ODE Methods for Computing One-Dimensional Self-Motion Manifolds

2025-07-29

AI Summary

This work addresses the global, exact computation of one-dimensional self-motion manifolds (SMMs) for redundant manipulators. To overcome challenges—including SMM multi-connectivity, non-convexity, and the absence of natural redundancy in non-redundant subsystems—we propose a numerical continuation method based on ordinary differential equations (ODEs). We design an explicit fixed-step ODE integrator that jointly incorporates initial-condition search and redundancy-induction mechanisms, enabling automatic tracing of all isolated SMM branches without post-hoc inverse-kinematics optimization. Our method is the first to support generalized configurations including prismatic joints. Extensive validation on canonical robotic platforms demonstrates high accuracy, strong robustness, and computational efficiency. The results expose fundamental limitations of conventional SMM algorithms regarding topological completeness and task adaptability, establishing a verifiable, globally consistent framework for representing the solution space—thereby advancing redundancy resolution, obstacle-avoidance planning, and human–robot collaboration.

Technology Category

• Intelligent Robots: Motion and Path Planning
• Search and Optimization: Mixed Discrete/Continuous Search
• Planning, Routing, and Scheduling: Mixed Discrete/Continuous Planning

Application Category

• Responsible Web: Machine-in-the-loop, human agency and autonomy
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

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.

Problem

• Computing global inverse kinematics solutions for redundant manipulators
• Inducing redundancy in non-redundant manipulators for specific tasks
• Finding disconnected components of self-motion manifolds accurately

Innovation

• ODE formulation for 1D self-motion manifolds
• Inducing redundancy in non-redundant manipulators
• Searching for disconnected SMM components