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Standard Dual Quaternion Optimization and Its Applications in Hand-Eye Calibration and SLAM

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Abstract

Several common dual quaternion functions, such as the power function, the magnitude function, the 2-norm function, and the kth largest eigenvalue of a dual quaternion Hermitian matrix, are standard dual quaternion functions, i.e., the standard parts of their function values depend upon only the standard parts of their dual quaternion variables. Furthermore, the sum, product, minimum, maximum, and composite functions of two standard dual functions, the logarithm and the exponential of standard unit dual quaternion functions, are still standard dual quaternion functions. On the other hand, the dual quaternion optimization problem, where objective and constraint function values are dual numbers but variables are dual quaternions, naturally arises from applications. We show that to solve an equality constrained dual quaternion optimization (EQDQO) problem, we only need to solve two quaternion optimization problems. If the involved dual quaternion functions are all standard, the optimization problem is called a standard dual quaternion optimization problem, and some better results hold. Then, we show that the dual quaternion optimization problems arising from the hand-eye calibration problem and the simultaneous localization and mapping (SLAM) problem are equality constrained standard dual quaternion optimization problems.

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Acknowledgements

I am thankful to Chen Ling, Ziyan Luo, and Zhongming Chen for the discussion on the standard dual quaternion optimization, to Wei Li for the discussion on the hand-eye calibration, to Jiantong Cheng for the discussion on the SLAM, to Guyan Ni for introducing Jiantong Cheng to me, and to Chen Ouyang and Jinjie Liu for Figs. 1 and 2. I would like to thank two anonymous referees who carefully read my manuscript and gave very helpful comments.

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Authors and Affiliations

  1. Department of Mathematics, School of Science, Hangzhou Dianzi University, Hangzhou, 310018, Zhejiang, China

    Liqun Qi

  2. Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong, China

    Liqun Qi

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Correspondence to Liqun Qi.

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This work was partially supported by Hong Kong Innovation and Technology Commission (InnoHK Project CIMDA).

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Qi, L. Standard Dual Quaternion Optimization and Its Applications in Hand-Eye Calibration and SLAM. Commun. Appl. Math. Comput. 5, 1469–1483 (2023). https://doi.org/10.1007/s42967-022-00213-1

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  • DOI: https://doi.org/10.1007/s42967-022-00213-1

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