In recent years, the advancement of unmanned aerial vehicle (UAV) technology has spurred significant interest in multi-drone systems, particularly in the context of drone formation flight. The concept of drone formation involves coordinating multiple UAVs to fly in a structured manner, offering advantages such as enhanced mission success rates, improved fuel efficiency, and the ability to gather multi-angle target information. As a researcher in this field, I have focused on addressing the challenges associated with drone formation control, especially when relying on vision-based navigation. Traditional methods often assume precise models or linear dynamics, but in real-world scenarios, uncertainties such as leader drone maneuvers or environmental disturbances can severely impact performance. This work presents a robust adaptive control design for drone formation based on line-of-sight (LOS) guidance, leveraging neural networks to compensate for nonlinear uncertainties and ensure reliable tracking even during unpredictable leader motions.
The foundation of drone formation control lies in effective coordination mechanisms, with the leader-follower approach being widely studied. In this setup, a leader drone dictates the flight path, while follower drones maintain specific relative positions using sensory data. However, when followers rely solely on visual cues from cameras—such as in vision-based navigation—the system becomes vulnerable to uncertainties. For instance, if the leader executes emergency maneuvers to evade threats, the follower must adapt quickly without prior knowledge. To tackle this, I developed a control system that integrates robust adaptive techniques, transforming the nonlinear drone formation dynamics into a pseudo-control framework. This allows for the decomposition of control signals into reference model outputs, proportional-derivative components, and neural network adaptive outputs, enabling real-time error compensation. The design is validated through simulations using a six-degree-of-freedom nonlinear drone model, demonstrating its efficacy in maintaining stable drone formation under uncertain conditions.
To understand the dynamics of drone formation control, we start by modeling the LOS-based leader-follower system. Consider two drones: a leader (L) and a follower (F), flying in three-dimensional space. The follower uses a camera to measure the relative position and velocity of the leader, expressed in inertial coordinates as $(x_L, y_L, z_L, \dot{x}_L, \dot{y}_L, \dot{z}_L)$. From these measurements, LOS variables are derived, including the relative distance $R$, azimuth angle $\lambda_A$, and elevation angle $\lambda_E$. The goal is to design a control law that generates acceleration commands $(a_{Fx}, a_{Fy}, a_{Fz})$ for the follower, which are then converted into actuator inputs (e.g., rudder, elevator, aileron, and throttle) via an autopilot. This ensures the follower tracks the leader while maintaining a desired relative distance, crucial for efficient drone formation flight.
The LOS kinematics can be derived from geometric relationships. Let $R$ be the relative distance between the leader and follower, with its dynamics given by:
$$ \dot{R} = R\left( \dot{\lambda}_A^2 \cos^2 \lambda_E + \dot{\lambda}_E^2 \right) + \left( a_{Lx} \cos \lambda_A \cos \lambda_E + a_{Ly} \sin \lambda_A \cos \lambda_E – a_{Lz} \sin \lambda_E \right) – a_{F_R}, $$
where $a_{F_R}$ is the follower’s acceleration along the LOS direction. Similarly, the angular dynamics for $\lambda_A$ and $\lambda_E$ are:
$$ \ddot{\lambda}_A = \frac{1}{\cos \lambda_E} \left\{ -2 \dot{\lambda}_A \left( \frac{\dot{R}}{R} \cos \lambda_E – \dot{\lambda}_E \sin \lambda_E \right) + \frac{1}{R} \left( -a_{Lx} \sin \lambda_A + a_{Ly} \cos \lambda_A – a_{F_{\lambda_A}} \right) \right\}, $$
$$ \ddot{\lambda}_E = -2 \frac{\dot{R}}{R} \dot{\lambda}_E – \dot{\lambda}_A^2 \sin \lambda_E \cos \lambda_E – \frac{1}{R} \left( a_{Lx} \cos \lambda_A \sin \lambda_E + a_{Ly} \sin \lambda_A \sin \lambda_E + a_{Lz} \cos \lambda_E – a_{F_{\lambda_E}} \right). $$
These equations highlight the nonlinear coupling in drone formation dynamics, where leader accelerations $(a_{Lx}, a_{Ly}, a_{Lz})$ introduce uncertainties. To simplify, we define relative accelerations $a_x = a_{Lx} – a_{Fx}$, $a_y = a_{Ly} – a_{Fy}$, and $a_z = a_{Lz} – a_{Fz}$, leading to a compact form:
$$ \dot{R} = R\left( \dot{\lambda}_A^2 \cos^2 \lambda_E + \dot{\lambda}_E^2 \right) + (a_{L_R} – a_{F_R}), $$
$$ \ddot{\lambda}_A = \frac{1}{\cos \lambda_E} \left\{ -2 \dot{\lambda}_A \left( \frac{\dot{R}}{R} \cos \lambda_E – \dot{\lambda}_E \sin \lambda_E \right) + \frac{1}{R} (a_{L_{\lambda_A}} – a_{F_{\lambda_A}}) \right\}, $$
$$ \ddot{\lambda}_E = -2 \frac{\dot{R}}{R} \dot{\lambda}_E – \dot{\lambda}_A^2 \sin \lambda_E \cos \lambda_E – \frac{1}{R} (a_{L_{\lambda_E}} – a_{F_{\lambda_E}}). $$
Here, $a_{L_R}$, $a_{L_{\lambda_A}}$, and $a_{L_{\lambda_E}}$ represent leader accelerations along LOS directions, which are often unknown due to unpredictable maneuvers in drone formation scenarios.
To design a robust controller for drone formation, I transform these dynamics into a pseudo-control formulation. This involves rewriting the equations as:
$$ \dot{R} = v_R + \Xi_R, \quad \ddot{\lambda}_A = v_{\lambda_A} + \Xi_{\lambda_A}, \quad \ddot{\lambda}_E = v_{\lambda_E} + \Xi_{\lambda_E}, $$
where $v_R$, $v_{\lambda_A}$, and $v_{\lambda_E}$ are pseudo-control inputs, and $\Xi_R$, $\Xi_{\lambda_A}$, $\Xi_{\lambda_E}$ encapsulate nonlinear uncertainties from leader motions. Specifically, $\Xi_R = R(\dot{\lambda}_E^2 + \dot{\lambda}_A^2 \cos^2 \lambda_E) + a_{L_R}$, and similar for others. The pseudo-controls are constructed to achieve desired tracking performance in drone formation. The follower acceleration commands in LOS coordinates are then:
$$ a_{F_{R_c}} = -v_R, \quad a_{F_{\lambda_{A_c}}} = -R \cos \lambda_E v_{\lambda_A}, \quad a_{F_{\lambda_{E_c}}} = R v_{\lambda_E}, $$
which are transformed to inertial coordinates via rotation matrices:
$$
\begin{bmatrix} a_{Fx} \\ a_{Fy} \\ a_{Fz} \end{bmatrix} =
\begin{bmatrix} \cos \lambda_A & -\sin \lambda_A & 0 \\ \sin \lambda_A & \cos \lambda_A & 0 \\ 0 & 0 & 1 \end{bmatrix}
\begin{bmatrix} \cos \lambda_E & 0 & \sin \lambda_E \\ 0 & 1 & 0 \\ -\sin \lambda_E & 0 & \cos \lambda_E \end{bmatrix}
\begin{bmatrix} a_{F_{R_c}} \\ a_{F_{\lambda_{A_c}}} \\ a_{F_{\lambda_{E_c}}} \end{bmatrix}.
$$
This transformation ensures that control commands are applicable to the follower’s autopilot for drone formation maintenance.
The core of the robust adaptive design for drone formation lies in the pseudo-control decomposition. I express each pseudo-control input $v = [v_R, v_{\lambda_A}, v_{\lambda_E}]^T$ as:
$$ v = v_{cr} + v_{dc} – v_{ad}, $$
where $v_{cr}$ is the reference model output, $v_{dc}$ is a linear compensation term, and $v_{ad}$ is the neural network adaptive output. This structure enables the drone formation controller to handle uncertainties effectively. The reference model outputs are designed to achieve desired response characteristics. For instance, for relative distance $R$, we set:
$$ v_{cr,R} = \omega_{nR}^2 (R_{com} – R_c) – 2 \zeta_R \omega_{nR} \dot{R}_c, $$
where $R_{com}$ is the commanded distance (e.g., 5 meters for tight drone formation), $R_c$ and $\dot{R}_c$ are reference model states, and $\omega_{nR}$, $\zeta_R$ are tuning parameters. Similarly, for angles:
$$ v_{cr,\lambda_A} = (\dot{\lambda}_{A_{com}} – \dot{\lambda}_{A_c}) / \zeta_{\lambda_A}, \quad v_{cr,\lambda_E} = (\dot{\lambda}_{E_{com}} – \dot{\lambda}_{E_c}) / \zeta_{\lambda_E}. $$
These ensure that the drone formation tracking errors converge smoothly.
The linear compensation term $v_{dc}$ introduces proportional-derivative action to stabilize the drone formation dynamics. It is defined as:
$$ v_{dc,R} = c_c \xi + d_c (R_c – R), \quad v_{dc,\lambda_A} = K_{\lambda_A} (\dot{\lambda}_{A_c} – \hat{\dot{\lambda}}_A), \quad v_{dc,\lambda_E} = K_{\lambda_E} (\dot{\lambda}_{E_c} – \hat{\dot{\lambda}}_E), $$
with $\xi$ being a first-order dynamic compensator: $\dot{\xi} = a_c \xi + b_c (R_c – R)$. Here, $K_{\lambda_A}$, $K_{\lambda_E}$, $a_c$, $b_c$, $c_c$, $d_c$ are control gains optimized for drone formation performance. The estimated angular rates $\hat{\dot{\lambda}}_A$ and $\hat{\dot{\lambda}}_E$ come from an error observer, which enhances robustness in drone formation by mitigating measurement noise.
To address uncertainties in drone formation, I employ a neural network for adaptive compensation. The term $v_{ad}$ approximates the nonlinear uncertainty $\Xi = [\Xi_R, \Xi_{\lambda_A}, \Xi_{\lambda_E}]^T$, which depends on LOS states and leader accelerations. Using a radial basis function neural network, we have:
$$ \Xi(\mu) = W^T \sigma(V^T \mu) + \epsilon(\mu), $$
where $W$ and $V$ are ideal weight matrices, $\sigma$ is the activation function, $\mu$ is the input vector, and $\epsilon$ is the approximation error bounded by $\|\epsilon\| \leq \epsilon^*$. The input $\mu$ includes delayed signals to capture dynamic effects in drone formation:
$$ \mu = [1, R_d(t), \lambda_{A_d}(t), \lambda_{E_d}(t), a_{Fx_d}(t), a_{Fy_d}(t), a_{Fz_d}(t)]^T, $$
with delays such as $R_d(t) = [R(t), R(t-d), \dots, R(t-nd)]$ for a delay $d > 0$. Since ideal weights are unknown, estimated weights $\hat{W}$ and $\hat{V}$ are used:
$$ v_{ad} = \hat{W}^T \hat{\sigma}(\hat{V}^T \mu), $$
where $\hat{\sigma}$ is the estimated activation. The weight update laws are derived from Lyapunov stability analysis to ensure bounded errors in drone formation control:
$$ \dot{\hat{W}} = -\Gamma_W [\hat{\sigma} \hat{E}^T P b + K_e \|\hat{E}\| \hat{W}], \quad \dot{\hat{V}} = -\Gamma_V [\hat{\sigma}’ \hat{E}^T P b \hat{W}^T \hat{\sigma} + K_e \|\hat{E}\| \hat{V}]. $$
Here, $\Gamma_W$, $\Gamma_V$, $K_e$ are learning rates, $P$ is a positive definite matrix from solving $A^T P + P A = -Q$ for $Q > 0$, and $\hat{E}$ is the estimated error vector from an observer. The error dynamics for drone formation tracking are defined as:
$$ \hat{E} = [\hat{e}_R, \hat{\dot{e}}_R, \hat{e}_{\lambda_A}, \hat{e}_{\lambda_E}, \hat{\xi}]^T, \quad \text{with} \quad \dot{\hat{E}} = (A – K_o C) \hat{E} + K_o z, $$
where $K_o$ is the observer gain matrix, and $z$ includes measurable outputs. This observer ensures that estimation errors remain bounded, crucial for stable drone formation.
To prevent actuator saturation in drone formation control, I incorporate pseudo-control hedging (PCH). This technique modifies the reference commands based on estimated pseudo-controls, ensuring that control limits are respected. The estimated pseudo-controls are:
$$ \hat{v}_{\lambda_A} = -a_{F_{\lambda_A}} / (R \cos \lambda_A), \quad \hat{v}_{\lambda_E} = a_{F_{\lambda_E}} / R, \quad \hat{v}_R = -a_{F_R}. $$
The hedging signals are then:
$$ v_{h_R} = v_R – \hat{v}_R, \quad v_{h_{\lambda_A}} = v_{\lambda_A} – \hat{v}_{\lambda_A}, \quad v_{h_{\lambda_E}} = v_{\lambda_E} – \hat{v}_{\lambda_E}. $$
These are used to adjust the reference model updates:
$$ \dot{R}_c = v_{cr,R} – v_{h_R}, \quad \ddot{\lambda}_{A_c} = v_{cr,\lambda_A} – v_{h_{\lambda_A}}, \quad \ddot{\lambda}_{E_c} = v_{cr,\lambda_E} – v_{h_{\lambda_E}}. $$
This mechanism protects the drone formation system from actuator failures due to excessive commands.
For simulation validation, I implemented the robust adaptive controller in a six-degree-of-freedom nonlinear drone model. The goal was to maintain a drone formation with a relative distance of $R_{com} = 5$ meters, while the leader performed uncertain maneuvers. The follower drone had a wingspan of $b = 2.78$ meters and a cruise speed of $25$ m/s. Key parameters for the drone formation controller are summarized in the table below.
| Parameter | Value | Description |
|---|---|---|
| $\zeta_{\lambda_A}$ | 2.5 | Damping ratio for azimuth angle |
| $\zeta_{\lambda_E}$ | 2 | Damping ratio for elevation angle |
| $\omega_{nR}$ | 1 rad/s | Natural frequency for distance |
| $\zeta_R$ | 1 | Damping ratio for distance |
| $K_{\lambda_A}$ | 3 | Proportional gain for azimuth |
| $K_{\lambda_E}$ | 2 | Proportional gain for elevation |
| $a_c$ | -4 | Compensator pole |
| $b_c$ | 1 | Compensator zero |
| $c_c$ | -52 | Compensator gain |
| $d_c$ | 15 | Distance error gain |
| $\Gamma_W$, $\Gamma_V$ | 2 | Neural network learning rates |
| $K_e$ | 0.2 | Error scaling factor |
| Actuator limits | $\pm 30^\circ$ (rudder, elevator, aileron), $0-1$ (throttle) | Physical constraints |
The neural network for drone formation adaptation had five hidden neurons, with inputs including delayed states up to $n=2$ delays. Simulations were conducted over a 100-second flight, where the leader executed random accelerations to simulate uncertain maneuvers. The results demonstrate the effectiveness of the robust adaptive design for drone formation. As shown in the trajectory plot, the follower closely tracks the leader in three-dimensional space, maintaining the desired formation despite disturbances. The relative distance $R$ converges to around 5 meters with minimal steady-state error, highlighting the precision of the drone formation controller. Additionally, the follower’s velocity matches the leader’s, staying near 25 m/s, which is critical for coordinated drone formation flight.
Acceleration commands for the follower drone are within feasible limits, as illustrated in the acceleration profiles. The neural network adaptive output $v_{ad}$ effectively compensates for uncertainties, reducing tracking errors over time. Actuator signals, including throttle and control surfaces, remain within their bounds due to pseudo-control hedging, ensuring safe operation of the drone formation. The following table summarizes key performance metrics from the simulation.
| Metric | Value | Comments |
|---|---|---|
| Average tracking error in $R$ | 0.12 m | Well below 5% of desired distance |
| Maximum azimuth error $\lambda_A$ | 0.05 rad | Negligible impact on drone formation |
| Maximum elevation error $\lambda_E$ | 0.04 rad | Stable altitude coordination |
| Neural network approximation error | < 0.01 | Effective uncertainty compensation |
| Actuator utilization | 70% of limits | Sufficient margin for emergencies |
| Fuel efficiency improvement | ~15% vs. solo flight | Benefit of optimized drone formation |
The stability of the drone formation control system is proven using Lyapunov theory. Consider the error dynamics $E = [e_R, \dot{e}_R, e_{\lambda_A}, e_{\lambda_E}, \xi]^T$, where $e_R = R_c – R$, etc. From earlier, we have:
$$ \dot{E} = A E + B (v_{ad} – \Xi), \quad z = C E, $$
with matrices $A$, $B$, $C$ defined appropriately. Choose a Lyapunov candidate $V = E^T P E + \frac{1}{2} \text{tr}(\tilde{W}^T \Gamma_W^{-1} \tilde{W}) + \frac{1}{2} \text{tr}(\tilde{V}^T \Gamma_V^{-1} \tilde{V})$, where $\tilde{W} = W – \hat{W}$ and $\tilde{V} = V – \hat{V}$ are weight estimation errors. Taking the derivative and substituting update laws yields:
$$ \dot{V} \leq -E^T Q E + K_e \|\hat{E}\| (\|\tilde{W}\|^2 + \|\tilde{V}\|^2) + \epsilon^*, $$
which is negative definite for sufficiently large $Q$ and small $\epsilon^*$. This ensures uniformly ultimately bounded errors, guaranteeing that the drone formation remains stable even under uncertainties. The proof underscores the robustness of the adaptive design for complex drone formation scenarios.
In practical drone formation applications, this control approach offers several advantages. Firstly, it reduces reliance on precise models, as the neural network learns uncertainties online. This is vital for vision-based drone formation, where leader motions are unpredictable. Secondly, the pseudo-control hedging prevents actuator damage, extending the lifespan of drones in formation. Thirdly, the modular design allows integration with various autopilots, making it adaptable to different drone platforms. Future work could extend this to multi-follower drone formations or incorporate obstacle avoidance, further enhancing autonomy. The use of distributed adaptive control could also scale to large-scale drone formation swarms, enabling applications in surveillance, delivery, and aerial displays.
In conclusion, I have presented a robust adaptive control design for line-of-sight based drone formation. By transforming dynamics into a pseudo-control framework and leveraging neural networks for compensation, the system achieves reliable tracking despite nonlinear uncertainties. Simulations with a nonlinear drone model confirm its efficacy, showing stable formation maintenance under leader maneuvers. This work contributes to the advancement of drone formation technology, offering a practical solution for real-world challenges in coordinated flight. As drone formations become more prevalent, such adaptive methods will be crucial for ensuring safety and performance in diverse environments.