FMT$^{\mathrm{X}}$: Lazy Wavefront Search for Dynamic Replanning

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arXiv:2509.08521v2 Announce Type: replace-cross Abstract: FMT$^{*}$ plans efficiently in static worlds by expanding a cost-ordered wavefront and collision-checking lazily, but its single-pass unvisited rule cannot revise paths when obstacles change. We present FMT$^{\mathrm{X}}$, an anytime, asymptotically optimal generalization of that wavefront for dynamic replanning. A cost-improvement test replaces the unvisited set, allowing a node to be revisited for best-parent selection whenever a lower-cost potential connection is found. This induces implicit rewiring within the wavefront while preserving lazy collision checking. FMT$^{\mathrm{X}}$ retains the online densification of RRT$^{*}$ but defers its eager neighborhood rewiring to the cost-ordered wavefront, so a node is revisited only when the expansion reaches it. Obstacle updates orphan the affected subtree and reseed the wavefront. A direct cost push from each parent to its children propagates cost improvements through validated tree edges, thereby preserving descendant cost consistency, a property not guaranteed by implicit rewiring alone. We compare a fixed-graph batch variant, Dynamic FMT$^{*}$ (D-FMT$^{*}$), with D$^{*}$ Lite on identical PRM$^{*}$ graphs, and FMT$^{\mathrm{X}}$ with eager RRT$^{\mathrm{X}}$ and path-centric LLPT$^{*}$ in geometric and kinodynamic scenes, including partial observability. D-FMT$^{*}$ stays close on path quality at far fewer collision checks. FMT$^{\mathrm{X}}$ lies between LLPT$^{*}$ and RRT$^{\mathrm{X}}$ on repair effort while tracking the reliability and trajectory quality of …

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