The coordinated control of multiple unmanned aerial vehicles (UAVs), particularly in the context of drone formation flight, represents a frontier of significant theoretical and practical importance in robotics and autonomous systems. The ability for a group of drones to autonomously configure into and maintain a specific geometric pattern enables a multitude of advanced applications that are infeasible for a single agent. These applications span from large-scale aerial light shows and precision agricultural surveying to complex military reconnaissance missions and collaborative environmental monitoring. The core challenge lies in developing decentralized control strategies that are scalable, robust to individual failures, and capable of ensuring safe operation in close proximity.
Traditional approaches to drone formation control often simplify each UAV to a point-mass model, focusing primarily on the geometric convergence of agents to their designated positions within a formation pattern. While this simplification is useful for theoretical analysis, it neglects a critical aspect of real-world deployment: inter-agent collision avoidance. During the initial formation rendezvous phase, where drones converge from dispersed initial locations to their target formation shape, the risk of collision is highest. This phase demands a control policy that not only drives agents towards a consensus on their collective state (position and velocity) but also actively incorporates safety constraints to prevent physical overlap. This paper addresses this gap by integrating a consensus-based rendezvous strategy with an active collision avoidance mechanism, specifically leveraging the unique vertical mobility of quadrotor drones.
The foundational methodology for multi-agent coordination often draws from graph theory and consensus algorithms. In a drone formation context, each UAV is considered a node in a communication or sensing network. The topology of this network—defined by which agents can exchange information with which others—is represented by a graph \(G = (V, E)\), where \(V = \{1, 2, …, N\}\) is the set of nodes (drones) and \(E \subseteq V \times V\) is the set of edges (communication links). The flow of information is captured by the adjacency matrix \(A = [a_{ij}] \in \mathbb{R}^{N \times N}\), where \(a_{ij} > 0\) if node \(j\) can receive information from node \(i\), and \(a_{ij} = 0\) otherwise. A critical algebraic object derived from this graph is the Laplacian matrix \(L = [l_{ij}]\), where \(l_{ii} = \sum_{j \neq i} a_{ij}\) and \(l_{ij} = -a_{ij}\) for \(i \neq j\). The eigenvalues of \(L\) determine the convergence properties of consensus algorithms. A key result states that a system under a consensus protocol will reach a common state if and only if the associated graph contains a directed spanning tree, meaning there exists at least one root node (a leader) that has a direct or indirect path to all other nodes in the network.
We model the translational dynamics of each quadrotor drone \(i\) for the purpose of high-level path planning as a second-order integrator system. This is a standard and effective abstraction that separates the intricate attitude dynamics and low-level motor control from the outer-loop formation control. The model is given by:
$$ \dot{s}_i = v_i, \quad \dot{v}_i = u_i $$
where \(s_i = [x_i, y_i, z_i]^T \in \mathbb{R}^3\), \(v_i = [\dot{x}_i, \dot{y}_i, \dot{z}_i]^T \in \mathbb{R}^3\), and \(u_i \in \mathbb{R}^3\) represent the position, velocity, and control input (acceleration command) of the \(i\)-th drone, respectively. The control input \(u_i\) is the variable we design to achieve both rendezvous and safety.
The formation pattern is defined relative to a formation center or a virtual leader. Let \(s_c(t)\) and \(v_c(t)\) be the desired trajectory (position and velocity) of this formation reference. For each follower drone \(i\), we assign a constant offset vector \(\tilde{s}_i \in \mathbb{R}^3\). The control objective is to drive each drone such that \(s_i – \tilde{s}_i \to s_c\) and \(v_i \to v_c\) as \(t \to \infty\), thereby achieving and maintaining the desired geometric pattern defined by the set \(\{\tilde{s}_i\}\). The rendezvous problem is a specific instance where the drones start from arbitrary initial positions and must converge to this static formation pattern, i.e., \(s_c\) is a constant point.
We adopt a leader-follower framework within the consensus architecture. A single drone is designated as the physical leader (index 1), which has access to the reference trajectory \(s_c, v_c\). The communication topology is fixed and defined such that the leader broadcasts its state information to all followers, while the followers may have an arbitrary, possibly sparse, communication network among themselves. This topology ensures the existence of a directed spanning tree rooted at the leader, satisfying the fundamental condition for consensus under a fixed graph. The following table summarizes a sample topology for a 5-drone formation:
| Information Receiver (Follower) | Information Sources |
|---|---|
| Drone 2 | Leader (1), Drone 3 |
| Drone 3 | Leader (1), Drone 4 |
| Drone 4 | Leader (1), Drone 5 |
| Drone 5 | Leader (1), Drone 2 |
Based on this topology and the second-order dynamics, we propose the following distributed consensus-based control law for each follower drone \(i \in \{2, …, N\}\):
$$ u_i(t) = \dot{v}_c – \alpha\left[\left(s_i – \tilde{s}_i\right) – s_c + \gamma(v_i – v_c)\right] – \sum_{j=1}^N a_{ij}\left[\left(s_i – \tilde{s}_i\right) – \left(s_j – \tilde{s}_j\right) + \gamma(v_i – v_j)\right] $$
Here, \(\alpha > 0\) is a gain applied to the leader-follower error term (we set \(\alpha=1\) if the leader’s information is available, otherwise 0), \(\gamma > 0\) is a damping gain that couples position and velocity errors to ensure critical damping and avoid oscillations, and \(a_{ij}\) are the adjacency matrix elements defining the follower communication network. This control law has an intuitive interpretation: the first term \(\dot{v}_c\) is a feedforward acceleration; the second term drives the drone’s offset-corrected position and velocity towards the leader’s reference; and the third term drives consensus between the drone and its neighboring followers on their offset-corrected states. Under the stated topology conditions, Lyapunov stability analysis or algebraic graph theory can show that this protocol guarantees global asymptotic convergence:
$$ \lim_{t \to \infty} (s_i(t) – \tilde{s}_i) = s_c, \quad \lim_{t \to \infty} v_i(t) = v_c \quad \forall i. $$
This solves the pure drone formation rendezvous problem. However, the trajectories generated by this consensus law do not inherently guarantee collision avoidance. To address this, we exploit the vertical degree of freedom inherent to quadrotor drones. The core idea is to temporarily perturb the z-component of the target offset \(\tilde{s}_i\) during the rendezvous phase to create vertical separation between drones whose planned paths would otherwise bring them too close horizontally.
The collision avoidance strategy is integrated into the rendezvous process via the following algorithmic steps. First, we define a safety radius \(R\) for each drone (often modeled as a sphere). The condition for collision avoidance between drones \(i\) and \(j\) at any time is \(d_{ij}(t) > R_i + R_j\), where \(d_{ij}(t) = \lVert s_i(t) – s_j(t) \rVert\).
Step 1 – Trajectory Prediction: Using the consensus control law with the original offsets \(\tilde{s}_i^0\), simulate or calculate the predicted rendezvous trajectories \(s_i^0(t)\) for all drones over the expected maneuver time.
Step 2 – Minimum Distance Calculation: For every pair of drones \((i, j)\), compute the minimum predicted separation distance:
$$ d_{min,ij}^0 = \min_{t \in [0, T]} d_{ij}^0(t) $$
Step 3 – Collision Risk Detection: Identify all drone pairs for which \(d_{min,ij}^0 \le R_i + R_j + \delta\), where \(\delta\) is a small safety margin. These pairs are at risk of collision.
Step 4 – Vertical Offset Adjustment: For each colliding pair \((i, j)\), we adjust their formation offsets in the z-direction. A simple heuristic is applied: for an ordered pair (e.g., \(i < j\)), increment the z-offset of drone \(i\) by \(\Delta h\) and decrement the z-offset of drone \(j\) by the same amount. This creates a vertical separation of \(2\Delta h\) between their target positions. Let the adjusted z-offset be \(\tilde{z}_i^1 = \tilde{z}_i^0 + \Delta h_i\). The table below illustrates this process for two colliding drones:
| Drone | Original Z-Offset \(\tilde{z}^0\) | Adjustment \(\Delta h\) | Adjusted Z-Offset \(\tilde{z}^1\) |
|---|---|---|---|
| i | \(h_0\) | \(+\Delta h\) | \(h_0 + \Delta h\) |
| j | \(h_0\) | \(-\Delta h\) | \(h_0 – \Delta h\) |
Step 5 – Iterative Refinement: After adjustment, return to Step 1 to compute new trajectories \(s_i^1(t)\) with offsets \(\tilde{s}_i^1\). Re-evaluate \(d_{min,ij}^1\). This process forms a loop. To find the minimal necessary adjustment, a binary search (or bisection method) can be employed over the adjustment magnitude \(\Delta h_i\) for each involved drone. The optimization aims to find the smallest set of offset changes \(\{\Delta h_i^*\}\) such that \(d_{min,ij}^* > R_i + R_j + \delta\) for all pairs. The termination condition for the bisection on a single parameter is \(\lVert \Delta h_i^{(k)} – \Delta h_i^{(k-1)} \rVert \le \epsilon\), where \(\epsilon\) is a small tolerance.
Step 6 – Two-Phase Maneuver Execution: Once safe offsets \(\{\tilde{s}_i^*\}\) are found, the drone formation maneuver executes in two phases:
1. Safe Rendezvous: Drones use the consensus control law with the safe, adjusted offsets \(\tilde{s}_i^*\) to converge to the “vertically separated” formation points.
2. Vertical Recovery: After reaching these safe points and coming to a hover (a key advantage of quadrotors), each drone performs a simple vertical maneuver to transition to its final, original offset \(\tilde{s}_i^0\). Since this recovery is performed from a hover and drones are already spatially separated in XY, the collision risk is eliminated.
The efficacy of the proposed integrated strategy for drone formation rendezvous with collision avoidance is validated through numerical simulations in MATLAB. We consider a scenario with \(N=5\) quadrotors. Drone 1 is the leader, stationed at a fixed point \(s_c = [2, 2, 2]^T\) m. The desired 2D formation is a cross pattern, defined by the original planar offsets \(\tilde{s}_i^0 = [\tilde{x}_i^0, \tilde{y}_i^0, 0]^T\) for followers \(i=2,…,5\). The key simulation parameters are summarized below:
| Parameter | Symbol | Value |
|---|---|---|
| Consensus Gain | \(\alpha\) | 1 |
| Damping Gain | \(\gamma\) | 2 |
| Adjacency (Follower Net) | \(a_{ij}\) | 1 if connected, 0 otherwise |
| Safety Radius | \(R\) | 0.05 m |
| Initial Adjustment Step | \(\Delta h_e\) | 0.1 m |
| Optimization Tolerance | \(\epsilon\) | 0.001 m |
The initial positions of the followers are randomized. Applying the pure consensus algorithm (without collision avoidance) to the original offsets generates predicted trajectories. Analysis of the minimum pairwise distances reveals potential collisions. For instance, in one simulation run, the initial minimum distance between Drone 2 and Drone 5 was found to be \(d_{min,25}^0 = 0.0077 \text{ m} < 0.1 \text{ m} (=2R)\). This triggers the avoidance algorithm.
The bisection-based optimization converges to an optimal vertical adjustment of \(\Delta h_2^* = +0.0629\) m and \(\Delta h_5^* = -0.0629\) m for Drones 2 and 5, respectively, while other drones’ heights remain unchanged. The resulting minimum separation distances for all pairs, \(d_{min,ij}^*\), are all verified to be greater than the safety threshold of 0.1 m. The following table contrasts the critical distances before and after optimization:
| Drone Pair (i,j) | Min Distance \(d_{min,ij}^0\) (m) | Min Distance \(d_{min,ij}^*\) (m) | Collision Risk Resolved? |
|---|---|---|---|
| (2,3) | 1.3842 | 1.3853 | Safe / Safe |
| (2,4) | 0.3928 | 0.3947 | Safe / Safe |
| (2,5) | 0.0077 | 0.1000 | Collision / Safe |
| (3,4) | 1.4142 | 1.4142 | Safe / Safe |
| (3,5) | 1.0000 | 1.0000 | Safe / Safe |
| (4,5) | 0.4450 | 0.4489 | Safe / Safe |
The simulation plots clearly demonstrate the two-phase maneuver. In the first phase, all drones converge smoothly to their vertically adjusted targets. Drone 2, for example, converges to \([2, 1, 1.9371]^T\) m instead of its final target \([2, 1, 2]^T\) m. In the second phase, upon stabilization, Drone 2 executes a vertical ascent of 0.0629 m to reach its final position within the formation. 3D trajectory plots show that the paths of Drones 2 and 5 no longer intersect, and the spheres representing their safety volumes remain separate at the point of closest approach, confirming the success of the collision avoidance strategy. The entire system achieves stable formation rendezvous without any inter-agent collisions.
This research presents a pragmatic and effective solution to a critical sub-problem in autonomous drone formation control. By merging the strengths of graph-theoretic consensus algorithms for decentralized coordination with a strategic, optimization-based use of the vertical dimension for conflict resolution, we have developed a method that ensures safe rendezvous for quadrotor teams. The proposed two-phase strategy—safe convergence to vertically separated targets followed by a final vertical alignment—effectively decouples the collision avoidance problem from the geometric formation control problem, simplifying the overall solution. The numerical simulations confirm the theoretical validity and practical feasibility of the approach. Future work will focus on extending this method to dynamic formations (moving \(s_c(t)\)), incorporating robust control to handle communication delays and packet loss, and addressing more complex obstacle-rich environments. The transition from simulation to real-world implementation with physical quadrotors, accounting for aerodynamic disturbances and full nonlinear dynamics, is the logical and necessary next step in advancing this line of research for reliable multi-drone systems.