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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June 2012
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Potential of the Pseudo-Inverse Method as a Constrained Static Optimization for Musculo-Tendon Forces Prediction
Florent Moissenet,
e-mail: florent.moissenet@mailoo.org
Florent Moissenet
Rehazenter, Laboratoire d’Analyse du Mouvement et la Posture
, 1 rue André Vésale, L-2674 Luxembourg, Luxembourg
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Laurence Chèze,
Laurence Chèze
Université de Lyon, F-69622 Lyon, France; Laboratoire de Biomécanique et Mécanique des Chocs, UMR_T9406; Université Lyon 1, Villeurbanne;IFSTTAR,
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Raphaël Dumas
Raphaël Dumas
Université de Lyon, F-69622 Lyon, France; Laboratoire de Biomécanique et Mécanique des Chocs, UMR_T9406; Université Lyon 1, Villeurbanne;IFSTTAR,
Bron
Search for other works by this author on:
Florent Moissenet
Rehazenter, Laboratoire d’Analyse du Mouvement et la Posture
, 1 rue André Vésale, L-2674 Luxembourg, Luxembourg
e-mail: florent.moissenet@mailoo.org
Laurence Chèze
Université de Lyon, F-69622 Lyon, France; Laboratoire de Biomécanique et Mécanique des Chocs, UMR_T9406; Université Lyon 1, Villeurbanne;IFSTTAR,
Bron
Raphaël Dumas
Université de Lyon, F-69622 Lyon, France; Laboratoire de Biomécanique et Mécanique des Chocs, UMR_T9406; Université Lyon 1, Villeurbanne;IFSTTAR,
Bron
J Biomech Eng. Jun 2012, 134(6): 064503 (7 pages)
Published Online: June 13, 2012
Article history
Received:
April 7, 2011
Revised:
May 15, 2012
Posted:
May 28, 2012
Published:
June 13, 2012
Online:
June 13, 2012
Citation
Moissenet, F., Chèze, L., and Dumas, R. (June 13, 2012). "Potential of the Pseudo-Inverse Method as a Constrained Static Optimization for Musculo-Tendon Forces Prediction." ASME. J Biomech Eng. June 2012; 134(6): 064503. https://doi.org/10.1115/1.4006900
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