Dealing with time-varying recruitment and length in Hill-type muscle models

Hill-type muscle models are often used in muscle simulation studies and also in the design and virtual prototyping of functional electrical stimulation systems. These models have to behave in a sufficiently realistic manner when recruitment level and contractile element (CE) length change continuously. For this reason, most previous models have used instantaneous CE length in the muscle׳s force vs. length (F-L) relationship, but thereby neglect the instability problem on the descending limb (i.e. region of negative slope) of the F-L relationship. Ideally CE length at initial recruitment should be used but this requires a multiple-motor-unit muscle model to properly account for different motor-units having different initial lengths when recruited. None of the multiple-motor-unit models reported in the literature have used initial CE length in the muscle׳s F-L relationship, thereby also neglecting the descending limb instability problem. To address the problem of muscle modelling for continuously varying recruitment and length, and hence different values of initial CE length for different motor-units, a new multiple-motor-unit muscle model is presented which considers the muscle to comprise 1000 individual Hill-type virtual motor-units, which determine the total isometric force. Other parts of the model (F-V relationship and passive elements) are not dependent on the initial CE length and, therefore, they are implemented for the muscle as a whole rather than for the individual motor-units. The results demonstrate the potential errors introduced by using a single-motor-unit model and also the instantaneous CE length in the F-L relationship, both of which are common in FES control studies.