Idealized walking and running gaits minimize work BY MANOJ SRINIVASAN 1, * AND ANDY RUINA 2 1 Mechanical and Aerospace Engineering, Princeton University, Princeton, NJ 08544, USA 2 Theoretical and Applied Mechanics, Cornell University, Ithaca, NY 14853, USA Even though human legs allow a wide repertoire of movements, when people travel by foot they mostly use one of two locomotor patterns, namely, walking and running. The selection of these two gaits from the plethora of options might be because walking and running require less metabolic energy than other more unusual gaits. We addressed this possibility previously using numerical optimization of a minimal mathematical model of a biped. We had found that, for a given step-length, the two classical descriptions of walking and running, ‘inverted pendulum walking’ and ‘impulsive running’, do indeed minimize the amount of positive work required at low and high speeds respectively. Here, for the case of small step-lengths, we establish the previous results analytically. First, we simplify the two-dimensional particle trajectory problem to a one-dimensional ‘elevator’ problem. Then we use elementary geometric arguments on the resulting phase plane to show optimality of the two gaits: walking at low speeds and running at high speeds. Keywords: locomotion; energy; optimality; walking; running; optimal control 1. Introduction When people travel by foot from point A to point B, and are not rushed, they walk. When they are in a hurry, they run. Classically, human walking has been idealized as a gait in which the hip (or the body centre of mass) travels in a series of inverted pendulum circular arcs about the foot on the ground as shown in figure 1a (e.g. Alexander 1976). In one energy-consistent version of this idealization, only one foot is on the ground at a time, and the transition from one circular arc to the next is accomplished by the trailing leg pushing off just before the heel of the leading leg strikes the ground ( Kuo 2001; Kuo et al. 2005; Ruina et al. 2005; Srinivasan & Ruina 2006). Analogously, as shown in figure 1b, running is simply idealized as a series of parabolic free- flights of the body, interrupted by brief bounces. In idealized running, the work absorbed in the downward part of a bounce is regenerated on the way back up (Rashevsky 1948; Ruina et al. 2005; Srinivasan & Ruina 2006). Proc. R. Soc. A doi:10.1098/rspa.2007.0006 Published online * Author for correspondence (msriniva@princeton.edu). Received 28 May 2007 Accepted 16 June 2007 1 This journal is q 2007 The Royal Society