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.

翻译：缩放机器人设计的尺寸仍是一项根本性挑战。虽然生物系统遵循成熟等速与异速缩放定律，涉及质量、步频、速度及力矩等参数，但尚不清楚这些关系如何迁移至机器人系统。本文针对双足机器人，在腿长跨越三个数量级的范围内生成了类似的异速缩放定律。首先，我们梳理了文献中的腿式机器人研究，并提取了腿长（L）、体长、质量与速度之间的经验关系。数据显示，机器人质量更接近L²标度，与等速缩放预测的L³标度形成对比。随后，我们利用Drake平台，基于三种具有不同足部几何构型与控制策略的真实准被动髋关节驱动行走器变体，开展了受控仿真研究。我们评估了每种设计随腿长L缩放的性能表现。所有机器人的行走速度均遵循动力学相似性预期的L^(1/2)趋势。最小所需力矩更接近m*L标度，而非等速模型的m*L²标度。足部几何构型按L^1比例缩放。这些结果为机器人设计如何按异速规律缩放至不同尺寸，以及这种缩放与等速或生物缩放定律的差异提供了新见解。