Numerical instability of Hill-type muscle models

 Hill-type muscle models are highly preferred as phenomenological modelsfor musculoskeletal simulation studies despite their introduction almost acentury ago. The use of simple Hill-type models in simulations, instead ofmore recent cross-bridge models, is well justified since computationally‘light-weight’—although less accurate—Hill-type models have great valuefor large-scale simulations. However, this article aims to invite discussionon numerical instability issues of Hill-type muscle models in simulationstudies, which can lead to computational failures and, therefore, cannotbe simply dismissed as an inevitable but acceptable consequence ofsimplification. We will first revisit the basic premises and assumptions onthe force–length and force–velocity relationships that Hill-type models arebased upon, and their often overlooked but major theoretical limitations.We will then use several simple conceptual simulation studies to discusshow these numerical instability issues can manifest as practical compu-tational problems. Lastly, we will review how such numerical instabilityissues are dealt with, mostly in an ad hoc fashion, in two main areas ofapplication: musculoskeletal biomechanics and computer animation.  Hill-type muscle models are highly preferred as phenomenological modelsfor musculoskeletal simulation studies despite their introduction almost acentury ago. The use of simple Hill-type models in simulations, instead ofmore recent cross-bridge models, is well justified since computationally‘light-weight’—although less accurate—Hill-type models have great valuefor large-scale simulations. However, this article aims to invite discussionon numerical instability issues of Hill-type muscle models in simulationstudies, which can lead to computational failures and, therefore, cannotbe simply dismissed as an inevitable but acceptable consequence ofsimplification. We will first revisit the basic premises and assumptions onthe force–length and force–velocity relationships that Hill-type models arebased upon, and their often overlooked but major theoretical limitations.We will then use several simple conceptual simulation studies to discusshow these numerical instability issues can manifest as practical compu-tational problems. Lastly, we will review how such numerical instabilityissues are dealt with, mostly in an ad hoc fashion, in two main areas ofapplication: musculoskeletal biomechanics and computer animation.