4自由度非全对称并联机构的完整雅可比矩阵

少自由度并联机构完整雅可比矩阵为6x6矩阵,包括运动子矩阵和约束子矩阵两部分,由于前者的代数特征不能反映出机构的约束特性,在对此类机构进行运动学分析和几何参数优化设计时,必须建立完整的6x6雅可比矩阵.鉴于此,基于对偶螺旋理论,以4自由度机构2RPS-2UPS为例,给出非全对称少自由度并联机构完整雅可比矩阵的推导方法.首先,根据螺旋理论推导约束支链中的运动螺旋系和反螺旋系,并利用互易积获得约束子矩阵.其次,锁定每个支链中的主动关节,根据螺旋理论计算约束支链和全运动支链中新增反螺旋系,并利用互易积建立运动子矩阵.将约束子矩阵和运动子矩阵联立建立机构完整雅可比矩阵.最后,分析雅可比矩阵的秩,得到2RPS-2UPS并联机构产生奇异位形的条件.

COMPLETE JACOBIAN MATRIX OF A CLASS OF IN COMPLETELY SYMMETRICAL PARALLEL MECHANISMS WITH 4-DOF

The complete Jacobian matrix of a lower-mobility parallel mechanism is a six by six matrix that consists of the actuation sub-matrix and the constraint sub-matrix. As the alge-braic feature of the former one cannot account for the related constraint characteristics, modeling complete six by six Jaco-bian matrix becomes necessary in the kinematic analysis and optimal design of geometric parameters of this kind of parallel mechanisms. A method for deducing the complete Jacobian matrix of an incompletely symmetrical lower-mobility parallel mechanism is illustrated by taking 2RPS-2UPS as an example based on the theory of reciprocal screw. Firstly, the systems of twists and reciprocal screws of the constraint limbs are estab-lished based on the screw theory, then the constraint sub-matrix is obtained through the orthogonal product. Secondly, by lock-ing active joints of each limb, the system of additional recipro-cal screws of both constraint and unconstraint limbs is estab-lished, then the actuation sub-matrix is also obtained through the orthogonal product. By integrating these two sub-matrices properly, the complete Jacobian matrix of an incompletely symmetrical lower-mobility parallel mechanism can be finally set up. In the end, the singular conditions of 2RPS-2UPS paral-lel mechanism are analyzed by investigating the ranks of the Jacobian matrices.