Radau to the rescue: efficient and accurate discretization for optimal control simulations of running

Biomechanical simulations based on optimal control provide valuable insights into muscle-driven locomotion when incorporating muscle-tendon dynamics. However, the influence of discretization methods in direct collocation for musculoskeletal simulations remains unexplored. We systematically evaluated six implicit Runge-Kutta schemes across gait tasks, dimensionalities, discretizations, and collocation densities. Radau IIa consistently minimized dynamic errors and outperformed Lobatto IIIc and Euler due to its numerical stability. Repeated-measures ANOVA showed significant method-node interactions. Backward Euler was fastest but least dynamically consistent, while two-stage Radau IIa offered a practical trade-off between dynamic consistency and computational effort. This paper provides guidelines for efficient optimal control in musculoskeletal modeling with implicit muscle dynamics.