The coordinated control of multiple unmanned aerial vehicles (UAVs) into a cohesive drone formation represents a pivotal advancement in aerial robotics, enabling complex missions that single agents cannot accomplish. A core and challenging subtask within drone formation control is precise path tracking, where the entire group must follow a prescribed spatial trajectory while maintaining their relative configuration, even in the presence of environmental disturbances like wind. This article presents a comprehensive guidance and control strategy for drone formation path tracking, leveraging the theoretical framework of vector fields. The proposed method ensures stable convergence for both straight-line and circular reference paths under time-varying wind conditions, a common and disruptive factor in real-world flight.
The foundational element of any drone formation control architecture is a robust model for individual agent dynamics. We consider a kinematic model for a UAV operating in a horizontal plane with a constant altitude assumption. The dynamics of a UAV, denoted as the leader or a follower, under the influence of wind are described by:
$$ \dot{x} = V_a \cos\psi + W\cos\psi_w = V_g \cos\chi $$
$$ \dot{y} = V_a \sin\psi + W\sin\psi_w = V_g \sin\chi $$
$$ \dot{\chi} = \alpha (\chi^c – \chi) $$
Here, $(x, y)$ denotes the inertial position. $V_a$ and $\psi$ are the airspeed and its heading angle, while $W$ and $\psi_w$ represent the wind speed and direction. The crucial variables for path following are the ground speed $V_g$ and the course angle $\chi$, which result from the vector sum of airspeed and wind velocity. The last equation models the closed-loop response of the autopilot’s heading-hold loop, where $\chi^c$ is the commanded course and $\alpha > 0$ is the response constant. For follower UAVs in a drone formation, we incorporate an additional speed-hold loop:
$$ \dot{V}_g = \beta (V_g^c – V_g) $$
where $\beta > 0$ and $V_g^c$ is the ground speed command. This allows followers to modulate their speed for formation keeping. Key parameters for the agent dynamics are summarized below.
| Symbol | Definition |
|---|---|
| $V_a$, $\psi$ | Airspeed & Air-Referenced Heading |
| $V_g$, $\chi$ | Ground Speed & Course Angle |
| Wind Speed & Wind Direction | |
| $\alpha$ | Heading Loop Response Constant |
| $\beta$ | Speed Loop Response Constant |
The cornerstone of our approach is the Vector Field (VF) path-following method for a single UAV. A VF defines a desired course angle $\chi^d$ at every point in space, directing the vehicle towards and along the desired path. For a straight-line path $y = ax + b$, the cross-track error $e_{ly} = y – (ax + b)$ is used to generate the desired VF course:
$$ \chi^d_l(e_{ly}) = i \left[ \frac{\chi_{\infty}}{2} + \frac{2}{\pi} \tan^{-1}(k e_{ly}) \right] + \tan^{-1}(a) $$
where $i = \pm1$ sets the direction of travel along the line, $\chi_{\infty} \in (0, \pi/2]$ defines the maximum approach angle, and $k>0$ tunes the convergence rate. For a circular path of radius $R$ centered at $(x_c, y_c)$, the radial error $e_l = d – R$ (with $d$ being the distance to the center) defines the VF:
$$ \chi^d_l(e_l) = \gamma + j \left[ \frac{\pi}{2} + \tan^{-1}(k e_l) \right] $$
Here, $\gamma$ is the line-of-sight angle from the circle center to the UAV, and $j = \pm1$ determines the direction of rotation (clockwise/counter-clockwise). The control law for the UAV’s commanded course to follow this VF is designed using a Lyapunov-based sliding mode approach:
$$ \chi^c = \chi + \frac{1}{\alpha} \dot{\chi}^d – \frac{\kappa}{\alpha} \text{sat}\left(\frac{\tilde{\chi}}{\epsilon}\right) $$
where $\tilde{\chi} = \chi – \chi^d$ is the course error, $\kappa, \epsilon > 0$ are control gains, and $\text{sat}(\cdot)$ is a saturation function. The term $\dot{\chi}^d$ is derived from the time derivative of the VF equations. A significant challenge in wind is that the ground speed $V_g$ in $\dot{\chi}^d$ is not directly measurable from standard avionics. To address this, a ground speed observer is introduced for robust performance:
$$ \dot{\hat{V}}_g = -\Gamma \rho \tilde{\chi} \frac{\partial \chi^d}{\partial e} \frac{\partial e}{\partial \chi} – \sigma \Gamma \tilde{V}_g $$
where $\hat{V}_g$ is the estimated ground speed, $\tilde{V}_g = \hat{V}_g – V_g$, and $\Gamma, \rho, \sigma > 0$ are observer gains and tuning parameters. The estimated speed $\hat{V}_g$ is then used in the control law, making the single-UAV path follower robust to unknown, time-varying wind.
Extending this to a multi-agent drone formation requires a structured approach. We adopt a leader-follower architecture, a common and effective paradigm for drone formation control. In this setup, a designated leader tracks the desired global path using the single-agent VF method described above. The follower UAVs are tasked with maintaining a specified geometric offset relative to the leader’s body frame, defined by $(g_x, g_y)$. The formation tracking error for a follower in the inertial frame is:
$$ x_E = g_x + d_f \sin\left(\gamma_f – \frac{\pi}{2} – \chi_l\right) $$
$$ y_E = g_y + d_f \cos\left(\gamma_f – \frac{\pi}{2} – \chi_l\right) $$
where $d_f$ is the distance between leader and follower, and $\gamma_f$ is their relative bearing. The core innovation for the drone formation control is the design of a dual vector field for each follower. Unlike traditional methods, we decouple the control objectives into a course vector field and a speed vector field, enabling independent yet coordinated convergence of lateral and longitudinal errors.
The Course Vector Field dictates the desired follower course $\chi^d_f$ based on the lateral formation error $y_E$:
$$ \chi^d_f = \chi_l + \chi_{\infty} \frac{2}{\pi} \tan^{-1}(k_y y_E) $$
This field guides the follower to align its path with the leader’s, correcting lateral offsets. The Speed Vector Field dictates the desired follower ground speed $V^d_{gf}$ based on the longitudinal error $x_E$:
$$ V^d_{gf} = V_{gl} + V_{\infty} \frac{2}{\pi} \tan^{-1}(k_x x_E) $$
where $V_{gl}$ is the leader’s ground speed. This field controls the follower’s speed to close or maintain the along-track separation. The parameters governing these fields and the associated control laws are critical for drone formation stability.
| Parameter | Role in Formation Control | Typical Value |
|---|---|---|
| $k_x$, $k_y$ | Gains for speed/course vector field convergence | 0.2 |
| $\chi_{\infty}$ | Max course correction for lateral error | $\pi/4$ rad |
| $V_{\infty}$ | Max speed correction for longitudinal error | 5-10 m/s |
| $\kappa$ | Sliding mode control gain | 3 |
| $\epsilon$ | Boundary layer thickness for saturation function | 0.01 |
The corresponding control laws for the follower UAV that enforce these dual vector fields are:
$$ \chi^c_f = \chi_f + \frac{1}{\alpha_f} \dot{\chi}^d_f – \frac{\kappa}{\alpha_f} \text{sat}\left(\frac{\tilde{\chi}_f}{\epsilon}\right) $$
$$ V^c_{gf} = V_{gf} + \frac{1}{\beta_f} \dot{V}^d_{gf} + \frac{1}{\rho \beta_f} x_E – \frac{\kappa}{\beta_f} \text{sat}\left(\frac{\tilde{V}_{gf}}{\epsilon}\right) $$
where $\tilde{\chi}_f = \chi_f – \chi^d_f$, $\tilde{V}_{gf} = V_{gf} – V^d_{gf}$, $\alpha_f$ and $\beta_f$ are the follower’s autopilot constants, and $\rho > 0$ is a weighting parameter. The derivatives $\dot{\chi}^d_f$ and $\dot{V}^d_{gf}$ are computed from the defined vector fields and the leader’s states.
The stability of the resulting closed-loop drone formation system is paramount. We analyze it using Lyapunov theory. Consider the composite Lyapunov candidate function for a follower:
$$ W = \frac{1}{2} x_E^2 + \frac{1}{2\rho} \tilde{V}_{gf}^2 + \frac{1}{2} y_E^2 + \frac{1}{2\rho} \tilde{\chi}_f^2 $$
This function accounts for both longitudinal ($x_E$, $\tilde{V}_{gf}$) and lateral ($y_E$, $\tilde{\chi}_f$) errors. Taking its time derivative and substituting the error dynamics and the proposed control laws yields, after significant algebraic manipulation and application of bounds, an expression of the form:
$$ \dot{W} \leq – \frac{V_{\infty} x_E}{\pi} \tan^{-1}(k_x x_E) – \frac{\rho \kappa}{\epsilon} \tilde{V}_{gf}^2 – \frac{\rho \kappa}{\epsilon} \tilde{\chi}_f^2 + \text{(negative definite terms related to $y_E$)} $$
The analysis shows that $\dot{W}$ can be rendered negative definite provided the control gains satisfy certain conditions related to the formation geometry and UAV performance, such as:
$$ \min\left( \frac{\rho \kappa}{V_{gf}}, \frac{y_E}{2} \right) > \frac{\pi \epsilon \mu}{4 \chi_{\infty} k_y} $$
where $\mu$ is a constant derived from bounding functions in the stability proof. This proves that the formation tracking errors $(x_E, y_E, \tilde{\chi}_f, \tilde{V}_{gf})$ are globally uniformly ultimately bounded and converge asymptotically to zero under the proposed dual vector field guidance, ensuring stable drone formation path tracking.
To validate the efficacy of the proposed vector field method for drone formation control, extensive numerical simulations were conducted. A formation of four UAVs (one leader, three followers) was tasked with tracking straight-line and circular paths in both calm and windy conditions. The wind was modeled as a time-varying disturbance: $W(t) = 5 + 3\cos(0.1t)$ m/s, $\psi_w(t) = 135^\circ + \pi\sin(0.1t)$ rad. The proposed method was compared against two established baselines: a classical PID-based formation controller and a Model Predictive Control (MPC) scheme. The key performance metric was the formation tracking error, with a practical stability threshold set at 10 meters.
For the straight-line path, the results demonstrated the superior convergence and robustness of the VF method. While all controllers eventually stabilized, the VF-guided drone formation exhibited significantly smaller transient errors and faster settling time. The followers’ errors converged smoothly alongside the leader’s path acquisition. In contrast, the PID and MPC controllers showed larger oscillations in the follower-to-leader tracking errors, especially during the initial alignment phase under wind disturbance.
The circular path tracking tests further highlighted the advantages of the vector field approach. The simulation results for trajectory and error are summarized below, comparing average steady-state tracking errors.
| Control Method | Env. | Avg. Straight-Line Error (m) | Avg. Circular Path Error (m) |
|---|---|---|---|
| Vector Field (Proposed) | No Wind | 0.85 | 1.12 |
| Vector Field (Proposed) | With Wind | 2.31 | 2.97 |
| PID Formation Control | With Wind | 6.78 | 8.45 |
| MPC Formation Control | With Wind | 5.22 | 7.13 |
The proposed method’s trajectory was notably smoother, with the entire drone formation gracefully curving onto the circular path. The PID and MPC controllers caused the formation to first intercept the circle with a nearly straight heading, requiring an aggressive turn to establish tracking, which induced larger and more oscillatory errors. Under wind, the performance gap widened, with our method maintaining errors well below the 10m threshold, while the baselines frequently exceeded it. This conclusively validates the robustness and precision of the VF-based drone formation path tracker.
In conclusion, this article has presented a robust solution to the problem of path tracking for a drone formation operating in windy environments. The methodology is built upon a foundation of vector field guidance, extended innovatively into a dual-field structure for formation keeping. The integration of a ground speed observer renders the system adaptive to unknown wind disturbances without requiring prior knowledge of the wind field. A rigorous Lyapunov-based stability analysis guarantees the convergence of the formation tracking errors. Comprehensive simulation studies against benchmark controllers confirm the method’s effectiveness, demonstrating lower tracking errors, smoother transient responses, and superior wind rejection capabilities for both straight and curved paths. This work provides a solid theoretical and practical framework for reliable drone formation navigation, a critical step towards the deployment of autonomous UAV teams in real-world surveillance, delivery, and inspection missions. Future work will focus on extending the approach to three-dimensional paths, incorporating communication delays between formation members, and enabling autonomous reconfiguration and collision avoidance within the drone formation.