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. However, the Spring-mass model does not include active muscles and cannot explain the metabolic energy cost of running, whether on level ground or on a slope. Here we add explicit actuation and dissipation to the Spring-mass model, and show how they explain substantial active (and thus costly) work during human running, and much of the associated energetic cost. 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. With both work and rapid force costs, the model reproduces the energetics of human running at a range of speeds on level ground and on slopes. Although elastic return is key to energy savings, there are still losses that require restorative muscle work, which can cost substantial energy during running.

Running economy and lower extremity stiffness in endurance runners: A systematic review and meta-analysis

“…This simple model can reproduce the motion and forces of running remarkably well. Schroeder and Kuo (2021) proposed a running model based on SMM that combines elasticity with active actuation and passive dissipation (Actuated Spring-mass model). This model included two types of dissipation, one dissipating for collision and hysteresis losses, and the other including hysteresis to simulate imperfect energy return of tendons and other series elastic tissues ( Schroeder and Kuo, 2021 ).…”

“… Schroeder and Kuo (2021) proposed a running model based on SMM that combines elasticity with active actuation and passive dissipation (Actuated Spring-mass model). This model included two types of dissipation, one dissipating for collision and hysteresis losses, and the other including hysteresis to simulate imperfect energy return of tendons and other series elastic tissues ( Schroeder and Kuo, 2021 ). However, the model is not perfect either, for example, it neglects a swing leg, whose active motion may also cost energy ( Doke et al, 2005 ).…”

How do differences in Achilles’ tendon moment arm lengths affect muscle-tendon dynamics and energy cost during running?

“…Additionally, a short AT MA allows for a lower muscle fascicle shortening and lower shortening velocity for a given joint rotation during stance, which in turn reduces metabolic cost of activating a greater volume of muscle (17)(18)(19) and muscle energy cost because of the muscle's force-velocity relationship (9). What has not yet been considered in these proposed explanations for the energetic benefits of a short AT MA is that elastic strain energy storage does not come without a metabolic cost itself (20,21). Indeed, the additional elastic strain energy storage associated with a short AT MA is a result of higher muscle forces for a given joint moment required to stretch the Achilles tendon.…”

“…The measurement of muscle fascicle and Achilles tendon length change and velocity of the triceps surae is easily performed during running (35)(36)(37)(38)(39). From these measurements, combined with an estimate of muscle-tendon forces using inverse dynamics (40), AT energy storage and return can be quantified and the muscle energy cost of contraction during stance can be calculated (20). The main determinant of whole-body E run is the generation and maintenance of muscular force, to support and accelerate the body (7), which is influenced by the AT MA .…”

“…What has not yet been considered in these proposed explanations for the energetic benefits of a short AT MA is that elastic strain energy storage does not come without a metabolic cost itself ( 20 , 21 ). Indeed, the additional elastic strain energy storage associated with a short AT MA is a result of higher muscle forces for a given joint moment required to stretch the Achilles tendon.…”