ODE Methods for Computing One-Dimensional Self-Motion Manifolds

Redundant manipulators are well understood to offer infinite joint configurations for achieving a desired end-effector pose. The multiplicity of inverse kinematics (IK) solutions allows for the simultaneous solving of auxiliary tasks like avoiding joint limits or obstacles. However, the most widely used IK solvers are numerical gradient-based iterative methods that inherently return a locally optimal solution. In this work, we explore the computation of self-motion manifolds (SMMs), which represent the set of all joint configurations that solve the inverse kinematics problem for redundant manipulators. Thus, SMMs are global IK solutions for redundant manipulators. We focus on task redundancies of dimensionality 1, introducing a novel ODE formulation for computing SMMs using standard explicit fixed-step ODE integrators. We also address the challenge of ``inducing'' redundancy in otherwise non-redundant manipulators assigned to tasks naturally described by one degree of freedom less than the non-redundant manipulator. Furthermore, recognizing that SMMs can consist of multiple disconnected components, we propose methods for searching for these separate SMM components. Our formulations and algorithms compute accurate SMM solutions without requiring additional IK refinement, and we extend our methods to prismatic joint systems -- an area not covered in current SMM literature. This manuscript presents the derivation of these methods and several examples that show how the methods work and their limitations.