Beyond Geometry: Topologically Grounded and Resource-Efficient Hierarchical Planning for Real-Time 3D Navigation

ABSTRACT: Long-horizon ground robots answer thousands of navigation queries on persistent 3D maps, making repeated planning a computational burden. We present a topology-grounded framework combining seed-reachable surface extraction, persistent-homology room segmentation, and a compact Transition Graph (TG). Surface extraction removes more than 80% of states in furniture-rich scenes. Room segmentation selects its persistence threshold automatically from map statistics, without using query outcomes. Paired boundary portals, cached Dijkstra-tree paths, and a consistent directed landmark bound reduce online computation. On 100 Habitat-Matterport 3D (HM3D) scenes and 500 queries at 0.1 m resolution, TG with one portal per boundary achieves a 0.054 ms median query time, 2.52× faster than corrected surface ALT (A*, landmarks, and triangle inequality). Five portals reduce median geometric path overhead to 3.27%, with 2.24× speedup. All three configurations return 451/500 positive-length paths, all passing the surface-adjacency check. Directed search costs match independent Dijkstra references in a separate targeted validation. Earlier cross-dataset measurements provide additional context: 452× faster search than 3D2M with 100% versus 66% success, and an approximate Geekbench-normalized cross-platform pipeline speedup exceeding 10× over point cloud tomography (PCT). These results support topology-grounded planning for repeated queries on persistent maps.