mathematics Article Integrated Structure-Control Design of a Bipedal Robot Based on Passive Dynamic Walking Josué Nathán Martínez-Castelán † and Miguel Gabriel Villarreal-Cervantes * ,†   Citation: Martínez-Castelán, J.N.; Villarreal-Cervantes, M.G. Integrated Structure-Control Design of a Bipedal Robot Based on Passive Dynamic Walking. Mathematics 2021, 9, 1482. https://doi.org/10.3390/math9131482 Academic Editors: Theodore E. Simos and Charampos Tsitouras Received: 26 May 2021 Accepted: 19 June 2021 Published: 24 June 2021 Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affil- iations. Copyright: © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). Mecatronic Section, Postgraduate Department, Instituto Politécnico Nacional, CIDETEC, Mexico City 07700, Mexico; jmartinezc1317@alumno.ipn.mx * Correspondence: mvillarrealc@ipn.mx † These authors contributed equally to this work. Abstract: The design of bipedal robots is generally fulfilled through considering a sequential design approach, where a synergistic relationship between its structure and control features is not promoted. Hence, a novel integrated structure-control design approach is proposed to simultaneously obtain the optimal structural description, the torque magnitudes, and the on/off time intervals for the control signal input of a semi-passive bipedal robot. The proposed approach takes advantage of the natural dynamics of the system and the control signal activation/deactivation for generating stable gait cycles with minimum energy consumption. Consequently, the passive features of the semi-passive bipedal robot are included in the integrated structure-control design process through evaluating the system behavior along consecutive passive and semi-passive walking stages. Then, the proposed design approach is formulated as a nonlinear discontinuous dynamic optimization problem, where the solution search is carried out using the differential evolution algorithm due to the discontinuities of the semi-passive bipedal robot dynamics. The results of the proposal are compared with those obtained by a sequential design process. The integrated structure-control design achieves a reduction of 63.55% in the value of the performance function related to the synergy between the walking capability and energetic efficiency, with a reduction in the activation of the control and its magnitude of 95.41%. Keywords: structure-control design; optimization; passive bipedal walker; bipedal robots; differen- tial evolution 1. Introduction Artificial bipedal walkers are systems that can walk due to the alternated execution of single and double support phases of their legs. Then, the single support or swing phase is defined as the locomotion phase where only one foot is on the ground; conversely, the double support phase is described when both feet of the system is in contact with the walking surface [1]. There exist three types of bipedal machines that can develop stable gait cycles [2]. The first type studies the fully actuated bipedal robots that are mechatronic systems where a precise joint-angle control is required to produce bipedal locomotion. This approach has reached impressive results mainly associated with the control design of humanoid robots, for instance, the navigation and interaction of the ASIMO robot [3], the walking on a low friction floor of the humanoid robot HRP-3 [4], the walking on large obstacles of the humanoid robot HRP-2 [5], among others. Nevertheless, the high- frequency response of actuators and real-time control computation cause that these robots are energetically inefficient [6]. The second type addresses the passive bipedal walkers that can achieve stable gait cycles without any control input. Tad McGeer demonstrated in his seminal work [7] the importance of mechanical structure in bipedal machines. His work showed that a purely passive walker can develop stable gait cycles when the system is located over an inclined surface. Despite the energetic efficiency of this type of system, they are not versatile since it is not possible to actively modify its gait indicators, such as Mathematics 2021, 9, 1482. https://doi.org/10.3390/math9131482 https://www.mdpi.com/journal/mathematics