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Research Papers

# Distribution of Forces Between Synergistics and Antagonistics Muscles Using an Optimization Criterion Depending on Muscle Contraction Behavior

[+] Author and Article Information
Carlos Rengifo1

Institut de Recherche en Communications et Cybernétique de Nantes UMR 6597, Ecole Centrale de Nantes, Université de Nantes, 1 de la Noë, 44321 Nantes, Francecarlos.rengifo@irccyn.ec-nantes.fr

Yannick Aoustin, Franck Plestan, Christine Chevallereau

Institut de Recherche en Communications et Cybernétique de Nantes UMR 6597, Ecole Centrale de Nantes, Université de Nantes, 1 de la Noë, 44321 Nantes, France

Notation $T$ denotes matrix transposition.

1

Corresponding author.

J Biomech Eng 132(4), 041009 (Mar 19, 2010) (10 pages) doi:10.1115/1.4001116 History: Received November 23, 2009; Revised January 22, 2010; Posted January 27, 2010; Published March 19, 2010; Online March 19, 2010

## Abstract

In this paper, a new neuromusculoskeletal simulation strategy is proposed. It is based on a cascade control approach with an inner muscular-force control loop and an outer joint-position control loop. The originality of the work is located in the optimization criterion used to distribute forces between synergistic and antagonistic muscles. The cost function and the inequality constraints depend on an estimation of the muscle fiber length and its time derivative. The advantages of a such criterion are exposed by theoretical analysis and numerical tests. The simulation model used in the numerical tests consists in an anthropomorphic arm model composed by two joints and six muscles. Each muscle is modeled as a second-order dynamical system including activation and contraction dynamics. Contraction dynamics is represented using a classical Hill’s model.

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## Figures

Figure 10

Distribution of forces using criterion 21(α=1/3). Normalized musculotendon forces versus time. Desired force (dotted line). Musculotendon force (solid line). Maximal and minimal achievable musculotendon forces (dashed lines).

Figure 11

Distribution of forces using criterion 27. Normalized musculotendon forces versus time (s). Desired force (dotted line). Musculotendon force (solid line). Maximal and minimal achievable musculotendon forces (dashed lines).

Figure 12

Muscular activations versus time (s). Criterion 21 (dashed line). Criterion 27 (solid line).

Figure 13

Elbow position tracking error versus time (s). Criterion 21 (dashed line). Criterion 27 (solid line).

Figure 1

Scheme of a musculotendon unit

Figure 2

Hill-type model of a musculotendon unit (15)

Figure 3

Active force: the force-length and force-velocity relationships

Figure 4

Contraction dynamics: the output of the model is the musculotendon force ft and the external inputs are the activation level a and the musculotendon length lmt

Figure 5

Minimal normalized musculotendon force: it is obtained when the muscular activation is at its minimal value (a=10−6)

Figure 6

Maximal normalized musculotendon force: it is obtained when the muscular activation is at its maximal value (a=1)

Figure 7

Schematic representation of the anthropomorphic arm. The arm motion is made by six muscles: four monoarticular ones (m1, m2, m3, m4) and two biarticular ones (m5, m6).

Figure 8

Anthropomorphic arm block diagram. U=[u1⋯u6]T is the muscular excitations vector (and then the control input vector). A=[a1⋯a6]T is the muscular activations vector. Ft=[Ft1⋯Ft6]T is the musculotendon forces vector. Lmt=[lmt1⋯lmt6]T is the musculotendon fiber length vector. Γ=[Γ1 Γ2]T is the joint torques vector. q=[q1 q2]T is the joint positions vector.

Figure 9

The proposed musculoskeletal simulation strategy. U=[u1⋯u6]T is the muscular excitations vector (and then the control input vector). Ft=[Ft1⋯Ft6]T is the musculotendon forces vector. Lmt=[lmt1⋯lmt6]T is the musculotendon fiber length vector. Γ=[Γ1 Γ2]T is the joint torques vector. q=[q1 q2]T is the joint positions vector. Γd, Ftd, and qd, respectively, represent the desired values of the variables Γ, Ft, and q.

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