Online Synthesis of Control Barrier Functions with Local Occupancy Grid Maps for Safe Navigation in Unknown Environments

TL;DR

This work tackles safe navigation in unknown environments by online synthesis of a single, differentiable Control Barrier Function (CBF) derived from local occupancy grid maps. It formulates a steady-state thermal field-inspired CBF (SSTF-CBF) by imposing Dirichlet boundary conditions on obstacle and safety-region boundaries and solving a sparse Laplace equation to obtain a harmonic safety function $h(\mathbf{p})$. The CBF constraint $L_{\mathbf{f}}h(\mathbf{p}) + L_{\mathbf{g}}h(\mathbf{p})\mathbf{u} \ge -\alpha(h(\mathbf{p}))$ is enforced online via a quadratic program, enabling real-time safety guarantees with a single constraint regardless of obstacle count or shape. The method is validated in both Gazebo simulations and real-world TurtleBot experiments, achieving millisecond-scale synthesis times on $200 \times 200$ OGMs and demonstrating robust obstacle avoidance without collisions, thereby offering a scalable, perception-driven safety mechanism for unknown environments.

Abstract

Control Barrier Functions (CBFs) have emerged as an effective and non-invasive safety filter for ensuring the safety of autonomous systems in dynamic environments with formal guarantees. However, most existing works on CBF synthesis focus on fully known settings. Synthesizing CBFs online based on perception data in unknown environments poses particular challenges. Specifically, this requires the construction of CBFs from high-dimensional data efficiently in real time. This paper proposes a new approach for online synthesis of CBFs directly from local Occupancy Grid Maps (OGMs). Inspired by steady-state thermal fields, we show that the smoothness requirement of CBFs corresponds to the solution of the steady-state heat conduction equation with suitably chosen boundary conditions. By leveraging the sparsity of the coefficient matrix in Laplace's equation, our approach allows for efficient computation of safety values for each grid cell in the map. Simulation and real-world experiments demonstrate the effectiveness of our approach. Specifically, the results show that our CBFs can be synthesized in an average of milliseconds on a 200 * 200 grid map, highlighting its real-time applicability.

Figures (7)

• Figure 1: (Left) The robot's navigation environment, where orange regions denote obstacles and the blue region represents the robot's local sensing range. (Right) The real-time occupancy grid map generated during navigation, where black grids indicate obstacle-occupied areas while blue grids correspond to free space.
• Figure 2: Obstacle distribution and color map of the corresponding steady-state thermal field safety function $h(\mathbf{p})$, computed with $a=b=1$.
• Figure 3: A simple example of the domain definition for our SSTF-CBF is provided, clearly displaying the boundary conditions and the safety values that need to be calculated in transition domain $\mathcal{T}$. In this example, region $O$ and $\partial O$ are identical.
• Figure 4: A rough schematic illustrating the robot entering an unsafe region $\mathcal{U}_c$ after passing through $p_0 \in \partial \mathcal{C}$ where the gradient of $h(\mathbf{p})$ vanishes.
• Figure 5: (Left) Obstacle distribution and Turtlebot's trajectory during sequential navigation. (Right) Color map of SSTF-CBF generated by the entire map. The three target points are consistently represented by five-pointed stars in different colors across both figures.

Theorems & Definitions (8)

• Definition 1: Forward Invariant Sets ames_control_2019
• Definition 2: Control Barrier Functions li2023robust
• Proposition 1
• proof
• Proposition 2
• proof
• Proposition 3
• proof