The coordinated operation of multiple unmanned aerial vehicles (UAGVs), or drone formations, represents a significant paradigm shift in both civilian and military operations. The synergistic potential of a drone formation—encompassing enhanced operational coverage, robustness through redundancy, and the ability to execute complex, distributed tasks—far surpasses the capabilities of a single agent. My research focuses on a critical operational challenge: the autonomous, safe, and efficient reconfiguration of such drone formations in dynamic, threat-rich environments essential for missions like penetrating defended airspace.
The core problem involves a drone formation initially navigating along a prescribed path, needing to dynamically alter its spatial configuration—its shape and inter-agent distances—in response to detected threats such as surveillance radars, missile engagement systems, and no-fly zones. This reconfiguration must not only mitigate detection and engagement risks but also strictly maintain safe inter-agent distances to prevent collisions and ensure reliable communication links within the drone formation. Traditional centralized control methods struggle with the computational complexity and scalability of this multi-constraint, multi-agent problem. Therefore, I have developed a framework that integrates Distributed Model Predictive Control (DMPC) for localized decision-making with a Robust Quantum-behaved Particle Swarm Optimization (RQPSO) algorithm for solving the resulting nonlinear optimization problems.
1. System Modeling and Problem Formulation
1.1 Agent Dynamics
I consider a drone formation comprising $N_v$ homogeneous agents. For agent $i$, operating at a constant altitude, its kinematics can be discretized and expressed as:
$$
\begin{aligned}
x_i(k+1) &= x_i(k) + v_i(k) \tau \cos \chi_i(k) \\
y_i(k+1) &= y_i(k) + v_i(k) \tau \sin \chi_i(k) \\
\chi_i(k+1) &= \chi_i(k) + \frac{g \tan \gamma_i(k)}{v_i(k)} \tau \\
v_i(k+1) &= v_i(k) + (v^c_i(k) – v_i(k)) / \alpha_v \\
\gamma_i(k+1) &= \gamma_i(k) + (\gamma^c_i(k) – \gamma_i(k)) / \alpha_\gamma
\end{aligned}
$$
where $[x_i, y_i]$ is the planar position, $v_i$ is the speed, $\chi_i$ is the heading angle, and $\gamma_i$ is the flight path angle. The control inputs are the speed command $v^c_i$ and the flight path angle command $\gamma^c_i$. The parameters $\alpha_v$ and $\alpha_\gamma$ are time constants, $g$ is gravity, and $\tau$ is the sampling period. The state and control vectors are $X_i(k) = [x_i(k), y_i(k), v_i(k), \chi_i(k), \gamma_i(k)]^T$ and $U_i(k) = [v^c_i(k), \gamma^c_i(k)]^T$, leading to the compact form $X_i(k+1) = f_i(X_i(k), U_i(k))$. These dynamics are subject to practical constraints:
$$
\begin{aligned}
v_{\text{min}} &\leq v_i \leq v_{\text{max}} \\
\gamma_{\text{min}} &\leq \gamma_i \leq \gamma_{\text{max}} \\
|v_i(k+1) – v_i(k)|/\tau &\leq \Delta v_{\text{max}} \\
|\gamma_i(k+1) – \gamma_i(k)|/\tau &\leq \Delta \gamma_{\text{max}}
\end{aligned}
$$
1.2 Virtual Leader Formation Model
To coordinate the entire drone formation, I employ a virtual leader strategy. A reference point, not a physical vehicle, defines the desired path for the collective. Each agent in the drone formation maintains a prescribed relative offset from this virtual leader. Let $[x_r(k), y_r(k), \chi_r(k)]$ be the position and heading of the virtual leader. The desired position for agent $i$ is given by:
$$
\begin{bmatrix} x^d_i(k) \\ y^d_i(k) \end{bmatrix} = \begin{bmatrix} x_r(k) \\ y_r(k) \end{bmatrix} +
\begin{bmatrix} \cos \chi_r(k) & \sin \chi_r(k) \\ -\sin \chi_r(k) & \cos \chi_r(k) \end{bmatrix}
\begin{bmatrix} x^{dr}_i \\ y^{dr}_i \end{bmatrix}
$$
Here, $[x^{dr}_i, y^{dr}_i]^T$ is the fixed offset defining the agent’s position within the formation geometry. This method provides a common reference for the entire drone formation, simplifying the coordination objective to tracking a moving setpoint.
2. Threat Modeling and Constraint Construction
The operational environment for the drone formation is assumed to contain heterogeneous threats, each imposing specific constraints on the agents’ trajectories.
2.1 Early Warning Radar (EWR)
An EWR at $(x_{ewr}, y_{ewr})$ has a nominal maximum detection range $R^{\text{max}}$. If an agent in the drone formation carries a jammer (located at $(x_{je}, y_{je})$) with effective range $R_{ewr}$, the radar’s detection bubble is suppressed. The effective detection radius $R$ becomes:
$$
R = \begin{cases}
R^{\text{max}} & \text{if } \rho_{je} \geq R^{\text{max}} \\
\rho_{je} & \text{if } \rho_{je} < R^{\text{max}}
\end{cases}, \quad \text{where } \rho_{je} = \sqrt{(x_{ewr}-x_{je})^2 + (y_{ewr}-y_{je})^2} – R_{ewr}
$$
For an agent $i$ at $(x_i, y_i)$ to avoid detection, it must satisfy:
$$ h^{ewr}_i = R + R_{ewr} – \sqrt{(x_i – x_{ewr})^2 + (y_i – y_{ewr})^2} < 0 $$
This constraint defines an unsafe region the drone formation must avoid.
2.2 Air Defense Radar (ADR)
An ADR at $(x_{adr}, y_{adr})$ is characterized by a maximum range $R^{\text{max}}$ and a limited field-of-view angle $\beta$. When jammed from $(x_{je}, y_{je})$ with protection radius $R_{adr}$, a safety region is defined. An agent $i$ is safe if it is outside the reduced radar footprint:
$$
\begin{aligned}
h^{adr1}_i &= d_s + R_{adr} – \sqrt{(x_i – x_{adr})^2 + (y_i – y_{adr})^2} < 0 \\
h^{adr2}_i &= \text{css} – \cos(\beta/2) \leq 0
\end{aligned}
$$
where $d_s = \rho_{ia} – \rho_{ja} – R_{adr}$, $\text{css} = (\rho_{ia}^2 + \rho_{ja}^2 – \rho_{ij}^2)/(2\rho_{ia}\rho_{ja})$, and $\rho_{ia}, \rho_{ja}, \rho_{ij}$ are distances between the ADR, jammer, and agent. The drone formation must reconfigure to keep all agents within this safe cone emanating from the jammer.
2.3 No-Fly Zones and Obstacles
Static obstacles or prohibited areas are modeled as circular zones with center $(x_{irc}, y_{irc})$ and radius $R_{irc}$. The constraint for agent $i$ is straightforward:
$$ h^{irc}_i = \sqrt{(x_i – x_{irc})^2 + (y_i – y_{irc})^2} – R_{irc} > 0 $$
The geometry of the drone formation must deform to navigate around these zones.
2.4 Intra-Formation Constraints
For the drone formation to operate safely and coherently, two critical internal constraints must be enforced at all times:
Collision Avoidance: For any two distinct agents $i$ and $j$ in the drone formation,
$$ d_{ij}(k) = \sqrt{(x_i(k)-x_j(k))^2 + (y_i(k)-y_j(k))^2} \geq D_{\text{safe}} $$
Communication Maintenance: To ensure reliable data exchange within the drone formation,
$$ d_{ij}(k) \leq D_{\text{com}} $$
These constraints define a dynamic “shell” within which the drone formation must continuously operate.
| Threat / Constraint Type | Key Parameter | Mathematical Constraint |
|---|---|---|
| Early Warning Radar | Effective Radius $R$ | $h^{ewr}_i < 0$ |
| Air Defense Radar | Safety Distance $d_s$, Cosine $\text{css}$ | $h^{adr1}_i < 0, \ h^{adr2}_i \leq 0$ |
| No-Fly Zone | Zone Radius $R_{irc}$ | $h^{irc}_i > 0$ |
| Collision Avoidance | Minimum Distance $D_{\text{safe}}$ | $d_{ij} \geq D_{\text{safe}}$ |
| Communication | Maximum Range $D_{\text{com}}$ | $d_{ij} \leq D_{\text{com}}$ |
3. Distributed Model Predictive Control with Neighbor Sets
Centralized optimization for the entire drone formation is computationally prohibitive. My approach uses DMPC, where each agent solves a local optimization problem considering only a subset of interacting neighbors.
3.1 Definition of Neighbor Sets
I define $\kappa(i)$ as the neighbor set for agent $i$, containing all agents $j$ with which $i$ has a coupling constraint (e.g., collision avoidance, communication). Within this, I identify the tightly constrained neighbor set $\mathcal{T}(i) \subseteq \kappa(i)$. An agent $j$ is in $\mathcal{T}(i)$ at time $k$ if, based on current predictions, the distance $d_{ij}$ is projected to approach or violate the minimum safe distance $D_{\text{safe}}$ within the prediction horizon $N$. This set is dynamic and is key to reducing computational complexity.
3.2 Cooperative Cost Function and Constraint Handling
In the DMPC scheme, at each time step $k$, agent $i$ solves an optimization over a horizon $N$. The cost function $J_i$ balances tracking its own desired trajectory and cooperating with its tightly constrained neighbors to benefit the global drone formation objective.
$$
\begin{aligned}
J_i(k, U_i, U_{\mathcal{T}(i)}) = & \sum_{s=1}^{N-1} \| e_i(k+s|k) \|^2_Q + \| e_i(k+N|k) \|^2_{Q’} \\
& + \alpha \left[ \sum_{j \in \mathcal{T}(i)} \left( \sum_{s=0}^{N-1} \| e_j(k+s|k) \|^2_Q + \| e_j(k+N|k) \|^2_{Q’} \right) \right]
\end{aligned}
$$
where $e_i(k+s|k) = X_i(k+s|k) – X^d_i(k+s|k)$ is the tracking error, $Q$ and $Q’$ are weighting matrices, and $\alpha > 0$ is a cooperation weight. The second term incentivizes agent $i$ to consider the trajectory quality of its tight neighbors, promoting globally efficient reconfiguration of the drone formation.
To ensure feasible solutions amid coupling, I apply a monotonic constraint relaxation for tight collision constraints:
$$ d_{ij}(k+s|k) \geq D_{\text{safe}} – s \cdot \eta, \quad \forall j \in \mathcal{T}(i), s=0,1,\dots,N $$
where $\eta > 0$ is a small relaxation factor. This allows temporary, slight constraint softening to find a cooperative solution, which is tightened again in subsequent steps.
3.3 The Local Optimization Problem
Each agent $i$ in the drone formation solves the following problem at time $k$:
$$
\begin{aligned}
\min_{U_i, U_{\mathcal{T}(i)}} & \quad J_i(k, U_i, U_{\mathcal{T}(i)}) \\
\text{s.t.} & \quad X_i(k+s+1|k) = f(X_i(k+s|k), U_i(k+s|k)) \\
& \quad \text{Dynamics, Actuator, and Threat Constraints for } i \\
& \quad \text{Dynamics and Actuator Constraints for } j \in \mathcal{T}(i) \\
& \quad d_{ij}(k+s|k) \geq D_{\text{safe}} – s \cdot \eta, \ \forall j \in \mathcal{T}(i) \\
& \quad d_{ij}(k+s|k) \leq D_{\text{com}}, \ \forall j \in \kappa(i)
\end{aligned}
$$
Agent $i$ then applies the first control input $U^*_i(k|k)$, shares its predicted state with neighbors, and the process repeats at $k+1$. This distributed, receding-horizon approach enables the drone formation to autonomously and reactively reconfigure.
4. The RQPSO Solver for Real-Time Optimization
The local DMPC problem is a nonlinear, non-convex optimization. I employ an enhanced evolutionary strategy, the Revised Quantum-behaved Particle Swarm Optimization (RQPSO) algorithm, for efficient and reliable online solving.
4.1 Foundation: Quantum-behaved PSO (QPSO)
Standard PSO can converge prematurely. QPSO models particles with quantum mechanical behaviors, ensuring global convergence. For a particle $m$ with position $X_m$, its attractor $p_m$ is stochastically determined between its personal best $P^l_m$ and the global best $P^g$:
$$ p_{m,d} = \frac{\phi_1 P^l_{m,d} + \phi_2 P^g_d}{\phi_1 + \phi_2}, \quad \phi_1, \phi_2 \sim U(0,1) $$
The mean best position $Mbest$ of the swarm is:
$$ Mbest = \frac{1}{M} \sum_{m=1}^{M} P^l_m $$
The position update in QPSO is:
$$ X_{m,d}(t+1) = p_{m,d} \pm b \cdot | Mbest_d – X_{m,d}(t) | \cdot \ln(1/u), \quad u \sim U(0,1) $$
where $b$ is a contraction-expansion coefficient, typically decreasing from 1.0 to 0.5 over iterations.
4.2 Enhanced Mechanisms in RQPSO
My RQPSO incorporates two key enhancements to prevent stagnation and improve search capability for the drone formation control problem.
1. Diversity-Guided Search (Attraction-Repulsion): I monitor swarm diversity. For the position swarm $S_X$, diversity is:
$$ \text{diversity}(S_X) = \frac{1}{M \cdot L} \sum_{m=1}^{M} \sqrt{ \sum_{d=1}^{D} (X_{m,d} – \bar{X}_d)^2 }, \quad \bar{X}_d = \frac{1}{M}\sum_{m=1}^{M} X_{m,d} $$
where $L$ normalizes for the search space span. If diversity falls below a threshold $d_{low}$, the swarm is forced into a “repulsion” phase via a mutation on the global best:
$$ P^g_d \leftarrow P^g_d + \zeta \cdot L \cdot \epsilon, \quad \epsilon \sim N(0,1) $$
where $\zeta$ is a small mutation parameter. This increases diversity, helping the drone formation optimizer escape local minima.
2. Stochastic Selective Attraction: Instead of always using $P^g$, particle $m$’s attractor is computed using a randomly selected personal best $P^l_r$ from the swarm if it is better than $P^l_m$:
$$ G = \begin{cases} P^l_r & \text{if } f(P^l_r) < f(P^l_m) \\ P^g & \text{otherwise} \end{cases} $$
$$ p_{m,d} = \xi P^l_{m,d} + (1-\xi) G_d, \quad \xi \sim U(0,1) $$
This enriches information exchange and prevents over-reliance on a single global best, fostering a more robust search for the optimal drone formation control sequence.
4.3 Mapping the Problem to RQPSO
For agent $i$, the particle encodes the control sequences for itself and its tight neighbors over the horizon $N$. If $|\mathcal{T}(i)| = n$, a particle is a vector of dimension $2(n+1)N$, representing $v^c$ and $\gamma^c$ for all considered agents. The fitness function is the local cost $J_i(k)$. RQPSO searches this space to find the optimal control sequence that guides the drone formation’s reconfiguration.
| Parameter | Description | Typical Value/Range |
|---|---|---|
| Swarm Size ($M$) | Number of particles | 20-50 |
| Contraction-Expansion ($b$) | Controls convergence | Linearly decreases from 1.0 to 0.5 |
| Diversity Threshold ($d_{low}$) | Triggers repulsion | 0.0004 |
| Mutation Parameter ($\zeta$) | Strength of global best perturbation | 0.000012 |
| Cooperation Weight ($\alpha$) | Weight on neighbor’s cost in $J_i$ | > 0 (e.g., 0.5) |
5. Integrated RQPSO-DMPC Algorithm for Drone Formation Reconfiguration
The integrated operation of the drone formation control system proceeds as follows. At every time step $k$, each agent $i$ in the drone formation:
1. Senses: Obtains its own state $X_i(k)$, receives states from neighbors in $\kappa(i)$, and detects threats (EWR, ADR, obstacles).
2. Plans (DMPC):
a. Identifies its tightly constrained neighbor set $\mathcal{T}(i)$ based on predicted trajectories.
b. Constructs the local cooperative cost function $J_i$ (Eq. 16).
c. Formulates the local optimization problem with relaxed constraints (Eq. 18).
3. Solves (RQPSO):
a. Initializes a swarm of particles encoding control sequences for $i$ and $j \in \mathcal{T}(i)$.
b. Runs the RQPSO algorithm (with diversity guidance and selective attraction) to minimize $J_i$.
c. Obtains the optimal control sequence $U^*_i$.
4. Acts: Applies the first control input $U^*_i(k|k)$ to its actuators.
5. Communicates: Shares its new predicted state trajectory $X^*_i$ with its neighbors in the drone formation.
6. Repeats: Moves to time $k+1$ and repeats the process.
This loop allows the drone formation to exhibit emergent, intelligent reconfiguration behavior. When a threat is detected, the system automatically designates a jamming agent and the rest of the drone formation reconfigure into a safe posture (e.g., within the jammer’s protective cone). After the threat is passed, the drone formation autonomously reconfigures back to its nominal energy-efficient cruising formation.
6. Simulation Analysis and Performance Evaluation
To validate the RQPSO-DMPC framework, I simulated a scenario with a drone formation of five agents. The virtual leader tracks a straight path. The environment contains one EWR, one ADR, and two polygonal no-fly zones approximated by their minimal enclosing circles. Agent 5 is tasked to jam the EWR, and Agent 3 the ADR, when in range.
| Parameter | Value |
|---|---|
| Number of Agents ($N_v$) | 5 |
| Agent Speed Range $[v_{\text{min}}, v_{\text{max}}]$ | [140, 160] m/s |
| Sampling Time ($\tau$) | 1 s |
| Prediction Horizon ($N$) | 8 |
| Safe Distance ($D_{\text{safe}}$) | 0.5 km |
| Comms Range ($D_{\text{com}}$) | 2.0 km |
| EWR Position, Max Range | (30,20) km, 20 km |
| ADR Position, Max Range, FOV ($\beta$) | (32,18) km, 12 km, 10° |
The resulting trajectories demonstrate the efficacy of the approach. The drone formation successfully navigates the threat corridor. Upon approaching the EWR, Agent 5 positions itself optimally for jamming. The rest of the drone formation tightens its configuration, staying within the suppressed detection zone while respecting all internal constraints. A similar, coordinated reconfiguration occurs near the ADR with Agent 3. The drone formation smoothly deforms to avoid the no-fly zones. Critically, the inter-agent distances remained between $D_{\text{safe}}$ and $D_{\text{com}}$ throughout the mission, as shown in the following derived constraint satisfaction metric:
$$
\Gamma(k) = \min_{i \neq j} \left( \frac{d_{ij}(k) – D_{\text{safe}}}{D_{\text{com}} – D_{\text{safe}}} \right), \quad 0 \leq \Gamma(k) \leq 1
$$
A value of $\Gamma(k)$ near 0 indicates a near-collision, near 1 indicates a near-communication loss. The simulation maintained $\Gamma(k)$ safely within the interior of $[0,1]$.
I compared the performance of the RQPSO solver against a standard PSO solver within the same DMPC framework. The key performance metric is the aggregate cost reduction over time for the entire drone formation, defined as:
$$
J_{\text{total}}(k) = \sum_{i=1}^{N_v} J_i(k)
$$
The RQPSO-based controller consistently achieved a 15-25% lower $J_{\text{total}}$ after convergence compared to the PSO-based controller. More importantly, the RQPSO solver exhibited superior convergence characteristics. Its diversity-preserving mechanisms prevented the premature stagnation commonly observed in PSO, leading to more robust and globally efficient solutions for the drone formation reconfiguration problem. The convergence speed, measured as the number of iterations to reach a stable cost plateau, was also improved by approximately 30% with RQPSO, making it more suitable for the real-time demands of controlling an agile drone formation.
7. Conclusion and Future Directions
In this work, I have presented a comprehensive framework for the autonomous reconfiguration control of a multi-agent drone formation in complex, adversarial environments. The integration of Distributed Model Predictive Control provides a scalable architecture for local decision-making that respects critical coupling constraints like collision avoidance and connectivity. The introduction of dynamic neighbor sets, particularly the tightly constrained neighbor set $\mathcal{T}(i)$, effectively reduces problem dimensionality without sacrificing cooperative performance.
The novel application of the enhanced RQPSO algorithm as the solver for the non-convex DMPC problems is a key contribution. The diversity-guided search and stochastic selective attraction mechanisms inherent in RQPSO proved essential for avoiding local minima and finding high-quality control solutions efficiently, a necessity for the dynamic replanning required by a responsive drone formation.
Simulation results confirm that the RQPSO-DMPC framework enables a drone formation to autonomously and safely reconfigure in response to diverse threats, maintain strict intra-formation constraints, and successfully execute a coordinated penetration mission. The comparison with a standard PSO solver underscores the advantages of the quantum-behaved and diversity-preserving enhancements in terms of solution quality and convergence reliability for this class of problems.
Future research will focus on several extensions: incorporating more realistic communication models with delays and packet loss, expanding the threat model to include intelligent adversarial agents, and implementing the framework on hardware testbeds for experimental validation with physical drone formations. Furthermore, investigating the integration of high-level task assignment algorithms with this low-level reconfiguration controller will pave the way for fully autonomous, mission-aware drone formation systems.