Hyperbolic pseudoinverses for kinematics in the Euclidean group

TL;DR

This paper introduces hyperbolic pseudoinverses for robot kinematics in the Euclidean group, providing invariant alternatives to the Moore-Penrose pseudoinverse that handle singularities and varying joint configurations.

Contribution

It develops a family of hyperbolic pseudoinverses based on invariant bilinear forms, extending the tools for robot kinematic control beyond traditional methods.

Findings

Hyperbolic pseudoinverses are invariant under coordinate changes.

Conditions for existence relate to classical line involution problems.

The approach addresses singularities and non-standard joint configurations.

Abstract

The kinematics of a robot manipulator are described in terms of the mapping connecting its joint space and the 6-dimensional Euclidean group of motions $SE(3)$. The associated Jacobian matrices map into its Lie algebra $\mathfrak{se}(3)$, the space of twists describing infinitesimal motion of a rigid body. Control methods generally require knowledge of an inverse for the Jacobian. However for an arm with fewer or greater than six actuated joints or at singularities of the kinematic mapping this breaks down. The Moore-Penrose pseudoinverse has frequently been used as a surrogate but is not invariant under change of coordinates. Since the Euclidean Lie algebra carries a pencil of invariant bilinear forms that are indefinite, a family of alternative hyperbolic pseudoinverses is available. Generalised Gram matrices and the classification of screw systems are used to determine conditions for their existence. The existence or otherwise of these pseudoinverses also relates to a classical problem addressed by Sylvester concerning the conditions for a system of lines to be in involution or, equivalently, the corresponding system of generalised forces to be in equilibrium.