On the System-Theoretic Passivity Properties of a Hill Muscle Model

The paper describes passivity-related input-output properties of a human muscle and tendon system given by a Hill type dynamic model. For a model having muscle contraction velocity as the input and force as the output, it is shown that the system is passive during the concentric phase. Also, it is shown that a negative strict passivity margin exists for the eccentric phase if the length and lengthening velocity of the contractile element are assumed bounded. Estimates of this margin are given by means of two alternative formulas. Further, it is shown that the mapping from contraction velocity to deviation from equilibrium force is passive in both concentric and eccentric phases. The paper discusses how these findings and the passivity theorem can be used to design controllers for a machine coupled to the muscle model by feedback interconnection. The simple case of a proportional-derivative force feedback regulator is considered as an example. A simulation example is given where the transient response of the coupled system crosses the eccentric region.