Variational obstacle avoidance problem on riemannian manifolds

Abstract

In this letter we study variational obstacle avoidance problems on complete Riemannian manifolds. The problem consists of minimizing an energy functional depending on the velocity, covariant acceleration and a repulsive potential function used to avoid a static obstacle on the manifold, among a set of admissible curves. We derive the dynamical equations for extrema of the variational problem, in particular on compact connected Lie groups and Riemannian symmetric spaces. Numerical examples are presented to illustrate the proposed method.

Dynamic interpolation for obstacle avoidance on Riemannian manifolds

“…In order to extremize the functional J on the set Ω one needs to compare the value of J at a curve x ∈ Ω to the value of J at a nearby curvex ∈ Ω, using one-parameter admissible variations of x in Ω. We recently proved in [5] the following result.…”

“…In the absence of velocity constraints, the model studied in this example corresponds with a free planar rigid body. The trajectory planning without interpolation points for the obstacle avoidance problem of a planar rigid body was studied using a similar framework previously by the authors in [5] (Section V-A).…”

“…In this paper, we aim to generate trajectories interpolating prescribed points and avoiding multiple obstacles in the workspace via the study of a second order variational problem on a Riemannian manifold M by an extension of the results presented in [5] for variational obstacle avoidance without interpolation points. We call this problem dynamic interpolation for obstacle avoidance.…”

Variational dynamic interpolation for kinematic systems on trivial principal bundles

“…A variational approach to such problems on Riemannian manifolds and compact connected Lie groups is presented in [4]. The further extensions to nonholonomic systems using sub-Riemannian geometry is explained in [5], [6], whereas reduction techniques using Lie group symmetries is presented in [7].…”

Variational collision avoidance problems on Riemannian manifolds

“…In Crouch and Silva Leite [11] the authors have used it to develop a theory of generalized cubic polynomials for dynamic interpolation problems on Riemannian manifolds. More recently, Bloch, Camarinha and Colombo [3] have used these variational methods to solve obstacle avoidance problems on Riemannian manifolds. In this article, inspired by the recent work [3], we seek to extend this method to find necessary conditions for optimal trajectories of multiple agents on a Riemannian manifold that seek to achieve a specified configuration while avoiding collisions among themselves.…”