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

Abstract

Experimental results presented in the literature suggest that humans use a position control strategy to indirectly control force rather than direct force control. Modeling the muscle-tendon system as a third-order linear model, we provide an explanation of why an indirect force control strategy is preferred. We analyzed a third-order muscle system and verified that it is required for a faithful representation of muscle-tendon mechanics, especially when investigating critical damping conditions. We provided numerical examples using biomechanical properties of muscles and tendons reported in the literature. We demonstrated that at maximum isotonic contraction, for muscle and tendon stiffness within physiologically compatible ranges, a third-order muscle-tendon system can be under-damped. Over-damping occurs for values of the damping coefficient included within a finite interval defined by two separate critical limits (such interval is a semiinfinite region in second-order models). An increase in damping beyond the larger critical value would lead the system to mechanical instability. 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. Under such condition, combined with high muscle activation, oscillation of the system can be compensated only by active control.

An Agonist-Antagonist Pitch Production Model

“…From it, we can extract the damping ratio ζ and the undamped resonant frequency ω 0 , which are given by (2). If we plug in physiologically plausible parameters taken from the elbow muscles [11] into the SDM, we obtain the zero-pole diagram and corresponding impulse responses in Fig. 2.…”

“…To derive its resonant frequency ω 0 (4) we can use the impedance electro-mechanical analogy [1] to draw the equivalent electrical circuit, find its input impedance Z i (jω), and equate its imaginary part to 0 [7]. Unfortunately, there is no straightforward solution for the damping ratio ζ [11].…”

“…We can see that the system reaches critical damping only for k ≥ 8, when the two imaginary poles reach the real axis, and is underdamped over most of the parameter range. In fact, for k ≥ 8 the Hill model exhibits underdamped oscillatory behaviour independent of its damping c [11]. …”

“…The physiological plausibility of the proposed model is grounded on the assumption that we can group all of the muscles responsible for the production of the pitch into two equivalent opposing muscles, whilst still being small enough to merit the small muscle assumption in the Hill model. This has been common practice when modelling muscle systems [11] and is also justified by the correlation seen in the activation of the CT and the VOC [16].…”

“…Research suggests that the SDM model is too simple to capture the basic mechanics of muscle activation [10]. On the other hand, the Hill type model while offering improved modelling of muscle-tendon dynamics, exhibits underdamped behaviour when using physiologically plausible muscle parameters [11]. In this paper we propose an agonist-antagonist pitch production (A2P2) model [12] and analyse how it relates to recent results in physiological intonation modelling.…”