Variational dynamic interpolation for kinematic systems on trivial principal bundles
For researchers in geometric mechanics and robotics, this work provides a variational framework for trajectory planning on trivial principal bundles, but it is an incremental extension of existing geometric control methods.
This paper addresses the dynamic interpolation problem for locomotion systems on trivial principal bundles, generating trajectories through specified points by minimizing the squared L2-norm of covariant acceleration. The method is applied to the generalized Purcell's swimmer, a low Reynolds number microswimmer, demonstrating trajectory generation.
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.