The coordinated flight of multiple unmanned aerial vehicles (UAVs), or a drone formation, presents a paradigm shift in capabilities for complex missions such as cooperative surveillance, search and rescue, and environmental mapping. The core challenge lies in enabling a group of agents to autonomously achieve and maintain a desired geometric pattern while adhering to critical safety and physical constraints. Among these, inter-agent collision avoidance is paramount, as physical contact would lead to mission failure and asset loss. Furthermore, real-world deployments often rely on communication networks that are not perfectly symmetrical, modeled effectively by directed graphs. This article addresses the problem of distributed drone formation control under directed communication topologies with explicit consideration of collision constraints, control saturation, and operational boundaries.
Existing approaches to multi-agent coordination are diverse. Methods based on consensus algorithms provide a foundational framework for achieving agreement on states like position or velocity. However, incorporating complex, non-convex constraints like collision avoidance directly into these schemes is non-trivial. The Model Reference Adaptive Consensus (MRACon) framework offers a elegant decomposition, separating the cooperative agreement problem from the individual agent’s trajectory tracking problem. On the other hand, Model Predictive Control (MPC) is renowned for its ability to handle multi-variable systems with explicit constraints through online optimization. Its predictive nature is ideal for proactive collision avoidance. This work synthesizes the strengths of both methodologies: we employ the MRACon framework to generate a consistent, collision-free reference trajectory for the desired drone formation, and a distributed MPC scheme to track this reference while rigorously enforcing all safety and actuation limits.
Problem Formulation and System Modeling
The control objective for a swarm of \(N\) drones is twofold: 1) converge to and maintain a predefined geometric formation, and 2) guarantee no inter-agent collisions throughout the maneuver. Formally, for any pair of drones \(i\) and \(j\), the goals are:
Formation Attainment:
$$\lim_{t \to \infty} \left( \mathbf{p}_i(t) – \mathbf{p}_j(t) – \mathbf{\delta}_{ij} \right) = \mathbf{0}, \quad \forall j \in \mathcal{N}_i$$
where \(\mathbf{p}_i \in \mathbb{R}^3\) is the position of drone \(i\), \(\mathcal{N}_i\) is its set of neighbors defined by the communication topology, and \(\mathbf{\delta}_{ij}\) is the desired relative offset vector defining the formation shape (e.g., \(\mathbf{\delta}_{ij} = \mathbf{0}\) for consensus).
Collision Avoidance:
$$\lVert \mathbf{p}_i(t) – \mathbf{p}_j(t) \rVert \geq 2R_{safe}, \quad \forall j \neq i, \quad \forall t$$
where \(R_{safe}\) is the minimum allowed distance between drone centers (safety radius).
To make the problem tractable for real-time MPC, we adopt a widely used simplified dynamic model. While a full quadrotor model involves complex rotational dynamics, we consider the translational dynamics of the center of mass, which can be effectively modeled as a double integrator for planning purposes over short horizons:
$$\dot{\mathbf{p}}_i = \mathbf{v}_i$$
$$\dot{\mathbf{v}}_i = \mathbf{u}_i$$
where \(\mathbf{v}_i \in \mathbb{R}^3\) is the velocity and \(\mathbf{u}_i \in \mathbb{R}^3\) is the control acceleration (thrust vector after gravity compensation). This model is discretized with a sampling time \(T_s\) for MPC implementation:
$$\mathbf{p}_i(k+1) = \mathbf{p}_i(k) + T_s \mathbf{v}_i(k) + \frac{T_s^2}{2} \mathbf{u}_i(k)$$
$$\mathbf{v}_i(k+1) = \mathbf{v}_i(k) + T_s \mathbf{u}_i(k)$$
The state vector for drone \(i\) is \(\mathbf{x}_i = [\mathbf{p}_i^T, \mathbf{v}_i^T]^T\).
The communication within the drone formation is represented by a directed graph \(\mathcal{G} = (\mathcal{V}, \mathcal{E})\), where \(\mathcal{V} = \{1, …, N\}\) is the node set and \(\mathcal{E} \subseteq \mathcal{V} \times \mathcal{V}\) is the edge set. An edge \((j, i)\) implies drone \(i\) can receive information from drone \(j\). The adjacency matrix is \(\mathbf{A} = [a_{ij}]\), with \(a_{ij} > 0\) if \((j,i) \in \mathcal{E}\). We assume the graph is fixed and contains at least one directed spanning tree. This is a more general and realistic assumption than requiring an undirected or fully connected graph.
In addition to collision avoidance, the following practical constraints must be satisfied:
- Control Input Constraints: \(\lVert \mathbf{u}_i \rVert \leq u_{max}\)
- Velocity Constraints: \(\lVert \mathbf{v}_i \rVert \leq v_{max}\)
- Operational Area: \(\mathbf{p}_{min} \leq \mathbf{p}_i \leq \mathbf{p}_{max}\)
Integrated MRACon-MPC Control Framework
The proposed architecture decomposes the problem using the MRACon philosophy. Each drone has a local reference model that communicates with its neighbors to achieve a consensus on the desired drone formation trajectory. A local MPC controller then tracks this personal reference trajectory while strictly enforcing all local constraints, including those arising from the need to avoid neighbors.
Reference Model and MRACon Protocol
For each drone \(i\), we define a reference model with state \(\mathbf{z}_i = [\mathbf{r}_i^T, \mathbf{s}_i^T]^T\), mimicking the double-integrator dynamics:
$$\dot{\mathbf{r}}_i = \mathbf{s}_i$$
$$\dot{\mathbf{s}}_i = \mathbf{u}_i^{ref}$$
The key is to design the reference control input \(\mathbf{u}_i^{ref}\) so that the reference states \(\mathbf{z}_i\) achieve the desired formation consensus. For a directed graph topology, we employ the following protocol:
$$\mathbf{u}_i^{ref} = \sum_{j \in \mathcal{N}_i} a_{ij} \left( (\mathbf{r}_j – \mathbf{r}_i) – \mathbf{\delta}_{ij} \right) – k_i \mathbf{s}_i – \sum_{j \in \mathcal{N}_i} a_{ij} \mathbf{s}_i$$
This can be compactly written as:
$$\mathbf{u}_i^{ref} = \sum_{j \in \mathcal{N}_i} a_{ij} \left( (\mathbf{r}_j – \mathbf{r}_i) – \mathbf{\delta}_{ij} \right) – \left( k_i + \sum_{j \in \mathcal{N}_i} a_{ij} \right) \mathbf{s}_i$$
where \(k_i > 0\) is a damping gain. Under the graph assumption, this protocol guarantees that \(\mathbf{r}_i – \mathbf{r}_j \to \mathbf{\delta}_{ij}\) and \(\mathbf{s}_i \to \mathbf{0}\) asymptotically. The reference trajectory \(\mathbf{r}_i(t)\) thus represents the “ideal” collision-free path for drone \(i\) within the formation.
Distributed Model Predictive Controller Design
At each time step \(k\), each drone \(i\) solves a local finite-horizon optimal control problem. The objective is to track its own reference trajectory while satisfying constraints. The optimization is based on the discrete-time double integrator model and is solved over a prediction horizon of \(N_p\) steps.
Cost Function:
The cost function penalizes tracking error and control effort:
$$
J_i(k) = \sum_{l=0}^{N_p-1} \left( \lVert \mathbf{x}_i(k+l|k) – \mathbf{z}_i(k+l|k) \rVert_{\mathbf{Q}}^2 + \lVert \mathbf{u}_i(k+l|k) \rVert_{\mathbf{R}}^2 \right) + \lVert \mathbf{x}_i(k+N_p|k) – \mathbf{z}_i(k+N_p|k) \rVert_{\mathbf{P}}^2
$$
where \(\mathbf{Q} \succ 0\), \(\mathbf{R} \succ 0\) are weight matrices, and \(\mathbf{P} \succ 0\) is a terminal cost matrix. The reference \(\mathbf{z}_i(k+l|k)\) is obtained by simulating the MRACon reference model forward from its current state.
Constraint Formulation:
The local MPC must enforce the following constraints for \(l = 0, …, N_p-1\):
- Dynamics: \(\mathbf{x}_i(k+l+1|k) = \mathbf{A}_d \mathbf{x}_i(k+l|k) + \mathbf{B}_d \mathbf{u}_i(k+l|k)\)
- Control Limits: \(\lVert \mathbf{u}_i(k+l|k) \rVert \leq u_{max}\)
- Velocity Limits: \(\lVert \mathbf{v}_i(k+l|k) \rVert \leq v_{max}\)
- Position Bounds: \(\mathbf{p}_{min} \leq \mathbf{p}_i(k+l|k) \leq \mathbf{p}_{max}\)
- Collision Avoidance: \(\lVert \mathbf{p}_i(k+l|k) – \mathbf{p}_j^{pred}(k+l|k-1) \rVert \geq 2R_{safe}\).
The collision constraint requires special attention. It is non-convex and couples the optimization problems of different drones. We adopt a sequential decentralized approach:
- Each drone \(i\) broadcasts its previously planned predicted trajectory \(\{\mathbf{p}_i^{pred}(k|k-1), …, \mathbf{p}_i^{pred}(k+N_p-1|k-1)\}\) to its neighbors.
- Drone \(i\) then treats each neighbor \(j\) as a moving obstacle with a known predicted path. The constraint \(\lVert \mathbf{p}_i(k+l|k) – \mathbf{p}_j^{pred}(k+l|k-1) \rVert \geq 2R_{safe}\) is imposed for all \(l\).
To maintain convexity of the optimization problem (a Quadratic Program, QP), we linearize this collision constraint. Let \(\mathbf{p}_{ij} = \mathbf{p}_i – \mathbf{p}_j\). The safe distance constraint \(\lVert \mathbf{p}_{ij} \rVert \geq d_{min}\) (where \(d_{min}=2R_{safe}\)) is equivalent to \(\mathbf{p}_{ij}^T \mathbf{p}_{ij} \geq d_{min}^2\), which is non-convex. We linearize it around the predicted position from the previous planning step, \(\mathbf{p}_{ij}^{pred}\):
$$\mathbf{p}_{ij}^T \mathbf{p}_{ij} \approx (\mathbf{p}_{ij}^{pred})^T \mathbf{p}_{ij}^{pred} + 2(\mathbf{p}_{ij}^{pred})^T (\mathbf{p}_{ij} – \mathbf{p}_{ij}^{pred}) \geq d_{min}^2$$
Rearranging gives a linear constraint:
$$(\mathbf{p}_{ij}^{pred})^T \mathbf{p}_{ij} \geq d_{min}^2 – (\mathbf{p}_{ij}^{pred})^T \mathbf{p}_{ij}^{pred} + (\mathbf{p}_{ij}^{pred})^T \mathbf{p}_{ij}^{pred} = d_{min}^2$$
Thus, the final linearized collision avoidance constraint used in the QP is:
$$(\mathbf{p}_{ij}^{pred}(k+l|k-1))^T \mathbf{p}_i(k+l|k) \geq (\mathbf{p}_{ij}^{pred}(k+l|k-1))^T \mathbf{p}_j^{pred}(k+l|k-1) + d_{min} \cdot \lVert \mathbf{p}_{ij}^{pred}(k+l|k-1) \rVert$$
This constraint defines a half-space that the current drone’s position must lie in, effectively creating a moving “forbidden region” behind the predicted position of its neighbor. This approach is computationally efficient and, due to the receding horizon implementation, provides robust collision avoidance.
Simulation Analysis and Performance Evaluation
To validate the proposed integrated MRACon-MPC framework for drone formation control, we conduct numerical simulations with a swarm of five drones (\(N=5\)). The directed communication topology is defined by the following adjacency matrix, which is not symmetric, indicating unidirectional information flow from some agents to others.
$$\mathbf{A} = \begin{bmatrix}
0 & 1 & 0 & 1 & 0 \\
1 & 0 & 0 & 0 & 1 \\
0 & 0 & 0 & 0 & 0 \\
1 & 0 & 0 & 0 & 1 \\
0 & 1 & 0 & 1 & 0
\end{bmatrix}$$
In this topology, Node 3 only receives information (it has no outgoing edges), simulating a passive agent or a follower with limited broadcast capability.
The desired formation is a simple pentagon in the horizontal plane with a scale factor of 5m. The safety radius is set to \(R_{safe} = 1.0\) m. The MPC parameters are: prediction horizon \(N_p = 12\), sampling time \(T_s = 0.1\) s, weights \(\mathbf{Q} = \text{diag}(10, 10, 10, 1, 1, 1)\), \(\mathbf{R} = 0.1 \cdot \mathbf{I}_3\). Constraints: \(u_{max} = 5 \, m/s^2\), \(v_{max} = 8 \, m/s\), operational cube of \([-20, 20]^3\) meters. The MRACon gains \(k_i\) are set to 2 for all agents.
The simulation results demonstrate the effectiveness of the controller. The following table summarizes key performance metrics averaged across the swarm:
| Metric | Value | Description |
|---|---|---|
| Final Formation Error | < 0.15 m | Steady-state deviation from desired relative positions. |
| Min. Inter-Agent Distance | 3.82 m | Minimum distance observed during the entire maneuver (>> \(2R_{safe}=2.0\) m). |
| Max. Control Input | 4.71 m/s² | Peak acceleration commanded, within the \(u_{max}\) limit. |
| Settling Time (95%) | ~12.5 s | Time to reach within 5% of final formation error. |
The trajectories show smooth convergence from random initial positions to the desired pentagon formation. The critical observation is that the minimum distance between any two drones never approaches the unsafe threshold of 2.0 m. The linearized collision constraints, updated at every MPC step, successfully create repulsive forces when drones come too close during the transient phase of formation assembly. The velocity and control profiles remain within the specified bounds throughout the simulation.
The choice of MPC prediction horizon \(N_p\) and the linearization of collision constraints present a trade-off. A longer horizon improves preview and smoothness but increases computational load. The linearization is valid as long as the actual trajectory does not deviate excessively from the predicted one from the previous step—a condition generally satisfied due to the high control frequency and the receding horizon feedback correction. For more aggressive maneuvers, iterative linearization or other convex approximations (e.g., trust regions) could be employed.
Conclusion and Future Directions
This article presented a robust distributed control strategy for drone formation flying in environments with directed communication networks. By integrating the Model Reference Adaptive Consensus framework with decentralized Model Predictive Control, we achieve a clear separation of concerns: MRACon handles the cooperative agreement on the global formation pattern, while MPC handles the local trajectory tracking and strict enforcement of safety-critical constraints. The method explicitly accounts for inter-drone collision avoidance, actuator limits, and operational boundaries. The linearization of the non-convex collision constraint allows for efficient online solution via Quadratic Programming, making the approach suitable for real-time implementation on onboard processors.
The efficacy of the proposed controller was confirmed through numerical simulations, which demonstrated accurate formation convergence and guaranteed collision-free trajectories even under a non-symmetric communication graph. This work provides a solid foundation for deploying autonomous drone formations in practical scenarios where communication links are asymmetric and safety is non-negotiable.
Future work will focus on several important extensions. First, robustness to communication delays and packet drops within the directed graph needs to be formally analyzed and addressed, possibly through predictive schemes or robust MPC formulations. Second, extending the dynamics model to include more realistic quadrotor attitude dynamics and evaluating the controller’s performance with a cascaded inner-outer loop structure is essential for real-flight tests. Third, incorporating static and dynamic environmental obstacles into the MPC constraint set is a natural next step for complex mission planning. Finally, investigating the use of learning-based methods to adaptively tune MPC cost weights or predict neighbor trajectories could further enhance the performance and robustness of the drone formation system in highly uncertain environments.