PolyMerge: Compressing 3D Gaussian Splats with Polytope Coverings for Provably Safe Resource-Constrained Navigation

Autonomous robot navigation requires rapid construction of obstacle-points-separated convex regions for trajectory planning. When obstacles are represented as point clouds from LiDAR or depth cameras, these regions must be constructed directly from finite obstacle samples while providing suitable constraints for downstream optimization. However, existing corridor-generation methods often struggle to meet real-time, sensor-rate computational requirements. To address this bottleneck, we introduce RISP, a randomized algorithm that constructs convex polytopes from finite point-cloud data, and PathCover, which chains these polytopes along a reference path to form an overlapping corridor. We prove finite termination, sequential intersection, and complete path coverage with respect to the supplied finite point set. Under a probabilistic elimination condition, the sampling-and-elimination stage of RISP has expected O(n) time and unconditional worst-case O(n^2) time. Extensive benchmarks on synthetic and real-world LiDAR datasets demonstrate an order-of-magnitude speedup over state-of-the-art methods in both corridor generation as well as trajectory optimization, while closed-loop quadrotor simulation and a physical quadruped traversal demonstrate integration with downstream motion planners. These results establish corridor separation from the supplied point-cloud representation and practical implementation feasibility. Source code of the entire pipeline is available at https://github.com/kunalnk123690/PathCover.

Fast quadrotor flight requires safe obstacle avoidance under tight onboard compute limits. While 3D Gaussian Splatting (3DGS) provides a continuous, geometry-aware scene representation for perception-driven navigation, existing 3DGS safety filters use reduced-order models such as single- and double-integrators that ignore actuator limits and assume commanded accelerations are realized instantaneously. Building on an analytic collision cone barrier for 3DGS, we introduce a nonlinear, actuator-aware safety filter enforced through the full quadrotor dynamics. We derive a high-relative-degree collision cone exponential CBF and a backup CBF that preserves QP feasibility under input constraints using a forward-simulated backup policy. Compared with a state-of-the-art 3DGS safety filter, our approach reduces trajectory jerk by 47% and runs 2.25 times faster. We validate the method in simulation and on hardware for real-time navigation in cluttered, perception-derived environments.

Active perception in uncertain environments requires robots to navigate safely while acquiring informative observations to reduce map uncertainty. These objectives inherently conflict, as informative viewpoints often lie near uncertain regions with higher collision risk. To address this challenge, we develop a conflict-aware active perception and control framework for robotic systems operating in environments represented by 3D Gaussian Splatting (3DGS). Safety is enforced using a Control Barrier Function (CBF) derived from an Average Value-at-Risk AV@R collision-risk metric that accounts for geometric uncertainty and guarantees forward invariance of a safe set. To improve perception, we propose a risk-aware Expected Information Gain (EIG) formulation for selecting the next-best-view and introduce perception barrier functions that align the camera orientation with the local information-ascent direction. To obtain a tractable formulation for these conflicting safety and perception objectives, we propose a unified safety-critical, perception-aware quadratic program that enforces safety as a hard constraint while relaxing perception constraints through slack variables. Simulation results demonstrate that the proposed method improves both safety and information acquisition compared to existing 3DGS-based approaches.