FKP results of Stewart Gough parallel robot, value of control coordinates in each key point 101 Table 3-32 FKP results of Stewart Gough parallel robot, control valueL i , error of object function, time (second) and iterations in the key point 102 Table 3-33 The error control of control for Stewart Gough parallel robot 103 Table 3-34 The results of the objective function at 12 controlled trajectory points of 3RRR planar parallel robot in Matlab 107 Table 3-35 The results of the objective function at 24 controlled trajectory points of TPM robot

in Matlab 108 Table 3-36 The results of the objective function at 24 controlled trajectory points of Stewart Gough robot in Matlab 109 Table 3-37 The results of the objective function and solver time for each keypoint for 3RRR robot in FKP 112 Table 3-38 The results of the objective function and solver time for each keypoint for 3RRR robot in IKP 113 Table 3-39 The results of the objective function and solver time for each keypoint for 3PRRR robot in FKP 113 Table 3-40 The results of the objective function and solver time for each keypoint for 3PRRR robot in IKP 114 Table 3-41 The results of the objective function and solver time for each keypoint for Stewart Gough robot 6SPS in FKP 115 Table 3-42 The results of the objective function and solver time for each keypoint for Stewart Gough robot 6SPS in IKP 116

Table 3-43 The actual computational time of the three method 117 Table3-44 Trajectory tolerance of 3RRR parallel robot 118 Table3-45 Trajectory tolerance of Stewart Gough Robot 6SPS 118 Table3-46 Trajectory tolerance of TPM robotdetailed development of right limb 118 Table 4-1 Coordinates of points belong to experimental controlled trajectory 123 Table 4-2 Interface of declaration of IKP of parallel Scara robot 124 Table 4-3 IKP results of parallel Scara robot 125 Table 4-4 FKP results of parallel Scara robot 127 Table 4-5 The error of controlled trajectory of parallel Scara robot 128 Table 4-6 The value of intermediate velocity at the displacement point of variable 1 133 Table 4-7 The model of each segment of the variable 1 joint space (state variable t) 134 Table 4-8 The model of each segment of variable 1in the joint space (actual time variable ) 135 Table 4-9 The value of intermediate velocity at the displacement points of variable 2 141 Table 4-10 The model of each segment of variable 2in joint space (state variable t) 142 Table 4-11 The model of each segment of variable 2 in the joint space (actual time variable)… ……… 143 Table 4-12 The results of kinematic problems of parallel scara robot in AutoCAD 149 Table 4-13 Comparision results of kinematic problems of parallel scara robot in AutoCAD with

in novel method 150 Table 4-14 Comparision of the results of displacement control of novel method and Adams software 153 Table 4-15 Error of displacement control of novel method and Adams software 153 Table 4-16 Average experimental values and errors of controlled trajectory 171 Table 4-17 Average experimental value and error of angles 1&2 172 Table 4-18 Average experimental value and error of angles 1&2 173 Table 4-19 The average experimental value and error of translational joint in the original configuration L1&L 174 2

Table 4-20 The degree of accuracy of the formula of variable change 175 Table 5-1 Kinematic parameters of robot Fanuc S900W 208 Table 5-2 Extracted results of measured tolerances of built-up links in robot Fanuc S900W

209 Table 5-3 The results of dimension tolerances in built-up links of robot Fanuc S900W 210 Table 5-4 Extracted results of measured tolerances in built-up links of 3RRR parallel robot 211 Table 5-5 The tolerance results of built-up link dimensions of 3RRR parallel robot 213

Chapter 1 Introduction Chapter 1 Introduction 1.1 Methods for information initialization of robot

How to have a machine with the skills of human which can replace human to do whatever they want, it is a legitimate demand that attract many scientists Today, the machine is becoming more compact and smart because its function is not only determined by the hardware but also mainly by the software The software itself has many different levels from the lowest which is merely a hard command sequence to implement to advanced software which is more popular in the industry today expressed, it forms different layer of control according to situations Despite the origin of the software level, the nature of information given in appropriate time or more detailed is the result of the survey describing a control lever system The root of the lower level software no matter what nature is giving information to the appropriate time, or it is more detailed results from the survey give a model to describe the control system

This study only discusses about the robot, which is a special product of the mechatronics, and also talk about how to build or initiate information to control it, more specifically, the kinematic information, data ensuring the accuracy of the controlling process according to a predetermined trajectory

At current levels, the robot inputs often initialized according to the method as follow:

- Manual programming (code G);

- From the limit switch or sample displacement (adjust gauge, contact…)

- Programming by PC (APT or APT, or APT like);

- Retrieved from another system via external links (CAD/CAM);

- From artificial vision (Camera, sensor);

- From auditory (voice)

- From neurobiology (bioelectric impulses of the living body);

- From sensors equipped on the robot (encoder + teach-in technique)

Each method has its own advantages and disadvantages:

- Manual programming with G code is appropriated with simple program, easy to implement and easy to learn, but not available for complex interpolation, especially the curves, most commercial robot has straight or curve interpolation or a combination of such objects only

- The information provided by the limit switch is only suitable for simple systems

- Although Programming by PC can create targeted program (post processor) with minimal source code (processor), it only saves time writing large programs with a combination of standard commands, without description of non-traditional trajectories

- In case of using external links, geometry information of the trajectory is based on graphics;

- In all listed situations, data is entered into movement line of the stage controlled according to its trajectory in works space while the electric motor operate in accordance with the rules joints space, except for teach-in technique in which data is not required

to go through inverse kinematics problem because this data is measured directly by sensor in joints space, this stage is required in the remaining methods, an effective algorithm to convert data between the two spaces in either direction is needed

The teach-in technique does not guarantee full accuracy required for late stages in the case

of applications requiring high precision such as assembly On the other hand, solving inverse kinematics problem ensures high precision and also enables the mathematical model to intervene deeply in the control as follow:

- Select the solution configuration according to the technological constraints or optimization;

- Forecast errors by mathematical model;

- Serve as a basis for dynamics control and geometry dynamic simulation

They are the points which are not met by current kinematics data initialization methods

1.2 Robot kinematics, models and methods

1.2.1 Robot kinematics

At current level, most industrial robot control two circuits which are the displacement control circuit and the dynamic control circuit In which, the displacement control or kinematic control is the basis for the accuracy of the robot's operation, while the dynamic control is intended to improve the robot's performance

Data which control displacement circuit was derived from the kinematic problem; it is the survey result kinematics model of manipulator when knowing in advance the requirements and configuration of the manipulator Therefore, kinematics problem can be described as:

- Given the configuration or structure of the robot (kinematic diagram), given the data in the joint space, find the location and direction of the end-effector in the work space, this is forward kinematic problem (FKP)

Chapter 1 Introduction

- Given the robot configuration and data about the location and direction of the work space, find data about component trajectory represented in joint space to ensure the laws known motion, this is the inverse kinematic problem (IKP)

The kinematic problem possibly includes two phases:

- Modeling phase from mathematical model practicing system is to set the kinematic equation for the robot

- The phase of surveying model (equation) received

1.2.2 Modelling phase

Figure 1-1 Diagram of closed loop on robot arrm and parallel robot The general principle for modeling any robot is usually based on the closed loop vector For example, from the layout shown in Figure 1-1 the relationship of the reference system as a closed loop 1 can be mapped out as shown in Figure 1-2

X

A6 A5

O v

Figure 1-2 Closed loop vector

If we take original O0 point of the basic reference system for benchmarking description, object of description is terminal P of the working tools Since terminal of the tool P need to be concede with the processing trajectory of work piece, this relationship can be written as a vector loop equation as follows:

R E X T A A

A1 2 n  (1-1) According homogeneous transformations of execution stages, function of the joint variables is described by the synthetic matrix of transformations:

A

1

1 0

0

z z z z

y y y y

x x x x

n

p a s n

p a s n

p a s n

A  f q q q ; q q1, 2, ,q n joint variables; n, s, a are direction vectors, p

is location vector, Oxyz is the original coordinate system

The synthetic transformation matrix form:

1000

34 33 32 31

24 23 22 21

14 13 12 11

a a a a

a a a a

a a a a R E

R P

Chapter 1 Introduction Due to the orthogonal nature of the direction vector, only three of the cosine components can indicate direction independently Therefore, combining (1-3) and (1-4) received:

a p

a p

a p

a a

a a

a s

z y x y x x

(1-5)

This is the basic kinematic equation of the robot

For serial-structured robot or robot arm, mathematical relationship between the position and direction of the end-effector and designed coordinate system of each link matched by such rules as: Denavit - Hartenberg (DH) method, Sheth-Uicker method, Khalil-Kleinfinger method

[1], Screw Theory method combined with general axis shifting matrix technique or transmission matrix [2] This matrix can be defined by a variety of methods such as generalized co-ordinates method, the Craig matrix method

Unlike robot arm, for parallel robot, due to the presence of passive joints (the joints are not driven, usually have many degrees of freedom), closed-loop modeling often not applies the method of separating the joints into joints type 5 to avoid the complexity of the problem, it only defines two frames of reference including one fixed coordinate system set based on fixed platform and one mobile coordinate system set on the table of operation machine (or moving platform) without much generalized coordinates as in robot arm [3]

Figure 1-3 Parallel structured Robot

Figure 1-4 Kinematics circuit of one limb of parallel robot Parallel robot is also different from robot arm in the sense that it has independent loop circuit structures For example, for parallel robot with 6 limbs as shown in Figure 1-3, the 7th

link of operation (moving platform) is connected to fixed platform 0 (base platform) by a number of independent parallel loop circuit from 1 to 6

For multi-loop structures like parallel robot, closed-loop equations are set up in accordance with each independent loop Because all the limb is similar to each other when establishing kinematic equation the study of a kinematics branch based on any th

i limb will help to calculate

the analysis collection of constraint equation Due to the large number of loops the number of associated equations of parallel robot can vector be more than 6 which is the maximum number

of equations of the robot arm

DH method is very general and will become more complex when there are many closed loop Therefore, for the parallel robot, it is more convenient to use geometric method to establish kinematics equations[3].Vector loop equations written for each limb and all coordinate systems in closed-loop structure are defined by the geometric link Moreover, passive joint variables were also excluded in the kinematics equation when using geometric method

It is easy to see that in the modeling phase, despite the method used, the general principles

to model a robot is based on the closed loop vector, as a result, the kinematics equation after modeling is always a composition of nonlinear functions and transcendental function

After modeling the operator needs to distinguish and classify the available parameters in the mathematical model, basically these parameters can be divided into two groups:

(1) Fixed structure parameters group: this group of parameters describes the structure of the fixed platform (base platform) and moving platform, the length of prismatic joints, for

Chapter 1 Introduction example: in Stewart Gough parallel robot is the length of each limb, this parameter group is formed when synthesizing kinematics, it is no need to find the parameter in kinematics analysis problem

(2) Group of variables: it is the combination control variables and the auxiliary parameters, the control variables can be set by controlling the motor of the robot (prismatic or rotation), and the auxiliary parameters is used to distinguish the solutions because the problem can have many solutions They share the same control variables but different auxiliary parameters This group

of control variables should be found when analyzing inverse kinematics while the auxiliary parameters are always need to be found in both FKP and IKP

1.2.3 Model survey phase

Solving the FKP or IKP of robot is to solve a system of kinematic equations received in the modeling phase This is also the content of robot kinematics analysis problem

Because the kinematics system of robot after modeling is the nonlinear system formed by the equation constructed from sine and cosine function, there may be many difficulties encountered in solving the problem Methods of solving this equation system are varied, but they can be divided into two different groups:

(1) Group of Numerical Method: this method finds the values of group of solutions The result of this method is approximated to the tolerance It used with the assistance of computers The method gives generalized solution for all types of robots with accurate results necessary in accordance with technical requirements The numerical method strategies for IKP [4] can be broadly classified into one of the following categories:

- Jacobian Inverse Methods

- Newton-like Methods

- Style or mesh-based Inverse Kinematics

- Heuristics based Inverse Kinematics

- Genetic Algorithm based Inverse Kinematics

(2) Group of Analytical Method: this method finds the formula or analytic equation denoting the relationship between the value of the Cartesian Space and other parameters of the

DH parameters Group of Analytical Method finds out correct solution, but overall primary transformation process to extract analytic solution needs intuition and take advantage of structural characteristics in an ingenious way to achieve goals This procedure is not recommended to be applied widely in groups with different structural characteristics due to difficulties in generalizing [5]

Since there is no general method to find a solution in analytical form for a robot, so in academic perspective, people are especially interest in numerical method, this method gives result as real number Although the numerical method is abundant in quantity, they can be classified in two main points:

- The method using derivative (Newton Raphson, Taylor expansions );

- The methods that do use derivative (random search method )

It can be classified according to the second method:

- The methods achieved by solving original problem (all available methods);

- Method to solve equivalent problem in optimal form

In essence, the optimal method is a method that uses derivatives to solve robot kinematics problem in optimal form The method using derivative often has a small number of iterative loops to finish searching process faster than other methods

Kinematics problem as stated is divided into two kinds of problems, FKP and IKP; it has different properties on each type of robot:

- FKP of robot arm has unique solution;

- The IKP of robot arm has many solutions

- Both FKP and IKP of parallel robot have many solutions

The kinematics problem is the basis to control robot, generally it must meet the following requirements:

- Easy to understand, apply;

- High precision;

- Apply to various types of robot;

- Short iterative time (apply for online control);

- Allow sufficient observation of the response of the structure in the different cases;

- Allow to assign solution selection

The position of the kinematics problem in the control circuit is shown in the figure below:

Figure 1-5 Control Diagram in joint space

Chapter 1 Introduction Figure 1-6 Control Diagram in work space

1.2.4 An overview of methods for solving kinematic problems of parallel robot The idea of designing parallel robots started in 1947 when D Stewart constructed a flight simulator based on his parallel design [6] Then, Merlet introduced other types of parallel robots

in [7] Parallel manipulators have received increasing attention because of their high stiffness, high speed, high accuracy and high carrying capability [8] These parallel robots are used in real-life applications such as force sensing robots, fine positioning devices, and medical applications

[9] As in the case of conventional serial robots, kinematic analysis of parallel manipulators is also performed in two phases In forward or direct kinematics, the position and orientation of the mobile platform as the robot end effector is determined given the leg lengths This is done with respect to a base reference frame In inverse kinematics, the position and orientation of the mobile platform is used to determine actuator lengths It is known that unlike serial manipulators, inverse position kinematics for parallel robots is usually simple and straightforward In most cases joint variables (actuator displacements) may be computed independently using the given pose of the movable platform The solution to this problem is in most cases uniquely determined [10]

However, the forward kinematics of parallel manipulators is generally very complicated

It usually involves a set of highly coupled nonlinear equations, that in general there is no closed form and usually no unique solution for it [11] Lazard, Ronga and Mourrainhave established that the general 6-6 hexapod forward kinematics problem has 40 complex solutions using, respectively, Gröbner bases [12], Chern classes of vector bundles and explicit elimination techniques [13][14][15] However, the significant issue is the one of real solutions since they correspond to an effective manipulator posture (often called the assembly mode) and the number of real solutions is always equal to or less than the number of complex ones Recently, Dietmaier has proposed an algorithm which modifies a Gough platform configuration into one that features 40 real solutions [16][17] For the most general parallel manipulators, analytical approach also leads to a polynomial of degree 40, resulting in 40 distinct solutions [18][19] However, the challenging problem is not to find all the solutions, but to directly determine

a unique solution among all possible solutions [20] In many instances, some methods are irrelevant solutions or miss some of them [17] The methods for kinematics problem of parallel robot can be classified as follow:

(1) Numeric Methods:

• Secant Method [21][22]  one solution, Bisection and Secant methods can also be advocated with slower convergence However, Bisection can be guaranteed to find a solution

• Newton method [23][24]one solution

• Continuation method with homophony [25][26][27]  several solutions

• Interval analysis [28][29] all solutions or no answer

• Geometric Iterative Method [1][30][31] one solution The geometric iterative method has shown promises but it needs a proper initial guess

(2) Algebraic Methods:

• Dyallitic Elimination [32][33][34] several solutions

• Resultants method [35][36][37] several solutions

• Groebner bases [12]  all exact solutions

(3) Optimization Techniques:

• Genetic Algorithm [38][39]  several solutions

• Simulated Annealing [40][41] some solutions

• Hybrid Genetic Algorithm and Simulated Annealing [42][43][44] all solutions

• G3-PCX (the generalized generation gap with parent-centric crossover operator) [45][46]

 all solutions

In the mathematical literature, many methods can solve non-linear systems of parallel robot However, only a few have been successively implemented to solve the FKP of the general 6-6 parallel manipulator The FKP is considered a very difficult problem as reported by Ragahvan and Roth [47] The numeric iteration methods such as the very popular Newton method were naturally first implemented [23][48][49] These methods only converge on one real root which is valid from a mathematical point of view, however, from a robotics context, the solution does not always necessarily correspond to the effective manipulator posture Moreover,

in some instances, the method can even fail to compute the solution Genetic algorithms method also has been used for solving the forward kinematics of parallel manipulators Initial work was done by Boudreau and Turkkan who used a real coded genetic algorithm (RCGA) for solving the FKP of 3-RPR manipulators [38][39] where it was reported that genetic algorithms are more time consuming than Newton-Raphson’s method [24] However, due to the difficulty in implementation of Newton’s method in terms of determining the Jacobian and its inverse, genetic algorithms are preferred [50] Furthermore, Newton’s method also suffers from convergence to a local minimum when a good initial position in the search space is not provided

To determine all real roots, intervals method were implemented [51][52][53] These methods although not appropriate for real-time use but it has the advantage of being numerically certified (no roots can be missed and the solution can be computed with an arbitrarily prescribed accuracy) However, these methods are often plagued by the usual Jacobian computation problems [54] Using elimination theory, a unilabiate polynomial can be numerically computed

Chapter 1 Introduction out of a polynomial system by Dialytic or determinant algorithms [55][56][57][58] The continuation method can also be implemented to determine all roots [59][60][61] However, the simpler starting system might yield less solutions than the actual one and thus the solution path pursuit process might not lead to all solutions Use of iterative numerical procedures [62][63] and auxiliary sensors

[64] are the two commonly adopted schemes to further lead to a unique solution Numerical iteration is usually sensitive to the choice of initial values and nature of the resulting constraint equations The auxiliary sensors approach has practical limitations, such as cost and measurement errors Redirect solve kinematic robot to solve optimal form equivalent is a method of Pham Thanh Long has been introduced in [65][66] This method is essentially a method that uses the derivative to solve the optimal form of robot kinematics However, when applied to kinematics problems of parallel robot, precision of the results solutions of the IKP is not high and incompatible with forms of objective quaternary functions that contain transcendental elements due to the structure of the objective function irrational

Dasgupta, Husty and Wampler preferred the related resultants in a computer algebra environment to alleviate numeric instabilities [35][36][37] Although this approach can find the real solutions, in [17] it was demonstrated how extraneous solutions can be hidden in the polynomial leading coefficients Thus, a sole unilabiate polynomial even with high degree cannot be proven equivalent to a complete system of several polynomials Hence, in the case of parallel manipulator FKP systems, there exists only a few mathematical methods which can compute a truly equivalent system where the properties are preserved and include a unilabiate equation The various solving methods can lead to one, some or all the results The following of table will show advantages, disadvantages and limitations of each method:

Table 1-1 Advantages, disadvantages and limitations of some methods

solving parallel kinematic problem Method names Introduction Observations Example

Newton's

Method

- Introduced in 1972

- The Newton-Raphson method was first implemented with the advantage of its very rapid convergence But, it can converge to only one real root and numerical instabilities can easily make it to fail

- Have one solution

- Quadratic convergence

- Small computation times

- May not converge

- Needs convergence test

as the Kantorovich theorem

- λ ∈{0,…, 1}

- May miss solutions

- May add solutions

- Crossing solutions

- Needs iterative method

- Problem going from the SSM to the 6-6

- Perhaps all solutions

- Complex solutions may become real solutions

- Spurious solutions are

added

- Simpler parallel robots:

OK

- Problem: 40 solutions for the SSM

- Perhaps all solutions

- Spurious solutions are

added

- Requires elimination step with IKP

- Simpler parallel robots

- Problem: 40 solutions for the SSM

- Solving for Res(f,g,x1)

= 0 equivalent to det(M) =

0

- In certain instances, the head terms of the polynomials cancel

→ it adds one extraneous

- conversion to a Rational univeariate Representation

- All exact solutions

- But long computation

times

- Rational or integer coefficients

- Requires solving the univariate equation

- 36 solutions for the SSM

- 6-6 computation times: 1 min in Maple

- May find many solutions through repeated trials

instances

- All solutions

- Quadratic convergence

- Long computation times

- May not converge

- Jacobian inversion

- Accounts for imprecision

- Needs Newton's method

- On singularity free SSM: 5% failures

- Needs enclosure test as with the Kantorovich theorem

In addition to the mentioned methods, parallel robot kinematics field are also attracting the interest of many researchers around the world They have been trying to solve the problem

of parallel robot kinematics by different approaches such as developing new algorithms

[67][68][69][70][71], enhancing or improving existing algorithm [72][73][74][75][76][77][78] or combining the advantages of existing algorithms [79][80][81][82][83][84],….However, no matter how the forward kinematics problem may be solved, direct determination of a unique solution is still a challenging problem [85]

Chapter 1 Introduction

1.3 Research orientation

In an attempt to synthesize some typical methods in solving robot kinematics problem within the past 40 years, it can be seen that although there are many methods to solve the robot kinematics problem but each method can only solve for groups of robots with specific structures with limited degrees of freedom and certain accuracy There are methods with complex algorithm and difficult to implement in practice; there also are methods developed from other methods or a combination of a few different methods and most importantly none of them can

be applied to all kinds of different robot structures with random degrees of freedom and easy to apply in practice

Moreover, when solving the inverse kinematics problem of robot these cases may happen:

- There can be many different solutions;

- The homogeneous equations with nonlinear, transcendental form often do not give correct solutions;

- There may be an indeterminate solution since there are redundant links like statically indeterminate structure；

- There may be a mathematical solution, but this solution is not acceptable physically because the structural elements of the structure cannot meet this solution

In general, the bigger the number of degrees of freedom, the more difficult to solve robot kinematics problem, then to choose the control solutions require the removal of the inappropriate solutions based on the constraints of operational limit of the joint

In academic and applicability perspective, it is necessary to research the FKP and IKP, from that it can be seen that a general method to solve this problem for groups of robots, especially parallel robots is still very limited, we propose to develop a new algorithm with the following criteria:

- New algorithms for the parallel robot kinematics problem have high generality, advantages in execution time compared to other models

- Can be applied to solve different the robot structures with any number of degrees of freedom

- Simple, easy to practice and apply Compatible with popular software to assist to solve the problem

- Algorithm has to have high reliability and high accuracy results to ensure precise control Fast processing time to ensure that the robot responses to controller in real time

1.4 Subjects and research methods

The subject of the research is the kinematic problem of the parallel robot structure Focusing primarily on the FKP and IKP problem solving method, define the parameters to control the trajectory motion to ensure the required accuracy

The kinematic parameters were determined using mathematical models, then the numerical results were verified with simulation results and the results of experiments on a real robot model will ensure the objectivity of the material

1.5 Contents of the present thesis

Appearing to come from the objectives of the research, the content of the thesis includes the following chapters:

Chapter 1: Introduction

This chapter briefly introduces the robot kinematic problem, the modeling steps and the model survey The overview and analysis of the advantages and disadvantages of current typical methods in the field of robot kinematic problem solving On that basis, to suggest the research direction of the thesis

Chapter 2: In this chapter, we will present the mathematical basis of transforming robot kinematic problem into the optimal one, the new mathematical model of the robot kinematic optimization problem is proposed Algorithm diagrams and some factors affect the accuracy of the problem such as minimum shifts, differential calculations, etc Classification of parallel robots in a new way will also be introduced in this chapter The modeling of all parallel robots and the transformation of the mathematical model of the parallel robotic kinematic problem into the optimal one form will be presented at the end of the chapter

Chapter 3: In this chapter, the content of generalized reduced gradient algorithm and the optimal application of Solver of Microsoft-Excel (MS-Excel) are introduced The use of equivalent alternative configuration to downgrade the problem model into the form of standard optimal problem The application of the proposed method and robot kinematic surveying procedure for each type of parallel robot is presented Difficulties encountered during application and the solution The test of the reliability as well as the accuracy of the proposed method will be discussed in the next part The end part of this chapter compares the accuracy

as well as the solving time of the proposed method with some other methods

Chapter 4: The entire contents of this chapter are intended for simulations and experiments

on real robot models to test the accuracy and feasibility of the proposed method when applied

in practice The entire database of experimental design, from hardware to software, embedded software, will be covered in this chapter The last part of the chapter are experimental results

Chapter 1 Introduction and the comparison between experimental and numerical results

Chapter 5: Developing a new application of the proposed method in another area that is the field of robot design with the aim of determining the tolerance of the components to ensure the given precision of the end-effector and vice versa The three-step sequence is applied and used to both robot arm and parallel robot with two processes: forward and inverse will be all presented In the end of this chapter, the results of two simulation examples are presented to demonstrate the accuracy and effectiveness of this method

Chapter 6: The content of the chapter outlines the conclusions achieved when completing the research project and recommends some further research directions

Figure 1-7 present the subject of the overall research program

Figure 1-7 The subject of the overall research program