Manoj Srinivasan
Email address :   msriniva@princeton.edu


Ph.D. thesis:

Why walk and run: Energetic costs and energetic optimality in simple
mechanics-based models of a bipedal animal.
Manoj Srinivasan, Cornell University, 2006.

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Abstract.
This thesis is a model-based exploration of the classic hypothesis that animals locomote in a manner that minimizes the metabolic cost of the task.

First, we formulate perhaps the simplest mathematical model of a bipedal animal that is capable of an infinite variety of gaits --- including many types of walking, running, and skipping. The model, first described by Alexander (1980),  consists of a point-mass upper body and massless legs which are capable of performing work on the upper body when in contact with the ground.  We determine the positive and negative work required by the model to perform idealized versions of various familiar gaits. Approximating the total metabolic cost as being only due to positive and negative work, we find that inverted pendulum walking is preferable to, specifically, impulsive running at low speeds and impulsive running is preferable to inverted pendulum walking at higher speeds. Further, we find that skipping is always a little more energetically expensive than impulsive running.

We then ask a larger question: why do people choose to walk and run when their legs are, in theory, capable of an infinite variety of gaits? Using numerical optimization on the minimal model, we show that from among an infinite variety of gaits that the minimal model is capable of, inverted pendulum walking requires the least energy at low speeds. Impulsive running requires the least energy at high speeds. At a small range of intermediate speeds and large step lengths, a new gait we have dubbed "pendular run'' is found to be optimal.

Next, we provide an analytical proof of the energetic optimality for walking at low speeds and running at high speeds in an informal simplification of the minimal biped model.

Finally, we present simple models for the energetics of swinging the leg. Combining this simple leg-swing model with the previously derived model of the work done by the leg during stance, we find that as an animal moves faster, the ratio of the cost for swinging the leg to the cost of the work done during stance is approximately a constant, as has been shown in some experiments.


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