Inverse dynamics-based static optimization
Inverse dynamics-based static optimization is a method for estimating muscle-tendon forces from the measured (e.g. through gait analysis) kinematics of a given body part. It exploits the concepts of inverse dynamics and static optimization (in opposition to dynamic programming). Joint moments are obtained by inverse dynamics and then, knowing muscular moment arms, a static optimization process is carried on to evaluate optimal single-muscle forces for the system formula_1 which is an underdetermined system. General concepts. We can solve the inverse dynamics of a system to obtain joint torques and nonetheless be unable  to estimate the forces exerted by single muscles even knowing the exact geometry of our joints and muscles due to the redundancy of our system. Through an optimization approach we could find a way to understand how our central nervous system chooses its control strategies so as to optimize some aspects of movement production (e.g. minimizing metabolic cost). Dynamic equations of motion. We use here the matricial form of the equations of motion formula_2 in which we are considering a body part with formula_3 joints and formula_4 muscles. Then The vector of muscle-tendon torques can be further decomposed as  follows formula_17 in which The static optimization process. Once we obtain formula_15, suppose formula_18 known from anatomo-physiological studies and that we can't obtain analytically muscle-tendon forces due to the redundancy of the system. Then we hypothesize that the actual muscle forces minimize a given cost function, formula_24, subject to equality and inequality constraints. We have then to solve formula_25 Usually this is written as formula_26 in which The choice of the cost function. Our choice of the cost function is based on the supposed optimization mechanisms carried on by our CNS. It needs to be clinically validated , especially in unhealthy patients. In [Erdemir, 2007] a list of possible cost functions with a brief rationale and the suggested model validation technique is available. Clarification on the use of the maximum isometric force. Muscle contraction can be eccentric (velocity of contraction formula_28), concentric (formula_29) or isometric (formula_30). From muscle force-velocity characteristic we notice that muscle force in an eccentric contraction is higher than the maximum isometric force, why then do we use it as a constraint on muscle force? Mainly for two reasons: