{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,28]],"date-time":"2026-03-28T16:48:49Z","timestamp":1774716529103,"version":"3.50.1"},"reference-count":34,"publisher":"MDPI AG","issue":"11","license":[{"start":{"date-parts":[[2018,11,15]],"date-time":"2018-11-15T00:00:00Z","timestamp":1542240000000},"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":["51475046"],"award-info":[{"award-number":["51475046"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["51475047"],"award-info":[{"award-number":["51475047"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Sensors"],"abstract":"<jats:p>An extended robot\u2013world and hand\u2013eye calibration method is proposed in this paper to evaluate the transformation relationship between the camera and robot device. This approach could be performed for mobile or medical robotics applications, where precise, expensive, or unsterile calibration objects, or enough movement space, cannot be made available at the work site. Firstly, a mathematical model is established to formulate the robot-gripper-to-camera rigid transformation and robot-base-to-world rigid transformation using the Kronecker product. Subsequently, a sparse bundle adjustment is introduced for the optimization of robot\u2013world and hand\u2013eye calibration, as well as reconstruction results. Finally, a validation experiment including two kinds of real data sets is designed to demonstrate the effectiveness and accuracy of the proposed approach. The translation relative error of rigid transformation is less than 8\/10,000 by a Denso robot in a movement range of 1.3 m \u00d7 1.3 m \u00d7 1.2 m. The distance measurement mean error after three-dimensional reconstruction is 0.13 mm.<\/jats:p>","DOI":"10.3390\/s18113949","type":"journal-article","created":{"date-parts":[[2018,11,15]],"date-time":"2018-11-15T11:32:47Z","timestamp":1542281567000},"page":"3949","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":22,"title":["Simultaneous Robot\u2013World and Hand\u2013Eye Calibration without a Calibration Object"],"prefix":"10.3390","volume":"18","author":[{"given":"Wei","family":"Li","sequence":"first","affiliation":[{"name":"Institute of Information Photonics and Optical Communications, Beijing University of Posts and Telecommunications, Beijing 100876, China"}]},{"given":"Mingli","family":"Dong","sequence":"additional","affiliation":[{"name":"Key Laboratory of the Ministry of Education for Optoelectronic Measurement Technology and Instrument, Beijing Information Science and Technology University, Beijing 100192, China"}]},{"given":"Naiguang","family":"Lu","sequence":"additional","affiliation":[{"name":"Institute of Information Photonics and Optical Communications, Beijing University of Posts and Telecommunications, Beijing 100876, China"},{"name":"Key Laboratory of the Ministry of Education for Optoelectronic Measurement Technology and Instrument, Beijing Information Science and Technology University, Beijing 100192, China"}]},{"given":"Xiaoping","family":"Lou","sequence":"additional","affiliation":[{"name":"Key Laboratory of the Ministry of Education for Optoelectronic Measurement Technology and Instrument, Beijing Information Science and Technology University, Beijing 100192, China"}]},{"given":"Peng","family":"Sun","sequence":"additional","affiliation":[{"name":"Institute of Information Photonics and Optical Communications, Beijing University of Posts and Telecommunications, Beijing 100876, China"},{"name":"Key Laboratory of the Ministry of Education for Optoelectronic Measurement Technology and Instrument, Beijing Information Science and Technology University, Beijing 100192, China"}]}],"member":"1968","published-online":{"date-parts":[[2018,11,15]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Du, H., Chen, X., and Xi, J. (2017). Development and Verification of a Novel Robot-Integrated Fringe Projection 3D Scanning System for Large-Scale Metrology. Sensors, 17.","DOI":"10.3390\/s17122886"},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Shah, M., Eastman, R.D., and Hong, T. (2012, January 20\u201322). An overview of robot\u2013sensor calibration methods for evaluation of perception systems. Proceedings of the Workshop on Performance Metrics for Intelligent Systems, College Park, MD, USA.","DOI":"10.1145\/2393091.2393095"},{"key":"ref_3","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_4","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_5","doi-asserted-by":"crossref","first-page":"717","DOI":"10.1109\/70.326576","article-title":"Robot sensor calibration: Solving AX = XB on the Euclidean group","volume":"10","author":"Park","year":"1994","journal-title":"IEEE Trans. Robot. Autom."},{"key":"ref_6","unstructured":"Andreff, N., Horaud, R., and Espiau, B. (1999, January 8). On-line hand\u2013eye calibration. Proceedings of the Second International Conference on 3-D Digital Imaging and Modeling, Ottawa, ON, Canada."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"195","DOI":"10.1177\/027836499501400301","article-title":"Hand\u2013eye calibration","volume":"14","author":"Horaud","year":"1995","journal-title":"Int. J. Robot. Res."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"286","DOI":"10.1177\/02783649922066213","article-title":"Hand\u2013eye calibration using dual quaternions","volume":"18","author":"Daniilidis","year":"1999","journal-title":"Int. J. Robot. Res."},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Zhao, Z. (2011, January 9\u201313). Hand\u2013eye calibration using convex optimization. Proceedings of the 2011 IEEE International Conference on Robotics and Automation (ICRA), Shanghai, China.","DOI":"10.1109\/ICRA.2011.5979569"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"228","DOI":"10.1177\/02783640122067372","article-title":"Robot hand\u2013eye calibration using structure-from-motion","volume":"20","author":"Andreff","year":"2001","journal-title":"Int. J. Robot. Res."},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Schmidt, J., Vogt, F., and Niemann, H. (2005). Calibration-free hand-eye calibration: A structure-from-motion approach. DAGM., Springer.","DOI":"10.1007\/11550518_9"},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Ruland, T., Pajdla, T., and Kr\u00fcger, L. (2012, January 16\u201321). Globally optimal hand\u2013eye calibration. Proceedings of the 2012 IEEE Conference on Computer Vision and Pattern Recognition (CVPR), Providence, RI, USA.","DOI":"10.1109\/CVPR.2012.6247781"},{"key":"ref_13","unstructured":"Heller, J., Havlena, M., Sugimoto, A., and Pajdla, T. (2011, January 20\u201325). Structure-from-motion based hand\u2013eye calibration using L\u221e minimization. Proceedings of the 2011 IEEE Conference on Computer Vision and Pattern Recognition (CVPR), Colorado Springs, CO, USA."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"1027","DOI":"10.1109\/TPAMI.2015.2469299","article-title":"Globally optimal hand\u2013eye calibration using branch-and-bound","volume":"38","author":"Heller","year":"2016","journal-title":"IEEE Trans. Pattern Anal. Mach. Intell."},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Zhi, X., and Schwertfeger, S. (2017, January 24\u201328). Simultaneous hand\u2013eye calibration and reconstruction. Proceedings of the 2017 IEEE\/RSJ International Conference on Intelligent Robots and Systems (IROS), Vancouver, BC, Canada.","DOI":"10.1109\/IROS.2017.8205949"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"145","DOI":"10.1109\/LRA.2015.2506663","article-title":"Simultaneous hand\u2013eye and robot\u2013world calibration by solving the $ AX = YB $ problem without correspondence","volume":"1","author":"Li","year":"2016","journal-title":"IEEE Robot. Autom. Lett."},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Pachtrachai, K., Allan, M., and Pawar, V. (2016, January 9\u201314). Hand\u2013eye calibration for robotic assisted minimally invasive surgery without a calibration object. Proceedings of the 2016 IEEE\/RSJ International Conference on Intelligent Robots and Systems (IROS), Daejeon, Korea.","DOI":"10.1109\/IROS.2016.7759387"},{"key":"ref_18","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_19","unstructured":"Hirsh, R.L., DeSouza, G.N., and Kak, A.C. (2001, January 21\u201326). An iterative approach to the hand\u2013eye and base-world calibration problem. Proceedings of the 2001 IEEE International Conference on Robotics and Automation (ICRA), Seoul, Korea."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"031007\u20131","DOI":"10.1115\/1.4024473","article-title":"Solving the robot\u2013world\/hand\u2013eye calibration problem using the Kronecker product","volume":"5","author":"Shah","year":"2013","journal-title":"J. Mech. Robot."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"617","DOI":"10.1109\/70.704233","article-title":"Simultaneous robot\u2013world and hand\u2013eye calibration","volume":"14","author":"Dornaika","year":"1998","journal-title":"IEEE Trans. Robot. Autom."},{"key":"ref_22","first-page":"1530","article-title":"Simultaneous robot\u2013world and hand\u2013eye calibration using dual-quaternions and Kronecker product","volume":"5","author":"Li","year":"2010","journal-title":"Int. J. Phys. Sci."},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Heller, J., Henrion, D., and Pajdla, T. (2014, January 31). Hand\u2013eye and robot\u2013world calibration by global polynomial optimization. Proceedings of the 2014 IEEE International Conference on Robotics and Automation (ICRA), Hong Kong, China.","DOI":"10.1109\/ICRA.2014.6907313"},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Tabb, A., Khalil, M., and Yousef, A. (October, January 28). Parameterizations for reducing camera reprojection error for robot\u2013world hand\u2013eye calibration. Proceedings of the 2015 IEEE\/RSJ International Conference on Intelligent Robots and Systems (IROS), Hamburg, Germany.","DOI":"10.1109\/IROS.2015.7353795"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"569","DOI":"10.1007\/s00138-017-0841-7","article-title":"Solving the robot\u2013world hand\u2013eye (s) calibration problem with iterative methods","volume":"28","author":"Tabb","year":"2017","journal-title":"Mach. Vis. Appl."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"603","DOI":"10.1137\/0117057","article-title":"A note on Kronecker matrix products and matrix equation systems","volume":"17","author":"Neudecker","year":"1969","journal-title":"SIAM J. Appl. Math."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"381","DOI":"10.1145\/358669.358692","article-title":"Random sample consensus: A paradigm for model fitting with applications to image analysis and automated cartography","volume":"24","author":"Fischler","year":"1981","journal-title":"Commun. ACM"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"904","DOI":"10.1016\/j.jvcir.2014.02.013","article-title":"Structure and motion from line correspondences: Representation, projection, initialization and sparse bundle adjustment","volume":"25","author":"Zhang","year":"2014","journal-title":"J. Vis. Commun. Image Represent."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1145\/1486525.1486527","article-title":"SBA: A software package for generic sparse bundle adjustment","volume":"36","author":"Lourakis","year":"2009","journal-title":"ACM Trans. Math. Softw."},{"key":"ref_30","doi-asserted-by":"crossref","unstructured":"Wu, C. (July, January 29). Towards linear-time incremental structure from motion. Proceedings of the 2013 IEEE International Conference on 3D Vision (3DV), Seattle, WA, USA.","DOI":"10.1109\/3DV.2013.25"},{"key":"ref_31","doi-asserted-by":"crossref","unstructured":"Corke, P. (2017). Robotics, Vision and Control.: Fundamental Algorithms in MATLAB\u00ae, Springer. [2nd ed.].","DOI":"10.1007\/978-3-319-54413-7"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"9834","DOI":"10.1364\/OE.25.009834","article-title":"Modelling and calibration of depth-dependent distortion for large depth visual measurement cameras","volume":"25","author":"Sun","year":"2017","journal-title":"Opt. Express"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"2303","DOI":"10.1109\/TPAMI.2012.43","article-title":"An efficient hidden variable approach to minimal-case camera motion estimation","volume":"34","author":"Hartley","year":"2012","journal-title":"IEEE Trans. Pattern Anal. Mach. Intell."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"558","DOI":"10.1016\/j.isprsjprs.2010.06.003","article-title":"Close range photogrammetry for industrial applications","volume":"65","author":"Luhmann","year":"2010","journal-title":"ISPRS J. Photogramm. Remote Sens."}],"container-title":["Sensors"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1424-8220\/18\/11\/3949\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T15:29:46Z","timestamp":1760196586000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1424-8220\/18\/11\/3949"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,11,15]]},"references-count":34,"journal-issue":{"issue":"11","published-online":{"date-parts":[[2018,11]]}},"alternative-id":["s18113949"],"URL":"https:\/\/doi.org\/10.3390\/s18113949","relation":{},"ISSN":["1424-8220"],"issn-type":[{"value":"1424-8220","type":"electronic"}],"subject":[],"published":{"date-parts":[[2018,11,15]]}}}