Optimization of targeted movements

An algorithm was developed for the evaluation of targeted movements, between an initial and a final configuration, possibly with some intermediate configurations. The algorithm was primarily aimed at robotic and musculoskeletal movement simulations, and has been found efficient and reliable for problems with moderate numbers of displacement coordinates, even for complicated dynamic formulations, [

1

,

2

]. The result from the algorithm contains the configuration as function of time, but also a set of a priori unknown control forces needed to create the desired motion.

The algorithm is based on a temporal finite element interpolation of displacements and controls, with a third order Hermitian form for the displacement components and a linear interpolation for the control forces. This allows the governing differential equation of the movement to be satisfied at two collocation points in each of the time intervals.

With sufficient freedom in the description of the control forces, they are chosen to optimize some measure of the movement or the controls. A common criterion is to minimize the needed forces, by formulating a cost function, which is the sum of square-integrated force components. Another interesting possibility is to seek the smoothest movement, which leads to a cost function based on the jerk values for the displacements, following an idea by Flash and Hogan, [

3

]. Both criteria were introduced in the algorithm.

Examples have shown that the introduced optimization criteria strongly influence the obtained solutions. Even for a simple movement of, e.g., the arm seen in a sagittal view, the obtained movements, and therefore the needed forces, are evaluated as rather different, with different criteria. This observation is independent of whether the muscular forces are seen as a redundant set of individual forces, or are summed to resultant acting moments at the considered joints.

The paper will describe the developed algorithm, the studied optimization criteria, and the conclusions from a set of examples concerned with bio-mechanical targeted movement.