Allometric Scaling Laws for Bipedal Robots

TL;DR

This paper investigates how bipedal robot design parameters scale with size, revealing deviations from biological and isometric scaling laws through literature review and simulation studies.

Contribution

It establishes new allometric scaling laws for bipedal robots across various sizes, contrasting them with biological and isometric predictions.

Findings

Robot mass scales approximately with L^2, not L^3 as in isometric scaling.

Walking velocity scales with L^(1/2), consistent with dynamic similarity.

Minimum torque requirement scales with m*L, differing from isometric models.

Abstract

Scaling the design of robots up or down remains a fundamental challenge. While biological systems follow well-established isometric and allometric scaling laws relating mass, stride frequency, velocity, and torque, it is unclear how these relationships translate to robotic systems. In this paper, we generate similar allometric scaling laws for bipedal robots across three orders of magnitude in leg length. First, we conduct a review of legged robots from the literature and extract empirical relationships between leg length (L), body length, mass, and speed. These data show that robot mass scales more closely to L^2, in contrast to the L^3 scaling predicted by isometric scaling. We then perform controlled simulation studies in Drake using three variants of real quasi-passive, hip-actuated walkers with different foot geometries and control strategies. We evaluate the performance of each design scaled with leg length, L. Across all robots, walking velocity follows the expected L^(1/2) trend from dynamic similarity. Minimum required torque scales more closely with m*L than the isometric model of m*L^2. Foot geometry scaled proportionally with L^1. These results provide new insight into how robot designs allometrically scale to different sizes, and how that scaling is different from isometric or biological scaling laws.