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Projecting onto the Unit Dual Quaternion Set

Ziyang Li, Chunfeng Cui, Jiaxin Xie

TL;DR

The paper tackles the projection of dual quaternions onto the unit dual quaternion set under the $2^R$-norm, a problem central to robust rigid-body motion representations. It reformulates dual-quaternion constraints into an 8-dimensional real-vector problem with constraints $\|\overrightarrow{\tilde{q}_s}\|_2^2=1$ and $\overrightarrow{\tilde{q}_d}^\top \overrightarrow{\tilde{q}_s}=0$, and solves it via a KKT-based, case-aware algorithm (Algorithm 1). The key contributions include a rigorous existence proof for a global minimizer, a detailed case analysis yielding closed-form updates in several scenarios, and a numerically validated projection method that outperforms a prior PM-based approach in the $2^R$-norm objective. This work has practical impact for optimization tasks in robotics and computer vision where dual-quaternion representations and accurate projections are essential, such as hand-eye calibration and motion estimation.

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.

Projecting onto the Unit Dual Quaternion Set

TL;DR

The paper tackles the projection of dual quaternions onto the unit dual quaternion set under the -norm, a problem central to robust rigid-body motion representations. It reformulates dual-quaternion constraints into an 8-dimensional real-vector problem with constraints and , and solves it via a KKT-based, case-aware algorithm (Algorithm 1). The key contributions include a rigorous existence proof for a global minimizer, a detailed case analysis yielding closed-form updates in several scenarios, and a numerically validated projection method that outperforms a prior PM-based approach in the -norm objective. This work has practical impact for optimization tasks in robotics and computer vision where dual-quaternion representations and accurate projections are essential, such as hand-eye calibration and motion estimation.

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 -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.
Paper Structure (14 sections, 2 theorems, 53 equations, 2 figures, 3 tables, 1 algorithm)

This paper contains 14 sections, 2 theorems, 53 equations, 2 figures, 3 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminary
  3. Quaternions
  4. Dual quaternions
  5. First-order optimality conditions
  6. Project onto the unit dual quaternion set
  7. Problem setup
  8. The existence of the global minimum of \ref{['Objective Function']}
  9. The existence of the global minimum of \ref{['Objective Function']}
  10. The algorithm
  11. Numerical Examples
  12. Synthetic data
  13. Real-world data
  14. conclusion

Key Result

Theorem 2.3

Suppose that $x^\# \in \mathbb{R}^n$ is a local solution of const-opt, the functions $f$ and $c_i$ in const-opt are continuously differentiable, and the LICQ holds at $x^\#$. Then there is a Lagrange multiplier vector $\lambda^\#\in \mathbb{R}^{|\mathcal{E}| +|\mathcal{I}|}$, with components $\lambd

Figures (2)

  • Figure 1: A three-dimensional geometric analogy for the vector projection problem. Left panel (Case 1): The non-unique solutions $\overrightarrow{\tilde{q}_s}^{\#}$ lie on the blue circular intersection of the green unit sphere and a plane. Right panel (Case 2.1): The unique solution $\overrightarrow{\tilde{q}_s}^{\#}$ is the radial projection of the input vector $\vec{\tilde{a}}_s$ onto the green unit sphere.
  • Figure 2: Figures depict the cumulative distribution of norm errors ($E_R$) and orthogonality errors ($E_O$) across increasing problem dimensions.

Theorems & Definitions (4)

  • Definition 2.1: Active set
  • Definition 2.2: LICQ
  • Theorem 2.3: First-order necessary conditions
  • Proposition 3.1