Velocity-dependent cost function for the prediction of force sharing among synergistic muscles in a one degree of freedom model

Prediction of accurate and meaningful force sharing among synergistic muscles is a major problem in biomechanics research. Given a resultant joint moment, a unique set of muscle forces can be obtained from this mathematically redundant system using nonlinear optimization. The classical cost functions for optimization involve a normalization of the muscle forces to the absolute force capacity of the target muscles, usually by the cross-sectional area or the maximal isometric force. In a one degree of freedom model this leads to a functional relationship between moment arms and the predicted muscle forces, such that for constant moment arms, or constant ratios of moment arms, agonistic muscle forces increase or decrease in unison. Experimental studies have shown however that the relationship between muscle forces is highly task-dependent often causing forces to increase in one muscle while decreasing in a functional agonist, likely because of the contractile conditions and contractile properties of the involved muscles. We therefore, suggest a modified cost function that accounts for the instantaneous contraction velocity of the muscles and its effect on the instantaneous maximal force. With this novel objective function, a task-dependent prediction of muscle force distribution is obtained that allows, even in a one degree of freedom system, the prediction of force sharing loops, and simultaneously increasing and decreasing forces for agonist pairs of muscles.