Effects of Kinematic Parameters on Planar 3-RPR PM’s Motion Stability at Singular Configurations

Inspired by the case that a running parallelogram mechanism can traverse the singular configuration without losing motion stability, parallel manipulator's (PM) motion stability at singular configurations corresponding to the input kinematic parameters and joint velocities has been studied analytically, taking a planar 3-RPR PM as an example. First, the Lagrangian approach is applied to establish the analytical uncoupled quadratic dynamical equations for the 3-RPR PM without multipliers. Then, following Lyapunov's first approximation stability theorem, the PM's motion stability at singular configurations corresponding to the input kinematic parameters and joint velocities is investigated by analyzing the eigenvalues of the dynamical system at the singularities of the linear approximation. It has been shown that the motion stability of the 3-RPR dynamical system at the singularity can be efficiently improved by adjusting the input kinematic parameters and the joint velocities separately or jointly. Specifically, increasing the angular speed of the movable platform along a proper rotation direction and adjusting the joint velocities to let the absolute velocity center of the movable platform move far away from the intersecting point of the three driving legs to a certain distance, the motion stability of the dynamic system at the singular configuration can be substantially improved. This study demonstrates the feasibility of creating novel singularity avoidance methods that incorporate the kinematic parameters of the PM.