⚡ electrical engineering

Control Barrier Function only Formation Tracking in Multi-Agent Systems

This paper proposes a real-time control framework for heterogeneous multi-agent systems that utilizes a single Control Barrier Function-like constraint within a quadratic optimization setting to achieve formation tracking using only relative neighbor information, thereby eliminating the need for manual parameter tuning or a separate nominal controller.

Original authors: S. Saharsh, Pushpak Jagtap

Published 2026-06-25

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: S. Saharsh, Pushpak Jagtap

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 flock of birds, a school of fish, or a group of dancers moving in perfect unison. In the world of robotics, this is called formation tracking. The goal is to get a group of different robots (some might be fast, some slow, some drive like cars, some like drones) to move together in a specific shape while following a leader, all without crashing into each other or getting lost.

This paper proposes a new "brain" for these robot groups that makes this happen in real-time, even when the robots are complex and the environment is tricky. Here is how it works, broken down into simple concepts:

1. The Problem: The "Tug-of-War"

Usually, when you program robots to move in a formation, you have to balance two competing goals:

Goal A: Stay in the right shape (Formation).
Goal B: Don't crash or go off the map (Safety).

In traditional methods, these two goals often fight each other. The computer gets confused, trying to satisfy both, and sometimes the math breaks down (the system becomes "infeasible"), causing the robots to stop or behave erratically. It's like trying to drive a car while simultaneously trying to park it perfectly and avoid a wall, but the steering wheel is stuck.

2. The Solution: The "Magic Compass" (CBF)

The authors introduce a clever trick using something called a Control Barrier Function (CBF). Think of this as a "Magic Compass" that doesn't just tell the robot where to go, but also guarantees it stays safe while getting there.

The Innovation: Instead of treating "staying in formation" and "being safe" as two separate, fighting tasks, this paper combines them into one single rule.
The Analogy: Imagine you are walking through a crowded room. Usually, you have to worry about "reaching the exit" (formation) and "not bumping into people" (safety) separately. This new method gives you a single instruction: "Walk toward the exit, but keep your hand out to gently guide yourself away from people." It turns the whole process into one smooth, safe motion.

3. How the Robots "See" Each Other

The robots don't need a GPS satellite or a central computer telling them exactly where everyone is. They only need to look at their immediate neighbors.

The "Bearing" Concept: Instead of needing to know the exact distance or coordinates of a neighbor, each robot just needs to know the direction (the angle) to look at its neighbors.
The Metaphor: Imagine a group of people holding hands in a circle in the dark. They don't need to know how far apart they are; they just need to know which way their neighbor is standing. If everyone keeps looking at their neighbor at the correct angle, the whole circle forms itself. The paper uses this "looking at the angle" (bearing) to figure out where everyone should be.

4. The "Leader-Follower" Dance

The system uses a Leader-Follower setup:

The Leaders: A few robots (like the conductors of an orchestra) follow a pre-planned path. They set the pace and the general direction.
The Followers: The rest of the robots watch the leaders and their immediate neighbors. They adjust their speed and direction to match the "shape" defined by the angles.
The Magic: The paper proves that if every follower just correctly aligns with its two closest neighbors, the entire group automatically forms the perfect shape, even if the leaders are moving fast or changing direction.

5. The "One-Step" Optimization

The brain of the robot uses a mathematical tool called Quadratic Programming (QP). Think of this as a super-fast calculator that solves a puzzle every fraction of a second.

Old Way: The calculator tried to solve two different puzzles at once (Shape + Safety), which often led to a "No Solution" error.
New Way: The calculator solves one single puzzle. It asks: "What is the easiest, most efficient move that keeps me in the right shape AND guarantees I reach my target angle within a specific time?"
The Result: The robots move smoothly, adjusting their speed and direction instantly to stay in formation without ever needing to stop and recalculate complex safety rules.

6. The Proof (The Simulation)

The authors tested this on a computer simulation with:

2 Leaders: Moving along a path.
6 Followers: Some were simple "double integrator" robots (like a point moving in a straight line), and others were "differential drive" robots (like real cars or Roombas that turn by spinning wheels).
The Outcome: Despite the robots being different types and starting in random, messy positions, they all successfully snapped into the desired formation and tracked the leaders perfectly. The error in their angles was tiny (almost zero).

Summary

This paper presents a new way to control robot swarms. Instead of juggling separate rules for "stay in shape" and "don't crash," it creates a single, unified rule that handles both. By relying only on the direction of neighbors (not their exact location), the system is robust, works without GPS, and allows different types of robots to move together seamlessly, like a well-rehearsed dance troupe following a lead dancer.

Technical Summary: Control Barrier Function Only Formation Tracking in Multi-Agent Systems

Problem Statement
The paper addresses the formation tracking problem for heterogeneous multi-agent systems (MAS) characterized by non-linear dynamics. A primary challenge in this domain is the reliance on relative state information due to real-time communication constraints that prevent access to global state data. Existing approaches often struggle with scalability, adaptability in dynamic environments, or the infeasibility caused by conflicts between nominal formation controllers and real-time safety constraints. Specifically, the authors target a leader-follower framework where follower agents must maintain a unique, rigid formation defined by desired bearing vectors relative to their neighbors, while leaders follow predetermined trajectories. The system operates under the assumption of infinitesimal bearing rigidity, ensuring that the desired bearing configuration corresponds to a unique formation.

Methodology
The proposed solution utilizes a Quadratic Programming (QP) optimization framework driven by a novel Control Barrier Function (CBF) formulation. The methodology is structured as follows:

Problem Decomposition: The global formation tracking task is decomposed into local subtasks for each follower agent. Each follower identifies at least two reference nodes (leaders or other followers) with non-parallel desired bearings. The local objective is to align the agent's relative bearing vectors with the desired ones, which collectively ensures the global formation is achieved.
Bearing-Guided Target: For each follower, a "bearing-guided target" ( $p_{iT}$ ) is computed. This target represents the unique intersection point of lines defined by the desired bearing slopes of the reference nodes. The control objective is to drive the agent's position to this moving target while aligning its velocity vector with the target direction.
Reachability Control Barrier Function (Reach-CBF): The authors introduce a modified CBF that differs from standard formulations. While conventional CBFs guarantee the forward invariance of a safe set, this "Reach-CBF" ensures both the forward invariance of a safe set and the finite-time reachability of a target set within that safe set.
- The safe set is defined by the alignment between the agent's velocity vector and the bearing-guided target vector.
- The target set corresponds to a state where this alignment error is within a small, pre-defined tolerance.
Control Law Design: The control input is derived by solving a QP problem that minimizes the deviation from a nominal speed-tracking controller subject to the Reach-CBF constraint.
- The nominal controller handles speed regulation to match the leader's velocity.
- The Reach-CBF constraint enforces the directional alignment required for formation maintenance.
- This design decouples the control of velocity magnitude (handled by the nominal controller) and direction (enforced by the CBF), eliminating the need for a separate nominal formation controller and avoiding the infeasibility issues common in competing objective formulations.

Key Contributions
The paper outlines three primary contributions:

Novel Reach-CBF Formulation: The authors propose a CBF-like function that extends standard CBFs by guaranteeing finite-time reachability of a target set within a forward-invariant safe set, rather than just ensuring safety.
Single-Constraint Optimization Framework: A new CBF-QP-like framework is introduced that utilizes a single CBF constraint to handle formation tracking for heterogeneous non-linear multi-agent systems. This approach removes the need for manual parameter tuning or a separate nominal formation controller.
Leader-Follower Formation Tracking: The method successfully achieves formation tracking for systems with non-linear agent dynamics (including double integrators, differential-drive, and Ackermann-drive robots) within a leader-follower architecture using only relative bearing information.

Results
The proposed framework was validated through simulations involving a heterogeneous system of eight agents in a 2D space: two leaders and six followers.

System Configuration: The followers included agents modeled as double integrators and differential-drive robots with non-linear dynamics.
Scenario: Leaders followed predetermined trajectories, while followers were tasked with maintaining a specific rigid formation defined by bearing vectors.
Performance: The simulation demonstrated that the follower agents successfully tracked the moving formation. After 30 seconds, the bearing errors for all followers were negligible (ranging from approximately $-4.6 \times 10^{-5}$ to $1.87 \times 10^{-4}$ ). The trajectories confirmed that the agents achieved the desired formation shape while following the leaders.

Significance and Claims
The paper claims that its approach offers a unified, constraint-based safe control design that addresses the limitations of existing methods, particularly the conflict between formation objectives and safety constraints. By encoding the formation task itself as a CBF constraint, the method ensures that the system remains within a safe set while converging to the desired formation in finite time. The authors emphasize that the framework operates without global state information, relying solely on local relative measurements (bearings), making it suitable for GPS-denied environments. The work concludes by noting that future efforts aim to extend this approach to 3D aerial vehicles navigating cluttered environments with accelerating obstacles.