Efficient embedding of empirically-derived constraints in the ODE formulation of multibody systems: Application to the human body musculoskeletal system

We present a novel method for deriving the governing equations of the musculoskeletal system, a new class of multibody systems in which the constituent components are connected together via anatomical joints which behave differently compared with traditional mechanical joints. In such systems, the kinematics of the joints and the corresponding constraints are characterized experimentally. We generate the equations of motion of these complex systems in which the homogeneous transformation matrices become matrix-valued functions of the generalized coordinate vector due to the empirical expression of body coordinates as smooth functions of generalized coordinates. The detailed mathematical procedure is provided to derive each term of the equations of motion using the novel calculus for the efficient evaluation of the partial derivatives of matrix-valued functions with respect to a vector. The governing equations obtained using the presented technique are expressed with ordinary differential equations rather than algebraic differential equations while not suffering from any simplification in experimental data describing the kinematics of the system. We then apply this method to derive the equations of motion of the “Andrews’ squeezer mechanism” for the validation. Furthermore, we successfully use this technique to model the shoulder rhythm with empirically-derived constraints in a trajectory tracking problem.