The coordinated flight of multiple unmanned aerial vehicles, commonly referred to as a drone formation, represents a significant advancement over single-agent operations, offering enhanced efficiency, robustness, and capability for complex tasks. As a pivotal branch of cooperative control, the problem of drone formation control has attracted extensive research, leading to various methodologies categorized by their required measurement data: position-based, distance-based, and bearing-based control. Among these, bearing-based formation control is particularly promising for practical, distributed systems. It requires only locally measurable relative bearing (direction) information, typically obtainable from onboard sensors like cameras or directional antenna arrays, eliminating the need for external positioning systems like GPS. This makes it highly suitable for operations in GPS-denied or complex unknown environments.
However, a critical challenge persists in bearing-based drone formation tracking control: most existing distributed protocols require all agents to have prior knowledge of the desired collective velocity. This is a stringent requirement, especially in leader-follower architectures where follower drones may not have direct communication links to the leaders who dictate the formation’s motion. To address this fundamental limitation, my research focuses on a novel integration strategy. I propose a distributed drone formation control framework that synergistically combines a distributed velocity observer with a bearing-only formation controller. This integration elegantly solves the problem for followers lacking direct access to the leader’s velocity command. Furthermore, the controller is designed with inherent capabilities for formation scaling and rotation—essential maneuvers for adapting to dynamic environments and avoiding obstacles—by manipulating leader positions and incorporating a rotation matrix. The stability of the entire closed-loop system is rigorously proven using Lyapunov theory, and its effectiveness is validated through comprehensive numerical simulations.
Fundamental Theory and Problem Definition
The theoretical foundation for this work lies in graph theory and bearing rigidity. Consider a drone formation system consisting of n agents. Their interaction topology is modeled by an undirected graph $\mathcal{G} = (\mathcal{V}, \mathcal{E})$, where $\mathcal{V} = \{1, 2, …, n\}$ is the node set (drones) and $\mathcal{E} \subseteq \mathcal{V} \times \mathcal{V}$ is the edge set representing available bearing measurements between drones. For an edge $k$ corresponding to the pair $(i, j)$, the relative position and bearing vector are defined as:
$$e_{ij} = p_j – p_i, \quad g_{ij} = \frac{e_{ij}}{\|e_{ij}\|}$$
where $p_i \in \mathbb{R}^d$ is the position of drone $i$ ($d=2$ or $3$), and $\|\cdot\|$ denotes the Euclidean norm. A key mathematical tool in bearing-based control is the orthogonal projection matrix onto the orthogonal complement of $g_{ij}$:
$$P_{g_{ij}} = I_d – g_{ij} g_{ij}^T$$
which satisfies $P_{g_{ij}} = P_{g_{ij}}^T$, $P_{g_{ij}}^2 = P_{g_{ij}}$, and $P_{g_{ij}} g_{ij} = 0$. This projector is central to constructing control laws that are invariant to the distance $\|e_{ij}\|$.
The concept of bearing rigidity ensures that a drone formation can be uniquely determined up to a translation and scaling factor solely by its inter-agent bearing vectors. For a leader-follower structure, we typically designate the first two agents as leaders. A critical assumption (Assumption 1) for convergence is that the follower subgraph is bearing rigid, implying that the corresponding bearing Laplacian matrix $\mathcal{B}_{ff}$ is positive definite. This matrix is derived from the overall bearing rigidity matrix and encapsulates the bearing constraints of the formation.
The kinematic model for follower drones is a simple single-integrator model, where the control input directly dictates velocity:
$$\dot{p}_i = u_i, \quad i \in \mathcal{V}_f \text{ (follower set)}$$
The leaders are assumed to move with the desired formation velocity $v_d(t)$:
$$\dot{p}_i = v_d, \quad i \in \mathcal{V}_l \text{ (leader set)}$$
The core control objectives (Problem 1) for the drone formation are threefold:
- Tracking: The entire formation must move with the desired velocity $v_d$.
- Shape Acquisition & Maintenance: The drones must achieve and maintain a predefined geometric shape defined by a set of desired bearing vectors ${g_{ij}^*}$.
- Maneuvering: The formation must be able to scale (expand/contract) and rotate as a whole in a coordinated manner.
These objectives are captured by the convergence conditions:
$$\lim_{t \to \infty} g_{ij}(t) = R(t) g_{ij}^*, \quad \lim_{t \to \infty} \dot{p}_i(t) = v_d(t)$$
where $R(t) \in \mathbb{R}^{d \times d}$ is a rotation matrix (e.g., $R(\theta) = \begin{bmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{bmatrix}$ for 2D).
| Control Type | Required Measurement | Pros | Cons | Dependency on External Systems |
|---|---|---|---|---|
| Position-Based | Absolute or Relative Position ($p_i$, $e_{ij}$) | Mature theory, straightforward design. | Requires GPS or motion capture, less practical in denied environments. | High |
| Distance-Based | Inter-agent Distance ($\|e_{ij}\|$) | Requires only rangefinders, scale-invariant. | Collinear configurations cause ambiguity; may not uniquely define shape. | Medium |
| Bearing-Based (This Work) | Relative Direction ($g_{ij}$) | Only needs cameras/directional sensors; suitable for GPS-denied spaces. | Requires estimation of collective velocity; traditional methods need velocity pre-knowledge. | Low |
Methodology: Observer and Controller Synthesis
The proposed solution breaks down into two interconnected, distributed modules: a velocity observer and a bearing-based formation controller. The overarching architecture enables a robust and practical drone formation control system.
1. Distributed Finite-Time Velocity Observer
A primary innovation is the design of a distributed observer that allows follower drones to estimate the unknown desired formation velocity $v_d$. Let $\hat{v}_i$ be the estimate held by drone $i$. The observer protocol is:
$$
\dot{\hat{v}}_i = -k_p \sum_{j \in \mathcal{N}_i} (\hat{v}_i – \hat{v}_j) – k_i \cdot \text{sgn}\left( \sum_{j \in \mathcal{N}_i} (\hat{v}_i – \hat{v}_j) \right)
$$
For a follower drone $i$ that is a neighbor of a leader $l$, the term $\hat{v}_j$ in the sum corresponding to that leader is replaced by the true velocity $v_d$. The parameters $k_p > 0$ and $k_i > 0$ are observer gains. The signum function $\text{sgn}(\cdot)$ facilitates finite-time convergence of the estimation error $\tilde{v}_i = \hat{v}_i – v_d$. Defining the stacked error vector $\tilde{v} = [\tilde{v}_1^T, …, \tilde{v}_n^T]^T$, the dynamics can be written in compact form using the graph Laplacian matrix $L$:
$$
\dot{\tilde{v}} = -k_p (L \otimes I_d) \tilde{v} – k_i \cdot \text{sgn}((L \otimes I_d) \tilde{v})
$$
Stability Proof (Sketch): Consider the Lyapunov function candidate $V_1 = \frac{1}{2} \tilde{v}^T (L \otimes I_d) \tilde{v}$. Its time derivative yields:
$$
\dot{V}_1 \leq -k_p \lambda_{\min}(L^2) \|\tilde{v}\|^2 – k_i \| (L \otimes I_d) \tilde{v} \|_1
$$
where $\lambda_{\min}(\cdot)$ is the smallest eigenvalue. This inequality ensures that $\dot{V}_1$ is negative definite, guaranteeing that the observer error $\tilde{v}$ converges to zero in finite time. Thus, after a finite transient period, all drones have an accurate estimate: $\hat{v}_i \rightarrow v_d, \forall i$.
2. Integrated Bearing-Only Formation Controller
With a reliable velocity estimate $\hat{v}_i$ available, the formation control law for each follower drone is designed as:
$$
u_i = -k_1 \sum_{j \in \mathcal{N}_i} P_{R g_{ij}^*} \, g_{ij} + \hat{v}_i
$$
Here, $k_1 > 0$ is a control gain. This controller has an intuitive two-part structure:
- Shape Correction Term ($-k_1 \sum P_{R g_{ij}^*} g_{ij}$): This term drives the current bearing $g_{ij}$ towards the desired (and potentially rotated) bearing $R g_{ij}^*$. The projection matrix $P_{R g_{ij}^*}$ ensures the control effort is orthogonal to the desired bearing, which is the correct subspace for correcting bearing errors without affecting distances directly. This is the core mechanism for achieving and maintaining the desired drone formation shape and orientation.
- Velocity Tracking Term ($+\hat{v}_i$): This term commands the drone to move with the estimated formation velocity, ensuring the entire formation tracks the desired trajectory.
The scaling maneuver is achieved by dynamically changing the desired relative positions between the leaders, which implicitly re-scales the entire formation defined by the bearing constraints. The rotation maneuver is directly commanded by varying the rotation matrix $R(t)$ over time.
| Symbol | Definition | Dimension/Type |
|---|---|---|
| $p_i$, $v_i$, $u_i$ | Position, velocity, and control input of drone $i$. | $\mathbb{R}^d$ |
| $g_{ij}$, $g_{ij}^*$ | Actual and desired relative bearing from $i$ to $j$. | $\mathbb{S}^{d-1}$ (unit sphere) |
| $P_{g}$ | Orthogonal projection matrix: $I_d – g g^T$. | $\mathbb{R}^{d \times d}$ |
| $R(\theta)$ | Rotation matrix (2D example). | $\mathbb{R}^{d \times d}$ |
| $\hat{v}_i$, $\tilde{v}_i$ | Estimated velocity and estimation error for drone $i$. | $\mathbb{R}^d$ |
| $\mathcal{B}_{ff}$ | Follower bearing Laplacian matrix. | $\mathbb{R}^{(n-2)d \times (n-2)d}$ |
| $k_p$, $k_i$, $k_1$ | Observer and controller gain parameters. | $\mathbb{R}^+$ |
3. Closed-Loop Stability Analysis
The stability of the combined observer-controller system is analyzed using a cascaded systems approach. Define the bearing error for each edge as $\tilde{g}_{ij} = g_{ij} – R g_{ij}^*$. The closed-loop dynamics are separated into two subsystems:
Subsystem 1 (Observer Error): $\dot{\tilde{v}} = f_1(\tilde{v})$ (exponentially/finite-time stable).
Subsystem 2 (Formation Error): $\dot{\tilde{g}} = f_2(\tilde{g}, \tilde{v})$.
The time derivative of the bearing error, using the property $\dot{g}_{ij} = P_{g_{ij}} (v_j – v_i) / \|e_{ij}\|$, can be manipulated to avoid the unscaled distance term, leading to an expression suitable for analysis. Consider a Lyapunov function for the formation subsystem:
$$
V_2 = \frac{1}{2} \sum_{(i,j)\in\mathcal{E}} \|\tilde{g}_{ij}\|^2
$$
Analyzing $\dot{V}_2$ under the control law $u_i$ and using the property that $\tilde{v} \to 0$, we can show that:
$$
\dot{V}_2 \leq -\alpha V_2 + \beta \|\tilde{v}\|
$$
for some positive constants $\alpha, \beta$. This demonstrates that the formation error system is Input-to-State Stable (ISS) with respect to the observer error $\tilde{v}$. Since the observer error subsystem is globally asymptotically stable ($\tilde{v} \to 0$), by the cascaded systems theorem, the equilibrium $(\tilde{g}, \tilde{v}) = (0, 0)$ of the overall closed-loop system is globally asymptotically stable. This proves that the drone formation achieves its control objectives: $g_{ij} \to R g_{ij}^*$ and $\dot{p}_i \to v_d$.
Simulation Results and Performance Analysis
The performance of the integrated observer-based drone formation control strategy is validated through simulation scenarios involving rotation and combined scaling/obstacle avoidance. A team of six drones in a 2D plane ($d=2$) is considered, with a communication and bearing-sensing topology as described in the original work. Drones 1 and 2 are leaders.
| Parameter / Agent | Value / Initial Position ($p_i$) | Role |
|---|---|---|
| Control Gain ($k_1$) | 10 | Formation Controller |
| Observer Gains ($k_p$, $k_i$) | 1.0, 0.01 | Velocity Observer |
| Desired Velocity ($v_d$) | $[1.0, 0.5]^T$ | Formation Tracking |
| Drone 1 (Leader) | $[1.0, 1.0]^T$ | Leader |
| Drone 2 (Leader) | $[1.0, -1.0]^T$ | Leader |
| Drones 3-6 (Followers) | $[1,2]^T, [-2,1]^T, [-3,1]^T, [-2,-2]^T$ | Followers |
Scenario 1: Formation Rotation
In this test, the formation is required to achieve the desired shape, track the velocity $v_d$, and execute a $45^\circ$ rotation at $t=10$ seconds. The rotation matrix is set as $R = I_2$ for $t < 10$s and $R(45^\circ)$ for $t \geq 10$s. The simulation results confirm the controller’s efficacy. The drones successfully converge to the desired hexagonal pattern, move along the desired velocity vector, and seamlessly rotate the entire formation structure at the specified time while maintaining formation integrity. The bearing errors $\|\tilde{g}_{ij}\|$ converge to near zero and exhibit only a small, transient perturbation during the rotation before re-converging. Crucially, the velocity estimation errors $\|\tilde{v}_i\|$ for all followers converge to zero rapidly, demonstrating the observer’s success in providing accurate velocity information without direct measurement.
Scenario 2: Formation Scaling and Obstacle Passage
This scenario demonstrates the scalability and practical utility of the drone formation control system. The task is to navigate through a narrow passage (simulated as an obstacle field). The formation starts at its default scale. At $t=10$s, the leaders are commanded to move closer together for a short duration ($10 < t \leq 30$s), causing the entire formation to contract. This scaled-down formation safely passes through the constricted area. After clearing it ($t > 30$s), the leaders are commanded to return to their original separation, causing the formation to expand back to its default scale. The leaders’ velocities are modulated with sinusoidal components to achieve this smooth scaling maneuver. The results clearly show the coordinated scaling behavior: all follower drones adjust their positions proportionally to maintain the bearing constraints while the inter-leader distance changes. The formation successfully navigates the hypothetical obstacle course due to its scaling capability. Throughout this dynamic maneuver, the bearing errors and velocity estimation errors remain bounded and small, confirming the robustness and stability of the integrated control framework.
| Performance Metric | Rotation Scenario | Scaling Scenario | Interpretation |
|---|---|---|---|
| Bearing Error Convergence | Exponential decay to near-zero; small transient during rotation. | Maintained near-zero throughout scaling maneuver. | Controller effectively achieves and maintains desired formation shape. |
| Velocity Tracking Error | Follower velocities converge accurately to $v_d$. | Follower velocities track $v_d$ despite leader maneuver. | Observer successfully provides necessary velocity information. |
| Observer Error Convergence | Rapid finite-time convergence to zero. | Fast convergence and maintenance near zero. | Distributed observer is effective and stable. |
| Maneuver Execution | Precise 45° rotation at t=10s. | Smooth contraction and expansion as planned. | System reliably executes scaling and rotation commands. |
Conclusions and Future Directions
This work presents a significant advancement in distributed bearing-based drone formation control by solving the critical problem of unknown desired velocity. The proposed integration of a finite-time distributed velocity observer with a bearing-only formation controller creates a fully distributed, practical solution. The controller’s inherent support for formation scaling and rotation through leader manipulation and a rotation matrix significantly enhances the operational flexibility of the drone formation, enabling complex maneuvers like obstacle avoidance in constrained environments. The rigorous Lyapunov-based stability proof provides a solid theoretical guarantee for the system’s performance.
The simulation results validate all theoretical claims, demonstrating precise formation shape acquisition, accurate velocity tracking, and successful execution of rotational and scaling maneuvers. The performance is robust, with errors remaining bounded during dynamic changes.
Future research will focus on extending this framework to more realistic and challenging scenarios. This includes adapting the control laws for drones with more complex nonlinear dynamics (e.g., quadrotor models with attitude dynamics), incorporating robustness against measurement noise and communication delays, and developing obstacle avoidance strategies that are fully integrated with the bearing-based formation control logic. Exploring event-triggered communication to reduce network load and extending the theory to time-varying communication topologies are also promising directions for creating even more resilient and efficient autonomous drone formation systems.