{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,27]],"date-time":"2026-05-27T17:40:02Z","timestamp":1779903602376,"version":"3.53.1"},"reference-count":41,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2021,9,13]],"date-time":"2021-09-13T00:00:00Z","timestamp":1631491200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100003459","name":"Guizhou University","doi-asserted-by":"publisher","award":["National Important Project under grant No. 2020YFB1713300, No. 2018AAA0101803"],"award-info":[{"award-number":["National Important Project under grant No. 2020YFB1713300, No. 2018AAA0101803"]}],"id":[{"id":"10.13039\/501100003459","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100003459","name":"Guizhou University","doi-asserted-by":"publisher","award":["Guizhou Province Higher Education Project under grant No. [2020]005, No. [2020]009."],"award-info":[{"award-number":["Guizhou Province Higher Education Project under grant No. [2020]005, No. [2020]009."]}],"id":[{"id":"10.13039\/501100003459","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Robot manipulator trajectory planning is one of the core robot technologies, and the design of controllers can improve the trajectory accuracy of manipulators. However, most of the controllers designed at this stage have not been able to effectively solve the nonlinearity and uncertainty problems of the high degree of freedom manipulators. In order to overcome these problems and improve the trajectory performance of the high degree of freedom manipulators, a manipulator trajectory planning method based on a radial basis function (RBF) neural network is proposed in this work. Firstly, a 6-DOF robot experimental platform was designed and built. Secondly, the overall manipulator trajectory planning framework was designed, which included manipulator kinematics and dynamics and a quintic polynomial interpolation algorithm. Then, an adaptive robust controller based on an RBF neural network was designed to deal with the nonlinearity and uncertainty problems, and Lyapunov theory was used to ensure the stability of the manipulator control system and the convergence of the tracking error. Finally, to test the method, a simulation and experiment were carried out. The simulation results showed that the proposed method improved the response and tracking performance to a certain extent, reduced the adjustment time and chattering, and ensured the smooth operation of the manipulator in the course of trajectory planning. The experimental results verified the effectiveness and feasibility of the method proposed in this paper.<\/jats:p>","DOI":"10.3390\/e23091207","type":"journal-article","created":{"date-parts":[[2021,9,13]],"date-time":"2021-09-13T21:37:12Z","timestamp":1631569032000},"page":"1207","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":35,"title":["Trajectory Planning of Robot Manipulator Based on RBF Neural Network"],"prefix":"10.3390","volume":"23","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-8328-9959","authenticated-orcid":false,"given":"Qisong","family":"Song","sequence":"first","affiliation":[{"name":"College of Mechanical Engineering, Guizhou University, Guiyang 550025, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4759-6000","authenticated-orcid":false,"given":"Shaobo","family":"Li","sequence":"additional","affiliation":[{"name":"College of Mechanical Engineering, Guizhou University, Guiyang 550025, China"},{"name":"State Key Laboratory of Public Big Data, Guizhou University, Guiyang 550025, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Qiang","family":"Bai","sequence":"additional","affiliation":[{"name":"College of Mechanical Engineering, Guizhou University, Guiyang 550025, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1915-9487","authenticated-orcid":false,"given":"Jing","family":"Yang","sequence":"additional","affiliation":[{"name":"College of Mechanical Engineering, Guizhou University, Guiyang 550025, China"},{"name":"State Key Laboratory of Public Big Data, Guizhou University, Guiyang 550025, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Ansi","family":"Zhang","sequence":"additional","affiliation":[{"name":"College of Mechanical Engineering, Guizhou University, Guiyang 550025, China"},{"name":"State Key Laboratory of Public Big Data, Guizhou University, Guiyang 550025, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6368-5762","authenticated-orcid":false,"given":"Xingxing","family":"Zhang","sequence":"additional","affiliation":[{"name":"State Key Laboratory of Public Big Data, Guizhou University, Guiyang 550025, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Longxuan","family":"Zhe","sequence":"additional","affiliation":[{"name":"College of Mechanical Engineering, Guizhou University, Guiyang 550025, China"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2021,9,13]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Lefebvre, T., and Crevecoeur, G. (2020). On entropy regularized path integral control for trajectory optimization. Entropy, 22.","DOI":"10.3390\/e22101120"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"31","DOI":"10.1016\/j.paerosci.2018.07.001","article-title":"Obstacle avoidance in space robotics: Review of major challenges and proposed solutions","volume":"101","author":"Rybus","year":"2018","journal-title":"Prog. Aerosp. Sci."},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Omisore, O.M., Han, S., Al-Handarish, Y., Du, W., Duan, W., Akinyemi, T.O., and Wang, L. (2020). Motion and trajectory constraints control modeling for flexible surgical robotic systems. Micromachines, 11.","DOI":"10.3390\/mi11040386"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"181855","DOI":"10.1109\/ACCESS.2020.3028740","article-title":"Object detection recognition and robot grasping based on machine learning: A survey","volume":"8","author":"Bai","year":"2020","journal-title":"IEEE Access"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"627","DOI":"10.1016\/j.procs.2018.07.094","article-title":"Trajectory Planning of Redundant Manipulators Moving along Constrained Path and Avoiding Obstacles","volume":"133","author":"Chembuly","year":"2018","journal-title":"Procedia Comput. Sci."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"4077","DOI":"10.1007\/s00521-020-05515-0","article-title":"Adaptive control of manipulator based on neural network","volume":"33","author":"Liu","year":"2021","journal-title":"Neural Comput. Appl."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/s11432-020-3187-8","article-title":"Robust adaptive H \u221e control for networked uncertain semi-Markov jump nonlinear systems with input quantization","volume":"65","author":"Dong","year":"2022","journal-title":"Sci. China Inf. Sci."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"2","DOI":"10.1145\/3423184","article-title":"Fast Linear Interpolation","volume":"17","author":"Zhang","year":"2021","journal-title":"ACM J. Emerg. Technol. Comput. Syst."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"248","DOI":"10.1016\/j.mechmachtheory.2018.11.009","article-title":"Improved trajectory planning of an industrial parallel mechanism by a composite polynomial consisting of B\u00e9zier curves and cubic polynomials","volume":"132","year":"2019","journal-title":"Mech. Mach. Theory"},{"key":"ref_10","first-page":"1","article-title":"Multi-objective trajectory planning of humanoid robot using hybrid controller for multi-target problem in complex terrain","volume":"179","author":"Parhi","year":"2021","journal-title":"Expert Syst. Appl."},{"key":"ref_11","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_12","doi-asserted-by":"crossref","unstructured":"Kim, H., and Kim, B.K. (2020). Energy-optimal transport trajectory planning and online trajectory modification for holonomic robots. Asian J. Control, 1\u201316.","DOI":"10.1002\/asjc.2449"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"5005","DOI":"10.1109\/TNNLS.2019.2955400","article-title":"Six-DOF Spacecraft Optimal Trajectory Planning and Real-Time Attitude Control: A Deep Neural Network-Based Approach","volume":"31","author":"Chai","year":"2020","journal-title":"IEEE Trans. Neural Netw. Learn. Syst."},{"key":"ref_14","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_15","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_16","doi-asserted-by":"crossref","first-page":"158","DOI":"10.1016\/j.rcim.2019.02.009","article-title":"Online near time-optimal trajectory planning for industrial robots","volume":"58","author":"Kim","year":"2019","journal-title":"Robot. Comput. Integr. Manuf."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"3171","DOI":"10.1007\/s12206-021-0638-5","article-title":"Time-optimal trajectory planning of serial manipulator based on adaptive cuckoo search algorithm","volume":"35","author":"Zhang","year":"2021","journal-title":"J. Mech. Sci. Technol."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"911","DOI":"10.1007\/s12541-015-0119-9","article-title":"Trajectory planning for energy minimization of industry robotic manipulators using the Lagrange interpolation method","volume":"16","author":"Luo","year":"2015","journal-title":"Int. J. Precis. Eng. Manuf."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"183","DOI":"10.1007\/s12541-018-0021-3","article-title":"Trajectory Planning with Minimum Synthesis Error for Industrial Robots Using Screw Theory","volume":"19","author":"Liu","year":"2018","journal-title":"Int. J. Precis. Eng. Manuf."},{"key":"ref_20","first-page":"1","article-title":"Energy-Conscientious Trajectory Planning for an Autonomous Mobile Robot in an Asymmetric Task Space","volume":"101","author":"Bakshi","year":"2021","journal-title":"J. Intell. Robot. Syst. Theory Appl."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"379","DOI":"10.1007\/s10846-013-9982-8","article-title":"A fast and unified method to find a minimum-jerk robot joint trajectory using particle swarm optimization","volume":"75","author":"Lin","year":"2014","journal-title":"J. Intell. Robot. Syst. Theory Appl."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"1","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":"Rob. Auton. Syst."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"1711","DOI":"10.1109\/TASE.2020.2974771","article-title":"Planning Jerk-Optimized Trajectory with Discrete Time Constraints for Redundant Robots","volume":"17","author":"Dai","year":"2020","journal-title":"IEEE Trans. Autom. Sci. Eng."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1155\/2016\/9329131","article-title":"Optimal Trajectory Planning for Glass-Handing Robot Based on Execution Time Acceleration and Jerk","volume":"2016","author":"Duan","year":"2016","journal-title":"J. Robot."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"431","DOI":"10.1016\/j.rcim.2019.04.016","article-title":"A multi-objective trajectory planning method based on the improved immune clonal selection algorithm","volume":"59","author":"Chen","year":"2019","journal-title":"Robot. Comput. Integr. Manuf."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"429","DOI":"10.1016\/j.procir.2019.03.074","article-title":"A machine learning based energy efficient trajectory planning approach for industrial robots","volume":"81","author":"Yin","year":"2019","journal-title":"Procedia CIRP"},{"key":"ref_27","doi-asserted-by":"crossref","unstructured":"Zhang, X., Huang, Y., Rong, Y., Li, G., Wang, H., and Liu, C. (2021). Optimal trajectory planning for wheeled mobile robots under localization uncertainty and energy efficiency constraints. Sensors, 21.","DOI":"10.3390\/s21020335"},{"key":"ref_28","first-page":"1","article-title":"Research on Manipulator Tracking Control Algorithm Based on RBF Neural Network","volume":"1802","author":"Chang","year":"2021","journal-title":"IOP Conf. Ser. Earth Environ. Sci."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"213","DOI":"10.1016\/j.neucom.2021.03.033","article-title":"Adaptive bias RBF neural network control for a robotic manipulator","volume":"447","author":"Liu","year":"2021","journal-title":"Neurocomputing"},{"key":"ref_30","doi-asserted-by":"crossref","unstructured":"Lin, C.J., Sie, T.Y., Chu, W.L., Yau, H.T., and Ding, C.H. (2021). Tracking control of pneumatic artificial muscle-activated robot arm based on sliding-mode control. Actuators, 10.","DOI":"10.3390\/act10030066"},{"key":"ref_31","doi-asserted-by":"crossref","unstructured":"Yeh, Y.-L. (2021). A Robust Noise-Free Linear Control Design for Robot Manipulator with Uncertain System Parameters. Actuators, 10.","DOI":"10.3390\/act10060121"},{"key":"ref_32","doi-asserted-by":"crossref","unstructured":"Ayeb, A., and Chatti, A. (2021). Sliding Mode Control of Nonholonomic Uncertain Perturbed Wheeled Mobile Robot. Int. J. Robot. Autom., 36.","DOI":"10.2316\/J.2021.206-0338"},{"key":"ref_33","doi-asserted-by":"crossref","unstructured":"Al-Darraji, I., Piromalis, D., Kakei, A.A., Khan, F.Q., Stojmenovic, M., Tsaramirsis, G., and Papageorgas, P.G. (2021). Adaptive robust controller design-based rbf neural network for aerial robot arm model. Electronics, 10.","DOI":"10.3390\/electronics10070831"},{"key":"ref_34","first-page":"126344","article-title":"Adaptive control design for uncertain switched nonstrict-feedback nonlinear systems to achieve asymptotic tracking performance","volume":"408","author":"Xu","year":"2021","journal-title":"Appl. Math. Comput."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.heliyon.2018.e01053","article-title":"Development of an 8DOF quadruped robot and implementation of Inverse Kinematics using Denavit-Hartenberg convention","volume":"4","author":"Atique","year":"2018","journal-title":"Heliyon"},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"5799","DOI":"10.1007\/s10586-017-1538-4","article-title":"Trajectory tracking control of robot manipulator based on RBF neural network and fuzzy sliding mode","volume":"22","author":"Wang","year":"2019","journal-title":"Clust. Comput."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"112471","DOI":"10.1016\/j.cam.2019.112471","article-title":"Matrix recursive polynomial interpolation algorithm: An algorithm for computing the interpolation polynomials","volume":"373","author":"Messaoudi","year":"2020","journal-title":"J. Comput. Appl. Math."},{"key":"ref_38","doi-asserted-by":"crossref","unstructured":"Sheng, G., Gao, G., and Zhang, B. (2019). Application of improved wavelet thresholding method and an RBF network in the error compensating of an MEMS gyroscope. Micromachines, 10.","DOI":"10.3390\/mi10090608"},{"key":"ref_39","first-page":"1492","article-title":"A new computed torque control system with an uncertain rbf neural network controller for a 7-dof robot","volume":"27","author":"Wang","year":"2020","journal-title":"Teh. Vjesn."},{"key":"ref_40","doi-asserted-by":"crossref","unstructured":"Gao, L., Xiong, L., Lin, X., Xia, X., Liu, W., Lu, Y., and Yu, Z. (2019). Multi-sensor fusion road friction coefficient estimation during steering with lyapunov method. Sensors, 19.","DOI":"10.3390\/s19183816"},{"key":"ref_41","first-page":"1","article-title":"Model-free adaptive sliding mode control with adjustable funnel boundary for robot manipulators with uncertainties","volume":"92","author":"Wang","year":"2021","journal-title":"Rev. Sci. Instrum."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/23\/9\/1207\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T07:01:58Z","timestamp":1760166118000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/23\/9\/1207"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,9,13]]},"references-count":41,"journal-issue":{"issue":"9","published-online":{"date-parts":[[2021,9]]}},"alternative-id":["e23091207"],"URL":"https:\/\/doi.org\/10.3390\/e23091207","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,9,13]]}}}