Variational dynamic interpolation for kinematic systems on trivial principal bundles

TL;DR

This paper develops a variational approach to generate smooth trajectories for locomotion systems on trivial principal bundles, incorporating constraints and symmetries, demonstrated on a microswimming mechanism.

Contribution

It introduces a novel variational interpolation method on trivial principal bundles that accounts for nonholonomic constraints and group symmetries.

Findings

Successfully generates trajectories passing through specified points.

Applies the method to a microswimming mechanism, demonstrating practical utility.

Provides explicit formulas for the Riemannian connection on trivial bundles.

Abstract

This article presents the dynamic interpolation problem for locomotion systems evolving on a trivial principal bundle $Q$. Given an ordered set of points in $Q$, we wish to generate a trajectory which passes through these points by synthesizing suitable controls. The global product structure of the trivial bundle is used to obtain an induced Riemannian product metric on $Q$. The squared $L^2-$norm of the covariant acceleration is considered as the cost function, and its first order variations are taken for generating the trajectories. The nonholonomic constraint is enforced through the local form of the principal connection and the group symmetry is employed for reduction. The explicit form of the Riemannian connection for the trivial bundle is employed to arrive at the extremal of the cost function. The result is applied to generate a trajectory for the generalized Purcell's swimmer - a low Reynolds number microswimming mechanism.