On the relevance of structure preservation to simulations of muscle actuated movements

In this work, we implement a typical nonlinear Hill-type muscle model in a structure-preserving simulation framework and investigate the differences to standard simulations of muscle-actuated movements with MATLAB/Simulink. The latter is a common tool to solve dynamical problems, in particular, in biomechanic investigations. Despite the simplicity of the examples used for comparison, it becomes obvious that the MATLAB/Simulink integrators artificially loose or gain energy and angular momentum during dynamic simulations. The relative energy error of the MATLAB/Simulink integrators related to a very low actual muscle work can naturally reach large values, even higher than 100%. But also during periods with large muscle work, the relative energy error reaches up to 2%. Even in simulations with very small time steps, energy and angular momentum errors are still present using MATLAB/Simulink and can (at least partially) be responsible for phase errors in long-term simulations. This typical behaviour of commercial integrators is known to increase for more complex models or for computations with larger time steps, whose use is crucial for efficiency, especially in the context of optimal control simulations. In contrast to that, time-stepping schemes being derived from a discrete variational principle yield discrete analogues of the Euler-Lagrange equations and Noethers theorem. This ensures that the structure of the system is preserved, i.e. the simulation results are symplectic and momentum consistent and exhibit a good energy behaviour (no drift).