Unified Singularity Crossing of a 3-(rR)PS Metamorphic Parallel Mechanism through Dynamic Modeling

Metamorphic parallel mechanisms can change into multiple configurations with different motion types and mobility, which consequently yield different solutions of inverse dynamics when crossing singularity. Thus, a unified solution of inverse dynamics to cross singularity becomes important. This solution relies on the consistency condition, the first indeterminate form, and this paper proposes an additional condition by extending into the second indeterminate form. This paper presents unified dynamic models of a 3-(rR)PS metamorphic parallel mechanism to pass through singularities. The analysis is carried out on all three configurations of the 3-(rR)PS metamorphic parallel mechanism. The dynamic models are established using Lagrange formulation, and three conditions to cross singularities are employed. The first condition is based on the consistency condition where the uncontrollable motion should be reciprocal to the wrench matrix. The denominator of inverse Jacobian is its determinant whose value is zero at singularities. This singularity can be discarded by compensating the numerator to be zero. Both the numerator and denominator are null, and this indeterminate form becomes the second condition. Both conditions are sufficient for inverse dynamics of one configuration to pass through singularity, but not for other configurations. Therefore, the second indeterminate form is proposed to be the third condition to be fulfilled. Consequently, the 11th-degree polynomial is required for path planning. The results are presented and confirmed by ADAMS simulation.