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：机器人的放大或缩小设计仍是一项基本挑战。虽然生物系统遵循关于质量、步频、速度和扭矩的成熟等速与异速生长定律，但这些关系如何应用于机器人系统尚不明确。本文针对腿长横跨三个数量级的双足机器人，推导了类似的异速生长定律。首先，我们对文献中的有腿机器人进行综述，提取腿长(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的缩放比例，这与等速生长定律预测的L³比例不同。随后，我们在Drake环境中对三种采用不同足部几何结构与控制策略的真实准被动髋关节驱动步行器变体进行受控仿真研究。我们评估了每种设计随腿长L缩放的性能表现。在所有机器人中，步行速度遵循动力学相似性预期的L^(1/2)趋势。最小所需扭矩更接近m*L的缩放模型，而非等速模型中的m*L²。足部几何结构按L^1的比例缩放。这些结果为机器人设计如何以异速生长方式适应不同尺寸提供了新见解，并揭示了这种缩放与等速或生物生长定律的差异。