Pigeon-Inspired Intra-Formation Control

• Pigeon-like intra-formation control is a distributed strategy that mimics natural leader–follower cascades and local force-based interactions for cohesive multi-agent navigation.
• The approach integrates mathematical models with position-velocity feedback and delayed reaction dynamics to achieve collision avoidance and optimal formation tracking.
• Empirical results demonstrate efficient 3D obstacle negotiation, enhanced formation stability via Lyapunov methods, and reduced communication overhead in dense urban scenarios.

Pigeon-like intra-formation control refers to a class of distributed or semi-distributed multi-agent control algorithms that draw inspiration from the collective behaviors observed in pigeon flocks, specifically their mechanisms for maintaining dense, cohesive formations and adapting to obstacles and disturbances. The primary goal is to enable multi-robot or multi-UAV systems to navigate complex environments while ensuring collision-free, coordinated group motion—often under explicit communication and sensing constraints. These methods blend biological observation, optimal control, force/interaction models, and formation assignment strategies, producing a robust framework suitable for urban and cluttered three-dimensional spaces.

1. Biological Foundations and Core Architectural Models

Pigeon-inspired intra-formation control is motivated by empirical studies of avian flocking, which reveal two major regulatory strategies: local, delayed leader-follower interactions and fast, field-of-view (FOV)-limited collision/obstacle response. GPS-tracking studies demonstrate that in natural pigeon flocks, one or two individuals at the front serve as leaders, with each subsequent bird monitoring and aligning with a leader immediately ahead (often including explicit reaction delays) (Islam, 28 Mar 2026). This cascade-type leader-follower topology contrasts with global-all-to-all and unstructured interaction models typical of classical Boids.

Architecturally, real-pigeon-inspired controllers incorporate:

• Leader–follower hierarchies, where each agent’s reference trajectory is defined by its designated parent, with empirically determined delays.
• Local force-based interaction laws for immediate collision avoidance and cohesion, realized either via artificial potential fields or force superposition schemes derived from bio-inspired principles (Ahmadvand et al., 2024, S et al., 2023).
• Field-of-view constraints and obstacle-driven maneuvers, limiting each agent’s reaction set to those objects/obstacles within a biologically plausible conical region relative to its current heading (Ahmadvand et al., 1 Jul 2025, Ahmadvand et al., 2024).

2. Mathematical Formulation of Control Laws

Pigeon-like intra-formation control schemes formalize agent dynamics as either double-integrator point masses or full nonlinear UAV models but typically employ the following structural decomposition of the total control input: $u_i = u^f_i + u^c_i + u^o_i$ where:

• $u^f_i$ is the formation-keeping/centroid tracking term, often derived from centralized or semi-centralized assignment (e.g., centroidal Voronoi tessellations via probabilistic Lloyd’s algorithm (Ahmadvand et al., 1 Jul 2025, Ahmadvand et al., 2024)).
• $u^c_i$ is the inter-agent collision-avoidance force, typically a distributed spring–damper law or force field, mimicking biological spacing rules (Ahmadvand et al., 2024, S et al., 2023, Lin et al., 20 Nov 2025).
• $u^o_i$ is the obstacle-avoidance component, combining repulsion and rotational (tangential) maneuvers within a FOV-limited local sensor model (Ahmadvand et al., 1 Jul 2025, Ahmadvand et al., 2024).

Example: Probabilistic Lloyd's Algorithm for Centroidal Assignment

Given $N$ agents over region $Q \subset \mathbb{R}^3$, the cost

$$J(p_1, ..., p_N) = \int_{Q} \min_{i} f(\|q - p_i\|) p(q) dq$$

is minimized via a generalized, sample-based Lloyd’s update. Each agent moves towards the centroid of its Voronoi region, with the centralized algorithm periodically broadcasting updated centroids as formation waypoints (Ahmadvand et al., 1 Jul 2025, Ahmadvand et al., 2024). The actual coordination law is then defined via simple position and velocity feedback to these centroids.

Example: Local Spring–Damper Collision Avoidance

Each agent senses neighbors within a fixed radius $r_a$ and computes, for each,

$$u^c_{ij} = \begin{cases} - k_{c1}\left(\|p_{j} - p_{i}\| - r_s\right) \frac{p_{j} - p_{i}}{\|p_{j} - p_{i}\|} - k_{c2}(v_{j} - v_{i}), & \|p_{j} - p_{i}\| < r_a \ 0, & \text{otherwise} \end{cases}$$

Summing over all such $j$ gives the net repulsive/damping term $u^f_i$0 for $u^f_i$1 (Ahmadvand et al., 1 Jul 2025, Ahmadvand et al., 2024, Lin et al., 20 Nov 2025).

Example: Pigeon-Style Obstacle Avoidance

An obstacle $u^f_i$2 at position $u^f_i$3 is detected if within distance $u^f_i$4 and in a conical FOV of half-angle $u^f_i$5. Upon detection, the repulsive and rotational gradients: $u^f_i$6 are employed, where $u^f_i$7 and $u^f_i$8 are parameterized potential functions encoding radial and tangential avoidance (Ahmadvand et al., 2024, Ahmadvand et al., 1 Jul 2025).

3. Leader-Follower Optimal Control and Empirical Weight Identification

A data-driven approach to pigeon-inspired control utilizes measured GPS trajectories from real flocks to infer optimal control laws via inverse optimal control (IOC). Each follower is modeled as minimizing a quadratic tracking cost to its observed parent, including a trajectory delay,

$u^f_i$9

where $u^c_i$0 is the delayed state of the leader (Islam, 28 Mar 2026). The feedback control structure is

$u^c_i$1

with costate $u^c_i$2 evolving according to Pontryagin minimum principle conditions. By enforcing exact costate dynamics on observed data and minimizing control stationarity violations, weights $u^c_i$3 and $u^c_i$4 can be inferred, providing parameters for high-fidelity leader-follower tracking control in robotic swarms.

Notably, learned $u^c_i$5 values in real pigeon data indicate greater emphasis on minimizing control effort than position error, suggesting a biomimetic tendency for energy-efficient, smooth intra-formation motion (Islam, 28 Mar 2026).

4. Architectural Integration: Semi-Distributed and Hierarchical Schemes

Recent implementations blend centralized formation assignment, via periodic recomputation of optimal agent centroids (e.g., probabilistic Lloyd’s), with fully distributed local interaction laws. In highly scalable architectures such as BINC (Lin et al., 20 Nov 2025), agents form network-aligned clusters, each with a pigeon-like leader–follower intra-formation law: $u^c_i$6 where $u^c_i$7 encodes symmetric repulsion/alignment/attraction from visible neighbors, and $u^c_i$8 is a leader–follower (position plus consensus velocity) force calculated over a short chain of higher-level neighbors. Clusters remain within two hops, ensuring tight formation and fast intra-group command propagation.

Communication overhead is minimized by fusing control and routing information in HELLO/CMN messages, yielding per-agent bandwidth well below standard protocols (e.g., 0.8 Kbps at 1000 UAVs), and empirical results show improved radial-error maintenance and collision margins compared to standard Boids (Lin et al., 20 Nov 2025).

5. Three-Dimensional Maneuvers and Obstacle Negotiation

Pigeon-like control is extended to 3D environments by:

• Replacing planar FOV checks with conical sectors in velocity space.
• Employing 3D rotation matrices (pitch, yaw, optionally roll) to generate tangential avoidance actions in response to obstacles.
• Translating all collision and formation-keeping forces into three-dimensional vector spaces, guaranteeing navigational safety in urban scenarios with both static and moving obstacles (Ahmadvand et al., 1 Jul 2025, Ahmadvand et al., 2024).

Empirical demonstrations, e.g., 12-UAV tests over 140 seconds, show sustained safety distances, dynamic reconfiguration, and stable centroidal-Voronoi formation under simultaneous obstacle encroachment and multi-level agent interaction.

6. Stability, Convergence, and Robustness

Theoretical analysis leverages Lyapunov-based techniques:

• Formation-keeping, by descending a CVT or leader-follower tracking cost, ensures agents converge to centroids or delayed leader references.
• Collision avoidance implements energy-dissipating spring-damper mechanics; proper choice of gains ensures collision-free operation.
• Obstacle avoidance with combined radial/tangential (potential-based) terms prevents deadlock in local minima.

In networked multi-cluster scenarios, the alignment of control and communication topology permits provable invariance principles (LaSalle) showing that velocity and inter-agent distance converge to prescribed bounds (Lin et al., 20 Nov 2025). Empirical results confirm low radial error, tight formation, and efficient dynamic obstacle negotiation.

7. Implementation Guidelines and Performance Metrics

Parameter selection is driven by communication and sensing limits, physical vehicle dynamics, and desired safety margins. Representative values include:

• Sensing/interaction range $u^c_i$9 m, safety margin $u^o_i$0 m, FOV angle $u^o_i$1.
• Formation gains $u^o_i$2, $u^o_i$3.
• Collision avoidance $u^o_i$4, $u^o_i$5.
• Obstacle avoidance gains $u^o_i$6, damping $u^o_i$7.

Performance benchmarks across multiple works include:

• Guaranteed inter-agent separations above $u^o_i$8 despite urban clutter (Ahmadvand et al., 2024, Ahmadvand et al., 1 Jul 2025).
• 21.6% improved velocity variance convergence versus Boids, robust cluster preservation, and 6.36% improved obstacle margin in large-scale swarms (Lin et al., 20 Nov 2025).
• Empirically verified smooth shape recovery and distributed fault tolerance (S et al., 2023).

These results provide a comprehensive framework for implementing pigeon-like intra-formation control capable of high-density, resilient, and scalable multi-agent navigation in realistic scenarios.