Swing-twist decomposition in Clifford algebra

TL;DR

This paper derives and discusses swing-twist decomposition formulas within Clifford algebra, enabling efficient expression of 3D spinors as products of twist-free and swing-free components, with an optimized algorithm outperforming existing methods.

Contribution

It introduces a novel Clifford algebra-based derivation of swing-twist decomposition formulas and an optimized algorithm for practical implementation.

Findings

Derived explicit formulas for swing-twist decomposition in Clifford algebra.

Proposed an optimized decomposition algorithm with improved performance.

Demonstrated the algorithm's advantages over existing methods.

Abstract

The swing-twist decomposition is a standard routine in motion planning for humanoid limbs. In this paper the decomposition formulas are derived and discussed in terms of Clifford algebra. With the decomposition one can express an arbitrary spinor as a product of a twist-free spinor and a swing-free spinor (or vice-versa) in 3-dimensional Euclidean space. It is shown that in the derived decomposition formula the twist factor is a generalized projection of a spinor onto a vector in Clifford algebra. As a practical application of the introduced theory an optimized decomposition algorithm is proposed. It favourably compares to existing swing-twist decomposition implementations.