Is there a Trade-off between Economy and Task Goal Variability in Transfemoral Amputee Gait?

Background: Energy cost minimization has been widely accepted to regulate gait. Optimization principles have been frequently used to explain how individuals adapt their gait pattern. However, there have been rare attempts to account for the role of variability in this optimization process. Motor redundancy can enable individuals to perform tasks reliably while achieving energy optimization. However, we do not know how the goal-irrelevant and goal-relevant variability is regulated.. In this study, we investigated how unilateral transfemoral amputees regulate step and stride variability based on the task to achieve energy economy. Methods: Nine individuals with unilateral transfemoral amputation walked on a treadmill at speeds of 0.6,0.8,1.0,1.2 and 1.4 m/s using their prescribed passive prostheses. We calculated the step-to-step and stride-to-stride variability and applied goal equivalent manifold (GEM) based control to decompose goal-relevant and goal-irrelevant manifold. To quantify the energy economy, the energy recovery rate (R) was calculated based on potential energy and kinetic energy. Comparisons were made between GEM variabilities and commonly used standard deviation measurements. A linear regression model was used to investigate the trade-off between R and GEM variabilities. Results: Our analysis shows greater variability along the goal-relevant manifold compared to the goal-irrelevant manifold (p<.001). Moreover, our analysis shows lower energy recovery rate for amputee gait compared to nonamputee gait (at least 20% less at faster walking speed). We found a negative relationship between energy recovery rate and goal-irrelevant variability. Compared to the standard deviation measurements, the variability decomposed using GEM reflected the preferred walking speed and the limitation of the passive prosthetic device.Conclusion: Individuals with amputation cleverly leverage task redundancy, regulating step and stride variability to the goal equivalent manifold (GEM). This result suggests that task redundancy enables unilateral amputees to benefit from motor variability in terms of energy economy. The differences observed between prosthetic step and intact step support the development of prosthetic leg capable of enhancing positive work during the double support phase and of powered prosthesis controllers that allow for variability along the task space while minimizing variability that interferes with the task goal. This study provides a different perspective on amputee gait analysis and challenge the field to think differently about the role of variability.

Trial-to-trial dynamics and learning in a generalized, redundant reaching task

If humans exploit task redundancies as a general strategy, they should do so even if the redundancy is decoupled from the physical implementation of the task itself. Here, we derived a family of goal functions that explicitly defined infinite possible redundancies between distance ( D) and time ( T) for unidirectional reaching. All [ T, D] combinations satisfying any specific goal function defined a goal-equivalent manifold (GEM). We tested how humans learned two such functions, D/T = c (constant speed) and D·T = c, that were very different but could both be achieved by neurophysiologically and biomechanically similar reaching movements. Subjects were never explicitly shown either relationship, but only instructed to minimize their errors. Subjects exhibited significant learning and consolidation of learning for both tasks. Initial error magnitudes were higher, but learning rates were faster, for the D· T task than for the D/ T task. Learning the D/ T task first facilitated subsequent learning of the D· T task. Conversely, learning the D· T task first interfered with subsequent learning of the D/ T task. Analyses of trial-to-trial dynamics demonstrated that subjects actively corrected deviations perpendicular to each GEM faster than deviations along each GEM to the same degree for both tasks, despite exhibiting significantly greater variance ratios for the D/ T task. Variance measures alone failed to capture critical features of trial-to-trial control. Humans actively exploited these abstract task redundancies, even though they did not have to. They did not use readily available alternative strategies that could have achieved the same performance.

Inter-Trial Dynamics of Repeated Skilled Movements

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.