The coordination and control of multiple unmanned aerial vehicles (UAVs) operating in a cohesive group, known as a drone formation, represents a significant advancement in autonomous systems with profound implications for applications ranging from aerial surveying and precision agriculture to complex defense and security operations. The core challenge in sustaining an effective drone formation lies in maintaining precise relative positions and attitudes among individual vehicles amidst inherent uncertainties and environmental disturbances. While a single drone’s flight control involves managing communication, trajectory planning, and stabilization, the complexity escalates in a multi-agent drone formation where the additional constraints of formation keeping, collision avoidance, and coordinated path planning must be seamlessly integrated. This discussion focuses primarily on the critical problem of formation keeping, assuming a pre-planned flight path for the lead vehicle and designing controllers for follower vehicles to maintain geometric cohesion and achieve collaborative consistency within the drone formation.
The fundamental architecture for many drone formation control strategies is the leader-follower paradigm. In this model, one designated vehicle, the leader, follows a predefined trajectory, while the follower vehicles adjust their states based on the leader’s information and relative positioning to preserve the desired formation geometry. Consider a simplified two-dimensional horizontal plane representation. The relative motion between a leader (L) and a wingman or follower (W) can be effectively described in a rotating coordinate frame attached to the follower’s velocity vector. The key states include the vehicles’ velocities ($V_L$, $V_W$), flight path angles or headings ($\phi_L$, $\phi_W$), and their inertial position coordinates. The relative separation in this follower-centric frame is defined by the along-track ($x$) and cross-track ($y$) distances. A typical relative kinematics model in the horizontal plane is given by:
$$ \dot{x} = \dot{\phi}_W y – V_W + V_L \cos(\phi_L – \phi_W) $$
$$ \dot{y} = -\dot{\phi}_W x + V_L \sin(\phi_L – \phi_W) $$
For control purposes, we define the formation keeping error. Let $x_c$ and $y_c$ be the desired separation coordinates in the follower’s body frame, and $x$ and $y$ be the actual measured separations. The formation error vector $\mathbf{e} = [e_x, e_y]^T = [x_c – x, y_c – y]^T$ is the primary variable we aim to regulate to zero. The dynamics of this error are central to the controller design for the drone formation. It is often practical to analyze this error in the inertial frame, $\mathbf{E} = [E_x, E_y]^T$, where its magnitude is preserved ($||\mathbf{E}|| = ||\mathbf{e}||$), avoiding singularities that might occur in the rotating frame when separations are zero.
The drone formation control system operates as a closed-loop autonomous system. The core principle involves the formation controller onboard each follower processing the desired formation geometry, the relative state information (distance, speed, heading difference), and its own state. It then generates command signals for the follower’s autopilot loops—typically for speed, heading, and altitude—to minimize the formation error. The process is cyclical: the leader broadcasts its state; followers compute current relative distances; formation errors are calculated; control commands are generated and executed. This continuous feedback enables the robust maintenance of the drone formation.
The design of an effective controller for a drone formation is complicated by several factors. Small, high-speed UAVs often have inaccurately modeled dynamics. Furthermore, the flight environment introduces unknown disturbances. A significant, often neglected disturbance in close drone formation flight is aerodynamic coupling, particularly wingtip vortices, which can induce forces and moments on neighboring vehicles, destabilizing the formation. Conventional control methods like PID may struggle with these variable and uncertain dynamics. This necessitates the use of more advanced, adaptive control strategies that can adjust to unknown parameters or bounded disturbances in real-time, ensuring the resilience of the drone formation.
We delve into two primary adaptive control designs for drone formation keeping, differentiated by the nature of the uncertainty they address.
1. Adaptive Control with Unknown Aerodynamic Coupling Parameters
In this scenario, we consider a dominant disturbance arising from the aerodynamic vortex effects between UAVs in close proximity. The disturbance forces and moments affecting the follower’s speed and heading channels can be modeled as:
$$ \Delta f_v(\cdot) = \frac{\bar{q}S}{m} \Delta C_{D_y} \Delta y $$
$$ \Delta f_{\phi}(\cdot) = \frac{\bar{q}S}{mV} (\Delta C_{Y_y} \Delta y + \Delta C_{Y_z} \Delta z) $$
Here, $\Delta C_{D_y}$ and $\Delta C_{Y_y}$ are the incremental drag and side-force coefficient derivatives due to the leader’s vortex, $\bar{q}$ is dynamic pressure, $S$ is reference area, $m$ is mass, and $V$ is velocity. $\Delta y$ and $\Delta z$ are lateral and vertical separations. The key assumption is that the model structure is known, but the coupling parameters $\Delta C_{D_y}$ and $\Delta C_{Y_y}$ are unknown constants or slowly varying. The control objective for the drone formation is to maintain $\mathbf{E} \rightarrow 0$ despite these unknown parameters.
Differentiating the inertial-frame error $\mathbf{E}$ twice and incorporating the vehicle’s autopilot dynamics (approximated as first-order for speed and heading) and the disturbance models yields:
$$ \ddot{\mathbf{E}} = \mathbf{C}\mathbf{G} \begin{bmatrix} V_{c} \\ \phi_{c} \end{bmatrix} – \mathbf{C}\mathbf{G} \begin{bmatrix} V \\ \phi \end{bmatrix} + \mathbf{D} + \mathbf{C} \begin{bmatrix} \Delta f_v \\ \Delta f_{\phi} \end{bmatrix} $$
Where $\mathbf{C}$ is a transformation matrix dependent on the desired formation geometry ($x_c$, $y_c$) and the follower’s heading $\phi_W$, $\mathbf{G} = \text{diag}(g_v, g_{\phi})$ contains the autopilot gains, and $\mathbf{D}$ contains terms related to the leader’s acceleration. The control inputs are the commanded speed $V_c$ and heading angle $\phi_c$ for the follower’s autopilot.
We define a combined error metric $\mathbf{S} = \dot{\mathbf{E}} + k_3 \mathbf{E}$, with $k_3 > 0$. The control law and the adaptive update law are designed together. Consider the following control input:
$$ \begin{bmatrix} V_{c} \\ \phi_{c} \end{bmatrix} = -(\mathbf{C}\mathbf{G})^{-1} \left( -(k_1+k_2)\mathbf{S} + \mathbf{D} + k_3\dot{\mathbf{E}} + \mathbf{C}\mathbf{G} \begin{bmatrix} -V \\ -\phi \end{bmatrix} \right) $$
The term $k_2$ is crucial for adaptation and is defined as $k_2 = \frac{\mathbf{r}^T \hat{\boldsymbol{\Delta}}_c}{||\mathbf{S}||}$, where $\mathbf{r} = ||\mathbf{C}|| [b_1, b_2]^T$ is a known scaling vector related to the disturbance model structure, and $\hat{\boldsymbol{\Delta}}_c = [\hat{\Delta c}_1, \hat{\Delta c}_2]^T$ are the estimates of the unknown aerodynamic parameters $[\Delta C_{D_y}, \Delta C_{Y_y}]^T$. The adaptive update laws for these estimates are:
$$ \dot{\hat{\Delta c}}_1 = ||\mathbf{S}|| \cdot ||\mathbf{C}|| b_1, \quad \dot{\hat{\Delta c}}_2 = ||\mathbf{S}|| \cdot ||\mathbf{C}|| b_2 $$
The stability of this adaptive controller for the drone formation is proven using Lyapunov theory. We choose a candidate Lyapunov function that includes both the tracking error and the parameter estimation error $\tilde{\boldsymbol{\Delta}}_c = \boldsymbol{\Delta}_c – \hat{\boldsymbol{\Delta}}_c$:
$$ V = \frac{1}{2} \mathbf{S}^T \mathbf{S} + \frac{1}{2} \tilde{\Delta c}_1^2 + \frac{1}{2} \tilde{\Delta c}_2^2 $$
Taking its time derivative and substituting the control and update laws leads to:
$$ \dot{V} \leq -k_1 ||\mathbf{S}||^2 \leq 0 $$
By Lyapunov’s direct method and Barbalat’s lemma, we can conclude that $\mathbf{S} \rightarrow 0$, which implies $\mathbf{E} \rightarrow 0$ and $\dot{\mathbf{E}} \rightarrow 0$. Therefore, the drone formation error converges to zero, and the geometric formation is maintained despite the initially unknown aerodynamic coupling parameters.
| Feature | Strategy 1: Unknown Parameters | Strategy 2: Unknown Bounded Disturbance |
|---|---|---|
| Disturbance Model | Known structure, unknown parameters (e.g., $\Delta C_{D_y}$) | Unknown structure, bounded magnitude |
| Core Mechanism | Online parameter estimation and cancellation | Robust control via discontinuous feedback based on error bounds |
| Control Law | Continuous, based on parameter estimates $\hat{\boldsymbol{\Delta}}_c$ | Contains a discontinuous ‘sign’ or smooth approximation term |
| Lyapunov Function | $V = \frac{1}{2}\mathbf{S}^T\mathbf{S} + \frac{1}{2}\tilde{\boldsymbol{\Delta}}_c^T\tilde{\boldsymbol{\Delta}}_c$ | $V = \frac{1}{2}\mathbf{\bar{E}}^T\mathbf{P}\mathbf{\bar{E}}$ |
| Key Stability Result | $\dot{V} \leq -k_1||\mathbf{S}||^2$, asymptotic convergence | $\dot{V} \leq -\frac{1}{2}\mathbf{\bar{E}}^T\mathbf{Q}\mathbf{\bar{E}}$, asymptotic convergence |
| Typical Performance | Smoother transient, efficient if model is accurate | More robust to unmodeled dynamics, may exhibit chattering |
2. Adaptive Robust Control for Bounded, Unstructured Disturbances
The second scenario addresses a more general and challenging case for practical drone formation flight: disturbances with completely unknown structure. These could encompass not only complex aerodynamic interactions but also wind gusts, sensor biases, or model discrepancies. We assume only that the net disturbance effects on the speed and heading channels, $\Delta f_v(\cdot)$ and $\Delta f_{\phi}(\cdot)$, are bounded. Specifically, we assume known nominal values $\Delta f_{vN}$, $\Delta f_{\phi N}$ and known bounds on the deviation: $|\Delta f_v – \Delta f_{vN}| \leq \tilde{f}_v$, $|\Delta f_{\phi} – \Delta f_{\phi N}| \leq \tilde{f}_{\phi}$.
The control objective remains $\mathbf{E} \rightarrow 0$. Inspired by sliding mode control principles, we design a controller that can overpower these bounded uncertainties. Starting from the same error dynamics $\ddot{\mathbf{E}} = \mathbf{C}\mathbf{G} \mathbf{u} + \mathbf{D’} + \mathbf{C} \boldsymbol{\Delta} \mathbf{f}$ (where $\mathbf{D’}$ absorbs known terms and $\boldsymbol{\Delta} \mathbf{f} = [\Delta f_v, \Delta f_{\phi}]^T$), we aim for a stable error dynamics: $\ddot{\mathbf{E}} + k_1 \dot{\mathbf{E}} + k_2 \mathbf{E} = 0$. This would be achieved ideally by the control law:
$$ \mathbf{u}_{ideal} = -(\mathbf{C}\mathbf{G})^{-1} \left( \mathbf{D’} + k_1 \dot{\mathbf{E}} + k_2 \mathbf{E} + \mathbf{C} \boldsymbol{\Delta} \mathbf{f} \right) $$
Since $\boldsymbol{\Delta} \mathbf{f}$ is unknown, we replace it with an estimate designed to guarantee robustness. We use the estimates:
$$ \hat{\Delta f}_v = \Delta f_{vN} – \tilde{f}_v \cdot \text{sign}( \mathbf{C}_v^T \mathbf{P} \mathbf{\bar{E}} ) $$
$$ \hat{\Delta f}_{\phi} = \Delta f_{\phi N} – \tilde{f}_{\phi} \cdot \text{sign}( \mathbf{C}_{\phi}^T \mathbf{P} \mathbf{\bar{E}} ) $$
Here, $\mathbf{\bar{E}} = [\mathbf{E}^T, \dot{\mathbf{E}}^T]^T$ is the extended state vector, $\mathbf{P}$ is a positive definite matrix solving a Lyapunov equation for the desired error dynamics matrix $\mathbf{A}$, and $\mathbf{C}_v$, $\mathbf{C}_{\phi}$ are columns of the $\mathbf{C}$ matrix. The $\text{sign}(\cdot)$ function can be approximated by a smooth saturation function $\text{sat}(\cdot/\delta)$ with a small boundary layer $\delta$ to mitigate chattering in the drone formation control signals. The actual control input becomes:
$$ \mathbf{u} = -(\mathbf{C}\mathbf{G})^{-1} \left( \mathbf{D’} + k_1 \dot{\mathbf{E}} + k_2 \mathbf{E} + \mathbf{C} \begin{bmatrix} \hat{\Delta f}_v \\ \hat{\Delta f}_{\phi} \end{bmatrix} \right) $$
With this control law, the closed-loop error dynamics can be written in state-space form:
$$ \dot{\mathbf{\bar{E}}} = \mathbf{A} \mathbf{\bar{E}} + \mathbf{C}_v (\hat{\Delta f}_v – \Delta f_v) + \mathbf{C}_{\phi} (\hat{\Delta f}_{\phi} – \Delta f_{\phi}) $$
Stability is again analyzed via Lyapunov method. Choosing $V = \frac{1}{2} \mathbf{\bar{E}}^T \mathbf{P} \mathbf{\bar{E}}$, its derivative becomes:
$$ \dot{V} = \frac{1}{2} \mathbf{\bar{E}}^T (\mathbf{A}^T\mathbf{P} + \mathbf{P}\mathbf{A}) \mathbf{\bar{E}} + \mathbf{\bar{E}}^T\mathbf{P}[\mathbf{C}_v (\hat{\Delta f}_v – \Delta f_v) + \mathbf{C}_{\phi} (\hat{\Delta f}_{\phi} – \Delta f_{\phi})] $$
Since $\mathbf{A}^T\mathbf{P} + \mathbf{P}\mathbf{A} = -\mathbf{Q} < 0$, and by construction of the estimates $\hat{\Delta f}$, the terms involving the disturbances satisfy:
$$ \mathbf{\bar{E}}^T\mathbf{P}\mathbf{C}_v (\hat{\Delta f}_v – \Delta f_v) \leq -|\mathbf{\bar{E}}^T\mathbf{P}\mathbf{C}_v| \tilde{f}_v + |\mathbf{\bar{E}}^T\mathbf{P}\mathbf{C}_v| \tilde{f}_v = 0 $$
A similar relation holds for the $\phi$-channel. Therefore, we obtain:
$$ \dot{V} \leq -\frac{1}{2} \mathbf{\bar{E}}^T \mathbf{Q} \mathbf{\bar{E}} < 0 \quad \text{for} \quad \mathbf{\bar{E}} \neq 0 $$
This proves that $\mathbf{\bar{E}} \rightarrow 0$, ensuring that the drone formation error $\mathbf{E}$ and its derivative converge to zero asymptotically. The formation geometry is thus maintained robustly against bounded, unstructured disturbances.
3. Simulation Analysis and Performance Evaluation
To validate the effectiveness of the proposed adaptive control schemes for drone formation flight, numerical simulations were conducted. A scenario involving three UAVs (one leader, two followers) in a horizontal ‘V’ formation was implemented over a 160-second flight. The leader’s trajectory consisted of three phases: straight-and-level flight, a coordinated turn, and another straight segment. This tests the formation’s ability to establish, maintain during maneuvers, and re-establish geometry. The initial conditions are summarized below:
| Parameter | Leader (UAV 1) | Follower 1 (UAV 2) | Follower 2 (UAV 3) |
|---|---|---|---|
| $x$ (m) | 0 | -8 | -8 |
| $y$ (m) | 0 | 7 | -7 |
| $V$ (m/s) | 80 | 60 | 70 |
| $\phi$ (deg) | 0 | 0 | 0 |
For the adaptive controller with unknown parameters (Case 1), the simulation results demonstrated excellent performance. The follower UAVs successfully synchronized their speed and heading with the leader within approximately 50 seconds, forming a stable drone formation. The response was smooth, with minimal overshoot or oscillation. During the leader’s turn at 80-96 seconds, the followers adjusted their trajectories precisely, maintaining the relative ‘V’ formation throughout the maneuver. The formation re-converged perfectly in the final straight segment. The parameter estimates $\hat{\Delta c}_1$ and $\hat{\Delta c}_2$ converged to steady-state values, actively compensating for the simulated aerodynamic coupling.
For the robust adaptive controller against bounded disturbances (Case 2), the drone formation was also successfully established and maintained. Convergence to the desired formation took slightly longer (approx. 60 seconds) and the transient phase showed more pronounced control activity, which is characteristic of robust controllers dealing with significant uncertainty bounds. However, once converged, the formation tracking was stable. During the leader’s turn, the followers maintained the formation effectively, proving the controller’s robustness. The use of a boundary layer in the sign function effectively prevented excessive chattering in the control commands, ensuring practical implementability for the drone formation.
| Performance Metric | Case 1: Parameter Adaptation | Case 2: Robust Control |
|---|---|---|
| Formation Convergence Time | ~50 seconds | ~60 seconds |
| Steady-State Tracking Error | Negligible (< 0.1m) | Negligible (< 0.1m) |
| Transient Smoothness | Very Smooth | Moderate, bounded oscillations |
| Robustness to Unmodeled Dynamics | High (for parameteric uncertainty) | Very High (for bounded disturbances) |
| Control Effort Characteristics | Continuous, moderate | Higher activity, smooth due to boundary layer |
4. Discussion and Concluding Perspectives
The pursuit of reliable and autonomous drone formation flight hinges on solving the formation keeping problem under real-world conditions of uncertainty and disturbance. This discussion has detailed two potent adaptive control frameworks to address this challenge. The first method provides an elegant solution when the disturbance can be parametrized, such as with aerodynamic coupling effects in close-formation flight. By integrating online parameter estimation with Lyapunov-based control synthesis, the system learns and compensates for the unknown interaction dynamics, leading to efficient and smooth formation convergence.
The second method offers a more general solution, requiring only that disturbances are bounded in magnitude. This robust adaptive approach is less dependent on a specific disturbance model and is therefore applicable to a wider range of environmental perturbations, making it highly suitable for complex, real-world drone formation operations where all sources of uncertainty cannot be perfectly modeled. The trade-off is often a slightly more aggressive control action and longer convergence time, but with guaranteed performance within the defined bounds.
The theoretical stability proofs, underpinned by Lyapunov’s direct method, provide strong guarantees for both approaches, ensuring that the formation tracking error $\mathbf{E}$ converges to zero asymptotically. This is the fundamental requirement for stable drone formation keeping. The simulation results corroborate the theoretical findings, illustrating the formation’s ability to establish, maintain during maneuvers, and recover its geometry.
Future directions in adaptive drone formation control are abundant. Extending these centralized leader-follower designs to distributed or decentralized architectures would enhance scalability and fault tolerance. Incorporating obstacle avoidance and dynamic formation reconfiguration (morphing drone formations) within the adaptive framework presents a significant challenge. Furthermore, integrating machine learning techniques with classical adaptive control could lead to systems that not only adapt to parameters but also learn and predict complex disturbance patterns, pushing the boundaries of what is possible with intelligent, collaborative drone formations.