An analytical approach to the problem of inverse optimization with additive objective functions: an application to human prehension

Abstract

We consider the problem of what is being optimized in human actions with respect to various aspects of human movements and different motor tasks. From the mathematical point of view this problem consists of finding an unknown objective function given the values at which it reaches its minimum. This problem is called the inverse optimization problem. Until now the main approach to this problems has been the cut-and-try method, which consists of introducing an objective function and checking how it reflects the experimental data. Using this approach, different objective functions have been proposed for the same motor action. In the current paper we focus on inverse optimization problems with additive objective functions and linear constraints. Such problems are typical in human movement science. The problem of muscle (or finger) force sharing is an example. For such problems we obtain sufficient conditions for uniqueness and propose a method for determining the objective functions. To illustrate our method we analyze the problem of force sharing among the fingers in a grasping task. We estimate the objective function from the experimental data and show that it can predict the force-sharing pattern for a vast range of external forces and torques applied to the grasped object. The resulting objective function is quadratic with essentially non-zero linear terms.

Fig. 1

Schematic representation of the handle. Here T stands for the external torque, F^l for the load force, $F_0^n\text{and}{F}_0^t$ for the normal and tangential components of the thumb force, $F_i^n\text{and}{F}_i^t$ for the normal and tangential components of the finger forces (i = 1, …, 4). The arrows define positive values of the forces

Fig. 2

An example of the normal forces of the index finger in different conditions. a The load force is fixed and equal to 12.5N, while the external torque and the total grip force are varied. The different symbols correspond to the values obtained for 100, 125, 150 and 175% of the natural grip force. b The load force and the external torque are varied, while the grip force is kept equal to 150% of the natural one. The different symbols correspond to different values of the load force: 12.5, 15.0, 17.5 and 20N

Fig. 3

The correlation between the experimental data and the solution of the direct optimization problem for the normal finger forces. Each point corresponds to a particular combination of the external torque, the load force and the grip force. The errors were computed as average absolute difference between the experimental data and the corresponding solutions of the direct optimization problem