The Slope Sum

How much energy does a quadruped robot need to walk a path? If the path crosses varied terrain — uphill, downhill, sidehill — the answer might depend on the specific sequence of slopes, the transitions between them, the gait adjustments at each boundary. It could be complicated.

It isn’t. Energy costs on slopes show near-linear dependence on slope angle, with lateral motion consistently costing more than longitudinal. More importantly, energy is additive across trajectory segments: the total cost of a multi-segment path equals the sum of segment costs. No interaction terms. No sequence effects. The energy of the whole is the sum of the parts.

This additivity enables path-level energy prediction using only onboard sensors — no prior map, no detailed terrain model. Measure the slope of each segment, look up the per-segment cost, add them. The prediction is accurate because the additivity is real, not an approximation.

The through-claim: additivity is the strongest possible simplification. It means the system has no memory across terrain transitions — each segment’s energy cost depends only on its own slope, not on what came before or what comes after. For a complex mechanical system with legs, joints, and gaits that adjust to terrain, this is surprising. It suggests that the gait controller is doing its job so well that it fully adapts within each segment, carrying no energetic debt forward. The additivity is not a property of the physics — it’s a property of the controller being good enough to make the physics effectively memoryless.