Analysis of parasitic motion with the constraint embedded Jacobian for a 3-PRS parallel manipulator

This study derives the equation of parasitic motion of 3-DoFs parallel manipulator from the velocity-level analytic-constraint equation and compares it with a well-known position-level geometric method. The velocity-level constraint is formulated based on the extended Jacobian, derived from the instantaneous motion space (IMS) and the instantaneous restriction space (IRS) for free motion and constraint. In contrast, the position-level constraint, adopted in previous studies, is geometrically obtained by analyzing the moving platform and limb motions. The velocity-level analytic-constraint matrix is used to further analyze the task-space motion. In this paper, the procedure of detecting and identifying the parasitic terms from the independent terms is introduced utilizing the property that comes from the virtue of analytic constraint and inverse rate kinematics algorithm. Then, an equation for the constraint-compatible task motion and a coupling relation between the parasitic and independent motions are derived during further analysis. The paper derived the parasitic motion from position constraint, and an algebraic equivalency with the velocity constraint is shown by taking the time derivative of a point-plane position level constraint for comparison. Finally, numerical simulations are provided to validate the proposed approach and demonstrate the effect of the constraints on the given input velocity within the entire rotational workspace.

本研究从速度层级解析约束方程出发，推导了3自由度并联机械手（3-DoFs Parallel Manipulator）的寄生运动方程，并将其与经典的位置层级几何方法进行对比。速度层级约束基于扩展雅可比矩阵构建，该矩阵由用于描述自由运动与约束的瞬时运动空间（Instantaneous Motion Space, IMS）和瞬时约束空间（Instantaneous Restriction Space, IRS）推导而来。与之相对，既往研究中采用的位置层级约束，则通过分析动平台与支链运动，以几何方式获取。本研究利用速度层级解析约束矩阵，进一步开展任务空间运动分析。本文借助解析约束特性与逆速率运动学算法，介绍了从独立运动项中检测并识别寄生项的流程。经进一步分析，推导得到了约束兼容的任务运动方程，以及寄生运动与独立运动间的耦合关系。本研究从位置约束出发推导寄生运动，并通过对点-平面位置层级约束求时间导数进行对比，证明其与速度约束在代数上等价。最终，本文通过数值仿真验证了所提方法的有效性，并展示了在整个旋转工作空间内，约束对给定输入速度的影响。