However, there are many more muscles than degrees of freedom in the human skeleton, and thus the muscle forces underlying a given motion cannot be uniquely calculated using rigid body dynamics. Since muscle forces are not directly measurable, these studies have been based on computational models. As a result, significant research effort has been dedicated to estimating muscle forces during normal (e.g., 1, 4, 20) and impaired (recently e.g., 13, 26, 31, 34) movement. 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.
The present approach lacks some of the major limitations of established methods such as static optimization and computed muscle control while remaining computationally efficient. Future work should focus on comparing the present approach to other approaches for computing muscle forces. 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. The formulation that used explicit contraction dynamics with muscle length as a state failed to converge in most cases. Problem formulation affected computational speed and robustness to the initial guess. 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. 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.
Four problem formulations were investigated for walking based on both a two and three dimensional model. This study sought to identify a robust and computationally efficient formulation for solving these dynamic optimization problems using direct collocation optimal control 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. Estimation of muscle forces during motion involves solving an indeterminate problem (more unknown muscle forces than joint moment constraints), frequently via optimization methods.
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