Collaborative Graph Exploration with Reduced Pose-SLAM Uncertainty via Submodular Optimization

TL;DR

This work tackles collaborative graph exploration in GPS-denied environments by separating exploration path planning from pose-graph reliability enhancement. It introduces a two-stage framework where VRP-based paths provide quick coverage and a subsequent loop-edge selection stage uses a $D$-optimal topology proxy $\mathbf{L}_{\gamma}$ to approximate SLAM uncertainty, enabling submodular optimization with guarantees. The core contributions include formulating a non-monotone submodular loop-edge selection objective with distance penalties, proving submodularity, applying $1/2$-guaranteed algorithms with ordering heuristics, and an MILP-based allocation for inter-robot loops; experiments on random graphs validate the topology-uncertainty link and compare algorithmic performance. The approach offers a scalable, principled path-planning tool for multi-robot SLAM with practical open-source implementations. The framework has potential impact for autonomous exploration in complex, GPS-denied environments by balancing efficiency and pose-estimation reliability.

Abstract

This paper considers the collaborative graph exploration problem in GPS-denied environments, where a group of robots are required to cover a graph environment while maintaining reliable pose estimations in collaborative simultaneous localization and mapping (SLAM). Considering both objectives presents challenges for multi-robot pathfinding, as it involves the expensive covariance inference for SLAM uncertainty evaluation, especially considering various combinations of robots' paths. To reduce the computational complexity, we propose an efficient two-stage strategy where exploration paths are first generated for quick coverage, and then enhanced by adding informative and distance-efficient loop-closing actions, called loop edges, along the paths for reliable pose estimation. We formulate the latter problem as a non-monotone submodular maximization problem by relating SLAM uncertainty with pose graph topology, which (1) facilitates more efficient evaluation of SLAM uncertainty than covariance inference, and (2) allows the application of approximation algorithms in submodular optimization to provide optimality guarantees. We further introduce the ordering heuristics to improve objective values while preserving the optimality bound. Simulation experiments over randomly generated graph environments verify the efficiency of our methods in finding paths for quick coverage and enhanced pose graph reliability, and benchmark the performance of the approximation algorithms and the greedy-based algorithm in the loop edge selection problem. Our implementations will be open-source at https://github.com/bairuofei/CGE.

Figures (6)

• Figure 1: The graph exploration with three robots in a $100m\times 100m$ 2D graph environment (light gray). The robots' exploration paths are colored in cyan, green, and blue, respectively, with the starting positions marked with red stars. The selected loop edges are colored in red, and the valid candidate loop edges are colored in black. The resulting exploration paths efficiently cover the whole graph, while forming a well-connected multi-robot pose graph topology to reduce SLAM uncertainty with informative and distance-efficient loop edges.
• Figure 2: The framework of the proposed method, which takes inputs of a graph representation of the environment $\mathcal{G}$ and robots' initial positions $\{v^{0}_{r}\}_{r\in \mathcal{R}}$, and finally outputs the robots' paths $\{\mathcal{P}_{r}\}_{r\in\mathcal{R}}$ that can cover the environment while resulting in a well-connected multi-robot pose graph.
• Figure 3: (a) The relationship between $\log\det(\mathbf{L}_{\gamma}^{-1})$ and $\log\det(\mathbb{I}^{-1})$ evaluated on a set of pose graphs derived from a $120m\times 120m$ graph environment; (b) The pose estimation uncertainty decreases as more loop edges are added into the collaborative pose graph.
• Figure 4: The objective gain of Problem \ref{['prob_edge_selection']} with the five algorithms in $50$ independent experiments. Note the results are sorted according to the objective value of the sGre algorithm for better visualization.
• Figure 5: (a) The objective improvement ratio of the dUSM and sGre algorithms w.r.t. the dGre algorithm; (b) The objective improvement ratio after adding the ordering heuristics to dGre and dUSM algorithms.

Theorems & Definitions (11)

• Definition 1: Submodularity
• Definition 2: Monotonicity
• Definition 3: abstracted pose graph
• Remark 1
• Remark 2
• Definition 4: Loop edge
• Remark 3
• Proposition 1
• proof
• Proposition 2