Vaulting mechanics successfully predict decrease in walk-run transition speed with incline

Abstract

There is an ongoing debate about the reasons underlying gait transition in terrestrial locomotion. In bipedal locomotion, the 'compass gait', a reductionist model of inverted pendulum walking, predicts the boundaries of speed and step length within which walking is feasible. The stance of the compass gait is energetically optimal-at walking speeds-owing to the absence of leg compression/extension; completely stiff limbs perform no work during the vaulting phase. Here, we extend theoretical compass gait vaulting to include inclines, and find good agreement with previous observations of changes in walk-run transition speed (approx. 1% per 1% incline). We measured step length and frequency for humans walking either on the level or up a 9.8 per cent incline and report preferred walk-run, walk-compliant-walk and maximum walk-run transition speeds. While the measured 'preferred' walk-run transition speed lies consistently below the predicted maximum walking speeds, and 'actual' maximum walking speeds are clearly above the predicted values, the onset of compliant walking in level as well as incline walking occurs close to the predicted values. These findings support the view that normal human walking is constrained by the physics of vaulting, but preferred absolute walk-run transition speeds may be influenced by additional factors.

Figure 1.

Stance according to the compass-gait model for (a) level and (b) incline. The CoM (grey circle) vaults over a massless rigid leg of constant length. Centripetal force requirement (F_cp,blue), the force necessary to keep the CoM on its circular arc. The component of weight in line with the leg (mg, red) decreases with leg angle (ϕ). Slope angle (α); instantaneous CoM velocity (v); mass (m) and leg length (L). In this example, net speed is the same on level and incline; however, the weight component along the leg is only sufficient to maintain compression throughout stance on the level (a), whereas leg tension is required during early stance (thus compass-gait walking is impossible) on the incline (b).

Figure 2.

Compass-gait limits of relative velocity and relative step length for level (solid black curve) and inclines (dashed and coloured curves), and various transition speeds. (a) Predicted and all observed transition speeds are reduced significantly on the incline. Preferred walk–run transition speeds (diamonds) fall somewhat below the limiting boundaries. Maximum walking speeds exceed the predicted boundaries (squares, a), but are only achieved with reduced vertical accelerations indicating compliant-walking. Walk–compliant-walk transition speeds, and their decrease with incline (circles, a) are well predicted by compass-gait mechanics. : multiples of the passive step frequency assuming a point mass pendular swing-leg of leg length. : normalized step length (step length/leg length). (b) A more extensive previous study [5] shows somewhat lower step length at preferred walk–run transition speed over a range of inclines, but (c) the change in relative velocity lies within the prediction if either or are treated as constant.