基于Grassmann线几何理论的并联髋关节试验机奇异位形研究

Singularity Analysis of a Parallel Hip Joint Simulator Based on Grassmann Line Geometry

并联髋关节试验机的核心运动模块采用3SPS+1PS空间并联机构,由定、动平台及连接两平台的3根SPS支链和1根PS支链构成,具有3转动自由度和1平动自由度。并联机构的奇异特性对其工作性能有着重要影响,因而须对该髋关节试验机的相关性质进行研究。建立基于Rodrigues参数的机构静力学模型,得到静力学转换矩阵的一般表达式,并推导出与6条用于Grassmann线几何分析的直线相对应的Plücker矢量。通过分析各线簇秩的空间分布特点,得到许多新颖的奇异位形,并推导出机构处于奇异位形时的各种约束方程。通过数值仿真,得到线簇秩从1到5的各奇异位形对应的奇异轨迹。髋关节试验机奇异位形的研究对其工作空间分析、机构尺寸优化和控制系统设计具有重要的意义。

A hip joint simulator with a 3SPS＋1PS spatial parallel manipulator as the core module is proposed.The spatial parallel manipulator has 4 degrees of freedom,including three rotations and one translation.It consists of a fixed base and a moving platform that is connected by three SPS-type active legs and one PS-type central strut.Singularity has great effects on working performance of parallel manipulator,so the singular characterizes of the hip joint simulator need to be studied.Kinematic model based on Rodrigues parameters is provided and a general formulation of static mechanical conversion matrix is deduced that allows one to determine Plücker vectors associated with six analytical lines for Grassmann analysis.Most singular configurations of the simulator are presented,and the expressions describing corresponding singularities are obtained in closed forms.Numerical simulations show the corresponding motion curves and surfaces of singular configurations with different linear variety ranks from 1 through 5,and the distribution characteristics of singular trajectories are studied respectively.The analysis of the singular configurations provided has great significance for trajectory planning,size optimization and control design of the simulator.