Multiple Walking Speed–frequency Relations are Predicted by Constrained Optimization

A person constrained to walk at a given speed v on a treadmill, chooses a particular step frequency f and step length d=v/f. Testing over a range of speeds generates a speed–frequency (v–f) relationship. This relationship is commonly posited as a basic feature of human gait. It is often further posited that this curve follows from minimum energy cost strategy. We observed that individuals walking under different constraint circumstances—walking to a range of fixed metronome frequencies (fixed f) or over a range of spaced markers (fixed d)—produce speed–frequency relations distinct from the constrained v relation. We show here that three distinct speed–frequency curves, similar to those observed, are predicted by the assumption that a walking person optimizes an underlying objective function F (v, f) that has a minimum at the preferred gait. Further, the metabolic cost of transport is a reasonable approximate candidate for the function F.