Walking on a moving surface: energy-optimal walking motions on a shaky bridge and a shaking treadmill can reduce energy costs below normal

Abstract

Understanding how humans walk on a surface that can move might provide insights into, for instance, whether walking humans prioritize energy use or stability. Here, motivated by the famous human-driven oscillations observed in the London Millennium Bridge, we introduce a minimal mathematical model of a biped, walking on a platform (bridge or treadmill) capable of lateral movement. This biped model consists of a pointmass upper body with legs that can exert force and perform mechanical work on the upper body. Using numerical optimization, we obtain energyoptimal walking motions for this biped, deriving the periodic body and platform motions that minimize a simple metabolic energy cost. When the platform has an externally imposed sinusoidal displacement of appropriate frequency and amplitude, we predict that body motion entrained to platform motion consumes less energy than walking on a fixed surface. When the platform has finite inertia, a massspring-damper with similar parameters to the Millennium Bridge, we show that the optimal biped walking motion sustains a large lateral platform oscillation when sufficiently many people walk on the bridge. Here, the biped model reduces walking metabolic cost by storing and recovering energy from the platform, demonstrating energy benefits for two features observed for walking on the Millennium Bridge: crowd synchrony and large lateral oscillations.

Walking crowds on a shaky surface: stable walkers discover Millennium Bridge oscillations with and without pedestrian synchrony

“…In a previous study, we had argued that humans could reduce their walking metabolic cost by optimally walking on a shaking bridge [10]. That model ignored stability considerations.…”

“…Estimating the metabolic cost for gaits observed in this current study, we find that the metabolic cost is lowest when the biped walks on solid ground and increases when the bridge starts shaking (figure 2e). We hypothesize that the short-timescale response of the humans will be governed by the constraint that they should not fall (thus increasing the cost), but if the subjects continued to walk on shaky bridges, they may learn to walk energy-efficiently [10], modifying their controller to be simultaneously energy-efficient and stable, perhaps using optimal feedback control [13].…”

“…Minimization of metabolic energy can approximately explain a variety of human walking behaviours [9]. We previously showed that walking in synchrony with lateral bridge oscillations lowers the walking energy cost, suggesting the usefulness of synchrony [10]. Here, by contrast, we use the human preference to walk without falling down (being stable) to model the observed phenomena [8].…”

Instantaneous Metabolic Cost of Walking: Joint-Space Dynamic Model with Subject-Specific Heat Rate

“…Several recent studies have demonstrated that MEE during walking can be predicted using joint-space kinematic and kinetic data [41,42]. However, due to the lack of an accurate MEE model in joint space, incomplete proxies for MEE, such as center-of-mass work [43,44], segmental work [43,45], modified total mechanical work [45], rate of normalized absolute joint moment impulses [43], mechanical work derived from experimental efficiencies [46,47], and other combinations [48], are often used as approximations in the literature.…”

Generating electricity while walking with a medial–lateral oscillating load carriage device

“…Previous work has shown that the metabolic cost of walking can be reduced by providing an M–L restorative force acting on the subject’s pelvis [19–21]. As well, modelling work has also shown that walking economy can be improved by having the walking surface oscillate and do work on the walking model [22]. Since the carried mass in our M–L load carriage device was oscillating out-of-phase, the M–L interaction force component was also out-of-phase, resembling a restorative force to the centreline similar to studies with elastically tethered subjects [19–21].…”