Passive Dynamics Explain Quadrupedal Walking, Trotting, and Tölting

Abstract

This paper presents a simplistic passive dynamic model that is able to create realistic quadrupedal walking, tölting, and trotting motions. The model is inspired by the bipedal spring loaded inverted pendulum (SLIP) model and consists of a distributed mass on four massless legs. Each of the legs is either in ground contact, retracted for swing, or is ready for touch down with a predefined angle of attack. Different gaits, that is, periodic motions differing in interlimb coordination patterns, are generated by choosing different initial model states. Contact patterns and ground reaction forces (GRFs) evolve solely from these initial conditions. By identifying appropriate system parameters in an optimization framework, the model is able to closely match experimentally recorded vertical GRFs of walking and trotting of Warmblood horses, and of tölting of Icelandic horses. In a detailed study, we investigated the sensitivity of the obtained solutions with respect to all states and parameters and quantified the improvement in fitting GRF by including an additional head and neck segment. Our work suggests that quadrupedal gaits are merely different dynamic modes of the same structural system and that we can interpret different gaits as different nonlinear elastic oscillations that propel an animal forward.

Effect of different head and neck positions on kinematics of elite dressage horses ridden at walk on treadmill

“…In normal walk there is a difference in the depth of the midstance dip in the vertical ground reaction force curve between forelimbs and hindlimbs (deeper for the hindlimbs). In a modeling study this could only be explained when taking the head dynamics into account (Gan et al, 2016). Further, at walk the horse moves with substantially less muscle activity in the lower back and abdomen (Wakeling et al, 2007;Zsoldos et al, 2010), and with larger lateral movement of the body center of mass, compared to trot (Buchner et al, 2000).…”

Optimal configuration of series and parallel elasticity in a 2D Monoped

“…This connection is readily apparent when a legged robot moves in a way that maximizes its energetic economy. Model based studies have shown that robots moving in this way take advantage of their natural dynamics [1], [2] (the intrinsic morphologically based motions that result from the interplay of gravity, inertia, and elasticity in the springs). For these models, springs play a key role in explaining the dynamics of legged systems.…”

Energy-Optimal Hopping in Parallel and Series Elastic One-Dimensional Monopeds

“…In nature, elastic elements are used for energy storage, as return springs, and to cushion impacts [1]. Model-based analyses have shown that compliant legs can explain the dynamics of human walking and running [2], as well as a wide variety of quadrupedal gaits, including walking, trotting, t€ olting, and galloping [3,4]. In all these cases, elastic energy storage enables the recycling of energy and improves energetic economy.…”