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Technical Briefs

Potential of the Pseudo-Inverse Method as a Constrained Static Optimization for Musculo-Tendon Forces Prediction

[+] Author and Article Information
Florent Moissenet

 Rehazenter, Laboratoire d’Analyse du Mouvement et la Posture, 1 rue André Vésale, L-2674 Luxembourg, Luxembourgflorent.moissenet@mailoo.org

Laurence Chèze, Raphaël Dumas

Université de Lyon, F-69622 Lyon, France; Laboratoire de Biomécanique et Mécanique des Chocs, UMR_T9406; Université Lyon 1, Villeurbanne;IFSTTAR, Bron

For more information, see https://simtk.org/home/kneeloads

J Biomech Eng 134(6), 064503 (Jun 13, 2012) (7 pages) doi:10.1115/1.4006900 History: Received April 07, 2011; Revised May 15, 2012; Posted May 28, 2012; Published June 13, 2012; Online June 13, 2012

Inverse dynamics combined with a constrained static optimization analysis has often been proposed to solve the muscular redundancy problem. Typically, the optimization problem consists in a cost function to be minimized and some equality and inequality constraints to be fulfilled. Penalty-based and Lagrange multipliers methods are common optimization methods for the equality constraints management. More recently, the pseudo-inverse method has been introduced in the field of biomechanics. The purpose of this paper is to evaluate the ability and the efficiency of this new method to solve the muscular redundancy problem, by comparing respectively the musculo-tendon forces prediction and its cost-effectiveness against common optimization methods. Since algorithm efficiency and equality constraints fulfillment highly belong to the optimization method, a two-phase procedure is proposed in order to identify and compare the complexity of the cost function, the number of iterations needed to find a solution and the computational time of the penalty-based method, the Lagrange multipliers method and pseudo-inverse method. Using a 2D knee musculo-skeletal model in an isometric context, the study of the cost functions isovalue curves shows that the solution space is 2D with the penalty-based method, 3D with the Lagrange multipliers method and 1D with the pseudo-inverse method. The minimal cost function area (defined as the area corresponding to 5% over the minimal cost) obtained for the pseudo-inverse method is very limited and along the solution space line, whereas the minimal cost function area obtained for other methods are larger or more complex. Moreover, when using a 3D lower limb musculo-skeletal model during a gait cycle simulation, the pseudo-inverse method provides the lowest number of iterations while Lagrange multipliers and pseudo-inverse method have almost the same computational time. The pseudo-inverse method, by providing a better suited cost function and an efficient computational framework, seems to be adapted to the muscular redundancy problem resolution in case of linear equality constraints. Moreover, by reducing the solution space, this method could be a unique opportunity to introduce optimization methods for a posteriori articulation of preference in order to provide a palette of solutions rather than a unique solution based on a lot of hypotheses.

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Copyright © 2012 by American Society of Mechanical Engineers
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Figures

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Figure 4

Estimated musculo-tendon forces (expressed in BW) using penalty-based, Lagrange multipliers and pseudo-inverse methods and EMG signals on a same muscle during 100% of gait cycle (the toe-off is indicated by a vertical line)

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Figure 1

From the left to the right: 2D knee musculo-skeletal model (adapted from Ref. [16]), segment parameters, joints kinematics and musculo-tendon geometry of the 3D lower limb musculo-skeletal model (adapted from Ref. [18])

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Figure 2

Isovalue curves of the cost functions in the (fpl,fh) plane with minimal cost function areas (dashed lines) and solutions for each optimization method (black cross): (a) penalty method (for three values of the penalty factor k), (b) Lagrange multipliers method, (c) pseudo-inverse method. The solution space line corresponding to the linear equality constraint ceq is also given (doubled dashed line).

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Figure 3

Isovalue curves of the cost function with level details of Lagrange multipliers method in the (λ,fpl) constraint plane with fh computed using Eq. 6

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