⚡ electrical engineering

Energy-Conserving Swarm Formation Control Using Pure Hamiltonian Mechanics and Active Braking

This paper proposes a novel Pure Hamiltonian 3D RK Swarm control framework that integrates anisotropic navigation potentials, active kinematic preview braking, and SO(3)-based formation projection to enable autonomous UAVs to navigate cluttered, non-flat terrains while strictly maintaining safety margins and preventing local minima failures.

Original authors: Santiago Garrido, Fernando Quevedo, Ramon Barber, Luis Moreno

Published 2026-09-28

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Santiago Garrido, Fernando Quevedo, Ramon Barber, Luis Moreno

Original paper licensed under CC BY 4.0 (https://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

The sky is no longer empty. It is becoming a crowded highway for small, autonomous aircraft, tasked with everything from searching for survivors in disaster zones to monitoring forests and delivering medical supplies. For these machines to work together effectively, they must fly in tight, coordinated groups, moving as a single unit while navigating a world that is rarely flat or empty. The challenge is immense: a swarm of drones must maintain a precise shape, avoid crashing into each other, and steer clear of trees, buildings, and uneven ground, all while reacting instantly to changes in their environment. Traditional methods for guiding these groups often rely on simple rules of attraction and repulsion, similar to how magnets pull and push. However, in complex, cluttered spaces, these simple rules frequently fail. The drones can get stuck in invisible traps, unable to find a way forward, or they can be pulled into collisions when the path narrows unexpectedly.

To solve this, researchers at the Universidad Carlos III de Madrid have developed a new way to guide drone swarms, one that treats the movement of the group not as a set of rigid commands, but as a flow of energy. Instead of forcing the drones to follow a pre-drawn map, the new system creates a smooth, invisible landscape of potential energy that guides the drones naturally. In this framework, the drones are like balls rolling across a surface; they move toward their goal because the terrain slopes downward in that direction, and they avoid obstacles because the ground rises sharply to form a barrier. This approach, rooted in the physics of how energy moves through a system, ensures that the drones never get stuck in dead ends. By combining this energy-based guidance with a clever safety mechanism that forces the lead drone to take a wide path around obstacles, the researchers have created a system where a group of nine drones can fly in a perfect V-shape over bumpy, rolling terrain without ever colliding with each other or the ground.

The core of this new method is a shift in how the drones perceive their world. Older systems often used simple mathematical fields that could create confusing dead ends, causing the drones to freeze or crash. The new system uses a more sophisticated mathematical structure, known as a navigation function, which guarantees that there is only one lowest point in the entire environment: the destination. This means the drones are mathematically certain to reach their goal without getting trapped. To make this work in the real world, where drones have weight and momentum, the researchers added a layer of active braking. Just as a car needs to brake before a sharp turn to avoid sliding off the road, the drones in this system apply a gentle, targeted force to slow down only when they are moving too fast toward a wall or an obstacle. This allows them to maintain their speed and agility in open spaces while stopping safely when they get too close to danger.

A critical innovation in this research is how the swarm handles the leader. In many systems, if the leader gets too close to an obstacle, the followers might be pulled into a collision because they are too tightly bound to the leader's exact path. The researchers solved this by giving the virtual leader a "wide berth" strategy. Before the leader even reaches a narrow gap between obstacles, it calculates a path that stays far away from the danger, creating a massive safety buffer of 6.5 meters around the obstacle. This early deflection ensures that the entire group, following in a rigid V-formation, has plenty of room to pass through without the outer drones clipping the sides of buildings or trees. The system also uses a dynamic rotation to keep the V-shape aligned with the direction of travel, allowing the formation to turn smoothly without breaking apart.

The researchers tested this system in a detailed computer simulation involving nine drones flying over a highly uneven, wave-like terrain with several tall, cylindrical towers scattered throughout the flight path. The results were precise and robust. The drones successfully maintained a strict safety distance of 2.5 meters from any obstacle and stayed 2.0 meters above the ground at all times, even as the terrain rose and fell beneath them. They also kept a safe distance of 0.9 meters from one another, preventing any internal collisions. When compared to older planning methods that rely on dividing the space into a grid, the new system was significantly faster and more reliable. While the grid-based methods took over 30 seconds to plan a single path in high detail, the new energy-based system calculated the path in just 0.002 seconds, allowing for real-time reactions. Furthermore, as the number of drones increased, the new system maintained a 100 percent success rate, whereas older methods saw their failure rates climb sharply as the group grew larger.

The study also explored how this system would handle rectangular buildings, which are more common in cities than round towers. By adjusting the mathematical shape of the safety barriers to match the flat sides and sharp corners of buildings, the system continued to perform flawlessly. The drones navigated around the corners of these structures without getting stuck or oscillating, proving that the approach works for various types of obstacles. The researchers noted that while these results were achieved through simulation, the next step is to move the system onto physical hardware. They plan to test the drones in real-world conditions, accounting for wind gusts, communication delays, and the imperfections of real sensors.

This work represents a significant step forward in making autonomous drone swarms safe and practical for complex missions. By treating the swarm as a single, energy-conserving entity that flows naturally around obstacles, the researchers have overcome the limitations of older control methods. The system's ability to maintain a rigid formation while dynamically avoiding traps and obstacles suggests a future where large groups of drones can operate safely in the most challenging environments, from dense forests to crowded city centers, without human intervention. The simulation results provide strong evidence that this approach is not just theoretically sound, but practically viable for the next generation of aerial robotics.

Technical Summary: Energy-Conserving Swarm Formation Control Using Pure Hamiltonian Mechanics and Active Braking

Problem Statement
The paper addresses the challenge of controlling autonomous multi-agent Unmanned Aerial Vehicle (UAV) swarms in unstructured, non-flat 3D environments. Traditional control methodologies, particularly standard Artificial Potential Fields (APFs), frequently fail in cluttered terrains due to the presence of spurious local minima, which can cause formations to stall or collide. Furthermore, existing approaches struggle to simultaneously satisfy three competing constraints: maintaining a rigid geometric formation (specifically a V-shape), tracking a highly agile moving target, and ensuring physical clearance over complex, undulating terrain and static obstacles. A specific failure mode identified is "wing-clipping," where a leader's tight turn around an obstacle forces trailing agents into collision zones due to momentum-driven overshoot and rigid offset tracking.

Methodology
The authors propose a Pure Hamiltonian 3D Rimon-Koditschek (RK) Swarm Control Framework. This approach models the swarm as a dynamic system within a pseudo-Hamiltonian dynamical framework, treating agents as particles with mass and momentum rather than simple kinematic velocity clamps.

The core methodology consists of four integrated layers:

Pseudo-Hamiltonian Dynamics with RK Potentials:
The system utilizes a modified Rimon-Koditschek navigation function, $\phi_i$ , which maps structural errors directly into smooth acceleration profiles. The potential is composite, combining an attractive term (weighted Euclidean distance to a target) and a repulsive term (obstacle boundaries).
- Anisotropic Vertical Scaling: The attractive potential includes a stiff penalty ( $\alpha_z = 8.0$ ) on vertical errors to ensure rapid altitude stabilization relative to the ground profile.
- Energy Exchange: The system evolves via Hamiltonian gradient flows, allowing agents to share kinetic energy states for agile flight, while introducing directional damping near boundaries to prevent instability.
Active Target Wide-Berth Deﬂection:
To prevent the trailing swarm from entering high-risk zones, the virtual target itself is subject to an active kinematic preview layer.
- The target evaluates obstacle proximity within an expanded activation zone (12.0 m beyond standard boundaries).
- It generates a massive lateral clearance buffer ( $d_{target\_extra\_buffer} = 6.5$ m) around obstacles, effectively "pre-deconﬂicting" narrow bottlenecks before the swarm arrives.
Coordinate-Free Rigid Formation Engine:
The formation is maintained without centralized communication graphs. A dynamic $SO(3)$ rotation matrix ( $R_z$ ) aligns the swarm's local coordinate system with the target's filtered heading vector. Rigid, parameterized spatial offsets are projected onto individual agents based on their indexing, ensuring a symmetric V-shape configuration that rotates and banks smoothly with the target.
Active Boundary Braking:
To counteract the momentum-driven overshoot inherent in Hamiltonian dynamics, a spatial decay braking force ( $F_{brake}$ ) is modulated by the normal gradient of the workspace topology. This force selectively drains kinetic energy only when an agent attempts to penetrate multi-tier safety zones (obstacle buffers and ground clearance), acting as a dissipative barrier without inducing sluggishness in open space.

Key Contributions

Pseudo-Hamiltonian RK Synthesis: The formalization of a 3D multi-agent potential field that eliminates local traps while preserving the energy-aware agility of mechanical systems.
Dynamic Target Deﬂection: The introduction of a "wide-berth" mechanism where the target actively steers away from obstacles with a large safety margin, protecting the integrity of the trailing formation.
Asymmetric V-Formation Rigid Offsets: A coordinate-free engine using dynamic rotation matrices to project rigid offsets, ensuring robust formation maintenance without centralized graphs.
Active Boundary Braking: A novel braking force that selectively dissipates kinetic energy based on the normal gradient of obstacles, preventing overshoot while maintaining high-speed agility.

Simulation Results
Extensive numerical simulations were conducted with $N=9$ agents navigating a highly undulating sinusoidal terrain ( $z = 5.0 \sin(0.07x) \cos(0.07y) + 5.0$ ) with five cylindrical tower obstacles.

Safety and Stability: The swarm strictly respected hard safety margins ( $d_{buffer} = 2.5$ m for obstacles, $z_{buffer} = 2.0$ m for terrain) and inter-drone separation limits ( $d_{spacing} = 0.9$ m). No collisions or local minima traps occurred.
Wing-Clipping Resolution: In a benchmark comparison against a standard APF without the extra target buffer, the standard approach resulted in a collision (clearance of -1.38 m) for outer wing agents. The proposed framework achieved a safe clearance of +2.61 m.
Formation Rigidity: The mean formation angle deviation remained under $3.1^\circ$ even during sharp turns, and inter-drone distances never dropped below the safety threshold.
Computational Efficiency: Unlike grid-based methods (Fast Marching Method and Fast Marching Square) which suffer from the "curse of dimensionality" and require seconds to compute paths at high resolutions, the proposed continuous, grid-free framework maintained a constant execution time of 2 ms, regardless of grid resolution.
Scalability: As swarm size increased from $N=3$ to $N=25$ , the proposed framework maintained a 100% collision-free success rate with negligible formation error growth, whereas grid-based master-path projection methods saw success rates drop to as low as 8% due to wing-clipping failures.

Significance and Claims
The paper claims that this framework provides a highly resilient architecture for autonomous multi-UAV operations in unstructured environments. By integrating topological safety (via RK potentials) with the dynamic agility of Hamiltonian mechanics and a proactive wide-berth target strategy, the system overcomes the limitations of traditional APFs and grid-based planners. The authors assert that the system successfully balances the need for strict geometric formation, rapid terrain following, and aggressive obstacle avoidance without relying on centralized communication or suffering from local minima.

Future Work
The authors outline a path toward physical hardware deployment, including:

Transitioning the control loops from MATLAB to C++ within a ROS 2 architecture for decentralized onboard execution.
Hardware-in-the-Loop (HIL) testing using PX4 and Gazebo to validate resilience against actuator saturation and communication latency.
Replacing analytical environment models with real-time perception systems (LiDAR/stereo cameras) to generate dynamic obstacle proximity functions.
Implementing adaptive damping controllers to reject wind gusts in outdoor flight.

Technical Summary: Energy-Conserving Swarm Formation Control Using Pure Hamiltonian Mechanics and Active Braking

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