Inverted pendular running: a novel gait predicted by computer optimization is found between walk and run in birds

Abstract

Idealized models of walking and running demonstrate that, energetically, walking should be favoured up to, and even somewhat over, those speeds and step lengths that can be achieved while keeping the stance leg under compression. Around these speeds, and especially with relatively long step lengths, computer optimization predicts a third, 'hybrid', gait: (inverted) pendular running (Srinivasan & Ruina 2006 Nature 439, 72-75 (doi:10.1038/nature04113)). This gait involves both walking-like vaulting mechanics and running-like ballistic paths. Trajectories of horizontal versus vertical centre of mass velocities-'hodographs'-over the step cycle are distinctive for each gait: anticlockwise for walk; clockwise for run; figure-of-eight for the hybrid gait. Both pheasants and guineafowl demonstrate each gait at close to the predicted speed/step length combinations, although fully aerial ballistic phases are never achieved during the hybrid or 'Grounded Inverted Pendular Running' gait.

Figure 1.

Idealized cartoons and hodographs, and example empirical (pheasant) hodographs of (a–c) walking, (d–f) inverted pendular running and (g–i) running. Black arrows relate to momentary impulses; dashed lines to passive vaulting; dotted lines to ballistic falling. Note that, while the cartoons and hodograph sketches for walking and inverted pendular running approach those for ideal, impulsive gaits, the running figure includes a qualitative realism of finite limb forces and stance paths (grey lines): running is not impulsive, stance periods are finite and the hodograph is not vertical during the leg impulse. In each case, the step cycle begins at the black circle, and the cartoons progress from left to right. +/− symbols indicate gain and loss of energy, respectively. Colour in the empirical hodographs show progression through time: green, orange, black.

Figure 2.

Forceplate-derived results for steady locomotion in guineafowl ((a) four individuals coded by symbol shape; (c,e) one individual) and pheasants ((b) three individuals coded by symbol shape; (d,f) one individual). Net hodograph sense (a,b) provides an objective—if arbitrary—metric for gait transition between predominantly walking (anticlockwise, blue) and running (clockwise, red). values indicate multiples of step frequency consistent with pendular swing-leg timing and duty factor = 0.5. The region shaded pink (between contours i and iv) is that predicted by numerical optimization to favour (energetically) inverted pendular running (Srinivasan 2006). The grey contour (ii) indicates the maximum boundary for a passive, stiff-limbed vaulter in which gravity provides sufficient centripetal acceleration of the CoM towards the foot to keep the leg under compression. The outer, black curve (iii) indicates the walk–run (run–walk) transition boundary predicted from energy minimization derived from collision mechanics while allowing limb tension. Hodographs from a single guineafowl (c,e) and pheasant (d,f) show the progression from midstances (black/green boundary) through single steps (order: green, orange, black). The right angle (c–f) gives a scale for the hodographs. Grounded inverted pendular running occurs near the grey walk-limit boundary, as indicated by the highlighted figure-of-eight hodographs ((e,f) providing a magnified view of the dashed box regions in (c,d)).