Projecting onto the Unit Dual Quaternion Set

TL;DR

This paper studies the projection onto the set of unit dual quaternions under the $2^R$-norm, providing a systematic analysis and an effective algorithm with numerical validation for applications in robotics and motion modeling.

Contribution

It introduces a comprehensive analysis of projections onto unit dual quaternion sets under the $2^R$-norm, including a novel algorithm and case distinctions.

Findings

The proposed algorithm effectively computes projections in various cases.

Numerical experiments validate the algorithm's accuracy and efficiency.

The study enhances the understanding of dual quaternion projections in practical applications.

Abstract

Dual quaternions have gained significant attention due to their wide applications in areas such as multi-agent formation control, 3D motion modeling, and robotics. A fundamental aspect in dual quaternion research involves the projection onto unit dual quaternion sets. In this paper, we systematically study such projections under the $2^R$-norm, which is commonly used in practical applications. We identify several distinct cases based on the relationship between the standard and dual parts in vector form, and demonstrate the effectiveness of the proposed algorithm through numerical experiments.