A simple model of a quadruped discovers single-foot walking and trotting as energy optimal strategies

Abstract

It is widely held that quadrupeds choose steady gaits that minimize their energetic cost of transport, but it is difficult to explore the entire range of possible footfall sequences empirically. We present a simple model of a quadruped that can spontaneously produce any of the 2300+ planar footfall sequences available to quadrupeds. Through trajectory optimization of a work and force-rate cost, and a large sample of random initial guesses, we provide stronger evidence towards the global optimality of symmetrical four-beat walking at low speeds and two beat running (trotting) at intermediate speeds. Using input parameters based on measurements in dogs (Canis lupus familiaris), the model predicts the correct phase offset in walking and a realistic walk-trot transition speed. It also spontaneously reproduces the double-hump ground reaction force profile observed in walking, and the smooth single-hump profile observed in trotting, despite the models lack of springs. However, the model incorrectly predicts duty factor at the slowest speeds, and does not choose galloping as globally optimal at high speeds. These limitations might point to missing levels of complexity that could be added to future quadrupedal models, but the present results suggest that massive legs, elastic components, head-neck oscillations and even multi-linked legs are not critical determiners for the optimality of natural quadrupedal walking and trotting.\n\nAuthor summaryWhy do quadrupedal mammals move in consistent ways, when so many options are available? We tackled this problem by determining energetically-optimal gaits using a simple computational model of a four-legged animal. The model can use virtually any pattern of movement (physics-permitting!) but selects natural movement strategies as energetically optimal. The similarities between the computer-based predictions and natural animal movement are striking, and suggest mammals utilize movement strategies that optimize energy use when they move.