The coordinated flight of multiple Unmanned Aerial Vehicles (UAVs), or drone formation, represents a significant advancement in aerial robotics, offering enhanced efficiency and capability for complex missions. The fundamental prerequisite for establishing any drone formation is the successful rendezvous—the simultaneous arrival of all UAVs at a designated assembly point with matching velocities and headings. For fixed-wing drones, which cannot hover and operate within a limited airspeed envelope, this poses a distinct challenge. Relying solely on speed adjustment is often insufficient, especially when drones start from disparate locations and times. This article synthesizes a strategy primarily based on real-time trajectory modulation, augmented by velocity tuning, to solve the multi-drone simultaneous arrival problem. The core of this strategy involves constructing a time-varying vector field that intelligently guides each drone, dynamically adjusting its flight path length to coordinate arrival times while ensuring terminal state alignment for seamless integration into a drone formation.
The process begins with a critical coordination step: determining the formation rendezvous time, $t_f$. Each drone, $i$, estimates its shortest possible time-to-go, $\hat{t}^i_{\text{straight}}$, to the assembly point assuming a direct path. The formation rendezvous time is then set as the maximum of these individual estimates, $t_f = \max_i(\hat{t}^i_{\text{straight}})$. This establishes the earliest feasible time for the entire drone formation to assemble. Drones with shorter estimated times must then elongate their paths to meet this common $t_f$.
The flight trajectory for each drone is conceptually divided into two segments: a Coarse Adjustment Segment and a Fine Adjustment Segment. The primary mechanism for controlling time-of-arrival is embedded within the Coarse Adjustment Segment using horizontal maneuvers—specifically, flying along arcs of circles. By varying the arc’s angle or radius, the total path length, and consequently the flight time, can be precisely controlled. The Fine Adjustment Segment involves only minor velocity corrections for final precision. The success of this drone formation rendezvous hinges on a real-time guidance law that can smoothly execute these path-length adjustments.
Mathematical Foundation and Time-Varying Vector Field Design
To implement the dynamic path adjustment, we employ a guidance method based on time-varying vector fields. Let $\vec{r}$ denote a drone’s position in a 2D plane (horizontal). The objective is to generate a vector field $\vec{f}(\vec{r}, t, \lambda)$ that drives the drone to follow a desired, time-evolving path $\Phi(t)$, ultimately ensuring arrival at the rendezvous point at $t_f$.
The construction is rooted in Lyapunov stability theory. We define a positive definite, continuously differentiable potential function $V_F(\vec{r}, t)$ such that $V_F = 0$ if and only if the drone is on the desired path $\Phi(t)$. A stable vector field can be constructed as:
$$
\vec{f}(\vec{r}, t, \lambda) = -\left[\frac{\partial V_F}{\partial \vec{r}} \Gamma(\vec{r}, t, \lambda)\right]^T + \Theta(\vec{r}, t, \lambda) + \Upsilon(\vec{r}, t, \lambda)
$$
where $\Gamma$ is a positive semi-definite matrix governing convergence, $\Theta$ is a term tangent to the desired path dictating the direction of travel, and $\Upsilon$ compensates for the explicit time-dependence of the path. The drone’s velocity command is set to follow this field: $\dot{\vec{r}}_d = \vec{f}(\vec{r}, t, \lambda)$.
Vector Field for Arc Segments (Coarse Adjustment)
During the Coarse Adjustment Segment, the desired path is an arc of a circle with a time-varying radius $R(t)$ and central angle $\eta(t)$. Let $\vec{r_c}$ be the circle’s center. Define $r = ||\vec{r} – \vec{r_c}||$. The potential function is $V_F = \frac{1}{2}(r – R(t))^2$. Let $\hat{r}_\nabla = (\vec{r} – \vec{r_c})/r$ be the unit radial vector, and $\hat{r}_\Delta$ be the unit tangent vector pointing along the desired flight direction on the arc.
Choosing $\Gamma = \mathbf{I}_2 / \chi$, $\Theta = \lambda_1 \hat{r}_\Delta / \chi$, and $\Upsilon = \lambda_2 \hat{r}_\nabla / \chi$, we derive the vector field for the arc:
$$
\vec{f}_{\text{arc}} = \frac{-\left(r – R(t) – \lambda_2\right) \hat{r}_\nabla + \lambda_1 \hat{r}_\Delta}{\chi(\vec{r}, t, \lambda)}
$$
where $\chi = \sqrt{(r – R(t) – \lambda_2)^2 + \lambda_1^2} / v_d$. The desired ground speed $v_d$ incorporates velocity adjustment based on the estimated time error: $v_d = v_{\text{nominal}} + k_T (\hat{t} – t_f)$. The desired heading command $\psi_d$ is derived from the components of $\vec{f}_{\text{arc}}$.
The key to time coordination lies in the evolution of the arc angle $\eta(t)$. We define a Lyapunov function $V_t = \frac{1}{2}(\hat{t} – t_f)^2$ based on the estimated time-to-go $\hat{t}$. To drive $V_t$ to zero, we set the update law for $\eta$ as:
$$
\dot{\eta} = -\lambda_3 (\hat{t} – t_f)
$$
with $\lambda_3 > 0$. The radius $R$ is related to the initial straight-line distance $d_{AB}$ and $\eta$ by $R = d_{AB} / (2n \sin(\eta/2))$ for $n$ arcs. The compensating term $\lambda_2$ is calculated to satisfy the vector field’s time-derivative condition: $\lambda_2 = -\lambda_3 \chi (d_{AB}/n) [\cos(\eta/2)/(2\sin(\eta/2))^2] (\hat{t} – t_f)$.
Vector Field for Straight Segments (Fine Adjustment)
In the Fine Adjustment Segment and for drones on direct paths, the desired path is a straight line. Let $r$ be the perpendicular distance from the drone to the line, and $\hat{r}_\nabla$ be the unit vector pointing perpendicularly towards the line. Let $\hat{r}_\Delta$ be the unit vector along the desired flight direction on the line. The potential is $V_F = \frac{1}{2}r^2$.
Setting $\Gamma = \mathbf{I}_2 / \chi$, $\Theta = \lambda_4 \hat{r}_\Delta / \chi$, and $\Upsilon = 0$, we obtain the straight-line vector field:
$$
\vec{f}_{\text{line}} = \frac{-r \hat{r}_\nabla + \lambda_4 \hat{r}_\Delta}{\chi(\vec{r}, t, \lambda)}
$$
where $\chi = \sqrt{r^2 + \lambda_4^2} / v_d$, and $v_d$ is similarly adjusted based on time error. The desired heading $\psi_d$ is computed accordingly. This field ensures the drone converges to and follows the straight path to the rendezvous point.
Stability and Convergence Analysis
The convergence of the drone to its time-varying desired path is guaranteed by the Lyapunov functions defined for each segment type. For the arc segment, with $V_m = \frac{1}{2}||\vec{r} – R(t)\hat{r}_\nabla||^2$, its derivative is:
$$
\dot{V}_m = -\frac{(r – R(t))^2}{\chi} \leq 0
$$
For the straight segment, with $V_m = \frac{1}{2}r^2$, its derivative is $\dot{V}_m = -r^2 / \chi \leq 0$. Furthermore, at the switch between path segments, the tracking error does not increase. Therefore, the drone’s path-following error converges to zero asymptotically. Combined with the update law for $\eta(t)$ which drives the estimated arrival time error $(\hat{t} – t_f)$ to zero, the overall system ensures the drone formation achieves simultaneous arrival.
Integrated Collision Avoidance for Drone Formation Rendezvous
A practical drone formation rendezvous system must incorporate collision avoidance. We address two types: drone-obstacle and drone-drone.
Drone-Obstacle Avoidance: The planned path is checked against terrain or no-fly zones. If a segment violates a minimum safe altitude $h_{\text{safe}}$, the path is modified by inserting new waypoints that navigate around the obstacle, often creating new, longer arcs for the Coarse Adjustment Segment. This inherently increases the flight time, which is automatically compensated for by the time-varying vector field.
Drone-Drone Avoidance: A rolling-horizon probabilistic collision detection is performed. For each drone, its possible future position region over a short time window (e.g., 4 seconds) is approximated as a polygon based on its current tracking error and speed bounds. If the probability of intersection between any two drones’ regions exceeds a threshold, an avoidance maneuver is triggered. The most effective and simple maneuver for a drone formation in this context is to create a vertical separation by commanding one drone to descend by a fixed offset (e.g., 30 meters) before returning to the formation altitude once the conflict is resolved.
Simulation and Performance Evaluation
The proposed methodology was validated using a high-fidelity, nonlinear dynamic model of a fixed-wing UAV. The control architecture translates the vector field’s desired velocity $(v_d)$ and heading $(\psi_d)$ into commands for throttle, aileron, and elevator. Simulations involved four drones starting from different positions and times, tasked with forming a drone formation at a common point and heading.
| Drone ID | Initial Position (m) | Initial Heading (rad) | Assembly Point (m) |
|---|---|---|---|
| UAV 1 | (0, 0, 0) | $\pi/3$ | (16000, 18000, 1000) |
| UAV 2 | (2000, 0, 0) | $\pi/3$ | (15800, 17800, 1000) |
| UAV 3 | (8000, 0, 0) | 1.0 | (15800, 18200, 1000) |
| UAV 4 | (14000, 0, 0) | $-\pi/2$ | (15600, 17600, 1000) |
The results demonstrated successful rendezvous. Drones with initially shorter time estimates (UAVs 1, 3, and 4) performed horizontal maneuvering (arcs) in their Coarse Adjustment Segments, while the drone with the longest initial estimate (UAV 2) flew a nearly direct path. All drones converged to the assembly area within a very small time window and aligned their speeds (approx. 52 m/s) and headings (approx. 0°). The terminal position errors within the drone formation were below 1.5%. The system was also tested under constant wind disturbance (10 m/s) and in mountainous terrain with altitude constraints, showing robustness with terminal errors below 10% and safe terrain clearance, respectively.
Parameter Sensitivity and Selection
The performance of the drone formation rendezvous system depends on key parameters:
| Parameter | Role | Effect | Recommended Value |
|---|---|---|---|
| $\lambda_1$, $\lambda_4$ | Path following gain (tangential) | High value: prioritizes direction over proximity, may cause offset. Low value: prioritizes proximity, may cause oscillatory approach. | Moderate values for smooth tracking. |
| $\lambda_3$ | Time coordination gain | Governs how aggressively the path ($\eta$) changes to correct time error. Too high causes instability; too slow causes poor time coordination. | 0.01 (provides stable adaptation). |
| $k_T$ | Velocity adjustment gain | Scales the speed correction based on time error. | 0.3 |
Simulations confirmed that setting $\lambda_3=0$ (non-time-varying field) leads to significant arrival time errors due to accumulated flight path deviations. Conversely, a very high $\lambda_3$ causes chattering in the vector field and poor path tracking.
Conclusion
This article presented a novel, integrated approach for solving the simultaneous arrival problem essential for initiating a coordinated drone formation. The method’s primary contribution is the fusion of trajectory elongation via horizontal maneuvering with speed tuning, mediated by a rigorously constructed time-varying Lyapunov vector field. This addresses the core limitation of fixed-wing drones—their restricted speed range—by making path length the primary control variable for timing. The vector field provides real-time, stable guidance that dynamically adjusts the path to meet a coordinated rendezvous time, $t_f$, while ensuring terminal alignment of velocity and heading. The inclusion of collision avoidance mechanisms enhances its practical applicability. Extensive simulation using a high-fidelity nonlinear drone model under various conditions (wind, terrain) validates the effectiveness and robustness of the approach. This work provides a solid theoretical and practical foundation for autonomous multi-drone system deployment, where reliable drone formation assembly from dispersed states is a critical enabling technology.