Communications on Applied Mathematics and Computation >

2023 , Vol. 5 >Issue 4: 1469 - 1483

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

ORIGINAL PAPERS

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

Expand
  • 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: 2022-07-24

  Revised date: 2022-08-24

  Online published: 2023-12-16

Supported by

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

Fold

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.

Key words： Standard dual quaternion functions; Dual quaternion optimization; Quaternion optimization; Hand-eye calibration; Simultaneous localization and mapping(SLAM)

Cite this article

Liqun Qi . Standard Dual Quaternion Optimization and Its Applications in Hand-Eye Calibration and SLAM[J]. Communications on Applied Mathematics and Computation, 2023 , 5(4) : 1469 -1483 . DOI: 10.1007/s42967-022-00213-1

References

1. Abaunza, H., Castillo, P., Victorino, A., Lozano, R.: Dual quaternion modeling and control of a quadrotor aerial manipulator. J. Intell. Robot. Syst. 88, 267-283 (2017)
2. Brambley, G., Kim, J.: Unit dual quaternion-based pose optimization for visual runway observations. IET Cyber Syst. Robot. 2, 181-189 (2020)
3. Bryson, M., Sukkarieh, S.: Building a robust implementation of bearing—only inertial SLAM for a UAV. J. Field Robot. 24, 113-143 (2007)
4. Bultmann, S., Li, K., Hanebeck, U.D.: Stereo visual SLAM based on unscented dual quaternion filtering. In: 2019 22th International Conference on Information Fusion (FUSION), pp. 1-8. IEEE (2019)
5. Cadena, C., Carlone, L., Carrillo, H., Latif, Y., Scaramuzza, D., Neira, J., Reid, I., Leonard, J.J.: Past, present and future of simultaneous localization and mapping: toward the robust-perception age. IEEE Trans. Rob. 32, 1309-1332 (2016)
6. Carlone, L., Tron, R., Daniilidis, K., Dellaert, F.: Initialization techniques for 3D SLAM: a survey on rotation and its use in pose graph optimization. In: IEEE International Conference on Robotics and Automation (ICRA), pp. 4597-4604. IEEE (2015)
7. Chen, Y., Qi, L., Zhang, X.: Color image completion using a low-rank quaternion matrix approximation. Pac. J. Optim. 18, 55-75 (2022)
8. Chen, Y., Xiao, X., Zhou, Y.: Low-rank quaternion approximation for color image processing. IEEE Trans. Image Process. 29, 1057-7149 (2019)
9. Cheng, J., Kim, J., Jiang, Z., Che, W.: Dual quaternion-based graph SLAM. Robot. Auton. Syst. 77, 15-24 (2016)
10. Daniilidis, K.: Hand-eye calibration using dual quaternions. Int. J. Robot. Res. 18, 286-298 (1999)
11. Flamant, J., Miron, S., Brie, D.: A general framework for constrained convex quaternion optimization. IEEE Trans. Signal Process. 70, 254-267 (2022)
12. Jia, Z., Ng, M.K., Song, G.: Robust quaternion matrix completion with applications to image inpaiting. Numer. Linear Algebra Appl. 26, e2245 (2019)
13. Kenright, B.: A beginners guide to dual-quaternions. In: 20th International Conference in Central Europe on Computer Graphics. Visualization and Computer Vision, Plzen, Czech (2012)
14. Leclercq, G., Lefévre, Ph., Blohm, G.: 3D kinematics using dual quaternions: theory and applications in neuroscience. Front. Behav. Neurosci. 7, 1-25 (2013)
15. Li, A., Wang, L., Wu, D.: Simultaneous robot-world and hand-eye calibration using dual-quaternions and Kronecker product. Int. J. Phys. Sci. 5, 1530-1536 (2010)
16. Li, W., Lv, N., Dong, M., Lou, X.: Simultaneous robot-world/hand-eye calibration using dual quaternion. Robot 40, 301-308 (2018) (in Chinese)
17. Peng, X., Shi, X., Gong, Y.: Dual-quaternion-based modelling and control for motion tracking of a tumbling target. J. Syst. Eng. Electron. 30, 985-994 (2019)
18. Qi, L., Ling, C., Yan, H.: Dual quaternions and dual quaternion vectors. Commun. Appl. Math. Comput. 4(4), 1494-1508 (2022)
19. Qi, L., Luo, Z.: Eigenvalues and singular values of dual quaternion matrices. Pac. J. Optim. (to appear)
20. Qi, L., Luo, Z., Wang, Q., Zhang, X.: Quaternion matrix optimization: motivation and analysis. J. Optim. Theory Appl. 193, 621-648 (2022)
21. Shiu, Y., Ahmad, S.: Calibration of wrist-mounted robotic sensors by solving homogeneous transform equation of the form AX = XB. IEEE Trans. Robot. Autom. 5, 16-27 (1989)
22. Tsai, R., Lenz, R.: A new technique for fully autonomous and efficient 3D robotic hand/eye calibration. IEEE Trans. Robot. Autom. 5, 345-358 (1989)
23. Wang, X., Yu, C., Lin, Z.: A dual quaternion solution to attitude and position control for rigid body coordination. IEEE Trans. Rob. 28, 1162-1170 (2012)
24. Wei, E., Jin, S., Zhang, Q, Liu, J., Li, X., Yan, W.: Autonomous navigation of Mars probe using X-ray pulsars: Modeling and results. Adv. Space Res. 51, 849-857 (2013)
25. Williams, S.B., Newman, P., Dissanayake, G.: Autonomous underwater simultaneous localization and map building. In: IEEE International Conference on Robotics and Automation (ICRA), pp. 1793-1798 (2000)
26. Zhuang, H.Q., Roth, Z., Sudhakar, R.: Simultaneous robot-world and tool/flange calibration by solving homogeneous transformation of the form AX = YB. IEEE Trans. Robot. Autom. 10, 549-554 (1994)
Options
Outlines

/

〈 〉

Copyright © Editorial By Communications on Applied Mathematics and Computation
沪ICP备09014157