Published on Wed Sep 08 2021

Convex Iteration for Distance-Geometric Inverse Kinematics

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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.

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