FMT$^{x}$: An Efficient and Asymptotically Optimal Extension of the Fast Marching Tree for Dynamic Replanning

Path planning in dynamic environments remains a core challenge in robotics, especially as autonomous systems are deployed in unpredictable spaces such as warehouses and public roads. While algorithms like Fast Marching Tree (FMT$^{*}$) offer asymptotically optimal solutions in static settings, their single-pass design prevents path revisions which are essential for real-time adaptation. On the other hand, full replanning is often too computationally expensive. This paper introduces FMT$^{x}$, an extension of the Fast Marching Tree algorithm that enables efficient and consistent replanning in dynamic environments. We revisit the neighbor selection rule of FMT$^{*}$ and demonstrate that a minimal change overcomes its single-pass limitation, enabling the algorithm to update cost-to-come values upon discovering better connections without sacrificing asymptotic optimality or computational efficiency. By maintaining a cost-ordered priority queue and applying a selective update condition that uses an expanding neighbor to identify and trigger the re-evaluation of any node with a potentially suboptimal path, FMT$^{x}$ ensures that suboptimal routes are efficiently repaired as the environment evolves. This targeted strategy preserves the inherent efficiency of FMT$^{*}$ while enabling robust adaptation to changes in obstacle configuration. FMT$^{x}$ is proven to recover an asymptotically optimal solution after environmental changes. Experimental results demonstrate that FMT$^{x}$ outperforms the influential replanner RRT$^{x}$, reacting more swiftly to dynamic events with lower computational overhead and thus offering a more effective solution for real-time robotic navigation in unpredictable worlds.