Communications on Applied Mathematics and Computation >

2025 , Vol. 7 >Issue 1: 228 - 238

DOI: https://doi.org/10.1007/s42967-023-00262-0

Motion, Dual Quaternion Optimization and Motion Optimization

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  • 1 Department of Mathematics, School of Science, Hangzhou Dianzi University, Hangzhou 310018, Zhejiang, China;
    2 Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong, China

Received date: 2022-12-26

  Revised date: 2023-02-09

  Accepted date: 2023-02-11

  Online published: 2025-04-21

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Abstract

We regard a dual quaternion as a real eight-dimensional vector and present a dual quaternion optimization model. Then we introduce motions as real six-dimensional vectors. A motion means a rotation and a translation. We define a motion operator which maps unit dual quaternions to motions, and a UDQ operator which maps motions to unit dual quaternions. By these operators, we present another formulation of dual quaternion optimization. The objective functions of such dual quaternion optimization models are real valued. They are different from the previous model whose object function is dual number valued. This avoids the two-stage problem, which causes troubles sometimes. We further present an alternative formulation, called motion optimization, which is actually an unconstrained real optimization model. Then we formulate two classical problems in robot research, i.e., the hand-eye calibration problem and the simultaneous localization and mapping (SLAM) problem as such dual quaternion optimization problems as well as such motion optimization problems. This opens a new way to solve these problems.

Key words： Motion; Unit dual quaternion; Motion operator; UDQ operator; Motion optimization; Hand-eye calibration; Simultaneous localization and mapping (SLAM)

Cite this article

Liqun Qi . Motion, Dual Quaternion Optimization and Motion Optimization[J]. Communications on Applied Mathematics and Computation, 2025 , 7(1) : 228 -238 . DOI: 10.1007/s42967-023-00262-0

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