The coordinated flight of multiple unmanned aerial vehicles (UAVs), or a drone formation, represents a significant leap in capability over single-agent operations. Such formations can cover larger areas, provide multiple perspectives, and offer inherent redundancy, making them ideal for complex tasks like aerial surveying, search and rescue, and precision delivery. However, the control of these drone formations presents substantial challenges. The system must not only enable a group of agents to achieve and maintain a desired geometric shape but also remain robust to internal failures, such as the loss of a member during a mission. This discussion explores a control framework designed to address these exact challenges, focusing on smooth convergence and adaptive reconfiguration.
The core objective in drone formation control is to achieve coordinated behavior where the states (position, velocity) of individual drones converge to values that collectively define the formation. Among various approaches, consensus theory provides a distributed and elegant solution. In a consensus-based drone formation, each vehicle updates its target state based on information exchanged with its neighbors according to a communication topology. While second-order consensus, which directly controls acceleration, is common, it can lead to abrupt changes in the acceleration command, posing challenges for the low-level flight controllers of physical drones. To promote smoother motion profiles, a third-order consensus algorithm is employed here, where the control input is the derivative of acceleration (jerk). This allows for gradual, continuous changes in acceleration, leading to more feasible and stable trajectories for the drone formation.
To establish the foundation, we begin with the kinematic model of a single quadrotor drone. Its translational motion in an inertial frame is described by its position, velocity, and acceleration vectors. For a drone labeled \(i\) in the formation, we define:
- Position: \(\mathbf{p}_i = (x_i, y_i, z_i)^T\)
- Velocity: \(\mathbf{v}_i = \dot{\mathbf{p}}_i = (\dot{x}_i, \dot{y}_i, \dot{z}_i)^T\)
- Acceleration: \(\mathbf{a}_i = \ddot{\mathbf{p}}_i = (\ddot{x}_i, \ddot{y}_i, \ddot{z}_i)^T\)
For formation control, it is often convenient to define a body-fixed frame attached to a designated leader drone. The target positions for all follower drones are then specified as constant offsets within this leader’s frame, simplifying the description of the drone formation shape.
The communication structure governing information flow is crucial for the consensus algorithm. It can be effectively modeled using graph theory. The formation is represented as a directed graph \(G = (V, E)\), where:
$$ V = \{v_1, v_2, …, v_n\} $$
is the set of nodes (drones), and \(E\) is the set of edges representing communication links. The adjacency matrix \(A = [a_{ij}]\) encodes this structure, where \(a_{ij} > 0\) if drone \(i\) receives information from drone \(j\), and \(a_{ij}=0\) otherwise. The associated Laplacian matrix \(L = [l_{ij}]\) is defined as:
$$ l_{ij} = \begin{cases} \sum_{k=1, k \ne i}^{n} a_{ik}, & \text{if } i = j \\ -a_{ij}, & \text{if } i \ne j \end{cases} $$
To facilitate both path following and reconfiguration, a hierarchical, leader-following structure is adopted. The drones are organized into tiers:
| Tier | Role | Information Source | Number of Drones |
|---|---|---|---|
| 1 | Formation Leader | External Command | 1 (\(u_1\)) |
| 2 | Primary Followers | Leader (\(u_1\)) | \(b_2\) (\(u_{21}, u_{22}, …\)) |
| 3 | Secondary Followers | Primary Followers | \(b_3\) (\(u_{31}, u_{32}, …\)) |
| … | … | … | … |
This structure ensures that the entire drone formation follows the path of the leader \(u_1\). For a sample 6-drone formation, the adjacency and Laplacian matrices are:
$$ A = \begin{bmatrix}
0 & 0 & 0 & 0 & 0 & 0 \\
a_{21} & 0 & 0 & 0 & 0 & 0 \\
a_{31} & 0 & 0 & 0 & 0 & 0 \\
0 & a_{42} & 0 & 0 & 0 & 0 \\
0 & a_{52} & 0 & 0 & 0 & 0 \\
0 & 0 & a_{63} & 0 & 0 & 0
\end{bmatrix}, \quad L = \begin{bmatrix}
0 & 0 & 0 & 0 & 0 & 0 \\
-a_{21} & a_{21} & 0 & 0 & 0 & 0 \\
-a_{31} & 0 & a_{31} & 0 & 0 & 0 \\
0 & -a_{42} & 0 & a_{42} & 0 & 0 \\
0 & -a_{52} & 0 & 0 & a_{52} & 0 \\
0 & 0 & -a_{63} & 0 & 0 & a_{63}
\end{bmatrix} $$
where \(a_{21}, a_{31}, …\) are positive coupling gains.
The heart of the control strategy is the third-order consensus protocol. For each follower drone \(i\), its target jerk (derivative of acceleration) is computed based on the discrepancies in position, velocity, and acceleration with its information neighbor \(j\):
$$ \dot{\mathbf{a}}_i(t) = -\sum_{j \in N_i} a_{ij} \left[ \gamma_1 (\mathbf{p}_i(t) – \mathbf{p}_j(t)) + \gamma_2 (\mathbf{v}_i(t) – \mathbf{v}_j(t)) + \gamma_3 (\mathbf{a}_i(t) – \mathbf{a}_j(t)) \right] $$
Here, \(N_i\) is the set of neighbors of drone \(i\), and \(\gamma_1, \gamma_2, \gamma_3 > 0\) are control parameters. This target jerk is then integrated using a trapezoidal discretization to obtain smooth target acceleration, velocity, and position commands for the drone’s low-level controller:
$$ \begin{aligned}
\mathbf{a}_i(t+\Delta t) &= \mathbf{a}_i(t) + \frac{\Delta t}{2}(\dot{\mathbf{a}}_i(t) + \dot{\mathbf{a}}_i(t+\Delta t)) \\
\mathbf{v}_i(t+\Delta t) &= \mathbf{v}_i(t) + \frac{\Delta t}{2}(\mathbf{a}_i(t) + \mathbf{a}_i(t+\Delta t)) \\
\mathbf{p}_i(t+\Delta t) &= \mathbf{p}_i(t) + \frac{\Delta t}{2}(\mathbf{v}_i(t) + \mathbf{v}_i(t+\Delta t))
\end{aligned} $$
The yaw angle \(\psi_i\) is simply set to match that of its information source: \(\psi_i = \psi_j\). The stability of this third-order drone formation system is determined by the eigenvalues of the matrix:
$$ \Gamma = \begin{bmatrix}
0_n & I_n & 0_n \\
0_n & 0_n & I_n \\
-\gamma_1 L & -\gamma_2 L & -\gamma_3 L
\end{bmatrix} $$
The system is stable if \(\Gamma\) has exactly three zero eigenvalues (corresponding to the leader’s dynamics) and all other eigenvalues have negative real parts.
A key innovation for robust operation is the variable-structure communication network. This allows the drone formation to reconfigure its information links in real-time if a member fails or severely deviates, maintaining the integrity of the remaining group. A set of pre-defined communication graphs \( \mathbb{G} = \{G_1, G_2, …, G_k\} \) is established. The switching logic is as follows:
| Failure Scenario | Reconfiguration Action |
|---|---|
| Drone in the last tier fails. | No change. The formation continues with one less member. |
| Drone \(u_{jw_j}\) in a middle tier \(j\) fails. | All drones in tier \(j+1\) that followed \(u_{jw_j}\) are reassigned to follow \(u_{j(w_j+1)}\) (or \(u_{j(b_j-1)}\) if \(w_j = b_j\)). If tier \(j\) is now empty, tier \(j+1\) drones connect to \(u_{(j-1)1}\). |
| The formation leader \(u_1\) fails. | The primary follower \(u_{21}\) is promoted to leader. All other primary followers now follow the new leader \(u_{21}\). |
This protocol ensures that the drone formation remains connected and can continue its mission with a degraded but stable configuration, significantly enhancing operational reliability.
The efficacy of the proposed third-order consensus with variable-structure is validated through numerical simulation of the 6-drone formation. The parameters are set as \(\gamma_1=2, \gamma_2=4, \gamma_3=6\) with a sampling time of 0.1s. The initial and desired relative positions in the leader’s frame are given below:
| Drone | Initial Offset (m) | Target Offset (m) |
|---|---|---|
| \(u_{21}\) | (8, 3, -5) | (-5, -5, 0) |
| \(u_{22}\) | (4, 5, 2) | (5, -5, 0) |
| \(u_{31}\) | (13, -18, 0) | (-10, -10, 0) |
| \(u_{32}\) | (-3, -4, 5) | (0, -10, 0) |
| \(u_{33}\) | (-8, -15, 2) | (10, -10, 0) |
A failure of drone \(u_{22}\) is simulated at \(t = 40s\). The performance is compared against a standard second-order consensus controller without reconfiguration capability. The results clearly demonstrate the advantages of the proposed method. The third-order controller brings the drone formation to the desired configuration in approximately 17 seconds, with less oscillatory transient behavior compared to the 21 seconds and more pronounced oscillations of the second-order controller. More critically, upon the failure of \(u_{22}\), the follower \(u_{33}\) (which relied on \(u_{22}\) for information) in the non-reconfigurable scheme drifts away, breaking the formation. In contrast, under the variable-structure protocol, the communication network instantly switches to a pre-defined alternative topology (e.g., \(u_{33}\) now follows \(u_{21}\)). After a minor and brief transient, the remaining five drones successfully re-stabilize into a new, coherent drone formation that continues to follow the intended path, with steady-state position errors on the order of \(10^{-8}\) meters.
In conclusion, the integration of third-order consensus with a variable-structure communication network presents a powerful framework for high-performance and fault-tolerant drone formation control. The third-order dynamics ensure smooth target trajectories that are easier for individual drone controllers to track, leading to faster and more stable formation convergence. The pre-planned, rule-based reconfiguration strategy provides a robust response to the loss of a formation member, allowing the surviving drones to autonomously reorganize and continue their coordinated mission. This combination of smooth control and inherent resilience is essential for deploying reliable autonomous drone formations in complex, real-world environments where system failures are a non-negligible risk. Future work may involve integrating obstacle avoidance and testing the strategy with more complex formation shapes and under communication constraints.