An indeterminate problem for an upper limb model with four biarticular muscles and its three modifications - analytical and numerical solution and sensitivity analysis

R. Raikova, Z. Ivanova, S. Angelova

Abstract: Analytical solutions of indeterminate problems formulated for models with more than one degree of freedom (DoF) that contain biarticular muscles, have been rarely reported. However, the analytical solution is exact and provides useful information regarding the dependence of optimal solution (predicted muscle forces) on the model parameters and on the optimisation function. In this study, we aimed: to compare four variants of an upper limb model in the sagittal plane with two DoF and different numbers of biarticular and monoarticular muscles (BASMOD – four biarticular muscles; MOD1 and MOD2 – two biarticular and two monoarticular muscles, MOD3 – four monoarticular muscles); to solve analytically and numerically the optimisation tasks; to perform sensitivity analysis. The analytical solution was obtained using the Lagrange multipliers method. It did not provide possible optimal solutions for muscle forces for MOD1, MOD2, and MOD3, but for BASMOD, such solutions exist for some combinations of the model parameters. The analytical solutions, when they exist, coincide with the numerical solutions. The results from the sensitivity analysis indicate that the muscle force moment arms and physiological cross-sectional areas can change not only the values of the predicted forces, but also the number and set of active muscles.

Series on Biomechanics, Vol.36 No.2 (2022), 3-17
DOI: 10.7546/SB.36.2022.02.01

Keywords: analytical versus numerical method; Optimization task; upper arm model

Date published: 2022-06-17