Maintaining precise path following for low altitude drones in time-varying wind environments presents significant challenges due to unpredictable aerodynamic disturbances. Conventional methods struggle with rapid wind shear variations characteristic of sub-600m operations, where meteorological phenomena like microbursts induce sudden velocity and directional shifts. We address this limitation by developing a novel vector field-based control framework integrating ground speed estimation for robust Bézier curve tracking.
Kinematic Modeling and Problem Formulation
For low altitude UAV operations, we consider the 2D kinematic model:
$$
\begin{cases}
\dot{x} = V_g \cos\chi \\
\dot{y} = V_g \sin\chi \\
\dot{\chi} = \alpha(\chi_c – \chi)
\end{cases}
$$
where $V_g$ denotes ground speed, $\chi$ represents heading angle, and $\alpha$ defines the response rate to heading commands $\chi_c$. Wind disturbance decomposition follows:
$$
\mathbf{W} = \underbrace{\begin{bmatrix} W_c \cos\psi_c \\ W_c \sin\psi_c \end{bmatrix}}_{\text{Constant}} + \underbrace{\begin{bmatrix} W_v(t) \cos\psi_v(t) \\ W_v(t) \sin\psi_v(t) \end{bmatrix}}_{\text{Time-Varying}}
$$
with $|W_v(t)| \leq W_{\max}$. The path-following error for Bézier curve $y_d = f(x)$ is defined as lateral deviation $e = y – f(x)$. Core assumptions include:
| Assumption | Description |
|---|---|
| A1 | Constant altitude and airspeed $V_a$ |
| A2 | Measurable heading angle $\chi$ |
| A3 | Bounded states: $V_{\min} \leq V_g \leq V_{\max}$, $|\mathbf{W}| \leq W_{\max}$ |
| A4 | Wind decomposable into constant and time-varying components |
Vector Field Guidance for Bézier Paths
Our guidance law generates desired heading $\chi_d$ using:
$$
\chi_d = \chi_p + \frac{\chi_\infty}{\pi} \tan^{-1}(ke) + i\pi
$$
where $\chi_p = \tan^{-1}(f'(x))$ is path-tangent angle, $k$ governs convergence rate, $\chi_\infty$ defines maximum approach angle, and $i\in\{0,1\}$ determines travel direction. The vector field properties ensure:
- As $e \to \infty$: $\chi_d \to \chi_p + \chi_\infty + i\pi$ (perpendicular approach)
- As $e \to 0$: $\chi_d \to \chi_p + i\pi$ (path-tangent following)
Robust Control Law Design
Constant Wind Conditions
With measurable $V_g$, the control law achieves asymptotic stability:
$$
\chi_c = \chi + \frac{\dot{\chi}_p}{\alpha} + \frac{1}{\alpha} \cdot \frac{k}{1+(ke)^2} \cdot \frac{2\chi_\infty}{\pi} \cdot \frac{V_g \sin(\chi – \chi_p)}{\cos\chi_p} – \frac{\kappa}{\alpha} \text{sat}\left(\frac{\tilde{\chi}}{\epsilon}\right)
$$
where $\dot{\chi}_p = \frac{f”(x)}{1+(f'(x))^2} V_g \cos\chi$. Lyapunov function $W = \frac{1}{2}e^2 + \frac{1}{2}\rho \tilde{\chi}^2$ confirms stability with $\tilde{\chi} = \chi – \chi_d$.
Time-Varying Wind with Ground Speed Estimation
For unknown time-varying wind, we design an estimator:
$$
\dot{\hat{V}}_g = \dot{V}_g – \tau \rho \tilde{\chi} \frac{k}{1+(ke)^2} \cdot \frac{2\chi_\infty}{\pi} \cdot \frac{\sin(\chi – \chi_p)}{\cos\chi_p}
$$
where $\dot{V}_g = \frac{\partial V_g}{\partial\chi} \left[ -\frac{\chi_\infty k V_g \sin(\chi – \chi_p)}{\pi(1+(ke)^2)} – \kappa \text{sat}\left(\frac{\tilde{\chi}}{\epsilon}\right) \right]$. The modified control law becomes:
$$
\chi_c = \chi + \frac{\dot{\chi}_p}{\alpha} + \frac{1}{\alpha} \cdot \frac{k}{1+(ke)^2} \cdot \frac{2\chi_\infty}{\pi} \cdot \frac{\hat{V}_g \sin(\chi – \chi_p)}{\cos\chi_p} – \frac{\kappa}{\alpha} \text{sat}\left(\frac{\tilde{\chi}}{\epsilon}\right)
$$
Global asymptotic stability is proven via Lyapunov function:
$$
W = \frac{1}{2}e^2 + \frac{1}{2}\rho \tilde{\chi}^2 + \frac{1}{2}\tau^{-1}\Theta^2, \quad \Theta = \hat{V}_g – V_g
$$
Simulation Analysis
Experimental Setup
We validate our method for low altitude UAV operations under wind profiles:
| Parameter | Value |
|---|---|
| $V_a$ | 13 m/s |
| $W_c$ | 6 m/s at $\frac{2\pi}{3}$ rad |
| $W_v(t)$ | $3\cos t$ m/s at $\pi\sin t$ rad |
| Bézier Points | [-25,-5], [0,20], [25,-25] |
| Control Gains | $\alpha=1.65$, $\kappa=\pi/2$, $k=0.01$ |
Path Following Performance
Under severe initial deviation (10m), our controller achieves convergence within 30s. Comparative results demonstrate:
| Metric | Proposed Method | Vector Field Baseline |
|---|---|---|
| Max Lateral Error | 1.0 m | 5.0 m |
| Steady-State Error | 0.1 m | Oscillating |
| Heading Error | 0.1 rad | 1.6 rad |
| Convergence Time | 15s | Not achieved |
Ground Speed Estimation
Estimation accuracy under initial value discrepancies:
| Initial $\Delta V_g$ (m/s) | Mean Error (m/s) | RMSE (m/s) |
|---|---|---|
| 0 | 3.0 | 2.7 |
| 2 | 4.5 | 4.3 |
| 3 | 6.9 | 7.1 |
| 4 | 7.5 | 7.6 |
Remarkably, the low altitude drone maintains trajectory tracking despite estimation errors, validating controller robustness.
Conclusion
This work establishes a comprehensive framework for low altitude UAV path following in time-varying winds. Key contributions include: 1) Bézier-adapted vector field guidance with provable convergence properties; 2) Lyapunov-stable control law with ground speed estimation for unknown wind disturbances; 3) Validation demonstrating <1m tracking error under severe wind shear. Future work will extend this methodology to 3D obstacle avoidance scenarios for urban low altitude drone operations.