Abstract
Bipedal animals experience ground reaction forces (GRFs) that pass close to the centre of mass (CoM) throughout stance, first decelerating the body, then re-accelerating it during the second half of stance. This results in fluctuations in kinetic energy, requiring mechanical work from the muscles. However, here we show analytically that, in extreme cases (with a very large body pitch moment of inertia), continuous alignment of the GRF through the CoM requires greater mechanical work than a maintained vertical force; we show numerically that GRFs passing between CoM and vertical throughout stance are energetically favourable under realistic conditions; and demonstrate that the magnitude, if not the precise form, of actual CoM-torque profiles in running is broadly consistent with simple mechanical work minimization for humans with appropriate pitch moment of inertia. While the potential energetic savings of CoM-torque support strategies are small (a few per cent) over the range of human running, their importance increases dramatically at high speeds and stance angles. Fast, compliant runners or hoppers would benefit considerably from GRFs more vertical than the zero-CoM-torque strategy, especially with bodies of high pitch moment of inertia—suggesting a novel advantage to kangaroos of their peculiar long-head/long-tail structure.
Highlights
Legs of running or hopping animals support body weight, and act to slow and accelerate the body horizontally over the course of every step
The resulting fluctuations in kinetic energy—which must impose some degree of energetic cost at the level of the muscle owing to the absence of perfectly elastic springs in biology—are perhaps perplexing
This analysis demonstrates that, with a distributed mass in the sagittal plane and perfect energy interchange between leg joints, non-zero CoMtorques offer an energetic advantage for supporting body weight in running or hopping
Summary
Analytical demonstration that through centre of mass forces are not always energetically optimal. Consider two extreme strategies for providing weight support over a symmetrical stance passing from 2F to þF (figure 1a). The first is totally ‘telescoping’, with forces always in line with the CoM and no CoMtorques. The second has purely vertical forces; there are CoM-torques but never horizontal forces. The vertical force profile—and related vertical work requirements—are assumed to be equivalent
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