Stewart并联机构运动Jacobian矩阵表达形式的多样性  

Diversity of Expression Forms of Jacobian Matrix of Stewart Parallel Mechanisms

作　　者：尤晶晶[1] 叶鹏达 徐帅 王澍声 黄宁宁 李成刚[3] YOU Jingjing;YE Pengda;XU Shuai;WANG Shusheng;HUANG Ningning;LI Chenggang(College of Mechanical and Electronic Engineering,Nanjing Forestry University,Nanjing 210037,China;School of Mechanical Engineering and Rail Transit,Changzhou University,Changzhou 213164,China;College of Mechanical and Electrical Engineering,Nanjing University of Aeronautics and Astronautics,Nanjing 210016,China)

机构地区：[1]南京林业大学机械电子工程学院,江苏南京210037 [2]常州大学机械与轨道交通学院,江苏常州213164 [3]南京航空航天大学机电学院,江苏南京210016

出　　处：《机械制造与自动化》2023年第6期85-88,122,共5页Machine Building & Automation

基　　金：国家自然科学基金资助项目(51405237);国家留学基金资助项目(201908320035)。

摘　　要：现有文献中Jacobian矩阵存在多种表达形式,不利于指标值的统一评估和对比。为此,通过剖析Stewart并联机构运动Jacobian矩阵的机理,找到引起上述现象的根本原因,用欧拉角及其导数描述动平台的转动速度并运用位置方程求导法推导出第1类形式的Jacobian矩阵;用角速度矢量描述动平台的转动速度,并运用环路方程法推导出第2类形式的Jacobian矩阵;将动平台上与静坐标系原点重合的点选为线速度基点,并运用环路方程法推导出第3类形式的Jacobian矩阵。运用角速度加法公式和线速度基点法,构建不同形式Jacobian矩阵之间的映射;运用矩阵理论推导各Jacobian矩阵行列式之间的关联。通过数值算例验证上述结论的正确性,并绘制机构的奇异曲面。所得结论为并联机构的型综合、型优化提供了理论指导。Since various expression forms of Jacobian matrix in existing literatures are not conducive to the unified evaluation and comparison of index values,the Jacobian matrix mechanism of Stewart parallel mechanisms is analyzed to identify the basic reasons.The Euler angles and their derivatives are adopted to describe the rotational velocity of the moving platform,and the first-type Jacobian matrix is derived based on the position equation derivation method.The angular velocity vector is applied to describe the rotational velocity of the moving platform,and the second-type Jacobian matrix is derived by the loop equation method.The point located on the moving platform coinciding with the origin of the fixed coordinate system is selected as the base point of the linear velocity,and the third-type Jacobian matrix is derived by the loop equation method.Base on the angular velocity addition formula and the linear velocity base point method,different Jacobian matrices mapping are constructed,and their relationships are deduced with the matrix determinant theory.The correctness of mapping matrices is verified by a numerical example,and the singular surfaces of a Stewart parallel mechanism are drawn.The conclusions provide theoretical guidance for the type synthesis and type optimization of parallel mechanisms.

关 键 词：并联机构 Jacobian矩阵 位置方程 环路方程 欧拉角 奇异 

分 类 号：TH122[机械工程—机械设计及理论]