We address the problem of fixed-wing drone formation flight along a prescribed path. In modern battlefield scenarios, swarms of fixed-wing drones are expected to operate in tight formations while accurately following complex trajectories. Existing methods, such as PID-based position control, vector field approaches, and previous moving-target strategies, often suffer from poor adaptability to varying path curvature and low accuracy in both path tracking and formation keeping. This paper presents a novel guidance method based on a front-positioned moving target, designed specifically for fixed-wing drones. Our method not only ensures stable single-drone path following but also extends naturally to multi-drone formation control. We provide a rigorous Lyapunov-based stability proof for the proposed guidance law under constant-curvature paths and demonstrate through extensive simulations that our approach outperforms three baseline methods in terms of tracking precision and formation rigidity, especially during coordinated turns on complex paths.
The remainder of this article is structured as follows. Section 1 describes the dynamics of fixed-wing drones and defines the formation and path-following problem. Section 2 details our proposed guidance method for both a single drone and a formation of drones. Section 3 presents the stability analysis using Lyapunov’s second method. Section 4 provides simulation results and comparisons. Section 5 concludes the paper with a summary of contributions and future perspectives.
1. Problem Formulation
1.1 Fixed-Wing Drone Dynamics
We consider the horizontal motion of a fixed-wing drone. The drone is modeled as a point mass with velocity v, heading angle ψ, and position P = [x, z]T in an inertial coordinate system. The planar kinematics are given by:
$$
\begin{aligned}
\frac{dx}{dt} &= v \cos\psi \\
\frac{dz}{dt} &= -v \sin\psi \\
\frac{dv}{dt} &= a_A \\
\frac{d\psi}{dt} &= \frac{a_N}{v}
\end{aligned}
\tag{1}
$$
Here, aA is the longitudinal acceleration (along the velocity vector) and aN is the lateral acceleration (perpendicular to the velocity vector). For fixed-wing drones, aN is the primary maneuvering control, typically generated by banking the aircraft.
1.2 Formation Definition
A formation of fixed-wing drones is defined with respect to a moving formation coordinate system (FCS). Let drone 0 be the leader, and drones i = 1, 2, …, n be followers. The origin of the FCS is placed at the leader’s position P0. The XF axis points along the leader’s velocity direction, and the ZF axis is perpendicular to the right. The transformation between inertial coordinates and FCS coordinates is:
$$
\begin{bmatrix} x_F \\ z_F \end{bmatrix} = \mathbf{L}_F \begin{bmatrix} x – x_0 \\ z – z_0 \end{bmatrix}, \quad \mathbf{L}_F = \begin{bmatrix} \cos\psi_0 & -\sin\psi_0 \\ \sin\psi_0 & \cos\psi_0 \end{bmatrix}
\tag{2}
$$
Each follower i has a desired position in the FCS, denoted by Pi,d,F = [xi,d,F, zi,d,F]T. The objective is to control all drones so that the leader accurately follows a given path s, and each follower maintains its desired relative position in the FCS.
2. Proposed Guidance Method
2.1 Single-Drone Path Following Using a Moving Target
We first develop a path-following law for a single fixed-wing drone. Let D be a fixed look-ahead distance. At any instant, we draw a circle of radius D centered at the drone’s position P0, called the target circle. This circle intersects the desired path s at up to two points. We select the intersection point that lies ahead along the path as the moving target point T0. If there is no intersection, we expand the circle progressively until a single tangent point is found, which becomes the critical target point T′0; the actual target is then taken as the intersection of the line P0T′0 with the original circle.
Once the target point is determined, we construct a command arc that passes through P0 and T0 and is tangent to the drone’s velocity vector at P0. The radius R of this arc is:
$$
R = \frac{D}{2\cos\eta_0}, \quad \text{where } \eta_0 = \frac{\pi}{2} – q_0
\tag{3}
$$
Here, q0 is the angle between the velocity vector and the line P0T0. The lateral acceleration command aNc0 is then the centripetal acceleration required to follow the arc:
$$
a_{Nc0} = \frac{2 v_0^2 \cos\eta_0}{D} = \frac{2 v_0^2 \sin q_0}{D}
\tag{4}
$$
Algorithm 1 summarizes the steps for computing aNc0 for a single drone.
| Step | Description |
|---|---|
| 1 | Input: path s, look-ahead distance D, drone position P0, velocity v0 |
| 2 | Construct target circle Ct with radius D centered at P0 |
| 3 | Compute intersections of Ct with s; let n be the number of intersections |
| 4 | if n ≥ 2 then select the forward-most intersection as T0 |
| 5 | else if n = 1 then set T0 = that intersection |
| 6 | else (n = 0) enlarge radius gradually until a single tangent point is found (critical circle); set T0 as the intersection of P0T′0 with the original circle |
| 7 | Compute angle q0 between v0 and P0T0 |
| 8 | aNc0 = 2v02 sin q0 / D |
| 9 | return aNc0 |
2.2 Extension to Multi-Drone Formation
For formation flight, the leader uses Algorithm 1 to track the path. Each follower i (i>0) is assigned a moving target point Ti that is derived from the leader’s target point T0 and the drone’s desired formation position Pi,d,F.
Let ψT0 be the path tangent angle at T0. We define a target coordinate system (TCS) with origin at T0, XT axis along the tangent direction, and ZT perpendicular to the right. The transformation from inertial to TCS is:
$$
\mathbf{L}_T = \begin{bmatrix} \cos\psi_{T0} & -\sin\psi_{T0} \\ \sin\psi_{T0} & \cos\psi_{T0} \end{bmatrix}
\tag{5}
$$
The follower’s target point in TCS is chosen to match its desired formation position:
$$
\mathbf{T}_{i,T} = \mathbf{P}_{i,d,F}
\tag{6}
$$
Transforming back to inertial coordinates gives:
$$
\mathbf{T}_i = \mathbf{L}_T^{-1} \mathbf{P}_{i,d,F} + \mathbf{T}_0
\tag{7}
$$
Now, the follower i treats Ti as a moving target and applies the same single-drone path-following logic. Its lateral acceleration command is:
$$
a_{Nci} = \frac{2 v_i^2 \sin q_i}{D_i}
\tag{8}
$$
where Di = |Ti − Pi| and qi is the angle between the follower’s velocity vector and the line PiTi.
To maintain longitudinal spacing, we also control the distance Di to a desired value Di,d. Under the assumption of a constant-curvature path with radius R0, the desired distance for follower i is:
$$
D_{i,d} = \frac{D (R_0 + z_{i,d,F})}{R_0}
\tag{9}
$$
For straight-line paths (R0 → ∞), the limit yields Di,d = D. The follower uses a target-distance controller to adjust its longitudinal acceleration aAci such that Di converges to Di,d.
Algorithm 2 describes the complete guidance for a follower drone.
| Step | Description |
|---|---|
| 1 | Input: T0, ψT0, path curvature radius R0 at T0, follower position Pi, velocity vi, desired formation position Pi,d,F |
| 2 | Compute target Ti using (7) |
| 3 | Di = |Ti − Pi| |
| 4 | Compute qi = angle between vi and PiTi |
| 5 | aNci = 2 vi2 sin qi / Di |
| 6 | Di,d = D (R0 + zi,d,F) / R0 |
| 7 | aAci = PID controller with setpoint Di,d and feedback Di |
| 8 | return aNci, aAci |
3. Stability Analysis
We prove local asymptotic stability of the proposed guidance law under a constant-curvature path and small errors. We consider a follower i in a polar coordinate system whose origin coincides with the center of the desired circular path. The state vector is X2 = [r, χ, vt, vn]T, where r is the radial distance, χ the angular position, vt the tangential speed (positive clockwise), and vn the radial speed (positive outward). The dynamics are:
$$
\dot{\mathbf{X}}_2 = \begin{bmatrix} \dot{r} \\ \dot{\chi} \\ \dot{v}_t \\ \dot{v}_n \end{bmatrix} = \begin{bmatrix} v_n \\ -\frac{v_t}{r} \\ a_t r – \frac{v_n v_t}{r} \\ -a_n r – \frac{v_t^2}{r} \end{bmatrix}
\tag{10}
$$
At the equilibrium (desired formation position Pi,d,F), we have:
$$
\mathbf{X}_{2,d} = \begin{bmatrix} R_i \\ \chi \\ v_t \\ 0 \end{bmatrix}
\tag{11}
$$
where Ri = R0 + zi,d,F is the radius of the follower’s desired trajectory. The lateral acceleration command (8) can be linearized near equilibrium. After simplifications using small-angle assumptions (|e| ≪ Ri, |e| ≪ Di), we obtain:
$$
a_{Nci} \approx \underbrace{\frac{2 v_t^2 \sin q_{i0}}{D_i}}_{\text{feedforward}} + K_p e + K_d \dot{e}
\tag{12}
$$
with
$$
K_p = \frac{2 v_t^2 \cos^2 q_{i0}}{D_i^2}, \quad K_d = \frac{2 v_t \cos q_{i0}}{D_i}
\tag{13}
$$
Here e = r − Ri is the radial position error. We define the Lyapunov function:
$$
V(\mathbf{X}_2) = K_p (R_i – r)^2 + v_n^2
\tag{14}
$$
Taking the time derivative and substituting the dynamics and the approximated control law yields:
$$
\dot{V} = -2 v_n (o + K_d v_n)
\tag{15}
$$
where o represents higher-order terms. In a sufficiently small neighborhood of the equilibrium, |o| < |K_d v_n|, and thus V̇ ≤ 0, with equality only at the equilibrium. Therefore, the system is locally asymptotically stable. The same proof applies to the leader by replacing the follower’s parameters with the leader’s.
4. Simulation Results
4.1 Simulation Model
We simulate fixed-wing drones using the 3-DOF point-mass dynamics with first-order lags on lateral and vertical acceleration commands, reflecting typical BTT-90 autopilot characteristics. The transfer functions are:
$$
G_N(s) = \frac{1}{0.5 s + 1}, \quad G_H(s) = \frac{1}{0.8 s + 1}
\tag{16}
$$
The longitudinal acceleration (for the leader) is regulated by a speed controller, while followers use a target-distance PID controller. Controller parameters are summarized in Table 1.
| Controller | Kp | Ki | Kd |
|---|---|---|---|
| Altitude PID | 0.5 | 0.01 | 1.2 |
| Speed PID (leader) | 1 | 0.001 | 1.414 |
| Target-distance PID (follower) | 1 | 0.001 | 1.414 |
4.2 Convergence Under Diverse Initial Conditions
We first test the method’s robustness with 10 followers initialized with random positions (within ±10 m in x, ±100 m in y), velocities (45–55 m/s), and headings (±45°). The path is a circle of radius 500 m. The leader flies at 50 m/s with D = 50 m. Formation positions are given in Table 2.
| Drone ID | xi,d,F / m | zi,d,F / m | Type |
|---|---|---|---|
| 0 | 0 | 0 | Leader |
| 1–5 | 0 | −50 | Follower |
| 6–10 | 0 | 50 | Follower |
Results show that all drones converge to their desired formation positions within 10 seconds. The steady-state error in both XF and ZF axes is less than 0.5 m. This demonstrates that the method is stable even when initial errors are not small, extending beyond the theoretical local region.
4.3 Comparison with Existing Methods on a Complex Path
We compare our method (adaptive moving target) against three existing methods:
- PID method: direct PID control of lateral and formation errors.
- Vector field method: using arctangent-based vector fields for heading command.
- Previous moving-target method: as described in [7] (same leader guidance but without curvature adaptation for followers).
The test path is a complex trajectory comprising multiple arcs and straight segments with curvatures ranging from 500 m to 2000 m. Three drones fly in a transversal formation (Table 3) with zero initial errors.
| Drone ID | xi,d,F / m | zi,d,F / m | Type |
|---|---|---|---|
| 0 | 0 | 0 | Leader |
| 1 | 0 | 100 | Follower |
| 2 | 0 | −100 | Follower |
Table 4 reports the RMS and maximum errors for the leader’s path following and for follower 1’s formation positions.
| Method | Leader lateral RMS / m | Follower XF RMS / m | Follower ZF RMS / m | Follower XF max / m | Follower ZF max / m |
|---|---|---|---|---|---|
| Adaptive moving target (ours) | 0.081 | 0.788 | 1.762 | 4.039 | 0.096 |
| PID | 2.540 | 0.964 | 3.109 | 5.491 | 7.525 |
| Vector field | 0.379 | 1.839 | 3.182 | 9.764 | 15.354 |
| Previous moving target [7] | 0.081 | 0.789 | 2.232 | 4.043 | 3.346 |
Our method achieves the smallest RMS and maximum errors in both leader tracking and follower lateral (ZF) position. The most notable improvement is in the ZF direction, where the error is almost an order of magnitude smaller than the other methods. The previous moving-target method performs similarly to ours in the leader tracking and longitudinal (XF) direction, but its lateral error is significantly larger due to the lack of curvature adaptation in the follower target distance. PID and vector field methods exhibit large steady-state and transient errors, especially during curved segments.
The superior performance of our method stems from the adaptive target distance calculation (equation (9)), which automatically adjusts the follower’s look-ahead distance according to the local path curvature. This ensures that the follower’s equilibrium position coincides exactly with the desired formation point regardless of the curvature, eliminating steady-state errors and reducing overshoot.
5. Conclusion
We have presented a novel formation path-following guidance method for fixed-wing drones based on a front-positioned moving target. The method features a single-drone path-following law that uses a dynamic target point on the desired path and a commanded arc to generate lateral acceleration. This law is extended to multi-drone formations by defining follower-specific target points that reflect the desired formation geometry and by adaptively adjusting the follower’s look-ahead distance according to the path curvature. A Lyapunov-based stability proof confirms local asymptotic stability under constant-curvature paths. Simulation results demonstrate that the method converges robustly from a wide range of initial conditions and significantly outperforms PID-based control, vector field methods, and a previous moving-target approach in both path-tracking accuracy and formation keeping, particularly on complex paths with varying curvature. The adaptive nature of our method makes it especially suitable for fixed-wing drone swarms operating in cluttered environments where precise lateral spacing is critical for collision avoidance.
Future work will integrate disturbance observers to handle wind gusts and model uncertainties, further enhancing the method’s robustness in real-world applications.