Elastic energy savings and active energy cost in a simple model of running

Abstract The energetic economy of running benefits from tendon and other tissues that store and return elastic energy, thus saving muscles from costly mechanical work. The classic “Spring-mass” computational model successfully explains the forces, displacements and mechanical power of running, as the outcome of dynamical interactions between the body center of mass and a purely elastic spring for the leg. Conversely, the Spring-mass model does not include active muscles and cannot explain the metabolic energy cost of running. Here we add explicit actuation and dissipation to the Spring-mass model, resulting in substantial active (and thus costly) work for running on level ground and up or down slopes. Dissipation is modeled as modest energy losses (5% of total mechanical energy for running at 3 m · s -1 ) from hysteresis and foot-ground collisions, that must be restored by active work each step. Even with substantial elastic energy return (59% of positive work, comparable to empirical observations), the active work could account for most of the metabolic cost of human running (about 68%, assuming human-like muscle efficiency). We also introduce a previously unappreciated energetic cost for rapid production of force, that helps explain the relatively smooth ground reaction forces of running, and why muscles might also actively perform negative work. Although elastic return is key to energy savings, there are still losses that require restorative muscle work, which can cost substantial energy during running. Author Summary Running is an energetically economical gait whereby the legs bounce like pogo sticks. Leg tendons act elastically to store and return energy to the body, thus saving the muscles from costly work with each running step. Although elasticity is known to save energy, it does not explain why running still requires considerable effort, and why the muscles still do any work at all. We use a simple computational model to demonstrate two possible reasons why. One is that small amounts of energy are lost when the leg collides with the ground and when the tendons are stretched, and muscles must restore that energy during steady running. A second reason is that muscles may perform work to avoid turning on and off rapidly, which may be even more energetically costly. The resulting muscle work, while small, may actually explain most of the energetic cost of running. Economy may be gained from elasticity, but running nonetheless requires muscles to do active work.

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