Evaluation of Direct Collocation Optimal Control Problem Formulations for Solving the Muscle Redundancy Problem.

Friedl De Groote,Allison L Kinney,Benjamin J Fregly,Anil V Rao

doi:10.1007/s10439-016-1591-9

Friedl De Groote, Allison L Kinney + Show 2 more

Open Access

https://doi.org/10.1007/s10439-016-1591-9

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Abstract

Estimation of muscle forces during motion involves solving an indeterminate problem (more unknown muscle forces than joint moment constraints), frequently via optimization methods. When the dynamics of muscle activation and contraction are modeled for consistency with muscle physiology, the resulting optimization problem is dynamic and challenging to solve. This study sought to identify a robust and computationally efficient formulation for solving these dynamic optimization problems using direct collocation optimal control methods. Four problem formulations were investigated for walking based on both a two and three dimensional model. Formulations differed in the use of either an explicit or implicit representation of contraction dynamics with either muscle length or tendon force as a state variable. The implicit representations introduced additional controls defined as the time derivatives of the states, allowing the nonlinear equations describing contraction dynamics to be imposed as algebraic path constraints, simplifying their evaluation. Problem formulation affected computational speed and robustness to the initial guess. The formulation that used explicit contraction dynamics with muscle length as a state failed to converge in most cases. In contrast, the two formulations that used implicit contraction dynamics converged to an optimal solution in all cases for all initial guesses, with tendon force as a state generally being the fastest. Future work should focus on comparing the present approach to other approaches for computing muscle forces. The present approach lacks some of the major limitations of established methods such as static optimization and computed muscle control while remaining computationally efficient.Electronic Supplementary MaterialThe online version of this article (doi:10.1007/s10439-016-1591-9 contains supplementary material, which is available to authorized users.

Highlights

Knowledge of muscle forces during healthy and impaired movement could facilitate the development of improved treatments for disorders affecting walking ability or improved training programs to increase athlete performance
This study evaluated four possible optimal control problem formulations for solving the muscle redundancy problem while taking muscle activation and contraction dynamics into account
All formulations converged from at least one initial guess for the simple model, the formulations that used explicit contraction dynamics failed to converge for all cases that were evaluated in this study (Tables 1, 2)

Summary

Introduction

Knowledge of muscle forces during healthy and impaired movement could facilitate the development of improved treatments for disorders affecting walking ability or improved training programs to increase athlete performance. Optimization methods have been used to resolve this redundancy by assuming that human movement is produced by optimizing some performance criterion.[24]. The numerical challenges arising from the use of optimization methods have led to a trade-off between computational efficiency and consistency with muscle physiology.[5] When the dynamics of muscle activation and contraction are modeled for consistency with muscle physiology, the resulting optimization problem is dynamic and challenging to solve due to the nonlinearity and stiffness of the equations describing muscle dynamics (i.e., muscle activation and contraction dynamics).[29]. The dynamic optimization problem is solved using direct shooting (e.g.,3,4,18–21). Direct shooting methods parametrize the controls, in this case muscle excitations, and solve for control parameters that optimize the cost function.

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