Model development for running. Stances are modelled for running at a range of speeds and stance periods treating the leg as a linear spring (though it is assumed that some constant proportion of the positive work demanded is due to muscle action). Impulsive stances (A: infinitely stiff, brief stance periods) minimise positive work but demand infinite power; finite but too-brief, too-stiff stances (B) demand excessive muscle activation to provide the power; intermediate stiffness (C) minimises muscle activation (resulting, at moderate speeds, in 0.1 s push-off, or a 0.2 s stance, matching work and power demands); too-compliant stances (D) result in excessive muscle activation to provide the positive work.

Model and empirical vertical forces for walking at a range of speeds and sizes. Results for an analytical approximation to the numerical walking model (black lines bounding ±1 s.d.) with 0.1 s crash and shove periods, based on empirical kinematic inputs. Measured vertical forces (red lines bounding ±1 s.d. for each group and speed bin) match well for adults, but poorly for children, especially smaller (younger) toddlers, which deviate considerably from the symmetrical inverted pendulum walking strategy. Sample sizes are shown in supplementary material Table S2.

Best-fit sine coefficients for walking vertical forces at a range of speeds and sizes. Shown with linear regressions against dimensionless speed (±95% CI) underlying in colour for adults (blue, A), older/larger children (pink, B) and younger/smaller children (green, C), and the regressions combined for comparison (D). The coefficients relate to the amplitudes of three sine waves which, when summed, minimise the root mean square error from the measured vertical ground reaction force. The example traces (E) (black line being the reconstructed curve; underlying grey the empirical curve being fitted) relate to a specific stance denoted by coloured symbols in B. The coefficient a₂ relates to a force–time bias, and increases with speed more rapidly with smaller walkers.

Model and empirical results for running stance periods. Running model results for muscle activation minimising stance durations (black line, A), cost contours and measurements (points) for adults (B) and children (C). Costs are derived for the running model of Fig. 2, for a range of speeds and step periods, calculating the activated muscle volume required to produce whichever is more demanding between mechanical work or power. Cost contours are presented normalised by the minimum value (of activated muscle volume) for each speed, with white contours indicating 5% boundaries above minimal (purple); red regions indicating greater than 20% above minimal required activation. Points denote empirical observations for undergraduates (grey points, B) and near-elite sprinters (white points, B; data from McGowan et al., 2012), and children with duty factor above 0.5 (black points, C) and below 0.5 (grey points, C). At moderate speeds, a stance period of 0.2 s is predicted to be optimal independent of leg length – for both adults and children – and this is close to empirical observation for running adults and children; at , the current model provides a much better prediction for children than simple dynamic similarity, which would suggest (blue outline cross) much briefer stances. High stance periods at high speeds are geometrically impossible if stance length exceeds double leg length; stance periods greater than swing result in no aerial phase (duty factor >0.5).

Results for alternative reductionist accounts for the vertical forces of bipedal walking. Underlying grey regions (A) denote the range of possible walking-like outcomes (symmetrical, with a broadly ‘M’-shaped profile) for the linear spring-mass model (or spring-loaded inverted pendulum or SLIP model) with appropriately tuned parameters. At low speeds, realistic forces can be found; at medium speeds, midstance forces are under-predicted; and no walking solutions can be found at high walking speeds (following Geyer et al., 2006). A simple, semi-mechanistic analytical model developed here on the assumptions and principles of Alexander provides remarkably good fits given only observed speed, leg length and duty factor (black lines, B, show ±1 s.d. using observed kinematic inputs; red lines show ±1 s.d. of empirical data for adults for each non-dimensional speed bin). Model midstance forces (black line, C) agree with measured minimum forces in the trough of the M and modelled maximum forces (grey line) agree with measured first peak (red points) and second peak (blue points), at least up to preferred walk-run transitions speeds (). However, the ‘Alexanderesque’ approach has limited mechanistic basis, and does not provide an account for why peak forces increase with walking speed.

Results for alternative accounts for stance durations in running. Empirical stance durations presented in non-dimensional form for running undergraduates (grey), sprinters (white), children (green) and children with a duty factor greater than 0.5 (black). Children are not dynamically similar to adults: their stance durations are disproportionately high. Stance durations found from spring-mass models with appropriate leg length, stance duration and non-dimensional leg stiffness provide a good match for adults (blue line) and children (red line). However, spring-mass models do not have a mechanistic basis, do not account for why leg stiffness should stay approximately constant across speed, and provide no account for the relatively more compliant legs of children. Theoretical mechanical work minimising running requires infinitely small stance durations (dashed lines) and infinite forces. Work-minimising gaits with a constrained maximum limb force would result in constant, minimal stance duration (grey line for adults), failing to account for higher stance durations at lower speeds.