Convex Iteration for Distance-Geometric Inverse Kinematics

TL;DR

This paper introduces CIDGIK, a convex iterative algorithm that efficiently solves inverse kinematics problems with complex workspace constraints by formulating them as a convex feasibility problem and using semidefinite programming.

Contribution

The work formulates inverse kinematics with workspace constraints as a convex feasibility problem and proposes CIDGIK, a novel convex iteration method that improves convergence speed and accuracy over traditional nonlinear methods.

Findings

CIDGIK achieves faster convergence than nonlinear optimization.

It provides more accurate solutions in obstacle-rich environments.

The formulation unifies robot geometry and obstacle constraints as linear matrix equations.

Abstract

Inverse kinematics (IK) is the problem of finding robot joint configurations that satisfy constraints on the position or pose of one or more end-effectors. For robots with redundant degrees of freedom, there is often an infinite, nonconvex set of solutions. The IK problem is further complicated when collision avoidance constraints are imposed by obstacles in the workspace. In general, closed-form expressions yielding feasible configurations do not exist, motivating the use of numerical solution methods. However, these approaches rely on local optimization of nonconvex problems, often requiring an accurate initialization or numerous re-initializations to converge to a valid solution. In this work, we first formulate inverse kinematics with complex workspace constraints as a convex feasibility problem whose low-rank feasible points provide exact IK solutions. We then present \texttt{CIDGIK} (Convex Iteration for Distance-Geometric Inverse Kinematics), an algorithm that solves this feasibility problem with a sequence of semidefinite programs whose objectives are designed to encourage low-rank minimizers. Our problem formulation elegantly unifies the configuration space and workspace constraints of a robot: intrinsic robot geometry and obstacle avoidance are both expressed as simple linear matrix equations and inequalities. Our experimental results for a variety of popular manipulator models demonstrate faster and more accurate convergence than a conventional nonlinear optimization-based approach, especially in environments with many obstacles.