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Research Papers

# A Rigorous Dynamical-Systems-Based Analysis of the Self-Stabilizing Influence of Muscles

[+] Author and Article Information
Melih Eriten, Harry Dankowicz

Department of Mechanical Science and Engineering, University of Illinois at Urbana-Champaign, Urbana, IL 61801

In the case that $A(t)=∂xf(x̃(t))$, where $f$ is not explicitly dependent on time and $x̃(t+T)=x̃(t)$ for all $t$, it follows thatDisplay Formula

$ddtf(x̃(t))=∂xf(x̃(t))⋅f(x̃(t))$
(33)
i.e., that $f(x̃(0))$ is an eigenvector with Floquet multiplier $1$ of the corresponding fundamental matrix. As this eigenvector corresponds to a perturbation in the direction of the reference time history, this introduces a drift along the reference time history but does not affect the local stability in directions transversal to the reference time history. In the event that the remaining Floquet multipliers are less than $1$ in magnitude, the reference time history is said to be orbitally asymptotically stable.

J Biomech Eng 131(1), 011011 (Nov 26, 2008) (8 pages) doi:10.1115/1.3002758 History: Received June 28, 2007; Revised July 18, 2008; Published November 26, 2008

## Abstract

In this paper, dynamical systems analysis and optimization tools are used to investigate the local dynamic stability of periodic task-related motions of simple models of the lower-body musculoskeletal apparatus and to seek parameter values guaranteeing their stability. Several muscle models incorporating various active and passive elements are included and the notion of self-stabilization of the rigid-body dynamics through the imposition of musclelike actuation is investigated. It is found that self-stabilization depends both on muscle architecture and configuration as well as the properties of the reference motion. Additionally, antagonistic muscles (flexor-extensor muscle couples) are shown to enable stable motions over larger ranges in parameter space and that even the simplest neuronal feedback mechanism can stabilize the repetitive motions. The work provides a review of the necessary concepts of stability and a commentary on existing incorrect results that have appeared in literature on muscle self-stabilization.

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## Figures

Figure 1

Shaded region in the figure identifies the (α,β) values that satisfy the inequalities given in Eq. 14

Figure 2

The rate of decay δ (left scale) toward the trivial equilibrium reference time history Δx(t)≡0 and the fraction Tfrac (right scale) of the period during which the real parts of the eigenvalues 24 are negative as functions of ε

Figure 3

A schematic of the two-link model of the human lower limb including attachment (A) and insertion points (I1 and I2) and a representative muscle model consisting active and passive elements. The muscle models specifically used in the simulations can be obtained by the following kPEE and kSEE values: CE: kPEE=0 and kSEE→∞, CE+SEE: kPEE=0, CE+PEE: kSEE→∞. Finally, the CE+PD model differs from the CE model in that the activation function E is an explicit function of state and time.

Figure 7

Regions in the k−c gain parameter space in which the reference time history is stable (solid) and unstable (white)

Figure 4

The time evolution of the activation function E(t) for the CE for each muscle model. Here, the CEi2 notation refers to the two-muscle model, where i=1 corresponds to the extensor and i=2 corresponds to the flexor.

Figure 5

An example of a nonrealizable activation function in the case of a single extensor muscle with only a contractile element

Figure 6

Variations in the magnitude of the largest-in-magnitude Floquet multiplier as a function of the stiffness kPEE of the parallel elastic element. In particular, for kPEE>11.25kNm−1 all Floquet multipliers lie within the unit circle in the complex plane. Values of kPEE beyond ≈60kNm−1 do not correspond to realizable activation functions.

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