On the Principled Description of Human Movements

TL;DR

This paper introduces a principled mathematical model of human movement based on optimal control of jerk, deriving a constant of motion that characterizes movement performance and relates to established laws like Fitts' law.

Contribution

It develops a novel formalism using optimal control theory to describe human movements, including a constant of motion analogous to energy in mechanics, and applies it to real movement data.

Findings

The model aligns with Fitts' law for rapid movements.

A constant of motion effectively characterizes movement performance.

Solutions within the model can exhibit oscillatory behaviors similar to tremor.

Abstract

While the use of technology to provide accurate and objective measurements of human movement performance is presently an area of great interest, efforts to quantify the performance of movement are hampered by the lack of a principled model that describes how a subject goes about making a movement. We put forward a principled mathematical formalism that describes human movements using an optimal control model in which the subject controls the jerk of the movement. We construct the formalism by assuming that the movement a subject chooses to make is better than the alternatives. We quantify the relative quality of movements mathematically by specifying a cost functional that assigns a numerical value to every possible movement; the subject makes the movement that minimizes the cost functional. We develop the mathematical structure of movements that minimize a cost functional, and observe that this development parallels the development of analytical mechanics from the Principle of Least Action. We derive a constant of the motion for human movements that plays a role that is analogous to the role that the energy plays in classical mechanics. We apply the formalism to the description of two movements: (1) rapid, targeted movements of a computer mouse, and (2) finger-tapping, and show that the constant of the motion that we have derived provides a useful value with which we can characterize the performance of the movements. In the case of rapid, targeted movements of a computer mouse, we show how the model of human movement that we have developed can be made to agree with Fitts' law, and we show how Fitts' law is related to the constant of the motion that we have derived. We finally show that solutions exist within the model of human movements that exhibit an oscillatory character reminiscent of tremor.