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

Bodily movements exhibit kinematic invariances, with the “one-third power law” relating velocity to curvature amongst the most established. 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. This article reviews prior power law calculation protocols, identifies suboptimal practices, before proposing solutions grounded in the kinematics literature and related fields of enquiry. Ultimately, we synthesise these 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 of modules useful in broader kinematic science applications. We propose that adoption of this protocol will eliminate spurious confirmation of the law and enable more sensitive quantification of recently noted power law divergences. These divergences have been linked to neurochemical disturbances arising from ingestion of dopaminergic drugs, and in neurological conditions such as Parkinson’s and autism.