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

TL;DR

This work addresses multi-robot navigation on an $m$-D manifold in $\mathbb{R}^n$ with flexible ordering, where traditional GVF methods struggle due to coupling in the propagation term. It proposes MOFM-Nav by introducing a higher-dimensional GVF (HGVF) with $m$ virtual coordinates, identifying a feasible set of $m-1$ auxiliary vectors to decouple the propagation terms, and redesigning the GVF to treat the manifold parameters as virtual coordinates for global convergence. Ordering-flexible coordination is achieved by having robots share the $m$ virtual coordinates with time-varying neighbors and a virtual target robot, enabling non-Euclidean metric-based interactions on the manifold. The approach is validated by extensive simulations on helicoid and 3-torus manifolds, including scenarios with robot breakdowns, demonstrating strong flexibility, adaptability, and robustness for on-manifold multi-robot coordination.

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.

Figures (7)

• Figure 2: First Simulation: Flexibility of the MOFM-Nav on the Helicoid Manifold with Different Initial Positions in Cases 1-2. (a)-(b) Trajectories of the $7$ robots controlled by the CGVF controller \ref{['MOFM_navigation']} under different initial positions. (c)-(d) Trajectories of the corresponding virtual coordinates $\omega_{i,j}$, where $i \in \mathcal{V}$ and $j = 1, 2, 3$. In (a)-(b), the blue and red triangles represent the initial and final positions of the robots, respectively, while the dashed lines indicate their trajectories. Similarly, in (c)-(d), the blue and red circles denote the initial and final positions of the virtual coordinates $\bm{\omega}_i$ for $i \in \mathcal{V}$, with the lines showing their trajectories.
• Figure 3: Temporal evolution of the on-manifold convergence errors $\|\Phi_i\|$, $i=1, \cdots, 7$, the condition of ordering-flexible coordination ${1}/{N}\sum_{i=1}^N\omega_{i,j}(t)-\omega_{\ast,j}, j=1, 2, 3$ in Definition \ref{['def_MOFM_navigation']}, and the relative distance between arbitrary two virtual coordinates $\|\bm\omega_i-\bm\omega_k\|, \forall i\neq k\in\mathcal{V}$ in Case 1 of Fig. \ref{['helicold_surface']} (a), (c) for example.
• Figure 4: Temporal evolution of the derivative of the virtual coordinates $\dot{\omega}_{i,j}, i\in\mathcal{V}, j=1,2, 3$, in Case 1 of Fig. \ref{['helicold_surface']} (a), (c) for example.
• Figure 5: Second Simulation: Adaptability to the higher-dimensional 3-Torus Manifold in Cases 1-2. (a) The $3$-D projection of the 3-torus manifold in \ref{['3_torus_equa']} with the fourth coordinate represented by varying colors. (b)-(c) The $3$-D projection of the first three coordinates of the $7$ robots, governed by the CGVF controller \ref{['MOFM_navigation']}, under different initial positions in Cases 1-2. (d)-(e) Side views of the subfigures (b)-(c). (f)-(g) Trajectories of the corresponding three virtual coordinates $\omega_{i,j}$, where $i \in \mathcal{V}$ and $j = 1, 2, 3$ in Cases 1-2. Here, the blue and red triangles, blue and red circles, and dashed colored lines have the same meanings as those in Fig. \ref{['helicold_surface']}.
• Figure 6: Third Simulation: Robustness to Some Robots Breakdown in Cases 1-2. (a)-(b) Trajectories of the $7$ robots when robots $i = 1, 4, 5$ and $i = 2, 3, 6$ suddenly breakdown, respectively. (c)-(b) Trajectories of the corresponding virtual coordinates $\omega_{i,j}, i \in \mathcal{V}$ and $j = 1, 2, 3$ under the scenariso of robots breakdown. Here, the blue and red triangles, blue and red circles, and dashed colored lines have the same meanings as those in Fig. \ref{['helicold_surface']}. The purple numbers represent the breakdown robots.

Theorems & Definitions (16)

• Definition 1
• Definition 2
• Definition 3
• Example 1
• Lemma 1
• proof
• Lemma 2
• proof
• Lemma 3
• proof