⚡ electrical engineering

A Closed-Form Dual-Barrier CBF Safety Filter for Holonomic Robots on Incrementally Built Occupancy Grid Maps

This paper presents a computationally efficient, closed-form dual-barrier control barrier function safety filter that ensures collision-free navigation for holonomic robots in incrementally built occupancy grid maps by simultaneously avoiding mapped obstacles and restricting entry into unexplored regions, as validated by zero-collision hardware experiments on a quadrotor.

Original authors: Himanshu Paudel, Basanta Joshi, Dhirendra Raj Madai, Alina Bartaula, Biman Rimal, Sanjay Neupane

Published 2026-05-07

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CC BY 4.0

Original authors: Himanshu Paudel, Basanta Joshi, Dhirendra Raj Madai, Alina Bartaula, Biman Rimal, Sanjay Neupane

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine a small, autonomous drone flying through a room it has never seen before. It's building a mental map of the room as it goes, cell by cell, using its cameras. The challenge is tricky: the drone needs to be brave enough to fly toward the unknown to learn more, but careful enough not to crash into walls it hasn't seen yet or fly into "black holes" where it has no idea what's there.

This paper presents a smart safety filter for such drones. Think of it as an invisible, super-fast bodyguard that sits between the drone's "brain" (which decides where to go) and its "legs" (the motors).

Here is how it works, broken down into simple concepts:

1. The Two "No-Go" Zones

The drone's safety guard enforces two strict rules simultaneously:

Rule A: Don't hit the furniture. If the drone sees a wall or a table on its map, it must stay a safe distance away.
Rule B: Don't step off the edge of the map. This is the unique part. If the drone is flying toward a corner of the room that hasn't been mapped yet, the guard says, "Stop! We don't know what's there. It could be a wall, a hole, or a cat." The drone is forbidden from entering areas where it has zero information.

2. The "Mathematical Rubber Band"

To enforce these rules without slowing the drone down too much, the authors use something called a Control Barrier Function (CBF).

The Analogy: Imagine the drone is a dog on a leash. The "leash" isn't a physical rope; it's a mathematical rubber band that gets tighter the closer the dog gets to a wall or an unmapped area.
The Magic: Usually, calculating how to pull the dog back without breaking the leash requires a heavy computer to solve a complex puzzle every fraction of a second. This paper found a shortcut. They figured out a "closed-form" solution.
The Result: Instead of solving a complex puzzle, the drone just solves a tiny, simple math problem (like a 2x2 grid) in a split second. This is so fast and light that it can run on a cheap, small computer (like a Raspberry Pi) attached to the drone, leaving plenty of power for the drone to actually see and map the room.

3. The "Smart Leash" (Adaptive Gain)

The safety filter is also clever about how tight it pulls the leash.

In the Unknown: If the drone is in a foggy, unmapped area, the filter loosens the leash slightly. It says, "Okay, we don't know what's there, but we need to explore, so let's get a little closer to the edge to see what's happening."
In the Known: If the drone is in a room it has already mapped perfectly, the filter tightens the leash. It says, "We know exactly where the walls are; stay back and be safe."
This allows the drone to be "adventurous" when necessary but "conservative" when safe, making exploration much faster.

4. The "Soft Landing" (The Fallback)

Sometimes, the two rules might conflict in a weird way (like if the wall and the edge of the map are pointing in the exact same direction). In these rare, confusing moments, the filter has a "soft" mode. Instead of getting stuck, it finds the least dangerous path possible, even if it means slightly bending the rules, just to keep the drone moving safely.

5. Real-World Proof

The team tested this on a real drone flying indoors.

The Result: The drone flew around for over two minutes, exploring a large area.
The Safety Record: Zero crashes. The drone never touched a wall or flew into the unknown.
The Efficiency: Because the safety filter was so fast and light, the drone could explore a much larger area (about 81 square meters) compared to a standard drone using older safety methods (which only covered about 56 square meters). The older method was too scared to go near the edges; this new filter was brave enough to go right up to the edge without crashing.

Summary

In short, the authors built a lightweight, super-fast safety net for drones. It uses simple math to keep the drone away from known walls and unknown voids. It's smart enough to relax its grip when exploration is needed and tightens it when safety is at risk. It runs on cheap hardware and proved in real life that it can keep a drone safe while letting it explore more than it could before.

Technical Summary: A Closed-Form Dual-Barrier CBF Safety Filter for Holonomic Robots on Incrementally Built Occupancy Grid Maps

Problem Statement
The paper addresses the challenge of safety-critical velocity control for holonomic robots (specifically quadrotors) operating in environments where maps are built incrementally via Simultaneous Localization and Mapping (SLAM). In such scenarios, the robot faces two simultaneous, often conflicting, safety requirements: avoiding mapped obstacles and avoiding unmapped regions where obstacle geometry is entirely unknown. Classical reactive controllers, such as Artificial Potential Fields (APF), often lack formal safety guarantees and can get trapped in local minima. Conversely, Model Predictive Control (MPC), while capable of handling constraints, imposes a computational burden that is often prohibitive for resource-constrained onboard computers (e.g., Raspberry Pi) running high-frequency perception and planning pipelines. The core problem is to design a safety layer that enforces both obstacle avoidance and frontier containment with minimal computational overhead, ensuring formal safety guarantees without modifying the underlying nominal controller.

Methodology
The authors propose a Dual-Barrier Control Barrier Function (CBF) safety filter that operates directly on occupancy grid maps. The methodology is structured as follows:

Dual Safety Constraints: The system defines two safe sets based on Signed Distance Fields (SDFs) derived from the occupancy grid:
- Obstacle Constraint ( $C_1$ ): Ensures the robot maintains a standoff distance from mapped obstacles.
- Frontier Constraint ( $C_2$ ): Prevents the robot from entering unmapped regions (beyond the explored frontier) where collision risks are unquantifiable.
Admissible Shaping Class: Instead of using raw SDF values, the authors compose the SDFs with a shaping function $T$ from an admissible class $\mathcal{T}$ (monotone, Lipschitz-continuous, sign-preserving, and bounded). They instantiate this class using the hyperbolic tangent function ( $\tanh$ ), which provides numerical stability and compact derivatives.
Closed-Form Solution: Rather than solving a general Quadratic Program (QP) at every control cycle, the authors derive an analytical solution for the dual-constraint projection. By enumerating the four Karush-Kuhn-Tucker (KKT) active-set cases, the filter reduces the computational task to solving at most a $2 \times 2$ linear system.
Fallback Mechanism: In the singular case where the gradients of the obstacle and frontier constraints align (making the hard QP infeasible), a "soft-QP" fallback is employed. This uses a bisection method to find the least-violating safe velocity with minimal slack.
Adaptive Gain Scheduling: The gain parameter ( $\gamma_2$ ) for the frontier constraint is scheduled online based on local uncertainty density. The constraint is relaxed in information-rich regions (allowing aggressive exploration) and tightened in well-mapped areas to ensure safety.
System Integration: The filter acts as a minimally invasive correction in velocity space, composing transparently with arbitrary nominal controllers (demonstrated here with an RRT*-APF controller).

Key Contributions
The paper makes three primary contributions to the field of robotic safety:

Dual-Barrier Formulation: It generalizes the existing single-barrier Occupancy Grid Map CBF (OGM-CBF) framework to a dual formulation. This simultaneously penalizes proximity to obstacles and excursion into unexplored space, utilizing a class-level analysis of shaping functions to ensure CBF conditions hold.
Closed-Form Dual Projection: The authors replace general-purpose QP solvers with an analytical enumeration of KKT cases. This reduces per-cycle computation to a small linear system solve, making the filter viable for execution on resource-constrained companion computers alongside heavy SLAM workloads.
Adaptive Frontier Gain: The introduction of an adaptive gain schedule for the frontier barrier allows the system to dynamically balance exploration efficiency and safety based on local map uncertainty, improving navigation performance without compromising formal guarantees.

Experimental Results
The system was validated through both simulation and hardware experiments on a custom quadrotor (CUAV Nano 7) equipped with a Raspberry Pi 4B, Intel RealSense D435i camera, and TFMini laser rangefinder, running PX4 v1.16 and ROS 2 Jazzy.

Simulation: Compared against an RRT*-APF baseline, the CBF-enabled approach achieved a significantly larger explored area (81.68 m² vs. 56.35 m²) while maintaining safety. The baseline controller exhibited "lazy" exploration due to conservative repulsion, whereas the CBF filter enabled "adventurous" navigation near boundaries.
Hardware: Three autonomous exploration runs were conducted in an indoor environment. The filter successfully prevented any obstacle contacts across all runs. The soft-QP fallback was never triggered, indicating the closed-form solution handled the constraints effectively. The minimum observed clearance was 0.25 m, and the filter intervened only when the nominal velocity would have violated safety margins.
Performance: The filter executed with minimal overhead, leaving sufficient compute budget for the perception and planning stack.

Significance and Claims
The paper claims that this approach provides a formal safety guarantee for robots operating in unknown, incrementally built environments without requiring modifications to the underlying navigation policy (including learning-based frameworks). By deriving a closed-form solution, the authors demonstrate that rigorous safety filtering is computationally feasible on low-cost hardware, addressing a critical bottleneck in deploying autonomous robots in real-world, unstructured settings. The work positions the dual-barrier CBF as a practical "safety wrapper" that resolves the tension between aggressive exploration and collision avoidance, ensuring that robots can explore efficiently while strictly adhering to safety constraints defined by both known obstacles and the limits of current knowledge.