A discrete variational approach to muscle wrapping in musculoskeletal optimal control simulations - Penner, Johann (Friedrich-Alexander-Universität Erlangen-Nürnberg, 2023)
This work shows the development of a simulation method for musculoskeletal models used for in-silico human posture experiments. Here, the application of discrete variational calculus to muscle wrapping problems allows the prediction of the action and path of muscles around joints, in conjunction with skeletal movements represented as a multibody system. Specifically, we tread the multibody system’s motion and the muscle’s action in a variational context. The resulting simulation model reflects the human anatomical structure with Hill-type muscle actuation. This muscle model requires the calculation of the muscle paths, their lengths, and changes in length during the movement, which are determined by the muscle wrapping problem. In the sequel, we refer to the combination of the multibody system and Hill-type muscle actuation as a musculoskeletal model and approximate the action of the muscle as a geodesic curve between the muscle’s origin and insertion. In order to solve the geodesic path problem and the skeletal equation of motion, we transfer principles of the calculus of variations to their discrete counterparts. The Hill-type muscle forces and the forces acting on the skeleton are discretized analogously. A vital advantage of the variational formulation is that the structure-preserving properties of the integrator enable the simulation to account for large, rapid changes in muscle paths at relatively moderate computational costs. In particular, the derived muscle wrapping formulation does not rely on special case solutions, has no nested loops, has a modular structure, and is entirely described by algebraic equations. Furthermore, we consider applications in which a musculoskeletal model is intended to perform specific movement tasks while information about the corresponding required muscle activities and forces are of interest. Modeling this simulation task as an optimal control problem and approximating its solution numerically is a well-suited procedure to obtain such information. The optimal control formulation in this work is based on the direct transcription method DMOCC and comprises the minimization of an objective function subject to the fulfillment of the discrete Euler-Lagrange equations. To solve this problem, we also focus on the numerical tools needed to solve a large-scale non-linear constrained optimization problem. Finally, we show simulation results for specific motion tasks and their ad-hoc validation through motion capture. <br