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 of human beings. 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’s law well reflects the nature of the mechanism in terms of the constraint forces from HKPs, and the existence of reaction forces at the spherical joints in it is tightly dependent on the number of kinematic chains. In comparison, the constraint forces and the reaction forces at spherical joints do not appear in the four models from the latter two methods, where the actuating torques can be minimised in a closed-form by virtue of the pseudo-inverse method. Meanwhile, the torques are smaller than those from the Newton-Euler’s law. 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, they outperform the first models significantly in computational speed, respectively, being minimally laborious and offering directly the simplest possible algorithms. 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.