Evidence for Composite Cost Functions in Arm Movement Planning: An Inverse Optimal Control Approach

Abstract

An important issue in motor control is understanding the basic principles underlying the accomplishment of natural movements. According to optimal control theory, the problem can be stated in these terms: what cost function do we optimize to coordinate the many more degrees of freedom than necessary to fulfill a specific motor goal? This question has not received a final answer yet, since what is optimized partly depends on the requirements of the task. Many cost functions were proposed in the past, and most of them were found to be in agreement with experimental data. Therefore, the actual principles on which the brain relies to achieve a certain motor behavior are still unclear. Existing results might suggest that movements are not the results of the minimization of single but rather of composite cost functions. In order to better clarify this last point, we consider an innovative experimental paradigm characterized by arm reaching with target redundancy. Within this framework, we make use of an inverse optimal control technique to automatically infer the (combination of) optimality criteria that best fit the experimental data. Results show that the subjects exhibited a consistent behavior during each experimental condition, even though the target point was not prescribed in advance. Inverse and direct optimal control together reveal that the average arm trajectories were best replicated when optimizing the combination of two cost functions, nominally a mix between the absolute work of torques and the integrated squared joint acceleration. Our results thus support the cost combination hypothesis and demonstrate that the recorded movements were closely linked to the combination of two complementary functions related to mechanical energy expenditure and joint-level smoothness.

Author Summary

To reach an object, the brain has to select among a set of possible arm trajectories that displace the hand from an initial to a final desired position. Because of the intrinsic redundancy characterizing the human arm, the number of admissible joint trajectories toward the goal is generally infinite. However, many studies have demonstrated that the range of actual trajectories can be limited to those that result from the fulfillment of some optimal rules. Various cost functions were shown to be relevant in the literature. A peculiar aspect of most of these costs is that each one of them aims at optimizing one specific feature of the movement. The necessary motor flexibility of everyday life, however, might rely on the combination of such cost functions rather than on a single one. Testing this cost combination hypothesis has never been attempted. To this aim we propose a reaching task involving target redundancy to facilitate the comparisons of different candidate costs and to identify the best-fitting one (possibly composite). Using a numerical inverse optimal control method, we show that most participants produced movements corresponding to a strict combination of two subjective costs linked to the mechanical energy consumption and the joint-level smoothness.