Choosing the Correct Generalized Inverse for the Numerical Solution of the Inverse Kinematics of Incommensurate Robotic Manipulators

TL;DR

This paper introduces a systematic approach using the Mixed Generalized Inverse for solving inverse kinematics in incommensurate robotic manipulators, ensuring robustness and property preservation across various system configurations.

Contribution

The paper proposes a novel, robust numerical method employing the Mixed Generalized Inverse for inverse kinematics applicable to all Jacobian types and robot configurations, including incommensurate systems.

Findings

The Mixed Generalized Inverse guarantees system property preservation.

Common GIs like Moore-Penrose can fail in incommensurate systems.

The proposed method outperforms traditional IK approaches in experiments.

Abstract

Numerical methods for Inverse Kinematics (IK) employ iterative, linear approximations of the IK until the end-effector is brought from its initial pose to the desired final pose. These methods require the computation of the Jacobian of the Forward Kinematics (FK) and its inverse in the linear approximation of the IK. Despite all the successful implementations reported in the literature, Jacobian-based IK methods can still fail to preserve certain useful properties if an improper matrix inverse, e.g. Moore-Penrose (MP), is employed for incommensurate robotic systems. In this paper, we propose a systematic, robust and accurate numerical solution for the IK problem using the Mixed (MX) Generalized Inverse (GI) applied to any type of Jacobians (e.g., analytical, numerical or geometric) derived for any commensurate and incommensurate robot. This approach is robust to whether the system is under-determined (less than 6 DoF) or over-determined (more than 6 DoF). We investigate six robotics manipulators with various Degrees of Freedom (DoF) to demonstrate that commonly used GI's fail to guarantee the same system behaviors when the units are varied for incommensurate robotics manipulators. In addition, we evaluate the proposed methodology as a global IK solver and compare against well-known IK methods for redundant manipulators. Based on the experimental results, we conclude that the right choice of GI is crucial in preserving certain properties of the system (i.e. unit-consistency).