{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,5]],"date-time":"2026-06-05T04:25:12Z","timestamp":1780633512755,"version":"3.54.1"},"reference-count":35,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2023,4,3]],"date-time":"2023-04-03T00:00:00Z","timestamp":1680480000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100012166","name":"National Key R&amp;D Program of China","doi-asserted-by":"publisher","award":["2020YFB1710905"],"award-info":[{"award-number":["2020YFB1710905"]}],"id":[{"id":"10.13039\/501100012166","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100012166","name":"National Key R&amp;D Program of China","doi-asserted-by":"publisher","award":["61903162"],"award-info":[{"award-number":["61903162"]}],"id":[{"id":"10.13039\/501100012166","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["2020YFB1710905"],"award-info":[{"award-number":["2020YFB1710905"]}],"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":["61903162"],"award-info":[{"award-number":["61903162"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>To address the time-optimal trajectory planning (TOTP) problem with joint jerk constraints in a Cartesian coordinate system, we propose a time-optimal path-parameterization (TOPP) algorithm based on nonlinear optimization. The key insight of our approach is the presentation of a comprehensive and effective iterative optimization framework for solving the optimal control problem (OCP) formulation of the TOTP problem in the (s,s\u02d9)-phase plane. In particular, we identify two major difficulties: establishing TOPP in Cartesian space satisfying third-order constraints in joint space, and finding an efficient computational solution to TOPP, which includes nonlinear constraints. Experimental results demonstrate that the proposed method is an effective solution for time-optimal trajectory planning with joint jerk limits, and can be applied to a wide range of robotic systems.<\/jats:p>","DOI":"10.3390\/e25040610","type":"journal-article","created":{"date-parts":[[2023,4,4]],"date-time":"2023-04-04T02:03:00Z","timestamp":1680573780000},"page":"610","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":8,"title":["A Cartesian-Based Trajectory Optimization with Jerk Constraints for a Robot"],"prefix":"10.3390","volume":"25","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-6231-3673","authenticated-orcid":false,"given":"Zhiwei","family":"Fan","sequence":"first","affiliation":[{"name":"State Key Laboratory of Robotics, Shenyang Institute of Automation, Chinese Academy of Sciences, Shenyang 110016, China"},{"name":"Institutes for Robotics and Intelligent Manufacturing, Chinese Academy of Sciences, Shenyang 110169, China"},{"name":"University of Chinese Academy of Sciences, Beijing 100049, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Kai","family":"Jia","sequence":"additional","affiliation":[{"name":"State Key Laboratory of Robotics, Shenyang Institute of Automation, Chinese Academy of Sciences, Shenyang 110016, China"},{"name":"Institutes for Robotics and Intelligent Manufacturing, Chinese Academy of Sciences, Shenyang 110169, China"},{"name":"SIASUN Robot & Automation Co., Ltd., Shenyang 110169, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Lei","family":"Zhang","sequence":"additional","affiliation":[{"name":"State Key Laboratory of Robotics, Shenyang Institute of Automation, Chinese Academy of Sciences, Shenyang 110016, China"},{"name":"Institutes for Robotics and Intelligent Manufacturing, Chinese Academy of Sciences, Shenyang 110169, China"},{"name":"SIASUN Robot & Automation Co., Ltd., Shenyang 110169, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Fengshan","family":"Zou","sequence":"additional","affiliation":[{"name":"State Key Laboratory of Robotics, Shenyang Institute of Automation, Chinese Academy of Sciences, Shenyang 110016, China"},{"name":"Institutes for Robotics and Intelligent Manufacturing, Chinese Academy of Sciences, Shenyang 110169, China"},{"name":"SIASUN Robot & Automation Co., Ltd., Shenyang 110169, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Zhenjun","family":"Du","sequence":"additional","affiliation":[{"name":"SIASUN Robot & Automation Co., Ltd., Shenyang 110169, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Mingmin","family":"Liu","sequence":"additional","affiliation":[{"name":"SIASUN Robot & Automation Co., Ltd., Shenyang 110169, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Yuting","family":"Cao","sequence":"additional","affiliation":[{"name":"State Key Laboratory of Robotics, Shenyang Institute of Automation, Chinese Academy of Sciences, Shenyang 110016, China"},{"name":"Institutes for Robotics and Intelligent Manufacturing, Chinese Academy of Sciences, Shenyang 110169, China"},{"name":"University of Chinese Academy of Sciences, Beijing 100049, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8324-6708","authenticated-orcid":false,"given":"Qiang","family":"Zhang","sequence":"additional","affiliation":[{"name":"School of Automation, Jiangsu University of Science and Technology, No. 666 Changhui Road, Zhenjiang 212100, China"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2023,4,3]]},"reference":[{"key":"ref_1","unstructured":"Mikolajczyk, T. (2012). Proceedings of the Advanced Materials Research II, Trans Tech Publications Ltd."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"127","DOI":"10.1007\/s10845-018-1433-8","article-title":"Literature review of Industry 4.0 and related technologies","volume":"31","author":"Oztemel","year":"2020","journal-title":"J. Intell. Manuf."},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Chiurazzi, M., Alcaide, J.O., Diodato, A., Menciassi, A., and Ciuti, G. (2023). Spherical Wrist Manipulator Local Planner for Redundant Tasks in Collaborative Environments. Sensors, 23.","DOI":"10.3390\/s23020677"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"269","DOI":"10.1007\/s11786-012-0123-8","article-title":"Trajectory Planning in Robotics","volume":"6","author":"Gasparetto","year":"2012","journal-title":"Math. Comput. Sci."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"521","DOI":"10.1007\/s12555-019-0703-3","article-title":"Time-optimal and Smooth Trajectory Planning for Robot Manipulators","volume":"19","author":"Zhang","year":"2021","journal-title":"Int. J. Control. Autom. Syst."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"645","DOI":"10.1109\/TRO.2018.2819195","article-title":"A New Approach to Time-Optimal Path Parameterization Based on Reachability Analysis","volume":"34","author":"Pham","year":"2018","journal-title":"IEEE Trans. Robot."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"3","DOI":"10.1177\/027836498500400301","article-title":"Time-Optimal Control of Robotic Manipulators Along Specified Paths","volume":"4","author":"Bobrow","year":"1985","journal-title":"Int. J. Robot. Res."},{"key":"ref_8","unstructured":"Kunz, T., and Stilman, M. (2012). Robotics: Science and Systems VIII, The MIT Press."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"1533","DOI":"10.1109\/TRO.2014.2351113","article-title":"A General, Fast, and Robust Implementation of the Time-Optimal Path Parameterization Algorithm","volume":"30","author":"Pham","year":"2014","journal-title":"IEEE Trans. Robot."},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Pham, H., and Pham, Q.C. (June, January 29). On the structure of the time-optimal path parameterization problem with third-order constraints. Proceedings of the 2017 IEEE International Conference on Robotics and Automation (ICRA), Singapore.","DOI":"10.1109\/ICRA.2017.7989084"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"888","DOI":"10.1109\/LRA.2017.2655580","article-title":"Essential Properties of Numerical Integration for Time-Optimal Path-Constrained Trajectory Planning","volume":"2","author":"Shen","year":"2017","journal-title":"IEEE Robot. Autom. Lett."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"735","DOI":"10.1109\/TMECH.2018.2810828","article-title":"Complete and Time-Optimal Path-Constrained Trajectory Planning With Torque and Velocity Constraints: Theory and Applications","volume":"23","author":"Shen","year":"2018","journal-title":"IEEE\/ASME Trans. Mechatronics"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"101969","DOI":"10.1016\/j.rcim.2020.101969","article-title":"Time-optimal tool motion planning with tool-tip kinematic constraints for robotic machining of sculptured surfaces","volume":"65","author":"Lu","year":"2020","journal-title":"Robot.-Comput.-Integr. Manuf."},{"key":"ref_14","first-page":"1","article-title":"Practical time-optimal trajectory planning for robots: A convex optimization approach","volume":"53","author":"Verscheure","year":"2008","journal-title":"IEEE Trans. Autom. Control."},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Xiao, Y., Dong, W., and Du, Z. (2012, January 5\u20138). A time-optimal trajectory planning approach based on calculation cost consideration. Proceedings of the 2012 IEEE International Conference on Mechatronics and Automation, Chengdu, China.","DOI":"10.1109\/ICMA.2012.6285102"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"1485","DOI":"10.1109\/TRO.2013.2277565","article-title":"Time-Optimal Path Following for Robots With Convex-Concave Constraints Using Sequential Convex Programming","volume":"29","author":"Debrouwere","year":"2013","journal-title":"IEEE Trans. Robot."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"1253","DOI":"10.1109\/TRO.2019.2920090","article-title":"Sequential Time-Optimal Path-Tracking Algorithm for Robots","volume":"35","author":"Nagy","year":"2019","journal-title":"IEEE Trans. Robot."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"103744","DOI":"10.1016\/j.robot.2021.103744","article-title":"A new approach to time-optimal trajectory planning with torque and jerk limits for robot","volume":"140","author":"Ma","year":"2021","journal-title":"Robot. Auton. Syst."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"491","DOI":"10.1109\/TAC.1986.1104317","article-title":"A dynamic programming approach to trajectory planning of robotic manipulators","volume":"31","author":"Shin","year":"1986","journal-title":"IEEE Trans. Autom. Control."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"521","DOI":"10.1109\/TRO.2018.2880120","article-title":"Nearly Optimal Path Following with Jerk and Torque Rate Limits Using Dynamic Programming","volume":"35","author":"Kaserer","year":"2019","journal-title":"IEEE Trans. Robot."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"2216","DOI":"10.1109\/LRA.2020.2970644","article-title":"Time Optimal Motion Planning and Admittance Control for Cooperative Grasping","volume":"5","author":"Kaserer","year":"2020","journal-title":"IEEE Robot. Autom. Lett."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"131","DOI":"10.1109\/TRO.2020.3010632","article-title":"A Bisection Algorithm for Time-Optimal Trajectory Planning Along Fully Specified Paths","volume":"37","author":"Barnett","year":"2021","journal-title":"IEEE Trans. Robot."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"1026","DOI":"10.1109\/TAC.2015.2466911","article-title":"Nonlinear Model Predictive Control for Constrained Output Path Following","volume":"61","author":"Faulwasser","year":"2016","journal-title":"IEEE Trans. Autom. Control."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"50","DOI":"10.1016\/j.sysconle.2017.02.001","article-title":"An optimal complexity algorithm for minimum-time velocity planning","volume":"103","author":"Consolini","year":"2017","journal-title":"Syst. Control. Lett."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"5200","DOI":"10.1109\/TII.2018.2851963","article-title":"An Efficient Iterative Learning Approach to Time-Optimal Path Tracking for Industrial Robots","volume":"14","author":"Steinhauser","year":"2018","journal-title":"IEEE Trans. Ind. Inform."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"3192","DOI":"10.1109\/TASE.2021.3111758","article-title":"A Sequential Algorithm for Jerk Limited Speed Planning","volume":"19","author":"Consolini","year":"2022","journal-title":"IEEE Trans. Autom. Sci. Eng."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"10399","DOI":"10.1109\/LRA.2022.3191813","article-title":"Time-Optimal Trajectory Planning With Interaction With the Environment","volume":"7","author":"Petrone","year":"2022","journal-title":"IEEE Robot. Autom. Lett."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"1253","DOI":"10.1007\/s00170-022-08796-y","article-title":"Time-optimal trajectory optimization of serial robotic manipulator with kinematic and dynamic limits based on improved particle swarm optimization","volume":"120","author":"Yang","year":"2022","journal-title":"Int. J. Adv. Manuf. Technol."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"88","DOI":"10.1115\/1.3143842","article-title":"Optimal Trajectory Generation for Robotic Manipulators Using Dynamic Programming","volume":"109","author":"Singh","year":"1987","journal-title":"J. Dyn. Syst. Meas. Control."},{"key":"ref_30","doi-asserted-by":"crossref","unstructured":"Slotine, J.J.E., and Yang, H.S. (1988, January 15\u201317). Improving the Efficiency of Time-Optimal Path-Following Algorithms. Proceedings of the 1988 American Control Conference, Atlanta, GA, USA.","DOI":"10.23919\/ACC.1988.4790076"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"790","DOI":"10.1109\/TRO.2019.2899212","article-title":"Optimal Time-Complexity Speed Planning for Robot Manipulators","volume":"35","author":"Consolini","year":"2019","journal-title":"IEEE Trans. Robot."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"15729","DOI":"10.1109\/TITS.2022.3145389","article-title":"Autonomous Driving on Curvy Roads Without Reliance on Frenet Frame: A Cartesian-Based Trajectory Planning Method","volume":"23","author":"Li","year":"2022","journal-title":"IEEE Trans. Intell. Transp. Syst."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"315","DOI":"10.1109\/TMECH.2021.3063627","article-title":"A Predictive Technique for the Real-Time Trajectory Scaling Under High-Order Constraints","volume":"27","author":"Faroni","year":"2022","journal-title":"IEEE\/ASME Trans. Mechatronics"},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/s12532-018-0139-4","article-title":"CasADi\u2014A software framework for nonlinear optimization and optimal control","volume":"11","author":"Andersson","year":"2019","journal-title":"Math. Program. Comput."},{"key":"ref_35","doi-asserted-by":"crossref","unstructured":"Fu, Z., Spyrakos-Papastavridis, E., Lin, Y.-H., and Dai, J.S. (August, January 31). Analytical Expressions of Serial Manipulator Jacobians and their High-Order Derivatives based on Lie Theory. Proceedings of the 2020 IEEE International Conference on Robotics and Automation (ICRA), Paris, France.","DOI":"10.1109\/ICRA40945.2020.9197131"}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/25\/4\/610\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T19:09:08Z","timestamp":1760123348000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/25\/4\/610"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,4,3]]},"references-count":35,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2023,4]]}},"alternative-id":["e25040610"],"URL":"https:\/\/doi.org\/10.3390\/e25040610","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,4,3]]}}}