On the Configurations of Closed Kinematic Chains in three-dimensional Space

TL;DR

This paper analyzes the configuration space of closed 3D kinematic chains with spherical joints, introducing new parameters for easier analysis and potential applications in motion planning and singularity detection.

Contribution

It introduces a novel parameterization of the configuration space of closed 3D kinematic chains using diagonal lengths, simplifying the analysis of their loop closure constraints.

Findings

New parameterization maps configurations to a cube of dimension n-3.

Numerical examples demonstrate the practical applicability of the method.

Potential applications include motion planning and singularity analysis.

Abstract

A kinematic chain in three-dimensional Euclidean space consists of $n$ links that are connected by spherical joints. Such a chain is said to be within a closed configuration when its link lengths form a closed polygonal chain in three dimensions. We investigate the space of configurations, described in terms of joint angles of its spherical joints, that satisfy the the loop closure constraint, meaning that the kinematic chain is closed. In special cases, we can find a new set of parameters that describe the diagonal lengths (the distance of the joints from the origin) of the configuration space by a simple domain, namely a cube of dimension $n-3$. We expect that the new findings can be applied to various problems such as motion planning for closed kinematic chains or singularity analysis of their configuration spaces. To demonstrate the practical feasibility of the new method, we present numerical examples.