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Computer optimization of a minimal biped model discovers walking and running


Although people's legs are capable of a broad range of muscle-use and gait patterns, they generally prefer just two. They walk, swinging their body over a relatively straight leg with each step, or run, bouncing up off a bent leg between aerial phases. Walking feels easiest when going slowly, and running feels easiest when going faster. More unusual gaits seem more tiring. Perhaps this is because walking and running use the least energy1,2,3,4,5,6,7. Addressing this classic1 conjecture with experiments2,3 requires comparing walking and running with many other strange and unpractised gaits. As an alternative, a basic understanding of gait choice might be obtained by calculating energy cost by using mechanics-based models. Here we use a minimal model that can describe walking and running as well as an infinite variety of other gaits. We use computer optimization to find which gaits are indeed energetically optimal for this model. At low speeds the optimization discovers the classic inverted-pendulum walk8,9,10,11,12,13, at high speeds it discovers a bouncing run12,13, even without springs, and at intermediate speeds it finds a new pendular-running gait that includes walking and running as extreme cases.

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Figure 1: Body motion in human gaits.
Figure 2: Point-mass biped model and its optimal solutions.
Figure 3: The regions in which each of the three collisional gaits are optimal.
Figure 4: Cost of transport versus speed.


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We thank J. Bertram for extensive discussions on related topics; J. Burns, A. Chatterjee, P. Holmes, A. Schwab, S. Strogatz, S. van Nouhuys and S. Walcott for editorial and technical comments. Author Contributions M.S. formulated the mechanical model and performed the numerical optimizations. This was partly informed by extensive discussions with A.R. during the writing of ref. 12, and from a course taught by A.R. The authors contributed equally to the analytic calculations, the interpretation of the results and to the writing of the paper.

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Correspondence to Manoj Srinivasan.

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Srinivasan, M., Ruina, A. Computer optimization of a minimal biped model discovers walking and running. Nature 439, 72–75 (2006).

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