{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,19]],"date-time":"2026-01-19T02:30:57Z","timestamp":1768789857398,"version":"3.49.0"},"reference-count":32,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2021,11,27]],"date-time":"2021-11-27T00:00:00Z","timestamp":1637971200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Robotics"],"abstract":"<jats:p>This work is aimed to demonstrate a multi-objective joint trajectory generation algorithm for a 7 degree of freedom (DoF) robotic manipulator using swarm intelligence (SI)\u2014product of exponentials (PoE) combination. Given a priori knowledge of the end-effector Cartesian trajectory and obstacles in the workspace, the inverse kinematics problem is tackled by SI-PoE subject to multiple constraints. The algorithm is designed to satisfy finite jerk constraint on end-effector, avoid obstacles, and minimize control effort while tracking the Cartesian trajectory. The SI-PoE algorithm is compared with conventional inverse kinematics algorithms and standard particle swarm optimization (PSO). The joint trajectories produced by SI-PoE are experimentally tested on Sawyer 7 DoF robotic arm, and the resulting torque trajectories are compared.<\/jats:p>","DOI":"10.3390\/robotics10040127","type":"journal-article","created":{"date-parts":[[2021,12,1]],"date-time":"2021-12-01T03:12:56Z","timestamp":1638328376000},"page":"127","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":12,"title":["Multi-Objective Swarm Intelligence Trajectory Generation for a 7 Degree of Freedom Robotic Manipulator"],"prefix":"10.3390","volume":"10","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-0812-0587","authenticated-orcid":false,"given":"Aryslan","family":"Malik","sequence":"first","affiliation":[{"name":"Aerospace Engineering Department, Embry\u2014Riddle Aeronautical University, Daytona Beach, FL 32114, USA"}]},{"given":"Troy","family":"Henderson","sequence":"additional","affiliation":[{"name":"Aerospace Engineering Department, Embry\u2014Riddle Aeronautical University, Daytona Beach, FL 32114, USA"}]},{"given":"Richard","family":"Prazenica","sequence":"additional","affiliation":[{"name":"Aerospace Engineering Department, Embry\u2014Riddle Aeronautical University, Daytona Beach, FL 32114, USA"}]}],"member":"1968","published-online":{"date-parts":[[2021,11,27]]},"reference":[{"key":"ref_1","unstructured":"Park, J., Choi, Y., Chung, W.K., and Youm, Y. (2001, January 21\u201326). Multiple tasks kinematics using weighted pseudo-inverse for kinematically redundant manipulators. Proceedings of the 2001 ICRA. IEEE International Conference on Robotics and Automation (Cat. No. 01CH37164), Seoul, Korea."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"245","DOI":"10.1109\/TSMC.1983.6313123","article-title":"Review of pseudoinverse control for use with kinematically redundant manipulators","volume":"SMC-13","author":"Klein","year":"1983","journal-title":"IEEE Trans. Syst. Man Cybern."},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Lynch, K.M., and Park, F.C. (2017). Modern Robotics, Cambridge University Press.","DOI":"10.1017\/9781316661239"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"269","DOI":"10.1007\/BF01386390","article-title":"A note on two problems in connexion with graphs","volume":"1","author":"Dijkstra","year":"1959","journal-title":"Numer. Math."},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Bottin, M., and Rosati, G. (2019). Trajectory optimization of a redundant serial robot using cartesian via points and kinematic decoupling. Robotics, 8.","DOI":"10.3390\/robotics8040101"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"286","DOI":"10.1016\/j.rcim.2017.10.005","article-title":"Time-optimal trajectory planning for hyper-redundant manipulators in 3D workspaces","volume":"50","author":"Xidias","year":"2018","journal-title":"Robot. Comput.-Integr. Manuf."},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Gu, S., Holly, E., Lillicrap, T., and Levine, S. (June, January 29). Deep reinforcement learning for robotic manipulation with asynchronous off-policy updates. Proceedings of the 2017 IEEE International Conference on Robotics and Automation (ICRA), Singapore.","DOI":"10.1109\/ICRA.2017.7989385"},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Gu, S., Holly, E., Lillicrap, T., and Levine, S. (2016). Deep reinforcement learning for robotic manipulation. arXiv.","DOI":"10.1109\/ICRA.2017.7989385"},{"key":"ref_9","unstructured":"Zhang, F., Leitner, J., Milford, M., Upcroft, B., and Corke, P. (2015). Towards vision-based deep reinforcement learning for robotic motion control. arXiv."},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Sasaki, H., Horiuchi, T., and Kato, S. (2017, January 19\u201322). A study on vision-based mobile robot learning by deep Q-network. Proceedings of the 2017 56th Annual Conference of the Society of Instrument and Control Engineers of Japan (SICE), Kanazawa, Japan.","DOI":"10.23919\/SICE.2017.8105597"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"177","DOI":"10.1049\/trit.2020.0024","article-title":"Multi-robot path planning based on a deep reinforcement learning DQN algorithm","volume":"5","author":"Yang","year":"2020","journal-title":"CAAI Trans. Intell. Technol."},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Xin, J., Zhao, H., Liu, D., and Li, M. (2017, January 20\u201322). Application of deep reinforcement learning in mobile robot path planning. Proceedings of the 2017 Chinese Automation Congress (CAC), Jinan, China.","DOI":"10.1109\/CAC.2017.8244061"},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Ruan, X., Ren, D., Zhu, X., and Huang, J. (2019, January 3\u20135). Mobile robot navigation based on deep reinforcement learning. Proceedings of the 2019 Chinese Control and Decision Conference (CCDC), Nanchang, China.","DOI":"10.1109\/CCDC.2019.8832393"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"104985","DOI":"10.1016\/j.compag.2019.104985","article-title":"Double-DQN based path smoothing and tracking control method for robotic vehicle navigation","volume":"166","author":"Zhang","year":"2019","journal-title":"Comput. Electron. Agric."},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Guo, Z., Huang, J., Ren, W., and Wang, C. (2019, January 26\u201328). A reinforcement learning approach for inverse kinematics of arm robot. Proceedings of the 2019 4th International Conference on Robotics, Control and Automation, Guangzhou, China.","DOI":"10.1145\/3351180.3351199"},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Phaniteja, S., Dewangan, P., Guhan, P., Sarkar, A., and Krishna, K.M. (2017, January 5\u20138). A deep reinforcement learning approach for dynamically stable inverse kinematics of humanoid robots. Proceedings of the 2017 IEEE International Conference on Robotics and Biomimetics (ROBIO), Macau, China.","DOI":"10.1109\/ROBIO.2017.8324682"},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Zhong, J., Wang, T., and Cheng, L. (2021). Collision-free path planning for welding manipulator via hybrid algorithm of deep reinforcement learning and inverse kinematics. Complex Intell. Syst., 1\u201314.","DOI":"10.1007\/s40747-021-00366-1"},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Xue, X., Li, Z., Zhang, D., and Yan, Y. (2019, January 12\u201314). A deep reinforcement learning method for mobile robot collision avoidance based on double dqn. Proceedings of the 2019 IEEE 28th International Symposium on Industrial Electronics (ISIE), Vancouver, BC, Canada.","DOI":"10.1109\/ISIE.2019.8781522"},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Collinsm, T.J., and Shen, W.M. (2017, January 22\u201324). Particle swarm optimization for high-DOF inverse kinematics. Proceedings of the 2017 3rd International Conference on Control, Automation and Robotics (ICCAR), Nagoya, Japan.","DOI":"10.1109\/ICCAR.2017.7942651"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"385","DOI":"10.1016\/j.mechmachtheory.2018.09.022","article-title":"Inverse kinematics of mobile manipulator using bidirectional particle swarm optimization by manipulator decoupling","volume":"131","author":"Ram","year":"2019","journal-title":"Mech. Mach. Theory"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"159622","DOI":"10.1109\/ACCESS.2020.3020318","article-title":"DPSO and Inverse Jacobian-Based Real-Time Inverse Kinematics With Trajectory Tracking Using Integral SMC for Teleoperation","volume":"8","author":"Khan","year":"2020","journal-title":"IEEE Access"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"748","DOI":"10.1016\/j.compeleceng.2015.05.019","article-title":"Integrated particle swarm optimization algorithm based obstacle avoidance control design for home service robot","volume":"56","author":"Lin","year":"2016","journal-title":"Comput. Electr. Eng."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"32341","DOI":"10.1109\/ACCESS.2021.3059714","article-title":"A General Robot Inverse Kinematics Solution Method Based on Improved PSO Algorithm","volume":"9","author":"Yiyang","year":"2021","journal-title":"IEEE Access"},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Bilbeisi, G., Al-Madi, N., and Awad, F. (2015, January 3\u20135). PSO-AG: A Multi-Robot Path Planning and obstacle avoidance algorithm. Proceedings of the 2015 IEEE Jordan Conference on Applied Electrical Engineering and Computing Technologies (AEECT), Amman, Jordan.","DOI":"10.1109\/AEECT.2015.7360565"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"949","DOI":"10.1007\/s10462-019-09683-x","article-title":"A meta-heuristic proposal for inverse kinematics solution of 7-DOF serial robotic manipulator: Quantum behaved particle swarm algorithm","volume":"53","author":"Dereli","year":"2020","journal-title":"Artif. Intell. Rev."},{"key":"ref_26","unstructured":"Robotics, R. (2019, August 10). Sawyer. Available online: https:\/\/sdk.rethinkrobotics.com\/intera\/Main_Page."},{"key":"ref_27","doi-asserted-by":"crossref","unstructured":"Malik, A., Lischuk, Y., Henderson, T., and Prazenica, R.J. (2021). Generating Constant Screw Axis Trajectories With Quintic Time Scaling For End-Effector Using Artificial Neural Network And Machine Learning, Accepted and presented, awaiting publishing.","DOI":"10.1109\/CCTA48906.2021.9658657"},{"key":"ref_28","first-page":"3319","article-title":"Using Products of Exponentials to Define (Draw) Orbits and More","volume":"175","author":"Malik","year":"2021","journal-title":"Adv. Astronaut. Sci."},{"key":"ref_29","doi-asserted-by":"crossref","unstructured":"Malik, A., Henderson, T., and Prazenica, R.J. (2021, January 11\u201315). Trajectory Generation for a Multibody Robotic System using the Product of Exponentials Formulation. Proceedings of the AIAA Scitech 2021 Forum, Virtual Event.","DOI":"10.2514\/6.2021-2016"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"1942","DOI":"10.1109\/ICNN.1995.488968","article-title":"Particle swarm optimization","volume":"Volume 4","author":"Eberhart","year":"1995","journal-title":"Proceedings of the IEEE International Conference on Neural Networks"},{"key":"ref_31","doi-asserted-by":"crossref","unstructured":"Yang, X.S., Cui, Z., Xiao, R., Gandomi, A.H., and Karamanoglu, M. (2013). Swarm Intelligence and Bio-Inspired Computation: Theory and Applications, Newnes.","DOI":"10.1016\/B978-0-12-405163-8.00001-6"},{"key":"ref_32","unstructured":"Shi, Y., and Eberhart, R.C. (1999, January 6\u20139). Empirical study of particle swarm optimization. Proceedings of the 1999 Congress on Evolutionary Computation-CEC99 (Cat. No. 99TH8406), Washington, DC, USA."}],"container-title":["Robotics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2218-6581\/10\/4\/127\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T07:36:52Z","timestamp":1760168212000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2218-6581\/10\/4\/127"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,11,27]]},"references-count":32,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2021,12]]}},"alternative-id":["robotics10040127"],"URL":"https:\/\/doi.org\/10.3390\/robotics10040127","relation":{},"ISSN":["2218-6581"],"issn-type":[{"value":"2218-6581","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,11,27]]}}}