Dynamics Modeling of Topologically Simple Parallel Kinematic Manipulators: A Geometric Approach

Abstract

Dynamics modeling is indispensable for the design and control of dexterous parallel kinematic manipulators/machines (PKM). Various modeling approaches proposed in the literature build upon the classical formulations for serial manipulators, and thus, inherit those modeling conventions that tend to be restrictive rather than user-friendly. Moreover, the special kinematic topology of PKM is treated either ad hoc or by resolving loop constraints using standard methods from multibody dynamics. Geometric formulations on the other hand, more precisely Lie group formulations, were developed over the last decades that provide a flexible and user-friendly approach to the modeling of robotic systems in general. A dedicated formulation for topologically simple PKM has not yet been proposed, however. Such a formulation is presented in this paper. The frame invariance of the geometric formulation gives rise to a modular modeling approach that further reduces the modeling effort. The equations of motion (EOM) in terms of task space coordinates as well as in actuator coordinates are presented for kinematically nonredundant and redundant topologically simple PKMs. A PKM is topologically simple if its moving platform is connected to the base by simple serial kinematic chains and if there are no other kinematic chains than these. The majority of PKMs are topologically simple, including fully parallel PKM. Applications of the EOM for dynamics simulation and model-based control are briefly discussed. The paper also provides a literature review of approaches to dynamics modeling of PKM.