{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,5]],"date-time":"2026-03-05T03:07:29Z","timestamp":1772680049854,"version":"3.50.1"},"reference-count":27,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2024,8,28]],"date-time":"2024-08-28T00:00:00Z","timestamp":1724803200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"National Natural Science Foundation of China","award":["62176112"],"award-info":[{"award-number":["62176112"]}]},{"name":"National Natural Science Foundation of China","award":["ZR2022MA030"],"award-info":[{"award-number":["ZR2022MA030"]}]},{"name":"National Natural Science Foundation of China","award":["319462208"],"award-info":[{"award-number":["319462208"]}]},{"name":"Natural Science Foundation of Shandong Province","award":["62176112"],"award-info":[{"award-number":["62176112"]}]},{"name":"Natural Science Foundation of Shandong Province","award":["ZR2022MA030"],"award-info":[{"award-number":["ZR2022MA030"]}]},{"name":"Natural Science Foundation of Shandong Province","award":["319462208"],"award-info":[{"award-number":["319462208"]}]},{"name":"Discipline with Strong Characteristics of Liaocheng University\u2014Intelligent Science and Technology","award":["62176112"],"award-info":[{"award-number":["62176112"]}]},{"name":"Discipline with Strong Characteristics of Liaocheng University\u2014Intelligent Science and Technology","award":["ZR2022MA030"],"award-info":[{"award-number":["ZR2022MA030"]}]},{"name":"Discipline with Strong Characteristics of Liaocheng University\u2014Intelligent Science and Technology","award":["319462208"],"award-info":[{"award-number":["319462208"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Dual algebra plays an important role in kinematic synthesis and dynamic analysis, but there are still few studies on dual quaternion matrix theory. This paper provides an efficient method for solving the QLY least squares problem of the dual quaternion matrix equation AXB+CYD\u2248E, where X, Y are unknown dual quaternion matrices with special structures. First, we define a semi-tensor product of dual quaternion matrices and study its properties, which can be used to achieve the equivalent form of the dual quaternion matrix equation. Then, by using the dual representation of dual quaternion and the GH-representation of special dual quaternion matrices, we study the expression of QLY least squares Hermitian solution of the dual quaternion matrix equation AXB+CYD\u2248E. The algorithm is given and the numerical examples are provided to illustrate the efficiency of the method.<\/jats:p>","DOI":"10.3390\/sym16091117","type":"journal-article","created":{"date-parts":[[2024,8,28]],"date-time":"2024-08-28T05:34:49Z","timestamp":1724823289000},"page":"1117","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":6,"title":["Solving the QLY Least Squares Problem of Dual Quaternion Matrix Equation Based on STP of Dual Quaternion Matrices"],"prefix":"10.3390","volume":"16","author":[{"ORCID":"https:\/\/orcid.org\/0009-0005-0057-7365","authenticated-orcid":false,"given":"Ruyu","family":"Tao","sequence":"first","affiliation":[{"name":"Research Center of Semi-Tensor Product of Matrices: Theory and Applications, College of Mathematical Sciences, Liaocheng University, Liaocheng 252000, China"}]},{"given":"Ying","family":"Li","sequence":"additional","affiliation":[{"name":"Research Center of Semi-Tensor Product of Matrices: Theory and Applications, College of Mathematical Sciences, Liaocheng University, Liaocheng 252000, China"}]},{"given":"Mingcui","family":"Zhang","sequence":"additional","affiliation":[{"name":"Research Center of Semi-Tensor Product of Matrices: Theory and Applications, College of Mathematical Sciences, Liaocheng University, Liaocheng 252000, China"}]},{"given":"Xiaochen","family":"Liu","sequence":"additional","affiliation":[{"name":"Research Center of Semi-Tensor Product of Matrices: Theory and Applications, College of Mathematical Sciences, Liaocheng University, Liaocheng 252000, China"}]},{"given":"Musheng","family":"Wei","sequence":"additional","affiliation":[{"name":"College of Mathematics and Science, Shanghai Normal University, Shanghai 200234, China"}]}],"member":"1968","published-online":{"date-parts":[[2024,8,28]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"535","DOI":"10.1115\/1.3591622","article-title":"Analysis of an offset unsymmetric gyroscope with oblique rotor using (3 \u00d7 3) matrices with dual-number elements","volume":"91","author":"Yang","year":"1969","journal-title":"Eng. Ind."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"141","DOI":"10.1016\/0094-114X(76)90006-9","article-title":"On the use of dual numbers, vectors and matrices in instantaneous, spatial kinematics","volume":"11","author":"Veldkamp","year":"1976","journal-title":"Mech. Mach. Theory"},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Angeles, J. (1998). The application of dual algebra to kinematic analysis. Computational Methods in Mechanical Systems: Mechanism Analysis, Synthesis, and Optimization, Springer.","DOI":"10.1007\/978-3-662-03729-4_1"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"3836","DOI":"10.3182\/20080706-5-KR-1001.00645","article-title":"Control of oriented mechanical systems: A method based on dual quater-nion","volume":"41","author":"Han","year":"2008","journal-title":"IFAC Proc. Vol."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"4086","DOI":"10.1109\/LRA.2020.2988407","article-title":"A dual quaternion-based approach for coordi-nate calibration of dual robots in collaborative motion","volume":"5","author":"Fu","year":"2020","journal-title":"IEEE Robot. Autom. Lett."},{"key":"ref_6","unstructured":"Dooley, J.R., and McCarthy, J.M. (1991, January 9\u201311). Spatial rigid body dynamics using dual quaternion components. Proceedings of the 1991 IEEE International Conference on Robotics and Automation, Sacramento, CA, USA."},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Bultmann, S., Li, K., and Hanebeck, U.D. (2019, January 2\u20135). Stereo visual SLAM based on unscented dual quaternion filtering. Proceedings of the 2019 22th International Conference on Information Fusion (FUSION), Ottawa, ON, Canada.","DOI":"10.23919\/FUSION43075.2019.9011391"},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Leclercq, G., Lef\u00e8vre, P., and Blohm, G. (2013). 3D kinematics using dual quaternions: Theory and applications in neu-roscience. Front. Behav. Neurosci., 7.","DOI":"10.3389\/fnbeh.2013.00007"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"181","DOI":"10.1049\/iet-csr.2020.0029","article-title":"Unit dual quaternion-based pose optimisation for visual runway observations","volume":"2","author":"Brambley","year":"2020","journal-title":"IET Cyber-Syst. Robot."},{"key":"ref_10","first-page":"257","article-title":"Eigenvalues and singular values of dual quaternion matrices","volume":"19","author":"Qi","year":"2023","journal-title":"Pac. J. Optim."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"1371","DOI":"10.1080\/01630563.2023.2254090","article-title":"Minimax principle for eigenvalues of dual quaternion Hermitian matrices and generalized inverses of dual quaternion matrices","volume":"44","author":"Ling","year":"2023","journal-title":"Numer. Funct. Anal. Optim."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"21","DOI":"10.1007\/s10915-024-02561-x","article-title":"A power method for computing the dominant eigenvalue of a dual quaternion Hermitian ma-trix","volume":"100","author":"Cui","year":"2024","journal-title":"J. Sci. Comput."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"1053","DOI":"10.1109\/TNN.2002.1031938","article-title":"A recurrent neural network for solving Sylvester equation with time-varying coefficients","volume":"13","author":"Zhang","year":"2002","journal-title":"IEEE Trans. Neural Netw."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"86","DOI":"10.1016\/j.cam.2006.05.028","article-title":"Sylvester Tikhonov-regularization methods in image restoration","volume":"206","author":"Bouhamidi","year":"2007","journal-title":"J. Comput. Appl. Math."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"78","DOI":"10.1007\/s00006-021-01180-1","article-title":"A real method for solving quaternion matrix equation X \u2212 AX^B = C based on semi-tensor product of matrices","volume":"31","author":"Ding","year":"2021","journal-title":"Adv. Appl. Clifford Algebr."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"689","DOI":"10.1007\/s00006-013-0384-6","article-title":"On a solution of the quaternion matrix equation and its applications","volume":"23","author":"Jiang","year":"2013","journal-title":"Adv. Appl. Clifford Algebr."},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Li, Y., Ding, W., Zhao, X., Wei, A., and Zhao, J. (2023). Direct methods of solving quaternion matrix equation based on STP. Matrix and Operator Equations and Applications, Springer Nature.","DOI":"10.1007\/16618_2023_47"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"425","DOI":"10.1016\/j.amc.2015.08.046","article-title":"Special least squares solutions of the quaternion matrix equation AX = B with applications","volume":"270","author":"Zhang","year":"2015","journal-title":"Appl. Math. Comput."},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"J\u00e4ntschi, L. (2023). Eigenproblem Basics and Algorithms. Symmetry, 15.","DOI":"10.3390\/sym15112046"},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Chen, Y., Wang, Q.W., and Xie, L.M. (2024). Dual quaternion matrix equation AXB = C with applications. Symmetry, 16.","DOI":"10.20944\/preprints202402.0316.v1"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"100141","DOI":"10.1016\/j.kjs.2023.10.008","article-title":"The solution of the dual matrix equation ATX + XTA = D","volume":"51","author":"Zeng","year":"2024","journal-title":"Kuwait J. Sci."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"1494","DOI":"10.1007\/s42967-022-00189-y","article-title":"Dual quaternions and dual quaternion vectors","volume":"4","author":"Qi","year":"2022","journal-title":"Commun. Appl. Math. Comput."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"1985","DOI":"10.1080\/03081087.2023.2223348","article-title":"The QLY least-squares and the QLY least-squares minimal-norm of linear dual least squares problems","volume":"72","author":"Wang","year":"2024","journal-title":"Linear Multilinear A"},{"key":"ref_24","first-page":"487","article-title":"Matrix realization of dual quaternionic electromagnetism","volume":"5","author":"Demir","year":"2007","journal-title":"Cent. Eur. J. Phys."},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Cheng, D., Qi, H., and Zhao, Y. (2012). An Introduction to Semi-Tensor Product of Matrices and Its Applications, World Scientific.","DOI":"10.1142\/8323"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"6450","DOI":"10.1002\/mma.8916","article-title":"Semi-tensor product of quaternion matrices and its application","volume":"46","author":"Fan","year":"2023","journal-title":"Math. Methods Appl. Sci."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"3009","DOI":"10.1109\/TAC.2012.2197074","article-title":"H-representation and applications to generalized Lyapunov equations and linear stochastic systems","volume":"57","author":"Zhang","year":"2012","journal-title":"IEEE Trans. Autom. 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