• CN:11-2187/TH
  • ISSN:0577-6686

›› 2009, Vol. 45 ›› Issue (4): 50-55.

• 论文 • 上一篇    下一篇

对称4自由度3R1T并联机构雅可比分析

李秦川;胡旭东;陈巧红;武传宇   

  1. 浙江理工大学浙江省现代纺织装备技术重点实验室
  • 发布日期:2009-04-15

Jacobian Analysis of Symmetrical 4-DOF 3R1T Parallel Mechanisms

LI Qinchuan;HU Xudong;CHEN Qiaohong;WU Chuanyu   

  1. Provincial Key Laboratory of Modern Textile Machinery, Zhejiang Sci-Tech University
  • Published:2009-04-15

摘要: 针对对称4自由度3R1T并联机构提出一种雅可比分析方法。首先运用位移群理论分析3R1T并联机构的自由度特性,得到动平台在空间的运动为四维位移流形,然后运用螺旋理论建立单个分支运动链的雅可比矩阵,该矩阵为6×5长方阵;再利用该类并联机构的自由度特性证明6×5的分支雅可比矩阵的第四行和第五行为冗余元素,删除其中之一则可把分支雅可比矩阵简化为5×5方阵,该方阵在机构非奇异位形下满秩。在选定并联机构的驱动副后,对每个分支简化后的5×5雅可比方阵求逆,再分别取出逆阵中对应于驱动副的行矢量构成一个4×5长方阵,由于该长方阵的第四列元素始终与0相乘为冗余信息,删除该列元素后得到一个4×4可逆方阵,对之求逆即得整个4自由度3R1T并联机构的雅可比矩阵。该方法简捷易行,可进一步应用于3R1T并联机构的性能分析和运动学设计。

关键词: 并联机构, 螺旋理论, 位移群理论, 雅可比矩阵

Abstract: A method for Jacobian analysis of 4-DOF 3R1T parallel mechanism is proposed. The mobility analysis of 3R1T parallel mechanisms is performed by using displacement group theory. The motion of the moving platform of a 3R1T parallel mechanism is a four dimensional displacement manifold. Then, the Jacobian matrix of single limb kinematic chain is based on the basis of screw theory, which is a 6×5 non-square matrix. On the basis of the mobility property, it can be proven that the fourth and the fifth rows of the limb Jacobian matrix are redundant. Deleting the fourth row or the fifth row of the limb Jacobian matrix leads to a 5×5 matrix, which is of full rank in non-singular configurations. After the actuation of the 3R1T parallel mechanism is determined, the inverse of the 5×5 matrix of each limb can be obtained. A 4×5 non-square matrix can be obtained by combining the four row vectors corresponding to the actuated pairs taken from the four inverses. The elements of the fourth column of the 4×5 non-square matrix are always multiplied by 0 and thus are redundant. Deleting the fourth column of the 4×5 non-square matrix yields a 4×4 invertible matrix, the inverse of which is the Jacobian matrix of the 4-DOF 3R1T parallel mechanism. This method is straightforward and can be further applied in performance analysis and kinematic design of the 3R1T parallel mechanism.

Key words: Displacement group theory, Jacobian matrix, Parallel mechanism, Screw theory

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