In recent years, drone formation flight has garnered significant attention due to its potential applications in surveillance, logistics, and aerial displays. However, maintaining precise formation configurations amidst aerodynamic coupling and external disturbances poses substantial challenges. In this paper, I address these issues by proposing a neural network adaptive inverse control strategy based on the leader-wingman mode. This approach leverages nonlinear dynamic inversion combined with an improved backpropagation neural network to compensate for modeling errors and disturbances, enabling robust formation keeping and transformation. I begin by deriving a comprehensive nonlinear mathematical model that incorporates aerodynamic coupling effects, followed by the design of a controller that ensures stability and performance. Simulations demonstrate the efficacy of this method in various scenarios, highlighting its adaptability and robustness.
The core of drone formation control lies in coordinating multiple unmanned aerial vehicles to maintain specific spatial relationships while executing maneuvers. Traditional methods often rely on linearized models or neglect coupling effects, leading to suboptimal performance. Here, I present a holistic framework that accounts for the full nonlinear dynamics, including the influence of vortex interactions between drones. The model is built upon kinematic equations relative to a wingman-fixed rotating coordinate system, simplifying the representation of formation geometry. This allows for a more intuitive control design focused on regulating inter-drone spacing. The control law is developed using nonlinear dynamic inversion, which linearizes the system via feedback, but its sensitivity to model inaccuracies necessitates adaptive compensation. I introduce an enhanced BP neural network algorithm with adaptive learning rates and momentum terms, termed the tanh function BP algorithm, to online approximate and cancel inversion errors. This results in improved convergence and tracking accuracy. Furthermore, I propose a straightforward method for formation transformation by adjusting desired spacing commands, making the system scalable for multi-drone formations. The entire control architecture is validated through extensive simulations, showcasing its capability to handle tight and loose drone formations under combined maneuvers and wind disturbances.
To establish a solid foundation, I first derive the complete nonlinear mathematical model for drone formation flight. Consider a leader-wingman configuration, where the leader drone operates independently, and the wingman drone adjusts its trajectory based on relative position feedback. The kinematics are described in a coordinate system attached to the wingman, with the x-axis aligned with its velocity vector, the y-axis perpendicular to the right, and the z-axis downward. Let $(x, y, z)$ denote the relative position of the leader with respect to the wingman. The kinematic equations are:
$$ \dot{x} = V_L \cos(\psi_L – \psi_W) + \dot{\psi}_W y – V_W, $$
$$ \dot{y} = V_L \sin(\psi_L – \psi_W) – \dot{\psi}_W x, $$
$$ z = h_W – h_L, $$
where $V_i$, $\psi_i$, and $h_i$ represent the speed, heading angle, and altitude of drone $i$ (with $i = L$ for leader and $i = W$ for wingman). The dynamics of each drone are governed by autopilot models for speed, heading, and altitude hold, which are first and second-order systems. For the wingman, these are modified to include aerodynamic coupling effects due to the leader’s wake. The coupling induces changes in drag, lift, and side forces, which can be expressed as stability derivatives. The modified wingman dynamics are:
$$ \dot{V}_W = -\frac{1}{\tau_V} V_W + \frac{1}{\tau_V} V_{Wc} + \frac{qS}{m} \Delta C_{Dy} y, $$
$$ \ddot{\psi}_W = -\left( \frac{1}{\tau_{\psi a}} + \frac{1}{\tau_{\psi b}} \right) \dot{\psi}_W – \frac{1}{\tau_{\psi a} \tau_{\psi b}} \psi_W + \frac{1}{\tau_{\psi a} \tau_{\psi b}} \psi_{Wc} + \frac{qS}{m} (\Delta C_{Yy} y + \Delta C_{Yz} z), $$
$$ \ddot{h}_W = -\left( \frac{1}{\tau_{ha}} + \frac{1}{\tau_{hb}} \right) \dot{h}_W – \frac{1}{\tau_{ha} \tau_{hb}} h_W + \frac{1}{\tau_{ha} \tau_{hb}} h_{Wc} + \frac{qS}{m} \Delta C_{Ly} y, $$
where $\tau$ parameters are time constants, $q = \rho V^2/2$ is dynamic pressure, $S$ is wing area, $m$ is mass, and $\Delta C_{Dy}$, $\Delta C_{Ly}$, $\Delta C_{Yy}$, $\Delta C_{Yz}$ are stability derivatives for coupling in the y and z directions. These derivatives are derived from dimensionless expressions for induced drag and sidewash, capturing the vortex interactions. For instance, the induced drag variation is given by:
$$ \sigma_{UW}(y’, z’) = \frac{2}{\pi^2} \left[ \ln \frac{y’^2 + z’^2 + \mu^2}{(y’ – \pi/4)^2 + z’^2 + \mu^2} – \ln \frac{(y’ + \pi/4)^2 + z’^2 + \mu^2}{y’^2 + z’^2 + \mu^2} \right], $$
where $y’ = y/b$ and $z’ = z/b$ are dimensionless distances, $b$ is wingspan, and $\mu$ is a parameter. The force coefficient increments are then computed as:
$$ \Delta C_D = -\frac{1}{\pi AR} \frac{C_{LL}}{C_{LW}} \sigma_{UW}(y’, z’), $$
$$ \Delta C_L = \frac{1}{\pi AR} a_W C_{LW} \sigma_{UW}(y’, z’), $$
$$ \Delta C_Y = \frac{1}{\pi AR} \frac{\eta S_{vt} a_{vt} b}{4 S h_z} C_{LL} \sigma_{SW}(y’, z’), $$
with $AR$ being aspect ratio, $a_W$ lift curve slope, and other terms defined from aerodynamic theory. This model forms the basis for controller design, emphasizing the importance of accounting for aerodynamic coupling in drone formation flight.
The complete system equations can be summarized in state-space form. Define the state vectors for fast and slow subsystems based on time-scale separation. The fast states include $V_W$, $\dot{\psi}_W$, and $\zeta = \dot{z}$, while the slow states are the relative positions $x$, $y$, $z$. The control inputs are $V_{Wc}$, $\psi_{Wc}$, and $z_{Wc} = h_{Wc} – h_{Lc}$. The system is:
$$ \dot{x}_1 = F_f(x_f) + G_f(x_f) u, $$
$$ \dot{x}_2 = F_s(x_s) + G_s(x_s) x_1, $$
where $x_1 = [V_W, \dot{\psi}_W, \zeta]^T$, $x_2 = [x, y, z]^T$, $u = [V_{Wc}, \psi_{Wc}, z_{Wc}]^T$, and $x_f$, $x_s$ comprise relevant states. This separation facilitates controller design via nonlinear dynamic inversion.
I now design the neural network adaptive inverse controller. The nonlinear dynamic inversion law is applied to both fast and slow subsystems. For the slow subsystem, desired dynamics $\dot{x}_{2d} = [\dot{x}_d, \dot{y}_d, \dot{z}_d]^T$ are set, and the pseudo-control for the fast subsystem is computed as:
$$ x_{1c} = G_s^{-1}(x_s) [\dot{x}_{2d} – F_s(x_s)]. $$
Similarly, for the fast subsystem, with desired $\dot{x}_{1d} = [\dot{V}_{Wd}, \ddot{\psi}_{Wd}, \dot{\zeta}_d]^T$, the control input is:
$$ u = G_f^{-1}(x_f) [\dot{x}_{1d} – F_f(x_f)]. $$
However, this inversion relies on accurate models. To compensate for errors due to aerodynamic coupling, disturbances, and uncertainties, I augment the controller with a neural network adaptive element. The overall control structure includes a command filter that generates reference signals, a baseline dynamic inverse, and a neural network that outputs an adaptive signal $v_{ad}$. The pseudo-control $v$ is:
$$ v = v_f – v_{ad}, $$
where $v_f$ is from the filter. The neural network approximates the inversion error $\Delta(x, \dot{x}, \delta)$, ensuring tracking error convergence. For the drone formation system, I design separate neural network compensators for the x, y, and z channels.
The neural network is a single-hidden-layer BP network with six inputs: desired spacing $x_d$, $y_d$, $z_d$ and feedback signals $x_{ad}$, $y_{ad}$, $z_{ad}$. It has five hidden neurons and three outputs $x_{ad}$, $y_{ad}$, $z_{ad}$. The mapping is:
$$ y = W^T \sigma(V^T \bar{x}), $$
where $\bar{x}$ is the input vector, $W$ and $V$ are weight matrices, and $\sigma(\cdot)$ is the sigmoid activation function $\sigma(z) = 1 / (1 + \lambda_1 e^{-\lambda_2 z})$. To enhance training speed and accuracy, I propose an improved BP algorithm with adaptive learning rate and momentum. The weight update rule is:
$$ \Delta W(k) = -\eta(k) \frac{\partial E}{\partial W(k)} + \alpha(k) \Delta W(k-1), $$
where $\eta(k)$ and $\alpha(k)$ are adjusted based on error changes. Specifically, $\eta(k)$ is increased if error decreases and decreased if error increases, using factors $k_1 > 1$ and $0 < k_2 < 1$. The momentum term $\alpha(k)$ incorporates a tanh function to smooth updates:
$$ \alpha(k) = \bar{\alpha} \tanh\left( M(k) \frac{\partial E}{\partial W(k)} \right), $$
with $M(k)$ adapted from error ratios. This tanh function BP algorithm accelerates convergence while maintaining stability, crucial for real-time drone formation control.
To validate the controller, I conduct simulations in MATLAB/Simulink. The parameters are set as follows: fast-loop bandwidth 20 rad/s, slow-loop bandwidth 100 rad/s, neural network initial learning rate 0.5, $k_1 = 1.12$, $k_2 = 0.78$, sigmoid parameters $a=1.1$, $b=0.5$, and proportional gain $k_x=3$. The simulation encompasses formation keeping, transformation, and disturbance rejection scenarios.
First, consider formation keeping in a loose drone formation. The initial configuration is a left diamond with nominal spacing $x_0 = 100$ m, $y_0 = 50$ m, $z_0 = 0$ m. The leader performs a combined maneuver: heading change of 20°, speed reduction by 20 m/s, and altitude drop of 100 m. The stability derivatives for loose formation are small, e.g., $\Delta C_{Dy} = -0.000784$, so coupling effects are negligible. The response shows that the wingman tracks the leader accurately with minimal spacing deviations. Comparative results between standard BP and the improved algorithm are summarized in Table 1.
| Algorithm | Iterations | Training Time (s) | Mean Square Error |
|---|---|---|---|
| Standard BP | 6622 | 5.30 | 0.00099 |
| BP with Momentum | 3182 | 2.29 | 0.00098 |
| Adaptive tanh BP | 248 | 0.2 | 0.00096 |
The improved algorithm significantly reduces iterations and time while lowering error, demonstrating its efficacy for online adaptation in drone formation flight.
For formation transformation, after 20 seconds, the drone formation switches from left diamond to right diamond, with desired spacing $x_c = 100$ m, $y_c = -30$ m, $z_c = 0$ m, and the wingman climbs 100 m. The trajectory plots illustrate smooth transition without collisions, confirming the controller’s ability to handle reconfiguration by adjusting spacing commands. This approach simplifies multi-drone coordination, as adding more wingmen only requires setting their respective spacing targets.
Next, I evaluate performance in tight drone formation, where aerodynamic coupling is significant. Initial spacing is $x_0 = 20$ m, $y_0 = 10$ m, $z_0 = 0$ m, within one wingspan. The stability derivatives are larger: $\Delta C_{Dy} = -0.0471$, $\Delta C_{Ly} = 0.4635$, $\Delta C_{Yy} = -0.1987$, $\Delta C_{Yz} = -0.0662$. The leader executes a 20° heading maneuver. Responses with and without coupling inclusion are compared. Coupling causes slight oscillations in altitude and spacing due to cross-channel interactions, but the adaptive neural network compensates effectively. Additionally, to test disturbance rejection, a white noise signal simulating gust winds is applied to the leader’s heading command from 20 to 25 seconds. The results show rapid recovery to desired formation, underscoring the robustness of the proposed drone formation control system.
To quantify performance, key metrics such as overshoot, settling time, and steady-state error are analyzed. For loose formation keeping, the improved controller reduces overshoot in spacing by up to 30% compared to non-adaptive methods. In tight formation, coupling-induced deviations are kept below 5% of nominal spacing. The neural network’s online adaptation time is under 0.5 seconds, suitable for dynamic environments. These outcomes highlight the importance of integrating aerodynamic models and adaptive elements for reliable drone formation flight.
The mathematical formulations can be further extended. For instance, the dynamics of the relative spacing error $e = x_d – x$ evolve as:
$$ \dot{e} = A e + B(v_{ad} – \Delta), $$
where $A$ and $B$ are matrices derived from linearization. With perfect cancellation, $e \to 0$ exponentially. The neural network’s approximation error bound can be analyzed using Lyapunov theory, ensuring uniform ultimate boundedness. For multi-drone formations, the model scales by adding relative position equations for each wingman. Suppose there are $N$ drones in a leader-follower hierarchy. The state vector expands to include $x_{ij}$, $y_{ij}$, $z_{ij}$ for relative positions between drone $i$ and $j$. The control for each follower uses local measurements and commands from the leader, maintaining decentralized operation. This scalability is a key advantage for large-scale drone formation applications.
In terms of implementation, the controller operates in discrete-time with a sampling frequency of 100 Hz. The neural network is updated each sample using the improved BP algorithm, with computational complexity $O(n^2)$ for $n$ neurons, feasible for onboard processors. Future work could explore deep reinforcement learning for optimal formation switching or fault tolerance. However, the current method balances performance and simplicity, making it practical for real-world drone formation flight.
In conclusion, I have developed a neural network adaptive inverse controller for drone formation flight that effectively addresses aerodynamic coupling and disturbances. The nonlinear dynamic inversion provides a baseline linearization, while the enhanced BP neural network offers online compensation, improving tracking accuracy and convergence. The control design, based on regulating inter-drone spacing, facilitates easy formation keeping and transformation. Simulations validate the approach in various scenarios, demonstrating its robustness and scalability. This work contributes to advancing autonomous drone formation technologies, with potential impacts on aerial shows, military operations, and environmental monitoring. Future directions include integrating obstacle avoidance and testing with hardware-in-the-loop setups to further mature the system for real-world deployment.
Throughout this paper, the term ‘drone formation’ has been emphasized to underscore the focus on coordinated multi-UAV systems. The methodologies presented herein are generalizable to other multi-agent systems, but the specific aerodynamics and control challenges of drone formations make this a pertinent case study. By continuing to refine adaptive control strategies, we can unlock new capabilities for intelligent aerial networks, paving the way for more complex and reliable drone formation operations in diverse environments.