Even though human legs allow a wide repertoire of movements, when people travel by foot they mostly use one of two locomotor patterns, namely, walking and running. The selection of these two gaits from the plethora of options might be because walking and running require less metabolic energy than other more unusual gaits. We addressed this possibility previously using numerical optimization of a minimal mathematical model of a biped. We had found that, for a given step-length, the two classical descriptions of walking and running, ‘inverted pendulum walking’ and ‘impulsive running’, do indeed minimize the amount of positive work required at low and high speeds respectively. Here, for the case of small step-lengths, we establish the previous results analytically. First, we simplify the two-dimensional particle trajectory problem to a one-dimensional ‘elevator’ problem. Then we use elementary geometric arguments on the resulting phase plane to show optimality of the two gaits: walking at low speeds and running at high speeds.
Solving the Optimal Trading Trajectory Problem Using a Quantum Annealer
