In recent years, the coordination of multiple unmanned aerial vehicles (UAVs), commonly referred to as drone formation, has emerged as a critical area in multi-agent systems. The ability to achieve precise and efficient drone formation is essential for applications ranging from surveillance and exploration to logistics and disaster response. However, traditional drone formation control methods often rely on accurate measurements of relative positions or distances, which can be challenging in environments where external positioning systems like GPS are unavailable or unreliable. This limitation has spurred interest in bearing-only approaches, where drone formation is defined and controlled using relative direction information alone. Such methods leverage visual sensors or wireless arrays to obtain bearing measurements, enabling robust drone formation in complex scenarios. In this article, we address the problem of finite-time stationary drone formation control under bearing constraints, proposing a novel control strategy that ensures convergence within a user-defined time frame while avoiding collisions among drones. Our approach integrates time-varying scaling gains and a leader-follower architecture, offering enhanced flexibility and global convergence guarantees. Through extensive simulations in Simulink and Gazebo, we demonstrate the efficacy of our method in achieving stable and rapid drone formation.
The fundamental challenge in bearing-based drone formation control lies in the nonlinear nature of direction vectors, which complicates convergence analysis. Existing studies have explored asymptotic and exponential convergence, but finite-time convergence—where the drone formation achieves its target configuration within a predetermined time—remains underexplored. Finite-time control is particularly desirable for time-sensitive missions, as it provides predictable performance and reduces energy consumption. Our work builds upon recent advances in bearing rigidity theory and Laplacian-based analysis to design a control law that relies solely on relative bearings. By introducing a time-dependent scaling function, we enable users to specify the convergence time, a feature not commonly available in prior drone formation strategies. Additionally, we ensure smooth control inputs by avoiding discontinuous signals, thereby improving practical implementation in real-world drone formation systems. This article presents a comprehensive framework, from theoretical derivations to simulation validations, highlighting the potential of our approach for advancing autonomous drone formation technologies.
To contextualize our contribution, we first review key concepts in graph theory and bearing rigidity, which underpin our drone formation control methodology. Consider a drone formation system comprising $N$ drones, indexed as $q_i$ for $i = 1, 2, \dots, N$. The interaction topology is represented by a graph $\mathcal{G} = \{\mathcal{Q}, \mathcal{E}, \mathcal{W}\}$, where $\mathcal{Q} = \{q_1, q_2, \dots, q_N\}$ is the node set, $\mathcal{E} \subseteq \{(q_i, q_j): q_i, q_j \in \mathcal{Q}\}$ is the edge set with $|\mathcal{E}| = m$ edges, and $\mathcal{W} = (w_{ij}) \in \mathbb{R}^{N \times N}$ is the weighted adjacency matrix with $w_{ij} = 1$ if $(i,j) \in \mathcal{E}$ and $w_{ij} = 0$ otherwise. The neighbor set of drone $q_i$ is denoted as $\mathcal{N}_i = \{q_j \in \mathcal{Q}: (q_i, q_j) \in \mathcal{E}\}$. We adopt a leader-follower structure, where $n_l$ drones are leaders and $n_f$ are followers, with $n_l + n_f = N$. The leaders follow predefined trajectories, while followers adjust their positions based on bearing information to achieve the desired drone formation.
The relative position vector between drones $q_i$ and $q_j$ is defined as $e_{ij} = p_i – p_j$, where $p_i \in \mathbb{R}^d$ is the position of drone $i$ in $d$-dimensional space (typically $d=2$ or $3$). The corresponding bearing vector is given by $g_{ij} = e_{ij} / \|e_{ij}\|$, provided $\|e_{ij}\| \neq 0$. Note that $e_{ij} = -e_{ji}$ and $g_{ij} = -g_{ji}$. The orthogonal projection matrix for $g_{ij}$ is $P_{g_{ij}} = I_d – g_{ij} g_{ij}^T$, which plays a crucial role in bearing-based control. For each directed edge $k \in \{1, 2, \dots, m\}$ corresponding to $(i,j)$, we define $e_k = p_i – p_j$ and $g_k = e_k / \|e_k\|$. The overall edge vector and bearing vector are concatenated as $e = (e_1^T, e_2^T, \dots, e_m^T)^T$ and $g = (g_1^T, g_2^T, \dots, g_m^T)^T$, respectively. Using the incidence matrix $H \in \mathbb{R}^{m \times N}$, we have $e = \bar{H} p$, where $\bar{H} = H \otimes I_d$ and $p = [p_1^T, p_2^T, \dots, p_N^T]^T$.
A key tool in our analysis is the bearing Laplacian matrix $\mathcal{B} \in \mathbb{R}^{dN \times dN}$, defined as:
$$
\mathcal{B}_{ij} =
\begin{cases}
0_{d \times d}, & i \neq j, (i,j) \notin \mathcal{E} \\
-P_{g_{ij}^*}, & i \neq j, (i,j) \in \mathcal{E} \\
\sum_{k \in \mathcal{N}_i} P_{g_{ik}^*}, & i = j
\end{cases}
$$
where $g_{ij}^*$ denotes the desired bearing for edge $(i,j)$. The bearing Laplacian can be partitioned based on leader and follower drones:
$$
\mathcal{B} = \begin{bmatrix}
\mathcal{B}_{ll} & \mathcal{B}_{lf} \\
\mathcal{B}_{fl} & \mathcal{B}_{ff}
\end{bmatrix}
$$
with $\mathcal{B}_{ff} \in \mathbb{R}^{d n_f \times d n_f}$. For the drone formation to be uniquely realizable, we assume that $\mathcal{B}_{ff}$ is positive semi-definite, a standard condition in bearing-based control.
The dynamics of each follower drone are modeled as a double integrator, reflecting its position and velocity evolution:
$$
\dot{p}_i = v_i, \quad \dot{v}_i = u_i, \quad i \in \mathcal{Q}_f
$$
where $v_i \in \mathbb{R}^d$ is the velocity and $u_i \in \mathbb{R}^d$ is the control input. Leaders are assumed to follow desired trajectories $p_i^*$ with constant velocity $v_l$, i.e., $p_i = p_i^*$ and $v_i = v_l$ for $i \in \mathcal{Q}_l$. Our control objective is to design $u_i$ for followers using only bearing measurements $g_{ij}$ such that the drone formation converges to the desired configuration $p_i^*$ within a finite time $T$, i.e., $p_i(t) = p_i^*$ for all $t \geq t_0 + T$, where $t_0$ is the initial time.
To achieve this, we propose a novel control law for follower drones:
$$
u_i = -\left( k_1 + k_2 \frac{\dot{x}}{x} \right) \sum_{j \in \mathcal{N}_i} (g_{ij} – g_{ij}^*), \quad (i,j) \in \mathcal{E}
$$
where $k_1 > 0$ and $k_2 > 0$ are control gains, and $x(t)$ is a time-varying scaling function defined as:
$$
x(t) =
\begin{cases}
\left( \frac{T^a}{(t_0 + T – t)^a} \right), & t \in [t_0, t_0 + T) \\
1, & t \in [t_0 + T, \infty)
\end{cases}
$$
with $a > 0$ being a design constant. The derivative of $x(t)$ is:
$$
\dot{x}(t) =
\begin{cases}
\frac{a}{T} x(t)^{1 + 1/a}, & t \in [t_0, t_0 + T) \\
0, & t \in [t_0 + T, \infty)
\end{cases}
$$
This scaling function ensures that the control input remains smooth and allows users to preset the convergence time $T$ for the drone formation. The term $\frac{\dot{x}}{x}$ acts as a time-dependent gain that amplifies the bearing error as $t$ approaches $T$, accelerating convergence without introducing discontinuities.
We now analyze the convergence properties of our control strategy. Define the position error for each drone as $\alpha_i = p_i – p_i^*$, and let $\alpha = [\alpha_1^T, \alpha_2^T, \dots, \alpha_N^T]^T$. Consider the Lyapunov function candidate:
$$
V = \frac{1}{2} \alpha^T \alpha
$$
Taking its derivative along the system trajectories and substituting the control law yields:
$$
\dot{V} = \alpha^T \dot{\alpha} = \alpha^T (\dot{p} – \dot{p}^*) = \alpha^T u_f
$$
where $u_f = [u_{n_l+1}^T, u_{n_l+2}^T, \dots, u_N^T]^T$ is the control input vector for followers. Using the bearing error expression, we can show:
$$
\dot{V} = -\left( k_1 + k_2 \frac{\dot{x}}{x} \right) \alpha^T \bar{H} (g – g^*)
$$
where $g^*$ is the desired bearing vector. By exploiting properties of the bearing Laplacian, we derive the inequality:
$$
\dot{V} \leq -\left( k_1 + k_2 \frac{\dot{x}}{x} \right) \frac{\lambda_{\min}(\mathcal{B}_{ff})}{2 \max_k \|e_k\|} \|\alpha\|^2
$$
under the assumption that $\mathcal{B}_{ff}$ is positive semi-definite. Let $\rho = \|\bar{H}\| (\|\alpha(t_0)\| + n s(t_0))$, where $s(t_0)$ is the scale of the drone formation at $t_0$. Then, defining $\lambda_1 = k_1 \lambda_{\min}(\mathcal{B}_{ff}) / \rho$ and $\lambda_2 = k_2 \lambda_{\min}(\mathcal{B}_{ff}) / \rho$, we obtain:
$$
\dot{V} \leq -\lambda_1 V – \lambda_2 \frac{\dot{x}}{x} V
$$
Integrating this differential inequality leads to:
$$
\|\alpha(t)\| \leq e^{-\lambda_1 t} x^{-\lambda_2} \|\alpha(0)\|, \quad t \in [t_0, t_0 + T)
$$
and $\|\alpha(t)\| = 0$ for $t \geq t_0 + T$. Thus, the drone formation converges to the desired configuration within finite time $T$. Furthermore, to ensure collision avoidance, we impose an initial condition constraint:
$$
\|\alpha(0)\| \leq \min_{i,j} \|p_i^* – p_j^*\| – \beta
$$
where $\beta > 0$ is a safety margin. This guarantees that inter-drone distances remain positive throughout the operation, i.e., $\|p_i – p_j\| \geq \beta$ for all $t$, thereby preventing collisions in the drone formation.
To validate our control approach, we conducted simulations in both Simulink and Gazebo environments. The desired drone formation was a cubic structure with 8 drones, where drones 1 and 2 served as leaders, and drones 3–8 were followers. The control parameters were set as $k_1 = 10$, $k_2 = 20$, $a = 3$, and $T = 20$ seconds for Simulink, and $k_1 = 2$, $k_2 = 3$, $a = 3$, $T = 20$ seconds for Gazebo. Initial positions and velocities were randomized within feasible bounds. The following table summarizes key parameters used in the simulations:
| Parameter | Description | Simulink Value | Gazebo Value |
|---|---|---|---|
| $k_1$ | Control gain 1 | 10 | 2 |
| $k_2$ | Control gain 2 | 20 | 3 |
| $a$ | Scaling exponent | 3 | 3 |
| $T$ (s) | Convergence time | 20 | 20 |
| $N$ | Number of drones | 8 | 8 |
| $n_l$ | Number of leaders | 2 | 2 |
| $d$ | Dimension | 3 | 3 |
The Simulink results demonstrated that the drone formation achieved the target cubic configuration within approximately 14 seconds, outperforming the preset $T = 20$ seconds. Bearing errors converged to near zero rapidly, and velocity tracking showed smooth transitions without overshoot. In Gazebo, which provides a more realistic physics-based environment, the drone formation exhibited similar convergence behavior, with errors settling around 13 seconds. The inclusion of dynamic obstacles and noise in Gazebo further highlighted the robustness of our method. Below, we present a comparative analysis of convergence metrics:
| Metric | Simulink Result | Gazebo Result |
|---|---|---|
| Convergence time (s) | ~14 | ~13 |
| Max bearing error (rad) | < 0.05 | < 0.07 |
| Velocity error norm (m/s) | < 0.1 | < 0.15 |
| Collision incidents | 0 | 0 |
These results confirm that our bearing-based finite-time control law effectively stabilizes the drone formation while ensuring safety and performance. The time-varying scaling gain $x(t)$ played a pivotal role in achieving user-defined convergence, a significant advancement over asymptotic methods. Moreover, the leader-follower structure facilitated global convergence, as leaders guided the overall drone formation topology. To illustrate the mathematical formulation of the error dynamics, we derive the closed-loop system equation. Let $\alpha_f = [\alpha_{n_l+1}^T, \dots, \alpha_N^T]^T$ denote the follower error vector. Then, from the control law, we have:
$$
\dot{\alpha}_f = -\left( k_1 + k_2 \frac{\dot{x}}{x} \right) \mathcal{B}_{ff} \alpha_f + \text{higher-order terms}
$$
Since $\mathcal{B}_{ff}$ is positive semi-definite, the system is exponentially stable in finite time. The scaling function modifies the eigenvalue distribution, ensuring that convergence accelerates as $t \to T$. This principle can be generalized to other multi-agent systems, underscoring the versatility of our approach for drone formation control.
In addition to convergence, we analyzed computational efficiency. Our control law requires only bearing measurements, which can be obtained from onboard cameras or directional antennas, reducing sensor costs. The algorithm’s complexity is $O(N^2)$ due to neighbor interactions, but for sparse networks typical in drone formation, it scales linearly with the number of edges. We implemented a distributed version where each drone computes its control input locally, enhancing scalability for large-scale drone formation. The following pseudocode summarizes the control algorithm for each follower drone:
Initialize: p_i, v_i, desired bearings g_{ij}^*, gains k_1, k_2, a, T
Loop at each time step t:
Measure relative bearings g_{ij} from neighbors j in N_i
Compute scaling function x(t) and its derivative dx/dt
Compute control input: u_i = -(k_1 + k_2 * (dx/dt)/x) * sum_{j in N_i} (g_{ij} - g_{ij}^*)
Update velocity and position using double integrator dynamics
If t >= t_0 + T, set u_i = 0 (formation achieved)
End Loop
Future work will focus on extending this method to dynamic drone formation where the desired configuration changes over time, such as in reconfigurable swarms. Incorporating adaptive gains to handle uncertainties in bearing measurements or external disturbances could further improve robustness. Additionally, integrating machine learning techniques for online optimization of parameters like $k_1$, $k_2$, and $a$ may enhance performance in unpredictable environments. Another direction is to explore heterogeneous drone formation with varied dynamics, such as mixing quadrotors and fixed-wing UAVs, where bearing constraints must account for different kinematic models.
In conclusion, we have presented a finite-time bearing-only control strategy for stationary drone formation. By leveraging time-varying scaling gains and a leader-follower architecture, our method achieves rapid convergence within a user-specified time while avoiding collisions. Theoretical analysis guarantees almost global stability under standard assumptions, and simulations in both Simulink and Gazebo validate practical effectiveness. This contribution advances the state-of-the-art in drone formation control, offering a flexible and reliable solution for applications requiring precise coordination under sensing constraints. As drone technologies continue to evolve, such bearing-based approaches will play a crucial role in enabling autonomous, scalable, and resilient multi-drone systems.