⚡ electrical engineering

Grey-Wolf-Optimized Obstacle-Aware Affine Formation Control for Quadrotor Swarms in Cluttered Three-Dimensional Space

This paper proposes a grey-wolf-optimized obstacle-aware affine formation control method for quadrotor swarms in cluttered 3D environments, which dynamically tunes controller gains and influence radii to balance formation tracking accuracy, obstacle clearance, and control effort while ensuring stability and collision avoidance.

Original authors: Mu Lan, Xiaoyue Deng

Published 2026-08-28

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Mu Lan, Xiaoyue Deng

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

In the sky above our cities and across rugged landscapes, a new kind of aviation is taking shape: swarms of small, unmanned drones working together as a single unit. Unlike a lone aircraft that flies a fixed path, these groups must move as a cohesive team, shifting their shape to squeeze through narrow gaps, expand to cover a wide area, or rotate to face a new direction. The challenge lies in keeping this formation intact while dodging obstacles like trees, buildings, or other drones. If the group moves too rigidly, it cannot navigate complex environments; if it reacts too frantically to avoid a crash, the formation breaks apart. Researchers have long sought a way to balance these needs, creating control systems that allow a swarm to flow around dangers without losing its collective identity. This delicate balance requires not just a plan for where to go, but a smart way to adjust the group's shape and speed in real time, ensuring every drone stays safe from both the environment and its neighbors.

A team of researchers at Nanjing University of Science and Technology has developed a new method to solve this problem for swarms of quadrotor drones flying in cluttered three-dimensional spaces. Their approach combines a geometric strategy for moving the group with a biological algorithm that fine-tunes the system's behavior. At the heart of their method is an "affine formation" concept, where a few leader drones dictate the overall movement—such as stretching, shrinking, or turning the entire group—while the rest of the drones automatically calculate their positions to maintain the shape. This allows the swarm to deform smoothly, like a flexible sheet, rather than breaking apart when it encounters an obstacle. However, simply deforming the shape is not enough to guarantee safety in a dense environment filled with spherical obstacles. The researchers added a layer of local awareness that pushes individual drones away from nearby dangers, but they faced a difficult question: how strong should these pushing forces be, and how far ahead should the drones look for trouble?

To answer this, the team turned to a computer program inspired by the social hierarchy and hunting behavior of grey wolves. This algorithm, known as the Grey Wolf Optimizer, acts as a virtual coach that tests thousands of different settings for the drone swarm's control system. It searches for the perfect combination of parameters—such as how aggressively a drone should steer away from an obstacle or how quickly it should return to its spot in the formation. The goal was to find a balance where the swarm tracks its intended path accurately, avoids collisions with obstacles and other drones, and does not waste energy making unnecessary sharp turns. The researchers ran extensive computer simulations with a swarm of eight drones navigating a corridor filled with eight spherical obstacles. They compared their optimized system against other methods, including one where the settings were chosen by hand and another that used a fixed, unchanging strategy for avoiding obstacles.

The results of these simulations revealed a clear trade-off rather than a perfect solution in every category. The grey wolf-optimized system allowed the swarm to follow its intended path with significantly higher accuracy than the other methods, reducing the average error in position to about 0.22 meters. It also kept the maximum deviation during the flight to under 0.6 meters, a marked improvement over the manually tuned system which drifted much further off course. Crucially, the swarm maintained a safe distance from the obstacles and from each other throughout the flight, never coming closer than 0.18 meters to an obstacle or 0.81 meters to another drone. However, this increased precision came at a cost: the optimized swarm used slightly more energy to maneuver than the manually tuned version, and it did not keep as large a safety buffer from obstacles as the more conservative, hand-tuned system did. The study suggests that while the algorithm does not make the drones invincible or infinitely efficient, it successfully finds a middle ground where the group moves with greater precision and flexibility than previous methods allowed.

The researchers emphasize that their findings are based on computer simulations and that the stability of the system relies on mathematical proofs that guarantee the drones will not crash even when the avoidance forces are active. They note that the algorithm does not replace the fundamental rules of flight but rather selects the best settings within a safe range. By treating the complex interaction of movement and avoidance as a single optimization problem, the team demonstrated that a swarm can be both agile and safe. The work highlights that in the future of autonomous flight, the ability to adapt the group's shape and tune its reaction to the environment in real time will be just as important as the speed of the individual drones. This approach offers a promising path for deploying drone swarms in real-world scenarios where the environment is unpredictable and the margin for error is small.

Technical Summary: Grey-Wolf-Optimized Obstacle-Aware Affine Formation Control for Quadrotor Swarms

Problem Statement
Affine formation control offers an efficient mechanism for unmanned aerial vehicle (UAV) swarms to execute complex maneuvers—including translation, rotation, scaling, and shearing—while preserving affine invariants. However, applying this control strategy in cluttered three-dimensional environments presents a significant challenge. Traditional approaches often rely on manually selected obstacle-avoidance parameters (such as gains and influence radii for artificial potential fields or barrier functions). These manual selections frequently result in a trade-off dilemma: conservative parameters ensure safety but lead to excessive formation deformation, energy-intensive detours, and poor tracking accuracy, while aggressive parameters improve tracking but risk unsafe transients and collisions. The paper addresses the need for a systematic method to tune these coupled parameters to balance formation tracking accuracy, obstacle clearance, inter-vehicle safety, and control effort in complex 3D spaces.

Methodology
The proposed framework integrates four core components:

Affine Formation Generation: A leader-follower architecture is employed where four affinely independent leaders define the global affine motion (translation, rotation, scaling, and shear). The remaining follower agents reconstruct their desired trajectories using affine coordinates, allowing the formation to adapt its shape globally.
Obstacle-Aware Deformation: To handle cluttered environments, the nominal affine reference is augmented with a smooth, obstacle-aware deformation. This includes:
- Global Deformation: Anisotropic scaling and lateral/vertical deformation terms are applied to the formation centroid based on the proximity of obstacles, creating a global tendency to compress or expand the formation before local avoidance takes over.
- Local Repulsion: Barrier-based repulsive terms are added to the control law for both obstacle avoidance and inter-UAV collision avoidance. These terms utilize safety functions that generate repulsive forces when agents approach obstacles or each other within specific influence radii.
Grey Wolf Optimizer (GWO) Parameter Tuning: A metaheuristic Grey Wolf Optimizer is utilized to automatically tune the controller's critical parameters: tracking gains ( $k_p, k_v$ $k_{p}, k_{v}$ ), obstacle and inter-UAV repulsion gains ( $k_o, k_c$ $k_{o}, k_{c}$ ), and their respective influence radii ( $\rho_o, \rho_c$ $ρ_{o}, ρ_{c}$ ). The GWO minimizes a composite performance index ( $J$ $J$ ) that weights:
- Final and maximum RMS formation tracking errors.
- Average control energy.
- Penalties for violating obstacle clearance and inter-UAV distance constraints.
- Rewards for maximizing minimum clearance (though the optimization prioritizes the composite balance).
Stability Analysis: The paper establishes theoretical guarantees separate from the optimization process.
- Nominal Stability: It is proven that the nominal affine tracking system (without avoidance terms) is asymptotically stable for positive stiffness and damping gains.
- Boundedness under Perturbation: The avoidance terms are treated as bounded inputs. Using input-to-state stability (ISS) arguments, the authors demonstrate that the closed-loop tracking error remains bounded during obstacle interactions, recovering nominal convergence once the agents leave the influence regions.

Key Contributions

Framework Development: The authors developed an obstacle-aware affine formation control framework specifically for quadrotor swarms in 3D cluttered environments, combining global shape adaptation with local safety feedback.
Optimization Integration: The introduction of GWO to jointly optimize controller gains and influence radii, moving away from manual tuning to a data-driven approach that balances competing objectives (accuracy vs. safety vs. energy).
Theoretical Rigor: The paper provides a stability analysis that decouples the metaheuristic optimization from closed-loop stability proofs, ensuring that the system remains stable within admissible parameter intervals regardless of the specific GWO solution found.
Comprehensive Evaluation: Through Python simulations involving eight quadrotors and eight spherical obstacles, the study compares the proposed method against manually tuned baselines (OAAF-BA), fixed-artificial potential field methods (Fixed-APF), and various ablation studies (removing scaling, local repulsion, or inter-UAV barriers).

Results
Simulations with eight quadrotors navigating a corridor with eight spherical obstacles yielded the following findings:

Tracking Accuracy: The GWO-optimized method (GWO-OAAF) achieved the lowest final RMS formation error (0.2173 m) and maximum RMS error (0.5845 m) among all collision-free methods, significantly outperforming the manually tuned OAAF-BA and Fixed-APF baselines.
Safety and Clearance: The method maintained positive obstacle clearance (minimum 0.1803 m) and inter-UAV separation (minimum 0.8102 m), confirming collision-free operation.
Trade-offs: The results highlighted a distinct trade-off. While GWO-OAAF improved tracking accuracy compared to both baselines, it resulted in higher control energy consumption than the manually tuned OAAF-BA (0.0780 vs. 0.0571) but lower energy consumption than Fixed-APF. Additionally, GWO-OAAF achieved smaller minimum obstacle clearances compared to the conservative OAAF-BA and Fixed-APF. This indicates the optimizer prioritized tracking precision over maximizing safety margins beyond the required threshold or minimizing energy relative to the manual baseline.
Ablation Insights: Removing local obstacle repulsion led to collisions; removing anisotropic scaling reduced flexibility in narrow passages; and removing inter-UAV barriers decreased internal safety distances. This confirms the necessity of all proposed components.

Significance and Claims
The paper claims that its primary significance lies in demonstrating that metaheuristic optimization can effectively enhance the adaptability of affine formation control in aerospace unmanned systems without compromising theoretical stability. The authors emphasize that the improvement is not due to a fundamental change in the stability mechanism but rather to the superior selection of coupled parameters that would be difficult to achieve through manual tuning.

The study concludes that the proposed method offers a practical "accuracy-clearance-effort" trade-off, achieving substantially better formation tracking and lower energy consumption than fixed-shape potential field methods, while maintaining safety. The authors modestly note that the current work is limited to a double-integrator model and spherical obstacles, suggesting that future work should address high-fidelity dynamics, actuator delays, sensor noise, and dynamic obstacles.

Technical Summary: Grey-Wolf-Optimized Obstacle-Aware Affine Formation Control for Quadrotor Swarms

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