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The Subspace Constrained Least Squares Solution of Unit Dual Quaternion Vector Equations and Its Application to Hand-Eye Calibration

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Abstract

In this paper, we study the solutions to two types of unit dual quaternion equations, namely \(\varvec{a}\check{x} = \check{x}\varvec{b}\) and \(\varvec{a}\check{x} = \check{z}\varvec{b}\). Due to the \(2\)-norm of the dual quaternion vector, there may exist multiple potential solutions for these equations. The main contribution of this study is the introduction of a novel formulation for subspace constrained least squares solutions to these two unit dual quaternion equations, along with the derivation of closed-form expressions for these solutions. We develop and implement numerical algorithms to address the robot-world and hand-eye calibration problems. Our findings demonstrate that the proposed subspace constrained least squares solution can avoid discussing the ambiguities associated with the non-uniqueness of signs that arise when mapping from rotation matrices to quaternions. Furthermore, we establish that when the transformation matrix equation related to the robot-world or hand-eye calibration problem possesses a solution, the corresponding unit dual quaternion is indeed a subspace constrained least squares solution to the equations \(\varvec{a}\check{x} =\check{x}\varvec{b}\) and \(\varvec{a}\check{x} = \check{z}\varvec{b}\), respectively. The experimental results demonstrate that the proposed subspace constrained least squares solutions are competitive when compared to existing solution methods.

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Data Availability

The data that support the findings of this study are available from the author upon reasonable request.

Notes

  1. All numerical experiments are implemented in MATLAB R2023a running on a computer with an Intel(R) Core(TM) i7 CPU @ 2.20GHz and 16GB of RAM.

  2. The Matlab code of comparison methods are downloaded from https://github.com/soheilsarabandi/Hand-Eye-Calibration.

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Acknowledgements

The authors are very grateful to two anonymous referees for their helpful comments and suggestions.

Funding

The first author’s research was partially supported by the National Natural Science Foundation of China Grant (No. 12271217, 12471307). M. Ng was supported by the National Key Research and Development Program of China under Grant 2024YFE0202900, HKRGC GRF 17300021, C7004-21GF and Joint NSFC-RGC N-HKU76921.

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  1. School of Mathematical Sciences, Jiangsu University, Zhenjiang, 212013, Jiangsu, China

    Hong Zhu

  2. Department of Mathematics, HongKong Baptist University, Kowloon Tong, 999077, Hong Kong, China

    Michael K. Ng

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Zhu, H., Ng, M.K. The Subspace Constrained Least Squares Solution of Unit Dual Quaternion Vector Equations and Its Application to Hand-Eye Calibration. J Sci Comput 103, 49 (2025). https://doi.org/10.1007/s10915-025-02866-5

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