{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T02:09:32Z","timestamp":1760148572921,"version":"build-2065373602"},"reference-count":30,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2023,5,23]],"date-time":"2023-05-23T00:00:00Z","timestamp":1684800000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100003141","name":"CONACYT Research Projects","doi-asserted-by":"publisher","award":["A1-S-31628","1030"],"award-info":[{"award-number":["A1-S-31628","1030"]}],"id":[{"id":"10.13039\/501100003141","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>In this work, we study the problem of designing control laws that achieve time-varying formation and flocking behaviors in robot networks where each agent or robot presents double integrator dynamics. To design the control laws, we adopt a hierarchical control approach. First, we introduce a virtual velocity, which is used as a virtual control input for the position subsystem (outer loop). The objective of the virtual velocity is to achieve collective behaviors. Then, we design a velocity tracking control law for the velocity subsystem (inner loop). An advantage of the proposed approach is that the robots do not require the velocity of their neighbors. Additionally, we address the case in which the second state of the system is not available for feedback. We include a set of simulation results to show the performance of the proposed control laws.<\/jats:p>","DOI":"10.3390\/e25060834","type":"journal-article","created":{"date-parts":[[2023,5,23]],"date-time":"2023-05-23T07:00:02Z","timestamp":1684825202000},"page":"834","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Formation and Flocking Control Algorithms for Robot Networks with Double Integrator Dynamics and Time-Varying Formations"],"prefix":"10.3390","volume":"25","author":[{"given":"Carlos","family":"Monta\u00f1ez-Molina","sequence":"first","affiliation":[{"name":"Departamento de Electr\u00f3nica y Telecomunicaciones, Divisi\u00f3n de F\u00edsica Aplicada, Centro de Investigaci\u00f3n Cient\u00edfica y de Educaci\u00f3n Superior de Ensenada, Ensenada 22860, Mexico"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Javier","family":"Pliego-Jim\u00e9nez","sequence":"additional","affiliation":[{"name":"Departamento de Electr\u00f3nica y Telecomunicaciones, Divisi\u00f3n de F\u00edsica Aplicada, Centro de Investigaci\u00f3n Cient\u00edfica y de Educaci\u00f3n Superior de Ensenada, Ensenada 22860, Mexico"},{"name":"Programa Investigadores por M\u00e9xico, Consejo Nacional de Ciencia y Tecnolog\u00eda, Mexico City 03940, Mexico"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3332-0839","authenticated-orcid":false,"given":"Rigoberto","family":"Mart\u00ednez-Clark","sequence":"additional","affiliation":[{"name":"Facultad de Ingenier\u00eda, Administraci\u00f3n y Ciencias Sociales, Universidad Aut\u00f3noma de Baja California, Mexicali 21100, Mexico"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2023,5,23]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"424","DOI":"10.1016\/j.automatica.2014.10.022","article-title":"A survey of multi-agent formation control","volume":"53","author":"Oha","year":"2015","journal-title":"Automatica"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"060202","DOI":"10.1088\/1674-1056\/27\/6\/060202","article-title":"Distance-based formation tracking control of multi-agent systems with double-integrator dynamics","volume":"27","author":"Wu","year":"2018","journal-title":"Chin. Phys. B"},{"key":"ref_3","first-page":"3547","article-title":"Leader-follower affine formation control of second-order nonlinear uncertain multi-agent systems","volume":"68","author":"Zhi","year":"2021","journal-title":"IEEE Trans. Circuits Syst. II Express Briefs"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"1729","DOI":"10.1007\/s12555-019-0759-0","article-title":"Leader-following formation control of second-order nonlinear systems with time-varying communication delay","volume":"19","author":"Wang","year":"2021","journal-title":"Int. J. Control. Autom. Syst."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"261","DOI":"10.1109\/TCST.2021.3062824","article-title":"Formation control with time-varying formations, bounded controls, and local collision avoidance","volume":"30","author":"Lippay","year":"2021","journal-title":"IEEE Trans. Control. Syst. Technol."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"429","DOI":"10.1109\/TCNS.2015.2507547","article-title":"Translational and scaling formation maneuver control via a bearing-based approach","volume":"4","author":"Zhao","year":"2017","journal-title":"IEEE Trans. Control. Netw. Syst."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"140","DOI":"10.1109\/TCNS.2019.2913619","article-title":"Distributed adaptive time-varying group formation tracking for multiagent systems with multiple leaders on directed graphs","volume":"7","author":"Hu","year":"2019","journal-title":"IEEE Trans. Control. Netw. Syst."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.ins.2016.05.043","article-title":"Time-varying formation control for second-order swarm systems with switching directed topologies","volume":"369","author":"Dong","year":"2016","journal-title":"Inf. Sci."},{"key":"ref_9","first-page":"774","article-title":"Distributed fault-tolerant time-varying formation control for second-order multi-agent systems with actuator failures and directed topologies","volume":"65","author":"Hua","year":"2018","journal-title":"IEEE Trans. Circuits Syst. II Express Briefs"},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Xu, Y., Zhao, S., Luo, D., and You, Y. (2018, January 17\u201319). Affine formation maneuver control of linear multi-agent systems with undirected interaction graphs with undirected interaction graphs. Proceedings of the IEEE Conference on Decision and Control (CDC), Miami, FL, USA.","DOI":"10.1109\/CDC.2018.8619540"},{"key":"ref_11","unstructured":"Tanner, H.G. (May, January 26). Flocking with obstacle avoidance in switching networks of interconnected vehicles. Proceedings of the IEEE International Conference on Robotics and Automation, New Orleans, LA, USA."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"401","DOI":"10.1109\/TAC.2005.864190","article-title":"Flocking for multi-agent dynamic systems: Algorithms and theory","volume":"51","year":"2006","journal-title":"IEEE Trans. Autom. Control"},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Liu, B., and Yu, H. (2009, January 6\u20138). Flocking in multi-agent systems with a bounded control input. Proceedings of the International Workshop on Chaos-Fractals Theories and Applications, Shenyang, China.","DOI":"10.1109\/IWCFTA.2009.35"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"293","DOI":"10.1109\/TAC.2008.2010897","article-title":"Flocking of multi-agents with a virtual leader","volume":"54","author":"Su","year":"2009","journal-title":"IEEE Trans. Autom. Control"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"1234","DOI":"10.1109\/TSMC.2020.3019469","article-title":"A ully distributed protocol for flocking of time-varying linear systems with dynamic leader and external disturbance","volume":"52","author":"Yazdani","year":"2022","journal-title":"IEEE Trans. Syst. Man Cybern. Syst."},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Wu, S., Pu, Z., Yi, J., Sun, J., Xiong, T., and Qiu, T. (2020, January 20\u201321). Adaptive flocking of multi-agent systems with uncertain nonlinear dynamics and unknown disturbances using neural networks. Proceedings of the 16th IEEE International Conference on Automation Science and Engineering (CASE), Hong Kong, China.","DOI":"10.1109\/CASE48305.2020.9216754"},{"key":"ref_17","unstructured":"Qing, Z., Xiaohao, X., Zhengquan, Y., and Zengqiang, C. (2013, January 26\u201328). Distance constrained flocking of multi-agents with time delay. Proceedings of the 32nd Chinese Control Conference, Xi\u2019an, China."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"061005","DOI":"10.1115\/1.4024903","article-title":"Flocking of multi-agent systems using a unified optimal control approach","volume":"135","author":"Wang","year":"2013","journal-title":"J. Dyn. Syst. Meas. Control"},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Dong, Y., Chen, J., and Huang, J. (2014, January 8\u201310). Flocking with connectivity preservation of multiple double-integrator systems subject to external disturbances. Proceedings of the IEEE Conference on Control Applications (CCA), Antibes, France.","DOI":"10.1109\/CCA.2014.6981513"},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Wellman, B.J., and Hoagg, J.B. (2017, January 24\u201326). A discrete-time flocking algorithm for agents with sampled-data double-integrator dynamics. Proceedings of the American Control Conference (ACC), Seattle, WA, USA.","DOI":"10.23919\/ACC.2017.7963137"},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Luo, X., Li, X., Li, X., and Guan, X. (2014, January 28\u201330). Distributed velocity observer based formation control for multi-agent systems. Proceedings of the 33rd Chinese Control Conference, Nanjing, China.","DOI":"10.1109\/ChiCC.2014.6896892"},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Keer, Z. (2017, January 19\u201321). Observer-based formation control of heterogeneous multi-agent systems without velocity measurements. Proceedings of the 32nd Youth Academic Annual Conference of Chinese Association of Automation (YAC), Hefei, China.","DOI":"10.1109\/YAC.2017.7967494"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"132205","DOI":"10.1007\/s11432-019-2815-7","article-title":"Observer-based self-triggered control for time-varying formation of multi-agent systems","volume":"24","author":"Chai","year":"2021","journal-title":"Sci. China Inf. Sci."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"1465","DOI":"10.1109\/TAC.2004.834433","article-title":"Information flow and cooperative control of vehicle formations","volume":"49","author":"Fax","year":"2004","journal-title":"IEEE Trans. Autom. Control"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"215","DOI":"10.1109\/JPROC.2006.887293","article-title":"Consensus and cooperation in networked multi-agent systems","volume":"95","author":"Fax","year":"2007","journal-title":"Proc. IEEE"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"935","DOI":"10.1109\/TAC.2010.2103415","article-title":"Synchronization of Networks of Nonidentical Euler-Lagrange Systems With Uncertain Parameters and Communication Delays","volume":"56","author":"Ortega","year":"2011","journal-title":"IEEE Trans. Autom. Control"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"71","DOI":"10.1109\/MCS.2007.338264","article-title":"Information consensus in multivehicle cooperative control","volume":"27","author":"Ren","year":"2007","journal-title":"IEEE Control Syst. Mag."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"1367","DOI":"10.1109\/TAC.2010.2041974","article-title":"Synchronization of Interconnected Systems With Applications to Biochemical Networks: An Input-Output Approach","volume":"55","author":"Scardovi","year":"2010","journal-title":"IEEE Trans. Autom. Control"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"1177","DOI":"10.1016\/j.automatica.2006.02.013","article-title":"Tracking control for multi-agent consensus with an active leader and variable topology","volume":"42","author":"Hong","year":"2006","journal-title":"Automatica"},{"key":"ref_30","unstructured":"Hale, J.K. (1969). Ordinary Diferential Equations, Wiley."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/25\/6\/834\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T19:40:14Z","timestamp":1760125214000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/25\/6\/834"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,5,23]]},"references-count":30,"journal-issue":{"issue":"6","published-online":{"date-parts":[[2023,6]]}},"alternative-id":["e25060834"],"URL":"https:\/\/doi.org\/10.3390\/e25060834","relation":{},"ISSN":["1099-4300"],"issn-type":[{"type":"electronic","value":"1099-4300"}],"subject":[],"published":{"date-parts":[[2023,5,23]]}}}