{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,13]],"date-time":"2026-05-13T03:24:07Z","timestamp":1778642647287,"version":"3.51.4"},"reference-count":27,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2022,2,26]],"date-time":"2022-02-26T00:00:00Z","timestamp":1645833600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["61873133, U1913208, 61873135"],"award-info":[{"award-number":["61873133, U1913208, 61873135"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"name":"National Key R&amp;D Program of China","award":["2018YFB1307601"],"award-info":[{"award-number":["2018YFB1307601"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Sensors"],"abstract":"<jats:p>A multi-robot collaboration system can complete more complex tasks than a single robot system. Ensuring the calibration accuracy between robots in the system is a prerequisite for the effective inter-robot cooperation. This paper presents a dual-robot system for orthopedic surgeries, where the relationships between hand-eye, flange-tool, and robot-robot need to be calibrated. This calibration problem can be summarized to the solution of the matrix equation of AXB=YCZ. A combined solution is proposed to solve the unknown parameters in the equation of AXB=YCZ, which consists of the dual quaternion closed-form method and the iterative method based on Levenberg\u2013Marquardt (LM) algorithm. The closed-form method is used to quickly obtain the initial value for the iterative method so as to increase the convergence speed and calibration accuracy of the iterative method. Simulation and experimental analyses are carried out to verify the accuracy and effectiveness of the proposed method.<\/jats:p>","DOI":"10.3390\/s22051861","type":"journal-article","created":{"date-parts":[[2022,2,27]],"date-time":"2022-02-27T20:48:33Z","timestamp":1645994913000},"page":"1861","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":24,"title":["Simultaneous Calibration of the Hand-Eye, Flange-Tool and Robot-Robot Relationship in Dual-Robot Collaboration Systems"],"prefix":"10.3390","volume":"22","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-5162-1665","authenticated-orcid":false,"given":"Yanding","family":"Qin","sequence":"first","affiliation":[{"name":"Tianjin Key Laboratory of Intelligent Robotics, College of Artificial Intelligence, Nankai University, Tianjin 300350, China"},{"name":"Institute of Intelligence Technology and Robotic Systems, Shenzhen Research Institute of Nankai University, Shenzhen 518083, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Pengxiu","family":"Geng","sequence":"additional","affiliation":[{"name":"Tianjin Key Laboratory of Intelligent Robotics, College of Artificial Intelligence, Nankai University, Tianjin 300350, China"},{"name":"Institute of Intelligence Technology and Robotic Systems, Shenzhen Research Institute of Nankai University, Shenzhen 518083, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Bowen","family":"Lv","sequence":"additional","affiliation":[{"name":"Tianjin Key Laboratory of Intelligent Robotics, College of Artificial Intelligence, Nankai University, Tianjin 300350, China"},{"name":"Institute of Intelligence Technology and Robotic Systems, Shenzhen Research Institute of Nankai University, Shenzhen 518083, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yiyang","family":"Meng","sequence":"additional","affiliation":[{"name":"Tianjin Key Laboratory of Intelligent Robotics, College of Artificial Intelligence, Nankai University, Tianjin 300350, China"},{"name":"Institute of Intelligence Technology and Robotic Systems, Shenzhen Research Institute of Nankai University, Shenzhen 518083, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Zhichao","family":"Song","sequence":"additional","affiliation":[{"name":"Tianjin Key Laboratory of Intelligent Robotics, College of Artificial Intelligence, Nankai University, Tianjin 300350, China"},{"name":"Institute of Intelligence Technology and Robotic Systems, Shenzhen Research Institute of Nankai University, Shenzhen 518083, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9664-4534","authenticated-orcid":false,"given":"Jianda","family":"Han","sequence":"additional","affiliation":[{"name":"Tianjin Key Laboratory of Intelligent Robotics, College of Artificial Intelligence, Nankai University, Tianjin 300350, China"},{"name":"Institute of Intelligence Technology and Robotic Systems, Shenzhen Research Institute of Nankai University, Shenzhen 518083, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2022,2,26]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"148","DOI":"10.1109\/TMRB.2020.2990286","article-title":"Technical and Functional Validation of a Teleoperated Multi-robots Platform for Minimally Invasive Surgery","volume":"2","author":"Leporini","year":"2020","journal-title":"IEEE Trans. Med. Robot. Bionics"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"22774","DOI":"10.1109\/ACCESS.2021.3055852","article-title":"Novel Implementation of Multi-Robot Space Exploration Utilizing Coordinated Multi-Robot Exploration and Frequency Modified Whale Optimization Algorithm","volume":"9","author":"Gul","year":"2021","journal-title":"IEEE Access"},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Wiles, A.D., Thompson, D.G., and Frantz, D.D. (2004). Accuracy assessment and interpretation for optical tracking systems. Medical Imaging 2004: Visualization, Image-Guided Procedures, and Display, Proceedings of the Society of Photo-Optical Instrumentation Engineers (SPIE), San Diego, CA, USA, 14\u201319 February 2004, International Society for Optical Engineering (SPIE).","DOI":"10.1117\/12.536128"},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Meng, Y., You, Y., Geng, P., Song, Z., Wang, H., and Qin, Y. (2021, January 27\u201331). Development of an intra-operative active navigation system for robot-assisted surgery. Proceedings of the 2021 IEEE Inter-national Conference on Robotics and Biomimetics, Sanya, China.","DOI":"10.1109\/ROBIO54168.2021.9739506"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"16","DOI":"10.1109\/70.88014","article-title":"Calibration of wrist-mounted robotic sensors by solving homogeneous transform equations of the form AX=XB","volume":"5","author":"Shiu","year":"1989","journal-title":"IEEE Trans. Robot. Autom."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"345","DOI":"10.1109\/70.34770","article-title":"A new technique for fully autonomous and efficient 3D robotics hand\/eye calibration","volume":"5","author":"Tsai","year":"1989","journal-title":"IEEE Trans. Robot. Autom."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"497","DOI":"10.1002\/rob.20082","article-title":"Hand to sensor calibration: A geometrical interpretation of the matrix equation AX=XB","volume":"22","author":"Fassi","year":"2005","journal-title":"J. Robot. Syst."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"286","DOI":"10.1177\/02783649922066213","article-title":"Hand-eye calibration using dual quaternions","volume":"18","author":"Daniilidis","year":"1999","journal-title":"Int. J. Robot. Res."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"031007","DOI":"10.1115\/1.4024473","article-title":"Solving the Robot-World\/Hand-Eye Calibration Problem Using the Kronecker Product","volume":"5","author":"Shah","year":"2013","journal-title":"J. Mech. Robot."},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Ackerman, M.K., Cheng, A., Shiffman, B., Boctor, E., and Chirikjian, G. (2013, January 3\u20137). Sensor calibration with unknown correspondence: Solving AX=XB using Euclidean-group invariants. Proceedings of the 2013 IEEE\/RSJ International Conference on Intelligent Robots and Systems, Tokyo, Japan.","DOI":"10.1109\/IROS.2013.6696518"},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Zhao, Z. (2011, January 3\u201313). Hand-eye calibration using convex optimization. Proceedings of the 2011 IEEE International Conference on Robotics and Automation, Shanghai, China.","DOI":"10.1109\/ICRA.2011.5979569"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"549","DOI":"10.1109\/70.313105","article-title":"Simultaneous robot\/world and tool\/flange calibration by solving homogeneous transformation equations of the form AX=YB","volume":"10","author":"Zhuang","year":"1994","journal-title":"IEEE Trans. Robot. Autom."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"22705","DOI":"10.1109\/ACCESS.2017.2781698","article-title":"Simultaneous Robot-World, Sensor-Tip, and Kinematics Calibration of an Underactuated Robotic Hand with Soft Fingers","volume":"6","author":"Tan","year":"2017","journal-title":"IEEE Access"},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Condurache, D., and Ciureanu, I.A. (2019, January 23\u201326). A novel solution for AX=YB sensor calibration problem using dual Lie algebra. Proceedings of the 2019 6th International Conference on Control, Decision and Information Technologies, Paris, France.","DOI":"10.1109\/CoDIT.2019.8820336"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"1369","DOI":"10.1109\/TMECH.2020.3019306","article-title":"Globally Optimal Symbolic Hand-Eye Calibration","volume":"26","author":"Wu","year":"2020","journal-title":"IEEE\/ASME Trans. Mechatron."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"012027","DOI":"10.1088\/1742-6596\/1939\/1\/012027","article-title":"Calibration of dual industrial robot system based on hand-eye calibration algorithm","volume":"1939","author":"Jin","year":"2021","journal-title":"J. Phys. Conf. Ser."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"211","DOI":"10.1016\/j.precisioneng.2017.05.009","article-title":"A novel calibration method for multi-robots system utilizing calibration model without nominal kinematic parameters","volume":"50","author":"Qiao","year":"2017","journal-title":"Precis. Eng."},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Wang, J., Wu, L., Meng, M.Q.-H., and Ren, H. (2014, January 14\u201318). Towards simultaneous coordinate calibrations for cooperative multiple robots. Proceedings of the IEEE\/RSJ International Conference on Intelligent Robots and Systems, Chicago, IL, USA.","DOI":"10.1109\/IROS.2014.6942592"},{"key":"ref_19","first-page":"413","article-title":"Simultaneous hand-eye, tool\u2013flange, and robot-robot calibration for comanipulation by solving the AXB=YCZ problem","volume":"32","author":"Wu","year":"2016","journal-title":"IEEE Robot. Autom. Lett."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"2729","DOI":"10.1017\/S0263574715000338","article-title":"Registration of a hybrid robot using the degradation-Kronecker method and a purely non-linear method","volume":"34","author":"Yan","year":"2016","journal-title":"Robotica"},{"key":"ref_21","unstructured":"Ma, Q., Goh, Z., and Chirikjian, G.S. (2016, January 12\u201324). Probabilistic Approaches to the AXB=YCZ Calibration Problem in Multi-Robot Systems. Proceedings of the 2016 Robotics: Science and Systems Conference, Ann Arbor, MA, USA."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"1497","DOI":"10.1007\/s10514-018-9744-3","article-title":"Probabilistic approaches to the AXB=YCZ calibration problem in multi-robot systems","volume":"42","author":"Ma","year":"2018","journal-title":"Auton. Robot."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"4086","DOI":"10.1109\/LRA.2020.2988407","article-title":"A Dual Quaternion-Based Approach for Coordinate Calibration of Dual Robots in Collaborative Motion","volume":"5","author":"Fu","year":"2020","journal-title":"IEEE Robot. Autom. Lett."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"1172","DOI":"10.1109\/TRO.2020.3043688","article-title":"Simultaneous Calibration of Multi-coordinates for a Du-al-Robot System by Solving the AXB=YCZ Problem","volume":"37","author":"Wang","year":"2021","journal-title":"IEEE Trans. Robot."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"21750","DOI":"10.1109\/JSEN.2021.3101299","article-title":"Improving Localization Success Rate of Three Magnetic Targets Using Individual Memory-Based WO-LM Algorithm","volume":"21","author":"Lv","year":"2021","journal-title":"IEEE Sens. J."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"110","DOI":"10.1063\/1.168600","article-title":"Damping\u2013undamping strategies for the Levenberg\u2013Marquardt nonlinear least-squares method","volume":"11","author":"Lampton","year":"1997","journal-title":"Comput. Phys."},{"key":"ref_27","first-page":"2","article-title":"A brief description of the Levenberg\u2013Marquardt algorithm implemented by Levmar","volume":"3","author":"Lourakis","year":"2005","journal-title":"Comput. Sci."}],"container-title":["Sensors"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1424-8220\/22\/5\/1861\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T22:28:13Z","timestamp":1760135293000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1424-8220\/22\/5\/1861"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,2,26]]},"references-count":27,"journal-issue":{"issue":"5","published-online":{"date-parts":[[2022,3]]}},"alternative-id":["s22051861"],"URL":"https:\/\/doi.org\/10.3390\/s22051861","relation":{},"ISSN":["1424-8220"],"issn-type":[{"value":"1424-8220","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,2,26]]}}}