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.

翻译：本文研究了多机器人导航问题，其中机器人在$n$维欧几里得空间中沿期望的$m$维（即$m$-D）流形运动，并保持一种灵活的空间顺序。我们考虑$m\geq 2$的情形，多机器人协调通过非欧几里得度量实现。然而，由于$m$维流形可由$n$个隐函数的零水平集表征，若辅助向量选择不当，则梯度向量场传播项的最后$m$个分量将与这些函数的偏导数形成强耦合。这种耦合不仅影响机器人在流形上的运动，也对进一步基于非欧几里得度量设计顺序可调的协调策略构成重大挑战。为解决该问题，我们首先确定了一组合适的辅助向量解，使得传播项的最后$m$个分量有效解耦为相同常数。随后，我们通过将$m$个流形参数视为额外的$m$个虚拟坐标，重新设计了协调梯度向量场算法，以增强奇点消除和全局收敛的优势。此外，我们允许每个机器人与其时变邻居及虚拟目标机器人共享$m$个虚拟坐标，实现了流形上顺序可调的运动协调，从而避免了使用欧几里得度量可能带来的复杂计算。最后，我们通过分别采用不同初始位置、高维流形及机器人故障的广泛仿真，验证了所提算法的灵活性、适应性与鲁棒性。