Walking on Shaky Bridges and Shaking Treadmills


Title:  Walking on a moving surface: energy-optimal walking motions on a shaky bridge and a shaking treadmill reduce energy costs below normal
Authors: Varun Joshi and Manoj Srinivasan
Journal: Proceedings of the Royal Society A, 20140662, 2015.
Article PDF: PDF and HTML versions are available free at the journal webpage. Here is a PDF with supplementary information.


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 point-mass upper body with legs that can exert force and perform mechanical work on the upper body. Using numerical optimization, we obtain energy-optimal 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 mass-spring-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.


Energy optimal walking on a shaky bridge (with Millennium Bridge parameters)

Energy optimal walking on a shaking treadmill

Video of the human-driven bridge motion


NS and MS were supported by NSF CMMI grant 1254842.


Gizmag. Swaying this way saves energy while walking.
Ohio State. Study examines phenomenon that caused UK’s Millennium Bridge to wobble.
(PhysOrg version.)