Probabilistic error bounds on constraint violation for empirical-analytical Lagrangian models of motion

Abstract

In contrast to many systems studied in the field of classical mechanics, models of animal motion are often distinguished in that they are both highly uncertain and evolve in a high-dimensional configuration space Q. Often it is either suspected or known that a particular motion regime evolves on or near some smaller subset $$Q_0\subseteq Q$$ . In some cases, $$Q_0$$ may itself be a submanifold of Q. A general strategy is presented in this paper for constructing empirical-analytical Lagrangian (EAL) models of the mechanics of such systems. It is assumed that the set $$Q_0\,\subseteq \,Q$$ is defined by a collection of unknown holonomic constraints on the full configuration space. Since the analytic form of the holonomic constraints is unknown, EAL models are defined that use experimental observations $$\{z_1,\ldots ,z_N\}\subseteq Q^N$$ to ensure that the approximate system models evolve near the underlying submanifold $$Q_0$$ . This paper gives a precise characterization of a probabilistic measure of the distance from the EAL model to the underlying submanifold.