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博士留学生学位论文

并联机器人运动学模型优化解析方法研究

作者姓名 TRANG THANH TRUNG(庄成忠)

学 科 专 业 机械制造及其自动化

指 导教师 李伟光 教授

所在学院 机械与汽车工程学院

论文提交日期 2017 年 11 月

2

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

分类号：TP242 学校代号：10561

学 号：201312800054

华南理工大学博士学位论文

并联机器人运动学模型优化解析方法研究

作者姓名：TRANG THÀNH TRUNG(庄成忠) 指导教师姓名、职称：李伟光教授/博士生导师

申请学位级别：工学博士 学科专业名称：机械制造及其自动化

研究方向：精密制造装备与现代控制技术

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 .

I

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

目标函数是 Rosenbrock-Banana 函数。经过很多试验，在非线性优化问题中 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。

其次，本文提出一种新的方式分类并联机器人，非棱柱并联机器人与棱柱并联机

器人，包括 3 种：非棱柱并联机器人（类型 1），棱柱并联机器人分成两种：主动棱柱

关节连接到固定平台的并联机器人（类型 2）和第二棱柱关节连接到固定平台的并联机

器人（类型 3）。本文描述所有并联机器人结构的模式化和将并联机器人运动学问题数

学模型转换成求解最优化问题的方式。对于类型1并联机器人，将初始数学模型转化为

最优问题时，目标函数为二次函数，因此直接应用广义简约求解运动学问题，而类型 2

和类型3的初始数学模型机器人是四次幂函数，与所提出的方法不相容。因此，本文提

出用同等代替结构将两类并联机器人（2 型和 3 型）的目标函数形式从四次幂函数降为

二次函数来解决这一问题，通过这种变换之后符合提出的方法。

第三，Excel 微软应用程序支持数学解析，并通过求解机器人的运动学问题，给出

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。

II

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。

关键词：并联机器人；运动学问题；优化问题；同等代替结构；广义简约梯度法；制

造设计误差。

III

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

IV

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

4

10

times) and has

shorter 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

V

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

VI

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

VII

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

VIII

Contents

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

IX

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