Prediction of Individual Muscle Forces Using Lagrange Multipliers Method — A Model of the Upper Human Limb in the Sagittal Plane: I. Theoretical Considerations

Abstract

Analytical solutions of indeterminate problems formulated for biomechanical models with more than one degree of freedom (DOF7rpar; are rarely found. This paper is an extension of the investigations of a 1 DOF model (Raikova, 1996, J. Biomechanics, 763-772) for a more complex 3 DOF model. The proposed model of the human upper limb is in the sagittal plane and includes ten muscle elements, four of them being two-joint ones. The formulated optimization task is solved analytically using the method of Lagrange multipliers. It is supposed that the optimization function is complex but can be approximated by a weighted sum of the squared muscle forces, where the nature of the weight factors of the muscles are unknown. The proposed computational algorithm for determination of the unknown individual muscle forces and joint reactions is easily implemented and may be extended without difficulties for more DOF and muscles. The aim is to establish a means of investigation of the possible weight coefficients for different modelled situations, which will help in searching for their physiological interpretation and analytical description.

Keywords:

• Modelling
• upper limb
• optimization
• indeterminate problem
• Lagrange multipliers