Comparisons of Inverse Dynamics Formulations in a Spatial Redundantly Actuated Parallel Mechanism Constrained by Two Point Contact Higher Kinematic Pairs

A spatial redundantly actuated parallel mechanism (RAPM) constrained by two point contact higher kinematic pairs (HKPs) has been designed, arising from the inspiration of mastication in human beings: the end effector is the lower jaw, the six kinematic chains are the primary chewing muscles, and the constraints at HKPs are the temporomandibular joints. In this paper, firstly, the constrained motions of the mechanism are described in detail; thereafter, five models are formulated by the well-known Newton-Euler's law, the Lagrangian equations, and the principle of virtual work, to explore its rigid-body inverse dynamics. The symbolic results show that the model structures based on these approaches are quite different: the model via the Newton-Euler law well reflects the nature of the mechanism in terms of the constraint forces from HKPs with six equations and eight unknowns, and the existence of reaction forces at the spherical joints is tightly dependent on the number of kinematic chains. In comparison, from the latter two methods, the constraint forces and the reaction forces at spherical joints do not appear in the four models in which there are only four equations and six unknowns. Further, by using the dynamics model of the non-redundantly actuated counterpart as the core in both the second models from the energy and virtual work-related methods, their computational cost is only about 16.7% and 36.63% of the two first models, respectively. Finally, the comparisons between the dynamics models of the RAPM and its counterpart clarify that the HKP constraints greatly alter the model structures and raise the technical difficulties.