(a) All subjects with the basic protocol (n= 28 subjects, indoor and outdoor, Drequired = 122 m). The time fraction of running is shown: the red dots are the raw data (each dot is a trial, all data points shown), the solid black line is the population median, and the pink band denotes 50% of the data centred around the median (25th–75th percentile). Pure walking dominates low prescribed speeds and pure running dominates high prescribed speeds, and most subjects use a walk–run mixture for intermediate speeds. (b) The data in the previous panel, decomposed into indoor and outdoor data. Median fractions are blue and pink solid lines, surrounded by the respective 50% bands in corresponding lighter shades. (c) Variants of the basic protocol overlaid on the 50% band from the first panel: experiments in which subjects had three trials per prescribed average speed for five different prescribed average speeds (red triangles), in which the trials had different random distances ranging between 70 and 250 m and different time allowed (blue circles), in which the subject travelled twice the distance, 244 m (black squares), and in which the subjects were allowed to arrive earlier than Tallowed (magenta triangles).
(a) Histograms of the speeds at which the subjects travelled during each of the 15 trials with different prescribed average speeds Vavg; pooled over the 10 subjects with GPS speed measurements, with histogram bin width = 0.1 m s−1. The histograms have a single peak for low and high prescribed average speed, but have two peaks for intermediate average speeds, suggesting a walk–run mixture. (b) The unfiltered GPS-derived speed as a function of time during the trial corresponding to Vavg = 2.39 m s−1 and Tallowed = 51 s, for five subjects. Notice the two speed plateaus in subjects 4–6. (c) Histograms of stride frequencies used during various trials, pooled over outdoor subjects with this pedometer-based data. Histogram bin width = 0.1 Hz. The horizontal scale of the histograms are such that the area within the histogram are the same across different prescribed speeds (proportional to total number of trials). The horizontal grey band across the histograms (a and c) indicating normal walking and running speeds or frequencies is simply to guide the eye, not meant to be a strict separator.
(a) Metabolic energy rate for walking and running as a function of speed, intersecting at speed v = VQ. (b) The combined ‘effective cost curve’ is shown in orange, by picking the gait that has the lower cost at every relevant speed: resting at v = 0, walking below VQ, and running above VQ. Walking at speed VB and running at VC for different fractions of time results in an average metabolic rate as given by the line BC. When BC is the unique common tangent to the two curves, switching between B and C results in a lower average metabolic rate than is possible by exclusively walking or running (in fact the lowest possible for this model). This lowering of cost is possible because of the ‘non-convexity’ of the effective cost curve. (c) Hypothetical metabolic rate curves for walking and running that result in a ‘convex’ effective cost curve, implying no direct energetic benefit from walk–run mixtures.
(a) Total metabolic rate as a function of walking and running speed, normalized by body mass, showing common tangent constructions for optimal walk–rest and walk–run mixtures. (b) Walk–run–rest fractions shown. As the average speed increases, the energy optimal strategy changes from a mixture of resting and walking to pure walking to a walk–run mixture to pure running. (c) Energy per unit distance for walking, running, walk–run mixtures and walk–rest mixtures. The mixture strategies (dotted lines) reduce the average energy per unit distance in the respective speed regimes, as shown. Note, the cost per distance of the mixtures is not a linear function of average speed, which is why the ‘common tangent construction’ is performed on the cost per time curves.
Effect of adding energy cost for transients: model predictions from numerical optimizations. (a) When travelling a finite distance overground, the extent of the walk–run transition regime depends on the distance travelled (Drequired = 120 and 30 m shown). Smaller distances have smaller transition regimes. (b) When having to remain on a constant-speed treadmill of finite length, the extent of the walk–run transition regime depends on the treadmill length. For long treadmills (e.g. 20 m), there is a substantial walk–run mixture regime. On very short treadmills (e.g. 1 m), essentially no walk–run mixture regime exists, and the transition from all walking to all running is sharp.