Scaling of the spring in the leg during bouncing gaits of mammals

Trotting, bipedal running, and especially hopping have long been considered the principal bouncing gaits of legged animals. We use the radial-leg spring constant [Formula: see text] to quantify the stiffness of the physical leg during bouncing gaits. The radial-leg is modeled as an extensible strut between the hip and the ground and [Formula: see text] is determined from the force and deflection of this strut in each instance of stance. A Hookean spring is modeled in-series with a linear actuator and the stiffness of this spring [Formula: see text] is determined by minimizing the work of the actuator while reproducing the measured force-deflection dynamics of an individual leg during trotting or running, and of the paired legs during hopping. Prior studies have estimated leg stiffness using [Formula: see text], a metric that imagines a virtual-leg connected to the center of mass. While [Formula: see text] has been applied extensively in human and comparative biomechanics, we show that [Formula: see text] more accurately models the spring in the leg when actuation is allowed, as is the case in biological and robotic systems. Our allometric analysis of [Formula: see text] in the kangaroo rat, tammar wallaby, dog, goat, and human during hopping, trotting, or running show that [Formula: see text] scales as body mass to the two-third power, which is consistent with the predictions of dynamic similarity and with the scaling of [Formula: see text]. Hence, two-third scaling of locomotor spring constants among mammals is supported by both the radial-leg and virtual-leg models, yet the scaling of [Formula: see text] emerges from work-minimization in the radial-leg model instead of being a defacto result of the ratio of force to length used to compute [Formula: see text]. Another key distinction between the virtual-leg and radial-leg is that [Formula: see text] is substantially greater than [Formula: see text], as indicated by a 30-37% greater scaling coefficient for [Formula: see text]. We also show that the legs of goats are on average twice as stiff as those of dogs of the same mass and that goats increase the stiffness of their legs, in part, by more nearly aligning their distal limb-joints with the ground reaction force vector. This study is the first allometric analysis of leg spring constants in two decades. By means of an independent model, our findings reinforce the two-third scaling of spring constants with body mass, while showing that springs in-series with actuators are stiffer than those predicted by energy-conservative models of the leg.