Modular and Recursive Kinematics and Dynamics for Parallel Manipulators

Constrained multibody systems typically feature multiple closed kinematic loops that constrain the relative motions and forces within the system. Typically, such systems possess far more articulated degrees-of-freedom (within the chains) than overall end-effector degrees-of-freedom. Thus, actuation of a subset of the articulations creates mixture of active and passive joints within the chain. The presence of such passive joints interferes with the effective modular formulation of the dynamic equations-of-motion in terms of a minimal set of actuator coordinates as well the subsequent recursive solution for both forward and inverse dynamics applications. Thus, in this paper, we examine the development of modular and recursive formulations of equations-of-motion in terms of a minimal set of actuated-joint-coordinates for an exactly-actuated parallel manipulators. The 3 RRR planar parallel manipulator, selected to serve as a case-study, is an illustrative example of a multi-loop, multi-degree-of-freedom system with mixtures of active/passive joints. The concept of decoupled natural orthogonal complement (DeNOC) is combined with the spatial parallelism inherent in parallel mechanisms to develop a dynamics formulation that is both recursive and modular. An algorithmic approach to the development of both forward and inverse dynamics is highlighted. The presented simulation studies highlight the overall good numerical behavior of the developed formulation, both in terms of accuracy and lack of formulation stiffness.