The median statistics and velocity dependence of multi-contact dry friction systems*

The ground-based locomotion of animals and robots typically occurs through sequences of multi-contact interaction between body/limbs and the ground. Over dry, rigid surfaces the interaction force of a contact is governed by Coulomb friction, which is independent of the contact speed. However, recent experiments and simulation have revealed that multi-legged animals and robots exhibit dynamics that are consistent with speed-dependent interaction forces when walking [1], [2]. In this work we provide a simple demonstration of how this speed dependence arises in multi-contact frictional systems with heterogeneous contact velocities. We first present experiments from a multi-contact frictional carousel: a system of ten wheels that all rotate at varied speeds and collectively drive the rigid rotation of a carousel. We find that over more than 700 experiments in varying wheel speeds the net speed of the system is determined by the median of the wheel velocities. A simple theoretical model of a multi-contact system with speed-independent forces is able to reproduce this result from first principles. Furthermore, by examining the cumulative distribution function of instantaneous contacts we are able to derive an effective viscosity for multi-contact friction related to the gradient of instantaneous contact velocities. We study this effective viscosity in perturbation experiments of our multi-contact frictional carousel. Ultimately these experimental and theoretical results provide insight into how speed-dependent dynamics arise from individual speed-independent forces of multi-contact locomotion.

[1] Chong, B., He, J., Li, S., Erickson, E., Diaz, K., Wang, T., Soto, D., & Goldman, D. I. (2023). Self-propulsion via slipping: Frictional swimming in multilegged locomotors. Proceedings of the National Academy of Sciences, 120(11), e2213698120.

[2] Zhao, D., Bittner, B., Clifton, G., Gravish, N., & Revzen, S. (2022). Walking is like slithering: A unifying, data-driven view of locomotion. Proceedings of the National Academy of Sciences of the United States of America, 119(37), e2113222119.