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Allometric Scaling Laws for Bipedal Robots

Naomi Oke, Aja Carter, Ben Gu, Steven Man, Cordelia Pride, Sarah Bergbreiter, Aaron M. Johnson

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.

Allometric Scaling Laws for Bipedal Robots

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.
Paper Structure (17 sections, 7 figures, 4 tables)

This paper contains 17 sections, 7 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Related Work
  3. Scaling Trends in Existing Robotics
  4. Simulation and Analysis Methods
  5. Simulation Results
  6. Mass, Frequency, and Velocity Scaling
  7. Torque Scaling
  8. Foot Scaling and Body Attitude
  9. Isometric Foot Scaling
  10. Feasible Foot Geometries
  11. Optimal Foot Geometries
  12. Discussion
  13. Mass Scaling
  14. Velocity Scaling and Dynamic Similarity
  15. Torque Demand Scaling
  16. ...and 2 more sections

Figures (7)

  • Figure 1: Two of the robots considered in this paper, the 15cm leg length Mugatu mugatu_kyle2023simplest and the 2.5cm leg length Zippy paper:man-zippy-2025.
  • Figure 2: Relationships between body length, leg length, and body mass across multiple existing robot morphologies.
  • Figure 3: Mugatu (Top) and Zippy (Bottom) robots in the Drake simulation (different scales for clarity).
  • Figure 4: Simulated forward velocity of robots across scales (see \ref{['tab:simresults']}).
  • Figure 5: Simulated minimum required torque of Zippy and Mugatu across scales (see \ref{['tab:simresults']}).
  • ...and 2 more figures