Dynamics and energetics of a class of bipedal walking systems Josep M. Font-Llagunes a,b, * , József Kövecses a a Department of Mechanical Engineering and Centre for Intelligent Machines, McGill University, 817 Sherbrooke Street West, Montréal, Québec, Canada H3A 2K6 b Department of Mechanical Engineering, Universitat Politècnica de Catalunya, Diagonal 647, 08028 Barcelona, Catalunya, Spain article info Article history: Received 2 September 2008 Received in revised form 14 April 2009 Accepted 17 May 2009 Available online 7 July 2009 Keywords: Bipedal locomotion Dynamics Energy analysis Walking systems Variable topology abstract The mechanical analysis of bipedal walking is a fundamental subject of research in biome- chanics. Such analysis is useful to better understand the principles underlying human loco- motion, as well as to improve the design and control of bipedal robotic prototypes. Modelling the dynamics of walking involves the analysis of its two phases of motion: (1) the single support phase, which represents finite motion; and (2) the impulsive motion of the impact that occurs at the end of each step (heel strike). The latter is an important event since it is the main cause of energy loss during motion and, in turn, it makes the topology of the system change. In this paper, we present a unified method to analyze the dynamics of both phases of walking. Emphasis is given to the heel strike event, for which we introduce a novel method that gives a complete decomposition of the dynamic equations and the kinetic energy of the system at topology change. As an application exam- ple, the presented approach is applied to a compass-gait biped with point feet. Based on this, the work includes a thorough analysis and discussions about the effect of the biped configuration and its inertial parameters on the dynamics and energetics of heel strike. Ó 2009 Elsevier Ltd. All rights reserved. 1. Introduction Increasing the level of understanding of the mechanics of bipedal walking is a major subject in the area of biomechanical engineering. Research in that field is useful for both applications related to human beings (e.g., design of rehabilitation or prosthetic devices), and also to develop better bipedal walking robots based on the principles underlying human locomotion. Modelling the dynamics of walking involves the study of the two phases defining the gait cycle, i.e., the single support phase, and the impulsive motion phase that occurs at heel strike. It was observed that during the single support phase, the motion of the stance leg is similar to that of an inverted pendulum and the swing leg also performs a pendulum-like motion rotating about the pelvis [1,2]. Thus, the motion of the swing leg is mainly passive in nature and it requires little muscle activity to be performed [3]. Based on these assumptions, simple models that provided useful information regarding the dynamics and the energetic aspects of bipedal locomotion were developed [4]. ‘‘Passive dynamic walking”, which was first introduced by McGeer [5,6], is based on the idea of passive pendulum-like motion during the single support phase. It refers to simple mechanical models that are able to walk down a slightly inclined walkway with no external control or actuation, i.e., gravity alone powers the motion. Several examples of such systems have been analyzed in the literature, both for 3-dimensional motion [7,8] and for 2-dimensional motion [9–13]. The work on this type of walkers was primarily motivated by the drive for energy efficiency and showed that it was possible to obtain orbitally stable limit cycles, with remarkably human-like motion, without any kind of actuation and control. 0094-114X/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.mechmachtheory.2009.05.003 * Corresponding author. Address: Department of Mechanical Engineering and Centre for Intelligent Machines, McGill University, 817 Sherbrooke Street West, Montréal, Québec, Canada H3A 2K6. E-mail addresses: font@cim.mcgill.ca (J.M. Font-Llagunes), jozsef.kovecses@mcgill.ca (J. Kövecses). Mechanism and Machine Theory 44 (2009) 1999–2019 Contents lists available at ScienceDirect Mechanism and Machine Theory journal homepage: www.elsevier.com/locate/mechmt