The rapid advancement of modern unmanned aerial vehicle (UAV) technology has ushered in an era where swarms composed of numerous small-scale drones are increasingly deployed for complex missions. These UAV drone collectives hold immense potential for applications ranging from environmental monitoring and large-scale surveillance to search and rescue operations in hazardous or inaccessible terrain. However, the control methodologies governing these swarms often struggle to address the multifaceted challenges presented by dynamic, cluttered environments. For a multi-UAV drone system, achieving and maintaining stability and convergence is paramount, especially when tasks and environmental conditions are in constant flux. As mission profiles grow more intricate, conventional coordination strategies frequently fail to adapt, leading to swarm instability, failed aggregation, and ineffective obstacle avoidance. Consequently, there is a critical need for robust swarm control frameworks that enable flexible, adaptive, and stable collective behavior to ensure mission success in unpredictable scenarios.
Existing strategies for UAV drone swarm coordination face significant challenges. Leader-follower approaches, while simplifying control design, lack the flexibility and scalability required for large-scale swarms in complex settings; the failure of a leader or a rapidly changing environment can lead to systemic instability. Bio-inspired methods, such as those based on behavioral rules like flocking, offer decentralization but often rely on fixed interaction radii or a static number of nearest neighbors. These fixed-topology or fixed-interaction-range models exhibit poor adaptability when the swarm must split to navigate obstacles, merge after a division, or change collective shape. The swarm’s performance degrades in dense, obstacle-rich environments where maintaining cohesion, avoiding collisions, and navigating simultaneously are required. The core issues manifest as poor convergence to target states, oscillatory or unstable trajectories during adaptive maneuvers, and an inability to smoothly transition between different collective objectives like grouped travel and dispersed obstacle negotiation.
This work directly addresses these shortcomings by proposing a novel, topology-driven adaptive control method for UAV drone swarms, inspired by the sophisticated collective behaviors observed in European starlings (Sturnus vulgaris). Starlings exhibit remarkable agility, cohesion, and adaptability in dense flocks, capable of swift, coherent maneuvers that involve splitting and merging around obstacles without collapsing into disorder. Key to this ability is their reliance on topological interaction rather than metric distance—each bird interacts with a fixed number of its closest neighbors, regardless of absolute distance. This principle ensures scalable and robust information transfer. Furthermore, their flocks demonstrate natural bifurcation-aggregation dynamics, seamlessly transitioning between a unified group and multiple sub-swarms in response to environmental pressures.
Our proposed framework, the Starling-inspired Bifurcation-Aggregation Adaptive Control, incorporates three foundational innovations to equip UAV drone swarms with similar capabilities. First, we abandon fixed-distance neighbor models in favor of an Adaptive Adjacency Modeling mechanism based on 3D Delaunay Triangulation. This constructs a dynamic, topology-based interaction network that inherently adapts to the spatial distribution of the swarm. Second, we introduce a novel Bifurcation Mechanism. This allows individual UAV drones to autonomously decide, based on local density, velocity alignment, and obstacle proximity, to trigger a splitting behavior, guiding a portion of the swarm along an alternative path. Third, we develop a Local Density-Driven Online Learning strategy for the weights of the fundamental interaction forces—attraction, repulsion, and alignment. This enables each UAV drone to fine-tune its control priorities in real-time based on its immediate surroundings. The synthesis of these components results in a control system that ensures stable convergence during aggregation, enables smooth and adaptive obstacle avoidance through controlled bifurcation, and maintains coherent motion throughout complex, multi-stage tasks.
Modeling Topology-Driven Bifurcation Inspired by Starling Flocks
1.1 Topological Neighbor Selection via Delaunay Triangulation
In bio-inspired swarm systems, the mechanism governing inter-agent interactions is crucial for emergent collective behavior. Contrary to the common assumption of interaction based on a fixed metric distance (e.g., all agents within a radius r), empirical studies of starling flocks reveal a topological rule: each bird interacts with a consistent number (approximately 6-7) of its nearest topological neighbors. This model is more robust to density fluctuations and spatial deformations. To embed this biological fidelity into our UAV drone swarm control, we employ a 3D Delaunay Triangulation-based method to define dynamic adjacency.
For a swarm of N UAV drones with positions $p_i(t) = [x_i(t), y_i(t), z_i(t)]^T \in \mathbb{R}^3$ at time \(t\), the Delaunay Triangulation of the point set $\{p_i(t)\}_{i=1}^N$ generates a tetrahedral mesh. A key property of this triangulation in 3D is the “empty circumsphere” condition, which results in a set of edges \(E_{DT}(t)\) that define a natural, proximity-based graph. The neighborhood \(N_i(t)\) for drone \(i\) is thus defined as:
$$ N_i(t) = \{ j \mid (p_i, p_j) \in E_{DT}(t) \} $$
This topology is inherently dynamic, updating as the swarm moves, which provides a realistic and computationally tractable model for distributed perception with limited, time-varying connectivity—mimicking the starling’s limited sensory focus.
To further enhance adaptability to density variations, we introduce a weighting factor \(\lambda_{ij}(t)\) that modulates the strength of the topological connection based on local spatial statistics:
$$ \lambda_{ij}(t) = \exp\left( -\sigma \bar{d}_i(t) \| p_i(t) – p_j(t) \| \right) \cdot \left( 1 + \gamma |\rho_i(t) – \rho_j(t)| \right) $$
where:
- \(\bar{d}_i(t) = \frac{1}{|N_i(t)|} \sum_{j \in N_i(t)} \| p_i(t) – p_j(t) \|\) is the average distance to neighbors.
- \(\rho_i(t) = V_i / |N_i(t)|\) is a measure of local spatial density (\(V_i\) is a local volume estimate).
- \(\sigma, \gamma > 0\) are tunable parameters.
This formulation ensures that connections are stronger between UAV drones in similar density regions and are attenuated by distance, promoting more biologically plausible and stable interactions during both dense aggregation and sparse, extended formations.
1.2 Information Transmission with Delay Compensation
Starlings achieve rapid, coordinated maneuvers through efficient local information transfer, primarily visual. In a UAV drone swarm, communication is subject to delays (\(\tau\)), packet loss, and noise. To mimic the robustness of biological systems, we model information transmission as a delayed, weighted fusion process with compensation. The state vector for drone \(i\) is defined as \(s_i(t) = [p_i(t)^T, v_i(t)^T, \psi_i(t)]^T \in \mathbb{R}^7\), encompassing position, velocity, and yaw. The state update from neighbor information is:
$$ s_i(t+1) = s_i(t) + \xi \sum_{j \in N_i(t)} \tilde{\omega}_{ij}(t) \left[ s_j(t – \tau_{ij}(t)) – s_i(t) \right] $$
where \(\xi > 0\) is the fusion rate. The effective weight \(\tilde{\omega}_{ij}(t)\) incorporates both the topological weight \(\lambda_{ij}(t)\) and a delay compensation factor \(\Phi(\tau_{ij}(t))\):
$$ \tilde{\omega}_{ij}(t) = \lambda_{ij}(t) \cdot \Phi(\tau_{ij}(t)), \quad \Phi(\tau) = e^{-\phi \tau}, \quad \phi > 0 $$
This compensation reduces the influence of highly delayed signals, enhancing the system’s robustness to network imperfections—a critical feature for real-world UAV drone deployments.
1.3 Bifurcation Decision Mechanism
The core of agile obstacle avoidance is the swarm’s ability to perform controlled splitting, or bifurcation. We model the trigger for this behavior as a multi-factor decision function evaluated locally by each UAV drone. The bifurcation decision variable \(\Delta_i(t)\) integrates local misalignment, density gradients, and obstacle proximity:
$$ \Delta_i(t) = k_1 (1 – \alpha_i(t)) + k_2 (1 – \gamma_i(t)) + k_3 \, d_{obs,i}^{-1}(t) + \epsilon_1 $$
The component terms are defined as follows:
- Velocity Alignment \(\alpha_i(t)\): Measures directional consensus among neighbors.
$$ \alpha_i(t) = \frac{1}{|N_i(t)|} \sum_{j \in N_i(t)} \frac{v_i(t) \cdot v_j(t)}{\|v_i(t)\| \|v_j(t)\|} $$ - Spatial Cohesion \(\gamma_i(t)\): Measures how tightly packed the local neighborhood is.
$$ \gamma_i(t) = \exp\left( -\frac{1}{|N_i(t)|} \sum_{j \in N_i(t)} \frac{d_0}{\| p_i(t) – p_j(t) \|} \right) $$ - Obstacle Proximity: $d_{obs,i}(t)$ is the distance to the nearest obstacle.
- Weights and Constant: \(k_1, k_2, k_3 > 0\) with \(k_1+k_2+k_3=1\), and \(\epsilon_1\) is a small stability constant.
This decision variable is compared against an adaptive threshold \(\theta_i(t)\):
$$ \theta_i(t) = \theta_0 + \mu_1 \tanh(\lambda_1 \rho_i(t)) + \mu_2 \sigma_{E,i}(t) $$
where \(\sigma_{E,i}(t)\) is an environmental perturbation index. The bifurcation state \(S_i(t)\) is then:
$$
S_i(t) =
\begin{cases}
1, & \text{if } \Delta_i(t) > \theta_i(t) \quad \text{(enter bifurcation mode)}, \\
0, & \text{otherwise} \quad \text{(remain in aggregation mode)}.
\end{cases}
$$
When \(S_i(t)=1\), the drone activates a dedicated bifurcation force, initiating a maneuver to guide itself and its topological followers away from the main swarm direction to navigate around an obstacle or toward a secondary target.
Cooperative Interaction Control for UAV Drones
2.1 UAV Dynamic Model
We consider a simplified 3D kinematic model for each UAV drone, suitable for high-level path planning and swarm control. The state includes position, ground speed, climb rate, and yaw angle. The dynamics are modeled as first-order systems to account for actuator response times:
| State Variable | Differential Equation | Description |
|---|---|---|
| Position \((x_i, y_i)\) | $\dot{x}_i = v_i \cos\psi_i$, $\dot{y}_i = v_i \sin\psi_i$ | Horizontal motion |
| Altitude \(z_i\) | $\dot{z}_i = \eta_i$ | Vertical motion |
| Ground Speed \(v_i\) | $\dot{v}_i = \frac{1}{\beta_v} (v_i^c – v_i)$ | Speed control with time constant \(\beta_v\) |
| Yaw Angle \(\psi_i\) | $\dot{\psi}_i = \frac{1}{\beta_{\psi}} (\psi_i^c – \psi_i)$ | Heading control with time constant \(\beta_{\psi}\) |
| Climb Rate \(\eta_i\) | $\dot{\eta}_i = \frac{1}{\beta_z}(z_i^c – z_i) – \frac{1}{\beta_{\eta}} \eta_i$ | Altitude/rate control |
Here, \(v_i^c\), \(\psi_i^c\), and \(z_i^c\) are the commanded speed, heading, and altitude, respectively, which are generated by the high-level swarm controller described next.
2.2 Social Force-Based Interaction Model
The commanded acceleration for each UAV drone is derived from a weighted sum of four virtual forces. These forces are calculated using the delayed-compensated neighbor states \(\tilde{p}_j, \tilde{v}_j\) and weights \(\tilde{\omega}_{ij}\) from Section 1.2.
1. Attraction Force \(F_i^{att}\): Drives the drone towards the swarm centroid or a target location.
$$ F_i^{att}(t) = c_{att} \sum_{j \in N_i(t)} w_{ij}^{att}(t) \frac{\| \tilde{p}_j(t) – p_i(t) \|^q}{\| \tilde{p}_j(t) – p_i(t) \|} (\tilde{p}_j(t) – p_i(t)) $$
where \(c_{att}>0\), \(q>1\) gives more weight to distant neighbors, and \(w_{ij}^{att}\) is the adaptive attraction weight.
2. Repulsion Force \(F_i^{rep}\): Prevents collisions with other UAV drones and obstacles.
$$ F_i^{rep}(t) = c_{rep} \sum_{j \in N_i(t)} w_{ij}^{rep}(t) \frac{\| p_i(t) – \tilde{p}_j(t) \|^r}{\| p_i(t) – \tilde{p}_j(t) \|} (p_i(t) – \tilde{p}_j(t)) + c_{obs} \sum_{k=1}^{M} \frac{\| p_i(t) – o_k \|^s}{p_i(t) – o_k} $$
where \(c_{rep}, c_{obs}>0\), \(r,s>2\) create strong short-range repulsion, and \(o_k\) is the position of the \(k\)-th obstacle.
3. Alignment Force \(F_i^{ali}\): Promotes velocity matching with neighbors.
$$ F_i^{ali}(t) = c_{ali} \sum_{j \in N_i(t)} w_{ij}^{ali}(t) (\| \tilde{v}_j(t) – v_i(t) \| + \epsilon) (\tilde{v}_j(t) – v_i(t)) $$
where \(c_{ali}>0\) and \(\epsilon\) is a small constant.
4. Bifurcation Force \(F_i^{bif}\): Activated when \(S_i(t)=1\). It generates a maneuver direction perpendicular to the current average swarm velocity.
$$ v_i^{avg}(t) = \frac{1}{|N_i(t)|} \sum_{j \in N_i(t)} \tilde{v}_j(t) $$
$$ n_i(t) = g_i(t) – \left( g_i(t) \cdot \hat{v}_i^{avg}(t) \right) \hat{v}_i^{avg}(t), \quad \hat{v}_i^{avg} = \frac{v_i^{avg}}{\| v_i^{avg} \|} $$
$$ F_i^{bif}(t) = \kappa_i(t) \cdot \frac{n_i(t)}{\| n_i(t) \| + \epsilon} $$
The vector \(g_i(t)\) can be a gradient towards a secondary target or away from an obstacle. This force enables the UAV drone to break from the main flow and initiate a bifurcation path.
2.3 Adaptive Weight Learning and Integrated Control
The performance of the UAV drone swarm hinges on the appropriate weighting of the four forces. We propose an online learning mechanism where each drone adapts its force weights \(w_i^{att}, w_i^{rep}, w_i^{ali}, w_i^{bif}\) based on local feedback. The core update rule for the primary three forces is:
$$
\begin{aligned}
w_i^{att}(t+1) &= w_i^{att}(t) + \zeta_1 \left( \| p_i(t) – p_{goal} \| – \delta_{att} \right) \\
w_i^{rep}(t+1) &= w_i^{rep}(t) + \zeta_2 \left( \frac{1}{\min_{j} \| p_i(t)-p_j(t)\|} – \delta_{rep} \right) \\
w_i^{ali}(t+1) &= w_i^{ali}(t) + \zeta_3 \left( v_i(t) \cdot v_i^{avg}(t) – \delta_{ali} \right)
\end{aligned}
$$
where \(\zeta_{1,2,3}\) are learning rates and \(\delta_{att}, \delta_{rep}, \delta_{ali}\) are desired setpoints for distance to goal, inter-agent separation, and velocity alignment, respectively.
Furthermore, these base weights are modulated by a density-distance coupling factor \(\iota_{ij}(t)\) applied to each neighbor-specific contribution:
$$ \iota_{ij}(t) = \frac{1}{1 + k_4 \, \rho_i(t) \cdot \| p_i(t) – p_j(t) \| } $$
This ensures that in dense regions, the influence of very close neighbors is tempered to prevent oscillatory behavior, enhancing overall stability for the UAV drone collective.
The final, integrated control input (commanded acceleration) for UAV drone \(i\) is:
$$ u_i(t) = F_i^{total}(t) = w_i^{att}F_i^{att} + w_i^{rep}F_i^{rep} + w_i^{ali}F_i^{ali} + S_i(t) \cdot w_i^{bif}F_i^{bif} $$
This control law is distributed, relying only on information from an agent’s dynamically defined topological neighborhood, making it highly scalable and robust for large UAV drone swarms.
2.4 Stability Analysis Outline
The stability of the swarm’s velocity consensus can be analyzed using a Lyapunov approach. Consider the average swarm velocity \(\bar{v} = \frac{1}{N}\sum_{i=1}^N v_i\) and the consensus error Lyapunov function \(V_v(t) = \frac{1}{2}\sum_{i=1}^N \| v_i(t) – \bar{v}(t) \|^2\). Under reasonable assumptions of bounded interaction weights, bounded bifurcation forces, and a connected interaction topology with a positive algebraic connectivity \(\lambda_2\), the time derivative of \(V_v\) can be bounded.
The alignment force contributes a stabilizing negative definite term proportional to \(-c_1 V_v\), where \(c_1\) depends on \(\lambda_2\) and the minimum alignment gain. The attraction, repulsion, and bifurcation forces act as bounded perturbations. Applying Young’s inequality, one can derive a differential inequality of the form:
$$ \dot{V}_v \leq -\frac{c_1}{2} V_v + c_2 \sum_{i=1}^{N} S_i(t) + C_0 $$
where \(C_0\) is a constant bound related to the magnitude of the attraction/repulsion forces, and the term \(c_2 \sum S_i(t)\) captures the destabilizing effect of active bifurcations.
This inequality leads to two key insights:
- In the absence of bifurcation (\(S_i(t)=0\)), the velocity error \(V_v\) converges exponentially to a small bound around zero, guaranteeing swift consensus.
- During finite-time bifurcation (\(S_i(t)=1\) for some drones), the error remains bounded. Once the bifurcation condition ceases (e.g., obstacle passed), the exponential convergence resumes. This proves that the proposed controller ensures the UAV drone swarm remains stable and recovers consensus after adaptive splitting maneuvers.
Simulation Results and Performance Analysis
To validate the proposed Starling-inspired Bifurcation-Aggregation Adaptive Control, extensive 3D simulations were conducted with a swarm of \(N=40\) UAV drones. The drones were initialized with random positions and velocities within a bounded airspace. Three spherical obstacles of varying sizes were placed along the swarm’s intended path to a primary goal. The simulation parameters are summarized below:
| Parameter | Value | Description |
|---|---|---|
| Number of UAV Drones (N) | 40 | Swarm Size |
| Simulation Steps | 700 | Total time = 70s (Δt=0.1s) |
| Speed Time Constant (\(\beta_v\)) | 2.0 | Controls speed response |
| Attraction Gain (\(c_{att}\)) | 0.8 | Strength of attraction force |
| Repulsion Gain (\(c_{rep}\)) | 1.2 | Strength of inter-agent repulsion |
| Obstacle Gain (\(c_{obs}\)) | 2.5 | Strength of obstacle repulsion |
| Alignment Gain (\(c_{ali}\)) | 0.9 | Strength of alignment force |
| Bifurcation Gain (\(\kappa\)) | 1.0 | Strength of bifurcation force |
3.1 Phase 1: Initial Aggregation and Cohesion
At the start of the simulation, the UAV drones are dispersed. Under the influence of the attraction, alignment, and adaptive weight learning, the swarm self-organizes from a disordered state into a cohesive, flying cluster. The Delaunay-based topology ensures stable neighbor connections despite the initial randomness. Within the first 15 seconds of simulation, the swarm converges to a stable formation with highly aligned velocities, demonstrating the effectiveness of the basic social force model with online learning in achieving collective order for the UAV drone team.
3.2 Phase 2: Adaptive Bifurcation for Obstacle Avoidance
As the cohesive swarm approaches the first large obstacle, drones on the leading edge detect the high obstacle proximity via their \(d_{obs,i}(t)\) sensor input. This increases their local bifurcation decision variable \(\Delta_i(t)\). For these drones, \(\Delta_i(t)\) surpasses the adaptive threshold \(\theta_i(t)\), triggering \(S_i(t)=1\). Consequently, the bifurcation force \(F_i^{bif}\) activates. This force, directed orthogonally to the swarm’s average velocity, pushes a subset of the swarm to the side. The dynamic topology allows this subgroup to temporarily form its own local interaction network while remaining globally connected through other drones. The swarm elegantly splits into two major streams that flow around both sides of the obstacle, perfectly mimicking the fluid bifurcation observed in starling flocks. Critically, no collisions occur between drones or with the obstacle, and the maneuver is executed smoothly without oscillatory instability.
3.3 Phase 3: Multi-Stage Target Guidance and Re-aggregation
After clearing the first obstacle, the sub-swarms have different spatial relationships to secondary target points. The attraction force weight \(w_i^{att}\) for drones in each subgroup is automatically adjusted by the online learner based on their distance to their respective assigned target. This causes the subgroups to steer towards different intermediate goals. During this phase, the swarm may encounter further smaller obstacles, triggering additional, localized bifurcation events. The density-distance coupling factor \(\iota_{ij}(t)\) plays a vital role here, preventing drones in the now-denser subgroups from getting too close and causing repulsion-induced jitter. Finally, as both subgroups approach a common final rendezvous point, the dominant attraction force to this common goal causes them to merge seamlessly back into a single, cohesive UAV drone swarm. The final state shows all drones aggregated at the target location with near-identical velocities, confirming successful convergence.
3.4 Quantitative Performance Metrics
The superiority of the proposed adaptive method is evident when compared to a baseline controller using fixed force weights and a static interaction radius. The following table highlights key performance metrics averaged over 10 simulation runs:
| Performance Metric | Proposed Adaptive Control | Fixed-Weight Static-Topology Control |
|---|---|---|
| Obstacle Collision Rate | 0% | 15% |
| Final Velocity Variance | 0.08 m²/s² | 0.65 m²/s² |
| Time to Final Aggregation | 62.3 s | >70 s (Incomplete) |
| Inter-Drone Distance Violations* | 1.2 per run | 8.7 per run |
| Control Effort Variance | Low | High (Oscillatory) |
*Number of times the minimum safe distance between any two UAV drones was breached.
The proposed method’s dynamic topology allows for efficient information flow during splitting and merging. The online weight learning enables the UAV drone swarm to balance cohesion, separation, and obstacle avoidance in real-time, leading to smoother trajectories, faster convergence, and zero collisions. In contrast, the fixed strategy either hesitates to bifurate, causing collisions, or bifurates chaotically, failing to re-aggregate properly. The velocity plots for the adaptive swarm show a rapid convergence to consensus after initialization and a quick recovery of consensus after each bifurcation event, visually corroborating the bounded stability proof. The control inputs for each drone remain smooth throughout the mission, indicating stable and efficient actuator usage.
Conclusion
This paper has presented a comprehensive, bio-inspired framework for the adaptive control of UAV drone swarms in complex, multi-obstacle environments. By drawing inspiration from the topological interaction, robust information transfer, and natural bifurcation-aggregation dynamics of starling flocks, we have developed a control system that addresses the critical limitations of conventional methods. The integration of Delaunay Triangulation-based dynamic adjacency, a local-state-driven bifurcation decision mechanism, and an online learning strategy for social force weights results in a highly adaptive and robust controller. This enables the UAV drone collective to perform stable aggregation, execute smooth and effective obstacle avoidance through controlled splitting, navigate multi-stage goals, and reliably re-aggregate, all while maintaining velocity consensus and collision-free operation.
Simulation results demonstrate clear advantages over fixed-topology and fixed-weight approaches in terms of success rate, convergence speed, and trajectory smoothness. The stability analysis provides a theoretical foundation for the observed robustness. The proposed framework is inherently distributed and scalable, making it suitable for real-world applications involving large-scale UAV drone swarms, such as autonomous surveillance in urban canyons, collective search in forested areas, or coordinated delivery in cluttered airspace. Future work will focus on implementing this algorithm on physical UAV drone platforms with real communication constraints, investigating its performance under severe sensor noise and packet loss, and extending the formulation to handle dynamic obstacles and adversarial environments.