{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,14]],"date-time":"2026-04-14T00:38:51Z","timestamp":1776127131166,"version":"3.50.1"},"reference-count":40,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2023,3,13]],"date-time":"2023-03-13T00:00:00Z","timestamp":1678665600000},"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":["11672290"],"award-info":[{"award-number":["11672290"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Sensors"],"abstract":"<jats:p>This paper proposes a novel trajectory planning algorithm to design an end-effector motion profile along a specified path. An optimization model based on the whale optimization algorithm (WOA) is established for time-optimal asymmetrical S-curve velocity scheduling. Trajectories designed by end-effector limits may violate kinematic constraints due to the non-linear relationship between the operation and joint space of redundant manipulators. A constraints conversion approach is proposed to update end-effector limits. The path can be divided into segments at the minimum of the updated limitations. On each path segment, the jerk-limited S-shaped velocity profile is generated within the updated limitations. The proposed method aims to generate end-effector trajectory by kinematic constraints which are imposed on joints, resulting in efficient robot motion performance. The WOA-based asymmetrical S-curve velocity scheduling algorithm can be automatically adjusted for different path lengths and start\/end velocities, allowing flexibility in finding the time-optimal solution under complex constraints. Simulations and experiments on a redundant manipulator prove the effect and superiority of the proposed method.<\/jats:p>","DOI":"10.3390\/s23063074","type":"journal-article","created":{"date-parts":[[2023,3,14]],"date-time":"2023-03-14T03:04:46Z","timestamp":1678763086000},"page":"3074","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":19,"title":["Time-Optimal Asymmetric S-Curve Trajectory Planning of Redundant Manipulators under Kinematic Constraints"],"prefix":"10.3390","volume":"23","author":[{"given":"Tianyu","family":"Liu","sequence":"first","affiliation":[{"name":"CAS Key Laboratory of On-Orbit Manufacturing and Integration for Space Optics System, Changchun Institute of Optics, Fine Mechanics and Physics, Chinese Academy of Sciences, Changchun 130033, China"},{"name":"School of Optoelectronics, University of Chinese Academy of Sciences, Beijing 100049, China"}]},{"given":"Jingkai","family":"Cui","sequence":"additional","affiliation":[{"name":"CAS Key Laboratory of On-Orbit Manufacturing and Integration for Space Optics System, Changchun Institute of Optics, Fine Mechanics and Physics, Chinese Academy of Sciences, Changchun 130033, China"},{"name":"School of Optoelectronics, University of Chinese Academy of Sciences, Beijing 100049, China"}]},{"given":"Yanhui","family":"Li","sequence":"additional","affiliation":[{"name":"CAS Key Laboratory of On-Orbit Manufacturing and Integration for Space Optics System, Changchun Institute of Optics, Fine Mechanics and Physics, Chinese Academy of Sciences, Changchun 130033, China"}]},{"given":"Siyuan","family":"Gao","sequence":"additional","affiliation":[{"name":"CAS Key Laboratory of On-Orbit Manufacturing and Integration for Space Optics System, Changchun Institute of Optics, Fine Mechanics and Physics, Chinese Academy of Sciences, Changchun 130033, China"}]},{"given":"Mingchao","family":"Zhu","sequence":"additional","affiliation":[{"name":"CAS Key Laboratory of On-Orbit Manufacturing and Integration for Space Optics System, Changchun Institute of Optics, Fine Mechanics and Physics, Chinese Academy of Sciences, Changchun 130033, China"}]},{"given":"Liheng","family":"Chen","sequence":"additional","affiliation":[{"name":"CAS Key Laboratory of On-Orbit Manufacturing and Integration for Space Optics System, Changchun Institute of Optics, Fine Mechanics and Physics, Chinese Academy of Sciences, Changchun 130033, China"}]}],"member":"1968","published-online":{"date-parts":[[2023,3,13]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"102025","DOI":"10.1016\/j.rcim.2020.102025","article-title":"An adaptive real-time NURBS curve interpolation for 4-axis polishing machine tool","volume":"67","author":"Ji","year":"2021","journal-title":"Robot. Comput.-Integr. Manuf."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"554","DOI":"10.1016\/j.cad.2008.01.015","article-title":"Development of an integrated look-ahead dynamics-based NURBS interpolator for high precision machinery","volume":"40","author":"Tsai","year":"2008","journal-title":"Comput.-Aided Des."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"612","DOI":"10.1016\/j.cad.2011.02.014","article-title":"The feedrate scheduling of NURBS interpolator for CNC machine tools","volume":"43","author":"Lee","year":"2011","journal-title":"Comput.-Aided Des."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"4382","DOI":"10.1177\/0954406220969734","article-title":"Optimum time-energy-jerk trajectory planning for serial robotic manipulators by reparameterized quintic NURBS curves","volume":"235","author":"Wu","year":"2021","journal-title":"Proc. Inst. Mech. Eng. Part C-J. Mech. Eng. Sci."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"102058","DOI":"10.1016\/j.rcim.2020.102058","article-title":"Local asymmetrical corner trajectory smoothing with bidirectional planning and adjusting algorithm for CNC machining","volume":"68","author":"Wang","year":"2021","journal-title":"Robot. Comput.-Integr. Manuf."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"101907","DOI":"10.1016\/j.rcim.2019.101907","article-title":"A tolerance constrained G(2) continuous path smoothing and interpolation method for industrial SCARA robots","volume":"63","author":"He","year":"2020","journal-title":"Robot. Comput.-Integr. Manuf."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"12","DOI":"10.1016\/j.cja.2021.06.011","article-title":"Path, feedrate and trajectory planning for free-form surface machining: A state-of-the-art review","volume":"35","author":"Sun","year":"2022","journal-title":"Chin. J. Aeronaut."},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Montalvo, V., Estevez-Ben, A.A., Rodriguez-Resendiz, J., Macias-Bobadilla, G., Mendiola-Santibanez, J.D., and Camarillo-Gomez, K.A. (2020). FPGA-Based Architecture for Sensing Power Consumption on Parabolic and Trapezoidal Motion Profiles. Electronics, 9.","DOI":"10.3390\/electronics9081301"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"43","DOI":"10.1016\/j.conengprac.2019.03.017","article-title":"Trajectory generation via FIR filters: A procedure for time-optimization under kinematic and frequency constraints","volume":"87","author":"Biagiotti","year":"2019","journal-title":"Control Eng. Pract."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"169","DOI":"10.1016\/j.conengprac.2017.06.015","article-title":"FIR filter-based online jerk-constrained trajectory generation","volume":"66","author":"Besset","year":"2017","journal-title":"Control Eng. Pract."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"1481","DOI":"10.1109\/TIE.2017.2726959","article-title":"Analysis of Embedded Prefilters in Motion Profiles","volume":"65","author":"Ha","year":"2018","journal-title":"IEEE Trans. Ind. Electron."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"42","DOI":"10.1109\/TRA.2002.807548","article-title":"Jerk-bounded manipulator trajectory planning: Design for real-time applications","volume":"19","author":"Macfarlane","year":"2003","journal-title":"IEEE Trans. Robot. Autom."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"264","DOI":"10.1109\/TMECH.2008.2012160","article-title":"Motion Profile Design to Reduce Residual Vibration of High-Speed Positioning Stages","volume":"14","author":"Li","year":"2009","journal-title":"IEEE-ASME Trans. Mechatron."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"102308","DOI":"10.1016\/j.rcim.2021.102308","article-title":"Global time-optimal B-spline feedrate scheduling for a two-turret multi-axis NC machine tool based on optimization with genetic algorithm","volume":"75","author":"Liang","year":"2022","journal-title":"Robot. Comput.-Integr. Manuf."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"309","DOI":"10.1016\/j.rcim.2012.08.002","article-title":"Time-optimal and jerk-continuous trajectory planning for robot manipulators with kinematic constraints","volume":"29","author":"Liu","year":"2013","journal-title":"Robot. Comput.-Integr. Manuf."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"6098","DOI":"10.1109\/TIE.2018.2874617","article-title":"Spline-Based Trajectory Generation for CNC Machines","volume":"66","author":"Mercy","year":"2019","journal-title":"IEEE Trans. Ind. Electron."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"530","DOI":"10.1016\/j.mechmachtheory.2017.11.006","article-title":"Optimal time-jerk trajectory planning for industrial robots","volume":"121","author":"Huang","year":"2018","journal-title":"Mech. Mach. Theory"},{"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":"1889","DOI":"10.1007\/s00170-014-6575-z","article-title":"A novel acc-jerk-limited NURBS interpolation enhanced with an optimized S-shaped quintic feedrate scheduling scheme","volume":"77","author":"Jahanpour","year":"2015","journal-title":"Int. J. Adv. Manuf. Technol."},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Lin, J., Somani, N., Hu, B., Rickert, M., and Knoll, A. (2018, January 1\u20135). An Efficient and Time-Optimal Trajectory Generation Approach for Waypoints Under Kinematic Constraints and Error Bounds. Proceedings of the 2018 IEEE\/RSJ International Conference on Intelligent Robots and Systems (IROS), Madrid, Spain.","DOI":"10.1109\/IROS.2018.8593577"},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Wang, F., Wu, Z., and Bao, T. (2022). Time-Jerk optimal Trajectory Planning of Industrial Robots Based on a Hybrid WOA-GA Algorithm. Processes, 10.","DOI":"10.3390\/pr10051014"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"127","DOI":"10.1016\/j.mechmachtheory.2019.03.019","article-title":"Smooth and time-optimal S-curve trajectory planning for automated robots and machines","volume":"137","author":"Fang","year":"2019","journal-title":"Mech. Mach. Theory"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"9582","DOI":"10.1109\/TIE.2018.2818669","article-title":"Near Time-Optimal S-Curve Velocity Planning for Multiple Line Segments Under Axis Constraints","volume":"65","author":"Lu","year":"2018","journal-title":"IEEE Trans. Ind. Electron."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"1385","DOI":"10.1016\/j.conengprac.2012.08.005","article-title":"FIR filters for online trajectory planning with time- and frequency-domain specifications","volume":"20","author":"Biagiotti","year":"2012","journal-title":"Control Eng. Pract."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"2724","DOI":"10.1109\/TMECH.2020.3045504","article-title":"Optimization of Generalized S-Curve Trajectories for Residual Vibration Suppression and Compliance with Kinematic Bounds","volume":"26","author":"Biagiotti","year":"2021","journal-title":"IEEE-ASME Trans. Mechatron."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"4707","DOI":"10.1109\/TMECH.2022.3164247","article-title":"Time-Optimal Pick-and-Throw S-Curve Trajectories for Fast Parallel Robots","volume":"27","author":"Hassan","year":"2022","journal-title":"IEEE\/ASME Trans. Mechatron."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"1892","DOI":"10.1109\/TCST.2020.2998094","article-title":"Optimization Process for Polynomial Motion Profiles to Achieve Fast Movement with Low Vibration","volume":"28","author":"Lee","year":"2020","journal-title":"IEEE Trans. Control Syst. Technol."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"103957","DOI":"10.1016\/j.mechmachtheory.2020.103957","article-title":"An approach for jerk-continuous trajectory generation of robotic manipulators with kinematical constraints","volume":"153","author":"Fang","year":"2020","journal-title":"Mech. Mach. Theory"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"149","DOI":"10.1504\/IJCAT.2016.074453","article-title":"A jerk-constrained asymmetric motion profile for high-speed motion stages to reduce residual vibration","volume":"53","author":"Li","year":"2016","journal-title":"Int. J. Comput. Appl. Technol."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"350","DOI":"10.1108\/AA-11-2021-0165","article-title":"An optimized cosine jerk motion profile with higher efficiency and flexibility","volume":"42","author":"Zhu","year":"2022","journal-title":"Assem. Autom."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"2499","DOI":"10.1109\/TIE.2009.2036032","article-title":"A Closed-Form Solution to Asymmetric Motion Profile Allowing Acceleration Manipulation","volume":"57","author":"Rew","year":"2010","journal-title":"IEEE Trans. Ind. Electron."},{"key":"ref_32","first-page":"527","article-title":"Nonprehensile Manipulation:A Trajectory-Planning Perspective","volume":"26","author":"Acharya","year":"2021","journal-title":"IEEE-ASME Trans. Mechatron."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"6789","DOI":"10.1007\/s00170-021-08033-y","article-title":"Time-optimal feedrate scheduling with actuator constraints for 5-axis machining","volume":"119","author":"Kornmaneesang","year":"2022","journal-title":"Int. J. Adv. Manuf. Technol."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"1993","DOI":"10.1109\/TMECH.2018.2835830","article-title":"Time-Optimal Freeform S-Curve Profile Under Positioning Error and Robustness Constraints","volume":"23","author":"Bai","year":"2018","journal-title":"IEEE\/ASME Trans. Mechatron."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"2287","DOI":"10.1007\/s00170-018-2382-2","article-title":"A new path-constrained trajectory planning strategy for spray painting robots\u2014Rev.1","volume":"98","author":"Trigatti","year":"2018","journal-title":"Int. J. Adv. Manuf. Technol."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"51","DOI":"10.1016\/j.advengsoft.2016.01.008","article-title":"The Whale Optimization Algorithm","volume":"95","author":"Mirjalili","year":"2016","journal-title":"Adv. Eng. Softw."},{"key":"ref_37","first-page":"206","article-title":"A complete S-shape feed rate scheduling approach for NURBS interpolator","volume":"2","author":"Du","year":"2015","journal-title":"J. Comput. Des. Eng."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.swevo.2019.03.004","article-title":"A comprehensive survey: Whale Optimization Algorithm and its applications","volume":"48","author":"Gharehchopogh","year":"2019","journal-title":"Swarm Evol. Comput."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"391","DOI":"10.1007\/s10107-004-0560-5","article-title":"An interior algorithm for nonlinear optimization that combines line search and trust region steps","volume":"107","author":"Waltz","year":"2006","journal-title":"Math. Program."},{"key":"ref_40","doi-asserted-by":"crossref","unstructured":"Liu, T., Zhu, M., and Chen, L. (2021, January 15\u201319). Trajectory Planning for Hyper-Redundant Manipulators Based on Lie Theory. Proceedings of the 2021 IEEE International Conference on Real-time Computing and Robotics (RCAR), Xining, China.","DOI":"10.1109\/RCAR52367.2021.9517595"}],"container-title":["Sensors"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1424-8220\/23\/6\/3074\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T18:53:50Z","timestamp":1760122430000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1424-8220\/23\/6\/3074"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,3,13]]},"references-count":40,"journal-issue":{"issue":"6","published-online":{"date-parts":[[2023,3]]}},"alternative-id":["s23063074"],"URL":"https:\/\/doi.org\/10.3390\/s23063074","relation":{},"ISSN":["1424-8220"],"issn-type":[{"value":"1424-8220","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,3,13]]}}}