Scale a bipedal robot up by doubling its leg length. Isometric scaling — the naive prediction — says mass should increase by a factor of eight (length cubed), torque requirements should increase by a factor of thirty-two (mass times length squared), and the design is a cube-square catastrophe. The bigger robot needs proportionally more torque per unit mass, so at some size the actuators can't keep up.
Oke, Carter, Gu, Man, Pride, Bergbreiter, and Johnson tested this prediction by surveying existing bipedal robots and running controlled simulations. The actual scaling doesn't follow the isometric exponents. Robot mass scales with leg length squared, not cubed. Torque scales with mass times length, not mass times length squared. The exponents are lower than predicted — larger robots are lighter and require less torque relative to their size than isometric scaling predicts.
The reason is that robots aren't isometric. Biological organisms, which allometric scaling was originally developed to describe, are made of tissue whose density and structure are roughly constant. A bigger animal is a scaled-up version of a smaller one, with the same material filling a larger volume. Robots are not. They're assembled from discrete components — motors, links, controllers — whose mass doesn't fill volume uniformly. A longer leg isn't a thicker leg in the same proportion. The structure is sparse where biology is dense.
This means the cube-square limit that constrains biological locomotion constrains robotic locomotion differently. Robots have more headroom at large scales than isometric scaling suggests, because the mass penalty for being bigger is milder. The foot geometry scales linearly with leg length — the contact patch grows in one dimension, not two — which further reduces the structural burden.
The scaling laws that describe biology describe robots only if the exponents are re-measured. The physics is the same. The engineering constraints are different. And the difference lives entirely in the exponents.