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Appl. Dyn. Syst."],"published-print":{"date-parts":[[2026,3,31]]},"abstract":"<jats:p>Abstract.<\/jats:p>\n                  <jats:p>In this paper, a new theoretical approach of gait generation is presented for the legged robots by using delay-coupling oscillators. First, a general theory of the central pattern generator (CPG) is constructed through the time delay coupled oscillators with a unidirectional ring network structure. Based on the symmetric Hopf bifurcation, the parameter condition for rhythm generation is obtained. The parameter group is divided into different regions by Hopf bifurcation curves, where the periodic solutions exhibit distinct spatiotemporal patterns in each region. Furthermore, to investigate the impact of these spatiotemporal patterns on the gait generation of multilegged robots, the proposed theoretical approach is applied to simulate various gaits for hexapod and biped robots using six FitzHugh\u2013Nagumo (FHN) oscillators and two Stuart\u2013Landau (SL) oscillators, respectively. The findings reveal that the rhythm signals generated by the CPG controller can prompt the robot to exhibit multiple stable gaits. The CPG controllers, composed of four and eight Van der Pol (VDP) oscillators, are designed to control the hip and knee joint movements of quadruped robots, which achieves the coordinated movement of a robot\u2019s leg joints. To this end, a large number of numerical simulations are provided to validate the correctness of the proposed theoretical approach.<\/jats:p>","DOI":"10.1137\/25m1739820","type":"journal-article","created":{"date-parts":[[2026,2,24]],"date-time":"2026-02-24T08:03:01Z","timestamp":1771920181000},"page":"518-553","source":"Crossref","is-referenced-by-count":1,"title":["A Theoretical Approach for Gait Generation of the CPG Controller by Using the Time-Delay Coupled Oscillators"],"prefix":"10.1137","volume":"25","author":[{"given":"Xiao","family":"Yu","sequence":"first","affiliation":[{"name":"Shanghai Research Institute for Intelligent Autonomous Systems, Tongji University, Shanghai 201210, People\u2019s Republic of China."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Zigen","family":"Song","sequence":"additional","affiliation":[{"name":"Corresponding author. School of Aerospace Engineering and Applied Mechanics, Tongji University, Shanghai, 200092 People\u2019s Republic of China; and Shanghai Institute of Aircraft Mechanics and Control, 100 Zhangwu Road, Yangpu District, Shanghai 200092, People\u2019s Republic of China."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Xiuting","family":"Sun","sequence":"additional","affiliation":[{"name":"School of Aerospace Engineering and Applied Mechanics, Tongji University, Shanghai, 200092 People\u2019s Republic of China; and Shanghai Institute of Aircraft Mechanics and Control, 100 Zhangwu Road, Yangpu District, Shanghai 200092, People\u2019s Republic of China."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Jian","family":"Xu","sequence":"additional","affiliation":[{"name":"School of Aerospace Engineering and Applied Mechanics, Tongji University, Shanghai, 200092 People\u2019s Republic of China; and Shanghai Institute of Aircraft Mechanics and Control, 100 Zhangwu Road, Yangpu District, Shanghai 200092, People\u2019s Republic of China."}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2026,2,24]]},"reference":[{"key":"ref1","unstructured":"S. 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Ind. Electron."},{"key":"ref37","doi-asserted-by":"publisher","DOI":"10.1142\/S0129065709002087"},{"key":"ref38","doi-asserted-by":"publisher","DOI":"10.1007\/s11571-013-9254-0"},{"key":"ref39","doi-asserted-by":"publisher","DOI":"10.1142\/S0218127415300402"},{"key":"ref40","doi-asserted-by":"publisher","DOI":"10.1007\/s11571-020-09575-9"},{"key":"ref41","doi-asserted-by":"publisher","DOI":"10.1007\/s11071-023-08783-2"},{"key":"ref42","doi-asserted-by":"publisher","DOI":"10.1137\/S1111111103419896"},{"key":"ref43","doi-asserted-by":"publisher","DOI":"10.1007\/BF00198086"},{"key":"ref44","first-page":"123","volume":"8","author":"Tee G.","year":"2005","journal-title":"New Zealand J. 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