Multibody Dynamics of Biomechanical Models for Human Motion via Optimization

Abstract

The human motion analysis, for gait or for most of other activities, relies mostly on the use of multibody formulations applied as kinematic or dynamic tools. In many biomechanical applications to gait analysis the choice between using direct or inverse dynamics to obtain the solution of the problem, even in pure kinematics, only depends on the personal preference of the user and not in any particular form of the data available or structure of the equations to be solved. In this work the structure of the equations of a multibody system are reviewed for direct and inverse dynamic analysis. It is shown that if the time dependencies of all degrees-of-freedom of the system are known the inverse dynamics is equivalent to a direct dynamics problem. This equivalence is particularly useful when the problem of the biomechanical analysis consists in finding the muscle forces in an over-actuated biomechanical model that leads to a prescribed motion, which is obtained by using video data acquisition or simply by designing such motion. The problem can then be solved by using optimization procedures in which the objective functions are physiological criteria and, eventually, a measure of matching the prescribed motion. If not used as part of the objective function the prescribed motion is introduced in the optimization problem as nonlinear constraints. The variables of the optimization problem are, for all type of analysis, the muscle forces, directly, or their corresponding muscle activations. It is shown that the natural choice for design variables of the optimal problem is the muscle activations. Two representations of the time history of the muscle actuation are tested in this work: the input sampling where the activations are found in a finite number of time instants and then linearly interpolated in between; the smooth exponential function approach where the actuation is described by a sum of exponential functions being the width and the size of the bumps of each of the functions the unknown quantities. Then the muscle forces are simply obtained by using a Hill type muscle model where the state of force-velocity and the forcelength relations are obtained directly from the kinematics of the biomechanical model. The methods presented in this work are demonstrated and discussed in the framework of two problems associated to the human locomotion apparatus.