A Comparative Study of the Configuration-Space and Actuator-Space Forward Dynamics of Closed-Loop Mechanisms Using the Lagrangian Formalism

This paper presents a comparative study between two related methods of formulating the equations of motion, within the Lagrangian framework, for closed-loop mechanisms. Such mechanisms encounter singularities not only at the boundaries of their workspaces, but also inside the workspaces. The latter kind of singularities are detrimental to the operation of the mechanisms and may lead to their mechanical failure. The primary objective of the paper is to investigate the ways in which these singularities impact the two formulations, and to establish a relation between them. A planar five-bar mechanism is used to illustrate that the singularities appearing in one formulation is a subset of those appearing in the other formulation. The second objective is to provide a qualitative analysis of the time-complexities of the respective formulations. A planar five-bar and the Stewart platform manipulator are used to study and compare the computational costs incurred in either of the formulations.