⚡ electrical engineering

Distributed 3D Leader-Follower Formation Control with Field-of-View Safety via Control Barrier Functions

This paper proposes a distributed 3D leader-follower formation control framework for multi-UAV systems that ensures vision-based safety by embedding a nominal tracking controller within a Control Barrier Function-based safety filter to guarantee the leader remains within the follower's field of view.

Original authors: Immanuel R. Santjoko, Richie R. Suganda, Miao Pan, Bin Hu

Published 2026-05-19

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CC BY 4.0

Original authors: Immanuel R. Santjoko, Richie R. Suganda, Miao Pan, Bin Hu

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 team of tiny drones flying together in a 3D space, like a school of fish or a flock of birds. In this setup, one drone is the Leader, and the others are Followers. The Followers' job is to stay in a specific formation relative to the Leader, but there's a catch: they can't see each other with eyes; they use cameras.

Here is the problem the paper solves: The "Blind Spot" Dilemma.

The Problem: The Camera Has a Tunnel Vision

Think of the camera on a follower drone like a flashlight beam or a tunnel. It has a limited width (Field of View) and a limited depth (it can't see things too close or too far away).

If the Leader drone suddenly darts to the side, speeds up, or stops abruptly, it might fly right out of the follower's "flashlight beam."

Without a safety system: The follower tries to chase the Leader exactly where it wants to go. If the Leader flies out of the camera's view, the follower goes "blind." It loses track of the Leader, and the formation breaks.
The Goal: The follower needs to stay in formation but must never let the Leader disappear from its camera.

The Solution: The "Safety Filter"

The authors built a smart control system that acts like a guardian angel or a traffic cop for the drones. They call it a "Control Barrier Function" (CBF), but you can think of it as a Safety Filter.

Here is how it works, step-by-step:

The "Dream" Plan (Nominal Controller): First, the drone calculates where it wants to go to maintain the perfect formation. This is the "ideal" path.
The Safety Check (The Filter): Before the drone actually moves, the Safety Filter checks: "If I go where I want to, will the Leader stay inside my camera's flashlight beam?"
- If Yes: The filter says, "Go ahead!" The drone moves exactly as planned.
- If No: The filter says, "Stop! That path will make you lose sight of the Leader." It then makes a tiny, careful adjustment to the drone's speed or direction. It changes the path just enough to keep the Leader visible, while still trying to stay as close to the original formation plan as possible.

The Analogy: Walking a Dog on a Leash

Imagine you are walking a dog (the Leader) on a leash, but you are wearing a blindfold and can only see the dog through a narrow tube held in front of your face (the Camera).

The Old Way (No Safety Filter): You try to walk in a perfect circle with the dog. If the dog suddenly runs to the left, you keep walking in your circle. The dog runs out of your tube's view. You are now lost and don't know where the dog is.
The New Way (With Safety Filter): You have a smart assistant guiding you. You want to walk in a circle, but the assistant sees the dog running toward the edge of your tube. The assistant gently pulls your arm, saying, "Don't walk straight; step a little to the right." You adjust your path slightly. You aren't walking the perfect circle anymore, but you keep the dog inside your tube so you never lose sight of it.

What They Tested

The researchers tested this idea in two ways:

Computer Simulations (Gazebo): They created a virtual cave with narrow tunnels and open rooms. They made the Leader drone do tricky things, like sudden stops and sharp turns.
- Result: The "Old Way" drones lost sight of the Leader when the Leader moved aggressively. The "New Way" drones successfully adjusted their paths to keep the Leader in view, even when the Leader tried to fly out of bounds.
Real Hardware (Crazyflie Drones): They flew actual tiny drones in a lab.
- Result: Just like in the simulation, the drones with the Safety Filter successfully tracked the Leader and kept it in the camera frame, even when the Leader made sudden, jerky movements. The drones without the filter crashed or lost the Leader.

The Bottom Line

This paper presents a new way to fly drone teams so they can stick together without losing sight of each other. It uses a mathematical "safety net" that automatically tweaks the drone's movements to ensure the Leader never disappears from the camera's view, balancing the need to follow a formation with the need to stay safe and visible.

Technical Summary: Distributed 3D Leader–Follower Formation Control with Field-of-View Safety via Control Barrier Functions

Problem Statement
The paper addresses the challenge of maintaining safe, vision-based 3D leader–follower formations (3D-LFF) for multi-UAV systems, particularly in GPS-denied environments where relative state estimation relies on onboard cameras. A critical constraint in such systems is the Field-of-View (FOV) and depth range of the follower's camera. If formation tracking commands drive the leader outside the follower's camera frustum, visibility is lost, leading to a failure in relative state estimation and potential mission failure. Existing position-based approaches often fail to rigorously model these 3D camera frustum constraints, while extending planar visibility methods to 3D UAV formations is nontrivial due to coupled 3D geometry and relative heading effects. The core problem is to design a distributed control framework that achieves accurate formation tracking while guaranteeing that the leader remains strictly visible within the follower's camera frustum at all times.

Methodology
The proposed solution is a hierarchical, perception-aware control architecture that integrates a nominal formation controller with a safety filter based on Control Barrier Functions (CBFs).

System Modeling: The authors derive a relative kinematic model for the leader–follower pair using spherical coordinates ( $L_{ij}, \phi_{ij}, \xi_{ij}$ ) augmented with a relative heading variable ( $\alpha_{ij}$ ). This model captures the geometry between the follower's body-fixed camera and the leader, expressed in the follower's body frame.
Nominal Controller: A distributed 3D-LFF tracking controller is designed using only locally available relative states and communicated leader states. This controller generates a nominal velocity command ( $u_i^{nom}$ ) intended to drive the formation error to zero exponentially.
Safety Filter (CBF-QP): To enforce visibility, the nominal command is processed through a Control Barrier Function–based Quadratic Program (CBF-QP).
- Safe Set Definition: The perception safe set is defined by the camera frustum constraints (horizontal/vertical FOV angles and depth limits $[x_{near}, x_{far}]$ ). These constraints are encoded as continuously differentiable functions ( $h_c$ ) in Cartesian coordinates.
- Optimization: At each control timestep, the QP minimizes the deviation between the safe command ( $u_i^{safe}$ ) and the nominal command ( $u_i^{nom}$ ) subject to the CBF constraints ( $\dot{h}_c + \kappa h_c \geq 0$ ). This ensures the forward invariance of the safe set, meaning if the leader starts within the frustum, it remains there.
- Robustness: The framework includes slack variables to handle control input bounds and unmodeled disturbances, ensuring feasibility even under aggressive maneuvers.

Key Contributions
The paper outlines three primary contributions:

Relative Kinematic Model: A 3D kinematic model in spherical coordinates that incorporates relative heading, enabling a distributed control law based on local relative measurements.
Real-Time Safety Filter: A CBF-QP safety filter that enforces 3D camera frustum constraints (FOV and depth) by minimally modifying nominal formation commands. This guarantees continuous visibility while preserving tracking performance whenever feasible.
Experimental Validation: Comprehensive validation using Gazebo simulations and Crazyflie 2.1 hardware experiments. The results demonstrate the framework's ability to track desired formations and strictly enforce FOV constraints, even when the desired formation conflicts with visibility limits or when the leader performs abrupt maneuvers.

Results
The proposed method was evaluated against a baseline controller without the safety filter (NoCBF) in two scenarios: a three-stage formation task and an abrupt leader motion task.

Three-Stage Task: In Stage 2, where the desired formation expansion pushed the leader toward the camera frustum boundary, the NoCBF baseline violated safety constraints, causing the leader to leave the FOV. In contrast, the CBF-equipped follower dynamically adjusted its velocity to keep the leader within the safe set, successfully resolving the conflict between formation objectives and perception safety.
Abrupt Motion: During a scenario involving sudden leader deceleration, the NoCBF baseline failed to prevent the follower from overshooting and losing the leader. The proposed framework successfully mitigated the overshoot, maintaining visibility.
Hardware Experiments: Physical flight tests with Crazyflie drones confirmed the simulation results. The CBF controller maintained the leader within the frustum during complex maneuvers (including a yaw-modulated lemniscate trajectory), whereas the baseline failed to maintain safety constraints during formation expansion phases.

Significance and Claims
The paper claims to provide a rigorous solution for perception-aware formation control in 3D space. Its significance lies in:

Guaranteed Visibility: Unlike previous methods that may rely on heuristics or fail to model 3D geometry fully, this approach uses CBFs to mathematically guarantee that the leader remains within the camera frustum (forward invariance).
Minimal Intervention: The CBF-QP formulation ensures that the safety filter only modifies the control input when necessary, preserving the nominal formation tracking performance as much as possible.
Practical Applicability: The successful validation on both simulation and physical hardware (Crazyflie) demonstrates that the approach is computationally feasible for real-time implementation on resource-constrained UAVs.

The authors note that while current experiments utilize motion capture to emulate ideal relative measurements, future work aims to integrate onboard perception to close the loop using raw image-based measurements. The paper does not claim to solve all multi-agent coordination problems but specifically targets the critical intersection of formation tracking and perception safety in 3D leader–follower configurations.