MOFM-Nav: On-Manifold Ordering-Flexible Multi-Robot Navigation

TL;DR

This paper introduces a novel multi-robot navigation algorithm on manifolds that maintains flexible spatial ordering, decouples complex interactions, and enhances robustness through a redesigned coordinated GVF approach.

Contribution

It proposes a new decoupling method for on-manifold navigation, a redesigned GVF algorithm, and a flexible ordering mechanism using virtual coordinates for multi-robot systems.

Findings

Effective decoupling of manifold parameters improves navigation stability.

Enhanced global convergence and singularity elimination.

Demonstrated robustness and flexibility in diverse simulation scenarios.

Abstract

This paper addresses the problem of multi-robot navigation where robots maneuver on a desired \(m\)-dimensional (i.e., \(m\)-D) manifold in the $n$-dimensional Euclidean space, and maintain a {\it flexible spatial ordering}. We consider $ m\geq 2$, and the multi-robot coordination is achieved via non-Euclidean metrics. However, since the $m$-D manifold can be characterized by the zero-level sets of $n$ implicit functions, the last $m$ entries of the GVF propagation term become {\it strongly coupled} with the partial derivatives of these functions if the auxiliary vectors are not appropriately chosen. These couplings not only influence the on-manifold maneuvering of robots, but also pose significant challenges to the further design of the ordering-flexible coordination via non-Euclidean metrics. To tackle this issue, we first identify a feasible solution of auxiliary vectors such that the last $m$ entries of the propagation term are effectively decoupled to be the same constant. Then, we redesign the coordinated GVF (CGVF) algorithm to {\it boost} the advantages of singularities elimination and global convergence by treating $m$ manifold parameters as additional $m$ virtual coordinates. Furthermore, we enable the on-manifold ordering-flexible motion coordination by allowing each robot to share $m$ virtual coordinates with its time-varying neighbors and a virtual target robot, which {\it circumvents} the possible complex calculation if Euclidean metrics were used instead. Finally, we showcase the proposed algorithm's flexibility, adaptability, and robustness through extensive simulations with different initial positions, higher-dimensional manifolds, and robot breakdown, respectively.