Third-Order Muscle Models: The Role of Oscillatory Behavior in Force Control

Abstract

This paper presents the analysis of a third-order linear differential equation representing a muscle-tendon system, including the identification of critical damping conditions. We analytically verified that this model is required for a faithful representation of muscle-skeletal muscles and provided numerical examples using the biomechanical properties of muscles and tendon reported in the literature. We proved the existence of a theoretical threshold for the ratio between tendon and muscle stiffness above which critical damping can never be achieved, thus resulting in an oscillatory free response of the system, independently of the value of the damping. Oscillation of the limb can be compensated only by active control, which requires creating an internal model of the limb mechanics. We demonstrated that, when admissible, over-damping of the muscle-tendon system occurs for damping values included within a finite interval between two separate critical limits. The same interval is a semi-infinite region in second-order models. Moreover, an increase in damping beyond the second critical point rapidly brings the system to mechanical instability.

Critical Damping Conditions for Third Order Muscle Models: Implications for Force Control

“…The presence of a tendon elastic element favors an oscillatory response to external disturbances that normally needs to be eliminated [9]. We demonstrated that a higher order model always exhibits an oscillatory free response and cannot be critically dampened for mechanical parameters included within the normal physiological range.…”

Therapist recognition of impaired muscle groups in simulated multi-joint hypertonia

“…The ratio between muscle stiffness and rigidity of the connective tissue has been reported to vary between 1:1 and 1:10 [21–24]. …”

Haptic recognition of dystonia and spasticity in simulated multi-joint hypertonia

“…(4) so to achieve five equally spaced reflex gains levels. Literature reported ratios between $K_m^{\mathit{\text{level}}}$ and α to vary between 1 and 10 [28–31]. Since we assumed the intrinsic muscle stiffness α to be $K_m^{\mathit{\text{level}}}/4$, the variation of $K_m^{\mathit{\text{level}}}$ following eq.…”