It is widely accepted that humans and animals minimize energetic cost while walking. While such principles predict average behavior, they do not explain the variability observed in walking. For robust performance, walking movements must adapt at each step, not just on average. Here, we propose an analytical framework that reconciles issues of optimality, redundancy, and stochasticity. For human treadmill walking, we defined a goal function to formulate a precise mathematical definition of one possible control strategy: maintain constant speed at each stride. We recorded stride times and stride lengths from healthy subjects walking at five speeds. The specified goal function yielded a decomposition of stride-to-stride variations into new gait variables explicitly related to achieving the hypothesized strategy. Subjects exhibited greatly decreased variability for goal-relevant gait fluctuations directly related to achieving this strategy, but far greater variability for goal-irrelevant fluctuations. More importantly, humans immediately corrected goal-relevant deviations at each successive stride, while allowing goal-irrelevant deviations to persist across multiple strides. To demonstrate that this was not the only strategy people could have used to successfully accomplish the task, we created three surrogate data sets. Each tested a specific alternative hypothesis that subjects used a different strategy that made no reference to the hypothesized goal function. Humans did not adopt any of these viable alternative strategies. Finally, we developed a sequence of stochastic control models of stride-to-stride variability for walking, based on the Minimum Intervention Principle. We demonstrate that healthy humans are not precisely “optimal,” but instead consistently slightly over-correct small deviations in walking speed at each stride. Our results reveal a new governing principle for regulating stride-to-stride fluctuations in human walking that acts independently of, but in parallel with, minimizing energetic cost. Thus, humans exploit task redundancies to achieve robust control while minimizing effort and allowing potentially beneficial motor variability.

We study inter-trial movement fluctuations exhibited by human participants during the repeated execution of a virtual shuffleboard task. Focusing on skilled performance, theoretical analysis of a previously-developed general model of inter-trial error correction is used to predict the temporal and geometric structure of variability near a goal equivalent manifold (GEM). The theory also predicts that the goal-level error scales linearly with intrinsic body-level noise via the total body-goal sensitivity, a new derived quantity that illustrates how task performance arises from the interaction of active error correction and passive sensitivity properties along the GEM. Linear models estimated from observed fluctuations, together with a novel application of bootstrapping to the estimation of dynamical and correlation properties of the inter-trial dynamics, are used to experimentally confirm all predictions, thus validating our model. In addition, we show that, unlike “static” variability analyses, our dynamical approach yields results that are independent of the coordinates used to measure task execution and, in so doing, provides a new set of task coordinates that are intrinsic to the error-regulation process itself.

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In this paper, we develop a class of discrete dynamical systems for modeling repeated, goal-directed, kinematically redundant human movements. The approach is based on a mathematical definition of movement tasks in terms of goal functions. Each goal function can give rise to an associated goal equivalent manifold (GEM), which contains all body states that exactly satisfy the task requirements. A hierarchical control scheme involving in-trial action templates and inter-trial stochastic optimal error correction is included to generate a nonlinear map for the repeated execution of the task. A simple throwing task is used to illustrate the underlying concepts and to develop a model problem for further study. The performance at the goal level, as measured by the root mean square error, is shown to result from factors that are measures of passive sensitivity, the magnitude of body fluctuations, the orientation of fluctuations with the GEM, and the stability properties of the inter-trial controller. The action of the inter-trial controller developed for our model system is simulated and is shown to agree with the mathematically predicted performance.

A simulation model is described for determining the distribution of asperity flash temperatures when rough surfaces undergo relative sliding. The asperities are assumed to deform elastically and to have coulomb friction at their tips. The spherical asperity model of Greenwood-Williamson is joined with the flash temperature approximation formulas developed by Kuhlmann-Wilsdorf. Two example applications illustrate the effect of sliding speed and material role reversal on mean flash temperature. The model is applied to scuffing tests on ground and polished roller specimens reported in the literature. The predicted flash temperature is found to vary inversely with the experimentally observed scuffing loads within each finish type. For the same rolling and sliding speeds, the ground specimens had a lower observed scuffing load and a higher predicted mean flash temperature than the smoother polished specimens.