How Many Equations of Motion Describe a Moving Human?

TL;DR

This paper investigates the complexity of human motion by analyzing real-world mobility data to determine the minimal set of differential equations needed to accurately describe human trajectories.

Contribution

It provides an empirical method to identify the maximum order of derivatives necessary for modeling human movement, showing higher derivatives are linearly dependent on lower ones.

Findings

Higher-order derivatives after acceleration are linearly dependent on previous derivatives.

The measure is robust against noise and differentiation methods.

Results impose constraints on differential equations modeling human kinematics.

Abstract

A human is a thing that moves in space. Like all things that move in space, we can in principle use differential equations to describe their motion as a set of functions that maps time to position (and velocity, acceleration, and so on). With inanimate objects, we can reliably predict their trajectories by using differential equations that account for up to the second-order time derivative of their position, as is commonly done in analytical mechanics. With animate objects, though, and with humans, in particular, we do not know the cardinality of the set of equations that define their trajectory. We may be tempted to think, for example, that by reason of their complexity in cognition or behaviour as compared to, say, a rock, then the motion of humans requires a more complex description than the one generally used to describe the motion of physical systems. In this paper, we examine a real-world dataset on human mobility and consider the information that is added by each (computed, but denoised) additional time derivative, and find the maximum order of derivatives of the position that, for that particular dataset, cannot be expressed as a linear transformation of the previous. In this manner, we identify the dimensionality of a minimal model that correctly describes the observed trajectories. We find that every higher-order derivative after the acceleration is linearly dependent upon one of the previous time-derivatives. This measure is robust against noise and the choice for differentiation techniques that we use to compute the time-derivatives numerically as a function of the measured position. This result imposes empirical constraints on the possible sets of differential equations that can be used to describe the kinematics of a moving human.