Allometric Scaling Laws for Bipedal Robots

Abstract

Legged robots operate across a wide range of physical scales, but how their designs should be adapted as size changes remains unclear. Here, we tackle this question in two ways. First, we survey existing legged robots to provide a broad context for the key scaling variables, robot mass m and leg length L. We find the surprising result that bipedal robot mass generally scales with the length squared, L^2, rather than the isometric prediction L^3. Then, to reduce the variance in design choices, we focus on a pair of previously developed bipeds that share the same quasi-passive morphology but differ by a factor of six in leg length, use different feet and controllers, and achieve different relative speeds. We reconstruct both robots in a common 3-D simulation environment and scale each design over leg lengths from 0.02 to 1.2 meters under both mass models (mass is proportional to L^2 and is proportional to L^). The controlled comparison shows that velocity follows dynamic similarity, velocity is proportional to L^{1/2}, across designs and mass models, while the torque needed to sustain walking follows that tau is proportional to mL. Consequently, torque scales approximately with L^3 when m is proportional to L^2 and L^4 when mass is proportional to L^3. A 3-D foot-shape sweep further shows that foot dimensions that permit walking scale approximately linearly with leg length, but the speed-maximizing shape and the mechanism by which each robot moves do not transfer by scaling alone. Overall, the results provide practical insights for rescaling legged systems that leverage natural body dynamics.