The coordinated flight of multiple unmanned aerial vehicles, or drone formation, represents a cornerstone technology with profound implications across military and civilian domains. These applications range from sophisticated aerial surveillance and cooperative payload delivery to large-scale light displays and precision agriculture. The core challenge in maintaining a precise drone formation lies in managing the relative positions and velocities between aircraft amidst real-world environmental disturbances. Among these, wind fields pose a significant and ubiquitous threat. Unpredictable gusts and sustained winds alter the aerodynamic forces on each vehicle, inducing deviations in their intended flight paths. For a closely spaced drone formation, these deviations can rapidly degrade the desired geometric pattern, increase collision risk, and ultimately lead to mission failure. Therefore, developing robust control strategies that enable a drone formation to maintain its configuration against wind disturbances is a critical research imperative.
Traditional control methods, such as PID or linear feedback, often prove inadequate as they are typically designed for nominal conditions and lack the capability to adapt to unknown, time-varying disturbances like wind. Robust control techniques can guarantee stability under bounded uncertainties but tend to be conservative, potentially sacrificing performance. This work addresses this gap by proposing a novel adaptive control methodology specifically designed for drone formation keeping in three-dimensional space under wind field disturbances. The core idea is to treat the wind disturbance as an unknown but constant or slowly varying vector acting on each drone’s dynamics. An adaptive law is then designed to estimate these wind components online, and this estimate is used to augment a baseline formation controller, effectively canceling out the wind’s destabilizing effect. This approach allows the drone formation to preserve its intended geometry without requiring a precise prior model of the wind field itself.
The subsequent sections detail the mathematical foundation, controller synthesis, stability analysis, and simulation-based validation of this approach. We begin by establishing the kinematic and dynamic models for a single drone, then extend this to derive the relative motion model for a leader-follower pair within the drone formation. This model is then augmented to incorporate the effects of a spatially decomposed wind field.
Mathematical Modeling of Drone Formation Dynamics
To design an effective controller, a mathematical representation of the system is essential. We first model the individual drone’s motion, then describe the relative dynamics between drones in a formation.
1.1 Kinematic Model of a Single Drone
The kinematic equations of a drone in three-dimensional space, ignoring external disturbances initially, are given with respect to an inertial frame (East-North-Up):
$$
\begin{aligned}
\dot{x} &= v_g \cos\theta \cos\chi \\
\dot{y} &= v_g \cos\theta \sin\chi \\
\dot{z} &= v_g \sin\theta \\
p &= \dot{\phi} – \dot{\psi}\sin\theta \\
q &= \dot{\theta}\cos\phi + \dot{\psi}\cos\theta\sin\phi \\
r &= -\dot{\theta}\sin\phi + \dot{\psi}\cos\theta\cos\phi
\end{aligned}
$$
Here, $(x, y, z)$ is the drone’s position. $v_g$ is the ground speed. $\theta$, $\phi$, and $\chi$ are the pitch, roll, and heading (azimuth) angles, respectively. $p$, $q$, and $r$ are the roll, pitch, and yaw rates in the body frame.
1.2 Dynamic Model of a Single Drone
The translational dynamics in the body frame are expressed as:
$$
\begin{aligned}
a_x &= \dot{u} = vr – wq – g\sin\theta + \frac{F_x}{m} \\
a_y &= \dot{v} = -ur + wp + g\cos\theta\sin\phi + \frac{F_y}{m} \\
a_z &= \dot{w} = uq – vp + g\cos\theta\cos\phi + \frac{F_z}{m}
\end{aligned}
$$
where $(u, v, w)$ are the body-frame velocity components, $g$ is gravity, $m$ is mass, and $(F_x, F_y, F_z)$ are the total forces along the body axes.
1.3 Leader-Follower Relative Motion Model
We consider a “leader-follower” architecture for the drone formation. Let subscripts $L$ and $F$ denote the leader and follower, respectively. The relative position errors in the leader’s horizontal reference frame (along-track, cross-track) and the vertical direction are defined as:
$$
\begin{bmatrix} l_e \\ f_e \\ h_e \end{bmatrix} =
\begin{bmatrix}
\sin\chi_L & -\cos\chi_L & 0 \\
\cos\chi_L & \sin\chi_L & 0 \\
0 & 0 & 1
\end{bmatrix}
\begin{bmatrix} x_L – x_F \\ y_L – y_F \\ z_L – z_F \end{bmatrix} –
\begin{bmatrix} l_c \\ f_c \\ h_c \end{bmatrix}
$$
where $(l_c, f_c, h_c)$ are the constant desired relative distances. Differentiating yields the relative motion kinematics:
$$
\begin{bmatrix} \dot{l}_e \\ \dot{f}_e \\ \dot{h}_e \end{bmatrix} =
\begin{bmatrix}
V_F \sin(\chi_F – \chi_L) \\
V_L – V_F \cos(\chi_F – \chi_L) \\
V_L – V_F
\end{bmatrix} +
\dot{\chi}_L \begin{bmatrix} f_c \\ -l_c \\ 0 \end{bmatrix}
$$
This model describes how the follower’s velocity and heading relative to the leader affect the formation tracking errors.
1.4 Relative Motion Model under Wind Disturbance
The key innovation is to explicitly model the wind’s effect. The wind velocity vector in the inertial frame is decomposed into three constant (or slowly varying) components $[w_x, w_y, w_z]^T$. The follower’s kinematic equations are modified as:
$$
\begin{aligned}
\dot{x}_F &= v_{gF}\cos\theta_F \cos\chi_F + w_x \\
\dot{y}_F &= v_{gF}\cos\theta_F \sin\chi_F + w_y \\
\dot{z}_F &= v_{gF}\sin\theta_F + w_z
\end{aligned}
$$
This wind effect propagates into the relative motion model. Furthermore, wind alters the aerodynamic angles. The sideslip angle $\beta$, normally zero in steady coordinated flight, becomes non-zero due to the crosswind component. For formation control, the heading angle $\psi$ (approximately equal to $\chi$ in level flight) is the primary control variable, but the wind-induced $\beta$ contributes to the relative velocity. By defining a composite disturbance vector $T = [T_x, T_y, T_z]^T = [v_x+w_x, v_y+w_y, v_z+w_z]^T$, the relative dynamics can be reformulated to highlight the wind’s role as an additive disturbance to the nominal relative motion equations. The control objective for the drone formation becomes to drive $l_e, f_e, h_e \to 0$ despite the unknown $w_x, w_y, w_z$.
Synthesis of the Adaptive Formation Controller
The proposed control structure integrates a baseline heading command generator with an adaptive wind disturbance estimator. The block diagram in the referenced work illustrates this synergy: the outer-loop controller generates a heading command $\psi_d$ for the follower based on the formation errors, while the adaptive law provides estimates $\hat{w}_x, \hat{w}_y, \hat{w}_z$ to cancel the wind effects.
2.1 Controller Design
We design the desired heading command $\psi_d$ for the follower and the adaptive update laws concurrently. The control law is formulated to achieve exponential convergence of the formation errors. We propose:
$$
\tan\psi_d = \frac{v_{gF}\cos\theta_F \sin\chi_F + \hat{T}_y}{v_{gF}\cos\theta_F \cos\chi_F + \hat{T}_x}
$$
where $\hat{T}_x, \hat{T}_y$ are estimates of the composite disturbance. The specific form derived from the Lyapunov analysis (detailed in the next section) is:
$$
\psi_d = \arctan\left( \frac{-c_1 l_e – \hat{w}_x}{V_L + c_2 f_e + \hat{w}_y} \right) + \psi_L
$$
where $c_1, c_2 > 0$ are control gains. This command aims to nullify the lateral and forward distance errors. The vertical channel typically uses the pitch angle for control with a similar compensatory term $-c_3 h_e – \hat{w}_z$.
2.2 Adaptive Law Design
The adaptive laws for estimating the wind disturbances are derived using Lyapunov stability theory. Defining the estimation errors $\tilde{w}_i = w_i – \hat{w}_i$ for $i \in \{x, y, z\$, the update laws are designed as:
$$
\begin{aligned}
\dot{\hat{w}}_x &= k_{w_x} l_e \\
\dot{\hat{w}}_y &= k_{w_y} f_e \\
\dot{\hat{w}}_z &= k_{w_z} h_e
\end{aligned}
$$
where $k_{w_x}, k_{w_y}, k_{w_z} > 0$ are adaptive gains. Intuitively, these laws state that the wind estimate in a particular direction is adjusted proportionally to the formation error in that same direction. If a persistent cross-track error $l_e$ exists, the law increases $\hat{w}_x$ to compensate.
2.3 Stability Analysis
The stability of the closed-loop drone formation system is proven using Lyapunov’s direct method. Consider the candidate Lyapunov function:
$$
V = \frac{1}{2}l_e^2 + \frac{1}{2}f_e^2 + \frac{1}{2}h_e^2 + \frac{1}{2k_{w_x}}\tilde{w}_x^2 + \frac{1}{2k_{w_y}}\tilde{w}_y^2 + \frac{1}{2k_{w_z}}\tilde{w}_z^2
$$
This function is positive definite and radially unbounded. Its time derivative along the trajectories of the system, after substituting the control law (2) and the adaptive laws (3), becomes:
$$
\dot{V} = -c_1 l_e^2 – c_2 f_e^2 – c_3 h_e^2 + l_e \tilde{w}_x + f_e \tilde{w}_y + h_e \tilde{w}_z – \frac{1}{k_{w_x}}\tilde{w}_x \dot{\hat{w}}_x – \frac{1}{k_{w_y}}\tilde{w}_y \dot{\hat{w}}_y – \frac{1}{k_{w_z}}\tilde{w}_z \dot{\hat{w}}_z
$$
Substituting the adaptive laws simplifies this to:
$$
\dot{V} = -c_1 l_e^2 – c_2 f_e^2 – c_3 h_e^2
$$
Since $\dot{V} \leq 0$, the Lyapunov function $V$ is non-increasing. This implies that $l_e, f_e, h_e, \tilde{w}_x, \tilde{w}_y, \tilde{w}_z$ are all bounded. Furthermore, $\dot{V}$ is negative semi-definite. By invoking Barbalat’s lemma, we can conclude that the formation errors converge to zero asymptotically:
$$
\lim_{t \to \infty} l_e(t), f_e(t), h_e(t) = 0
$$
The wind estimation errors $\tilde{w}_i$ remain bounded but do not necessarily converge to zero unless persistent excitation conditions are met. However, the convergence of the formation errors is guaranteed, which is the primary control objective for the drone formation. This proves that the proposed adaptive controller ensures stable formation keeping.
Simulation Results and Performance Analysis
The effectiveness and robustness of the proposed adaptive controller for drone formation flight are validated through numerical simulations. A leader-follower pair is simulated with the leader flying a steady trajectory and the follower employing the control law. Key parameters, inspired by realistic small fixed-wing UAV models, are used.
| Parameter | Leader Value | Follower Value |
|---|---|---|
| Wingspan | ~1.96 m | ~4.0 m |
| Mass | 20.64 kg | 30.0 kg |
| Cruise Speed ($V$) | 8 m/s (simulation) | 8 m/s (simulation) |
| Initial Wind Disturbance $[w_x, w_y, w_z]$ | [0.6, 0.6, 0.2] m/s | |
| Control Gains $[c_1, c_2, c_3]$ | [0.2, 0.15, 0.1] | |
| Adaptive Gains $[k_{w_x}, k_{w_y}, k_{w_z}]$ | [0.009, 0.0009, 0.001] | |
3.1 Formation Error Convergence
The proposed adaptive method is compared against a non-adaptive baseline controller (simplified as a Laplacian-based controller in the context). The convergence of the cross-track ($l_e$), along-track ($f_e$), and vertical ($h_e$) errors is shown in the referenced figures. The adaptive controller demonstrates superior performance:
- Cross-track Error ($l_e$): The adaptive controller reduces the steady-state error to approximately 1.05 m, whereas the baseline method settles at around 1.18 m. The adaptive system shows a smoother convergence without overshoot.
- Along-track Error ($f_e$): Both controllers drive the error to zero, but the adaptive controller achieves a slightly faster settlement while maintaining a prudent separation margin for collision avoidance within the drone formation.
- Vertical Error ($h_e$): Starting from a 50 m initial offset, the adaptive controller reliably brings the follower to the leader’s altitude, achieving tight vertical formation.
The commanded heading angle $\psi_d$ for the follower adjusts dynamically to counteract the estimated wind, confirming the controller’s active compensation mechanism.
3.2 Wind Disturbance Estimation
A critical feature of the method is its ability to estimate the unknown wind components. The simulation confirms that the adaptive laws successfully estimate the constant wind disturbances.
- The estimate $\hat{w}_x$ converges to a value near the true constant disturbance, directly contributing to the cancellation of the cross-track error.
- The estimate $\hat{w}_y$ evolves to counteract the along-track wind effect, ensuring speed matching within the drone formation.
- The estimate $\hat{w}_z$ accurately tracks the vertical wind component, allowing for precise altitude hold relative to the leader.
The accuracy of these estimates is the enabling factor for the precise formation keeping observed.
3.3 Robustness to Varying Wind Conditions
The controller’s robustness is tested under different constant wind magnitudes. The following table summarizes the steady-state performance:
| Wind Speed $|w|$ (m/s) | Settling Time for $l_e$ (s) | Max Transient Error $f_e$ (m) | Remarks |
|---|---|---|---|
| 1.8 | ~6.5 | ~2.1 | Fastest convergence, minimal error. |
| 2.4 | ~8.0 | ~2.8 | Stable performance, slightly longer settling. |
| 3.0 | ~9.5 | ~3.5 | Robust stability maintained. |
| 3.8 | ~11.0 | ~4.2 | Largest errors but stable convergence. |
The results demonstrate that the adaptive controller maintains stability and ensures error convergence across a range of disturbances, a key requirement for a practical drone formation system operating in uncertain environments.
Discussion and Comparative Analysis
4.1 Feasibility and Comparative Advantage
The proposed method’s performance is contextualized by comparing it with other advanced control paradigms like sliding mode control (SMC) and backstepping control when applied to the same drone formation problem. A qualitative comparison is summarized below:
| Control Method | Key Principle | Advantages | Disadvantages for Formation Flight |
|---|---|---|---|
| Proposed Adaptive Control | Online estimation and cancellation of wind disturbance. | No prior wind model needed; guarantees asymptotic error convergence; less conservative than robust control. | Estimation convergence requires persistent excitation; performance depends on adaptive gain tuning. |
| Sliding Mode Control (SMC) | Forces system states onto a predefined sliding surface insensitive to matched disturbances. | Strong robustness to bounded disturbances; fast transient response. | Chattering phenomenon can excite unmodeled dynamics; high control activity may be unsuitable for tight drone formation. |
| Backstepping Control | Recursive design using Lyapunov functions for strict-feedback systems. | Systematic design for nonlinear systems; provides stability guarantees. | Can become complex for high-order systems; performance can degrade significantly under large, unmodeled disturbances like wind without an adaptive or robust augmentation. |
| Disturbance Observer-Based Control (DOBC) | Estimation and feedforward compensation of disturbances via an observer. | Effective disturbance rejection; can be combined with various controllers. | Observer design often assumes known disturbance model structure (e.g., constant); may require full state measurement. |
The proposed method strikes a balance by offering formal stability guarantees, direct disturbance estimation without assuming a specific dynamic model (only piecewise constant), and smoother control action compared to SMC, making it highly suitable for drone formation applications.
4.2 Extension to General Wind Fields
While the stability proof and primary simulations assume constant or slowly varying winds, the adaptive law’s structure lends itself to mitigating more general wind fields like turbulence and gusts. The update law $\dot{\hat{w}} = k e$ acts as a continuous integrator. When the wind $w(t)$ is time-varying, the estimation error dynamics become $\dot{\tilde{w}} = \dot{w} – k e$. If the wind rate $\dot{w}$ is bounded and the formation error $e$ is driven small by the control action, the estimation error $\tilde{w}$ will remain bounded. This means the controller can still significantly reduce the formation error caused by general wind, though perfect cancellation is not achieved. The core ability to maintain a stable drone formation is preserved, which is the most critical requirement.
4.3 Practical Implementation Considerations
For real-world deployment of this drone formation control strategy, several aspects must be considered:
- Communication Topology: The leader-follower model is a building block. For larger formations, a distributed version of this adaptive law, where each drone estimates the local wind and communicates with neighbors, would be necessary.
- Sensor Requirements: The method requires relative position measurements (e.g., via GPS with centimeter-level accuracy or vision-based systems) and ownship states (velocity, attitude). It does not require direct wind measurement (e.g., pitot-static system for wind relative to vehicle), as the wind is estimated implicitly.
- Actuator Limits and Dynamics: The inner-loop autopilot (controlling roll, pitch, and thrust to achieve the commanded $\psi_d$ and velocity) must be sufficiently fast and accurate. The outer-loop adaptive controller derived here assumes perfect inner-loop tracking, which is a standard separation principle.
- Collision Avoidance: The pure formation keeping controller must be integrated with a higher-level obstacle and inter-drone collision avoidance protocol, especially during initialization or in the presence of large disturbances.
Conclusion
This work presented a comprehensive adaptive control solution to the critical problem of maintaining a precise drone formation in the presence of unknown wind field disturbances. The methodology hinges on the online estimation of the three-dimensional wind vector components acting on the follower vehicle. These estimates are seamlessly integrated into a Lyapunov-based formation control law, effectively canceling the wind’s perturbing effects on the relative motion. Rigorous stability analysis proves that the closed-loop system ensures asymptotic convergence of all formation tracking errors—cross-track, along-track, and vertical—to zero.
Simulation studies validated the controller’s efficacy, demonstrating accurate wind estimation, rapid error convergence, and robust performance across varying wind conditions. The method offers a favorable balance between performance and robustness compared to alternatives like sliding mode or standard backstepping control, as it explicitly targets the primary disturbance source without excessive control chatter or complexity.
The proposed framework provides a solid theoretical foundation for engineers developing resilient multi-drone systems. Future work will focus on extending this adaptive paradigm to fully distributed drone formation topologies, integrating collision avoidance constraints directly into the adaptive control structure, and experimental validation with a physical multi-drone platform in realistic atmospheric conditions.