Biological kinematics: a detailed review of the velocity-curvature power law calculation.

The 'one-third power law', relating velocity to curvature is among the most established kinematic invariances in bodily movements. Despite being heralded amongst the 'kinematic laws of nature' (Flash 2021, p. 4), there is no consensus on its origin, common reporting practice, or vetted analytical protocol. Many legacy elements of analytical protocols in the literature are suboptimal, such as noise amplification from repeated differentiation, biases arising from filtering, log transformation distortion, and injudicious linear regression, all of which undermine power law calculations. Recent findings of power law divergences in clinical populations have highlighted the need for improved protocols. This article reviews prior power law calculation protocols, identifies suboptimal practices, before proposing candidate solutions grounded in the kinematics literature. We evaluate these candidates via two simple criteria: firstly, they must avoid spurious confirmation of the law, secondly, they must confirm the law when it is present. Ultimately, we synthesise candidate solutions into a vetted, modular protocol which we make freely available to the scientific community. The protocol's modularity accommodates future analytical advances and permits re-use in broader kinematic science applications. We propose that adoption of this protocol will eliminate artificial confirmation of the law and facilitate more sensitive quantification of recently noted power law divergences, which are associated with neurochemical disturbances arising from dopaminergic drugs, and in conditions such as Parkinson's and autism.