Dynamic Stability of Parallel Manipulator at its Singularities Corresponding to Kinematic Parameters of Dynamic Systems

With the aid of the Liyapunov first approximate stability criterion, the dynamic stability condition for the 3-RPR parallel mechanism to realize a deterministic motion at singular configurations is deduced. Based on this condition, the distributions of the kinematic parameters including input velocities and accelerations of the system corresponding to the stable motion at its singular configuration are investigated then. It is found that for a given singular configuration, increasing input velocities and accelerations, the sub-distributions of eigenvalues with positive real parts have a tendency to shrink and, consequently, the motion stability at the singular configuration can be enhanced; adjusting input velocities and accelerations only can not necessarily get all negative real parts of the eigenvalues sharing a common intersection of the distributing subintervals and, normally, the additional adjustment of initial velocities of the particle system should be added. Besides, while the movable platform goes through the singular configuration, if the control law of the input parameters makes the instantaneous velocity center of the movable platform far away from the singular point, the platform is able to go through the singular configuration with high stability and strong capability to resist external disturbances. This research indicates the effectiveness to improve the motion stability of the dynamics system at singular configurations via adjusting the input kinematic parameters. From this, a singularity-free approach via adjusting the input kinematic parameters can be utilized to exclude singularities of parallel mechanisms dynamically in the joint trajectory planning stage without introducing either redundancy or active mass.