In modern aerial combat, fixed-wing UAV swarms have emerged as a transformative force, enabling high-altitude, long-endurance operations with coordinated formation flight. The ability to follow complex waypoint paths while maintaining precise formation geometry is critical for mission effectiveness. However, existing methods often suffer from poor adaptability to curved paths and low accuracy in both path-following and formation keeping. To address these challenges, we propose a novel guidance method based on a front-positioned moving-target strategy, specifically designed for fixed-wing UAV formations. This approach leverages the inherent dynamics of fixed-wing UAVs and provides a unified framework for single-UAV path-following and multi-UAV formation control. In this paper, we first derive a single-UAV path-following algorithm using a virtual target point moving along the desired path, then extend it to formation flight by carefully planning target points for each follower and adjusting the desired distance based on path curvature. We prove the asymptotic stability of the proposed method near the equilibrium point using Lyapunov’s second method, and validate its performance through numerical simulations. Compared to PID-based methods, vector-field methods, and a previous front-positioned moving-target method, our approach significantly improves tracking accuracy and formation stability, especially during complex turning maneuvers with varying curvature.
Problem Formulation
Fixed-Wing UAV Dynamics
We consider a fixed-wing UAV flying in the horizontal plane. Its motion is described by the following set of differential equations in an inertial coordinate system:
$$ \begin{aligned}
\frac{dx}{dt} &= v\cos\psi \\
\frac{dz}{dt} &= -v\sin\psi \\
\frac{dv}{dt} &= a_A \\
\frac{d\psi}{dt} &= \frac{a_N}{v}
\end{aligned} $$
Here, $[x, z]^T$ is the position vector, $v$ is the speed, $\psi$ is the heading angle (positive counterclockwise), $a_A$ is the longitudinal acceleration (thrust controlled), and $a_N$ is the lateral acceleration (main maneuvering input), typically produced by banking the fixed-wing platform. The fixed-wing UAV is underactuated; only $a_N$ provides fast maneuverability.
Formation Coordinate System
To describe the relative geometry of the formation, we define a moving formation coordinate system (FCS) attached to the leader (UAV 0). The origin $O_F$ coincides with the leader’s position $P_0$, the $X_F$ axis points along the leader’s velocity vector, and the $Z_F$ axis is perpendicular to the right. The formation speed $v_F$ and heading $\psi_F$ equal those of the leader. The transformation from inertial to FCS coordinates is:
$$ \begin{pmatrix} x_F \\ z_F \end{pmatrix} = \begin{bmatrix} \cos\psi_0 & -\sin\psi_0 \\ \sin\psi_0 & \cos\psi_0 \end{bmatrix} \begin{pmatrix} x – x_0 \\ z – z_0 \end{pmatrix} $$
where $[x_0, z_0]^T$ and $\psi_0$ are the leader’s position and heading. Each follower $i$ has a desired position $P_{i,d,F} = [x_{i,d,F}, z_{i,d,F}]^T$ in FCS.
Formation Path-Following Problem
The goal is to compute the control inputs $a_A$ and $a_N$ for each fixed-wing UAV so that:
- The leader tracks a given path $s$ (a continuous planar curve).
- Each follower maintains its desired position relative to the leader in FCS, i.e., its actual FCS position converges to $P_{i,d,F}$.
Single-UAV Path-Following Strategy
We first address the path-following of an individual fixed-wing UAV, which serves as the core for the leader and later for followers.
Front-Positioned Moving Target
As shown in the concept, a fixed look‑ahead distance $D$ is used. With the UAV’s current position $P$ as center and radius $D$, we draw a circle $C_t$. The intersection of $C_t$ with the path $s$ that lies ahead along the path direction is selected as the moving target $T$. The vector from $P$ to $T$ is $\mathbf{D} = T – P$, with magnitude $D = |\mathbf{D}|$. Let $q$ be the angle between the velocity vector $\mathbf{v}$ and $\mathbf{D}$:
$$ q = \angle(\mathbf{v}, \mathbf{D}) $$
The lateral acceleration command is derived by constructing a command arc through $P$ and $T$ that is tangent to $\mathbf{v}$ at $P$. The radius $R$ of this arc is:
$$ R = \frac{D}{2\cos\eta} = \frac{D}{2\sin q} $$
where $\eta = \pi/2 – q$. The required centripetal acceleration is then:
$$ a_{Nc} = \frac{2v^2 \sin q}{D} \tag{6} $$
If the path is far away and $C_t$ does not intersect $s$, we enlarge the radius to $D’$ until $C’_t$ is tangent to $s$, then use the tangent point as an intermediate target to compute the same formula.
Algorithm Summary for Single UAV
The procedure is summarized in Algorithm 1 (pseudocode omitted for brevity). The key point is that the control law (6) is simple and requires only the current position, velocity, and path geometry. It does not rely on a predefined model of the path curvature, yet it provides asymptotic convergence to a constant-curvature path.
Multi-UAV Formation Path-Following Method
We extend the single-UAV strategy to a formation by designing a virtual target point for each follower, based on the leader’s moving target and the desired formation geometry.
Target Point Planning for Followers
Let $T_0$ be the current moving target of the leader, and let $\psi_{T0}$ be the direction of the path tangent at $T_0$. We define a target coordinate system (TCS) with origin at $T_0$ and $X_T$ axis along $\psi_{T0}$. The rotation matrix from inertial to TCS is:
$$ L_T = \begin{bmatrix} \cos\psi_{T0} & -\sin\psi_{T0} \\ \sin\psi_{T0} & \cos\psi_{T0} \end{bmatrix} $$
We then place the follower $i$’s target point $T_i$ in TCS exactly at the same coordinates as its desired FCS position:
$$ T_{i,T} = P_{i,d,F} = \begin{pmatrix} x_{i,d,F} \\ z_{i,d,F} \end{pmatrix} $$
Transforming back to inertial coordinates gives:
$$ T_i = L_T^{-1} P_{i,d,F} + T_0 = \begin{bmatrix} \cos\psi_{T0} & \sin\psi_{T0} \\ -\sin\psi_{T0} & \cos\psi_{T0} \end{bmatrix} \begin{pmatrix} x_{i,d,F} \\ z_{i,d,F} \end{pmatrix} + \begin{pmatrix} x_{T0} \\ z_{T0} \end{pmatrix} \tag{10} $$
Thus, the set of target points forms a formation that moves with the path’s geometry. Each follower is guided to track its own moving target using the single-UAV law.
Lateral Formation Control
For follower $i$, the distance to its target is $D_i = |T_i – P_i|$, and the angle between its velocity and the line-of-sight is $q_i$. The lateral acceleration command follows:
$$ a_{Nci} = \frac{2 v_i^2 \sin q_i}{D_i} \tag{12} $$
This ensures that each fixed-wing UAV converges to follow the trajectory of its target point.
Longitudinal Formation Control
Since the target point $T_i$ moves independently of the follower’s state, the distance $D_i$ can vary. To maintain the desired longitudinal spacing, we introduce a desired distance $D_{i,d}$ that depends on the path curvature. Under the assumption that the path has constant curvature $R_0$, and using the ideal leader hypothesis, the follower’s expected trajectory radius is $R_i = R_0 + z_{i,d,F}$ (for right turns). The ratio of distances should equal the ratio of radii:
$$ \frac{D_{i,d}}{D} = \frac{R_i}{R_0} $$
Hence,
$$ D_{i,d} = \frac{D (R_0 + z_{i,d,F})}{R_0} \tag{15} $$
For straight paths ($R_0 \to \infty$), $\lim D_{i,d} = D$, consistent with the leader. A simple PID controller adjusts the longitudinal acceleration $a_{Aci}$ to make $D_i$ converge to $D_{i,d}$.
Algorithm for Follower
Algorithm 2 summarizes the complete follower control. The leader uses the single-UAV algorithm (Section 2) with a constant speed controller.
Stability Analysis
Lyapunov-Based Proof for Constant Curvature Paths
We prove local asymptotic stability of the proposed method under the following conditions: constant path curvature, ideal leader, and pure lateral formation (negligible longitudinal offset). We set up the follower dynamics in a polar coordinate system with origin at the center of the desired circular path. Let $r$ be the radial distance, $\chi$ the azimuth, $v_n$ the radial velocity, and $v_t$ the tangential velocity. The state vector is $\mathbf{X}_2 = [r, \chi, v_t, v_n]^T$. The dynamics are:
$$ \frac{d\mathbf{X}_2}{dt} = \begin{pmatrix} v_n \\ -\frac{v_t}{r} \\ a_t r – v_n v_t \\ -a_n r – v_t^2 \end{pmatrix} \tag{21} $$
where $a_n$ and $a_t$ are the normal and tangential acceleration inputs. At the desired formation point (equilibrium), we have $r = R_i$, $v_n = 0$, and $v_t = \text{const}$. After linearizing the control law about this equilibrium and using small-angle approximations, the lateral acceleration command can be expressed as:
$$ a_{Nci} = a_{Nci,0} + K_p e + K_d \dot{e} + o $$
where $e = r – R_i$ is the radial error, $K_p$ and $K_d$ are positive gains derived from the look-ahead parameters, and $o$ is a higher-order term. Choose the Lyapunov function:
$$ V = K_p e^2 + v_n^2 $$
Its derivative along the system trajectories becomes:
$$ \dot{V} = -2 v_n \left( K_p e + a_n – \frac{v_t^2}{r} \right) $$
Substituting the control law and simplifying, we obtain:
$$ \dot{V} = -2 v_n (K_d v_n + o) $$
Since $K_d > 0$ and $|o| < K_d |v_n|$ in a sufficiently small neighborhood of the equilibrium, $\dot{V} \le 0$, and $\dot{V}=0$ only at the equilibrium. Hence, by Lyapunov’s second method, the equilibrium is asymptotically stable. The same reasoning applies to the leader’s path-following by setting $z_{i,d,F}=0$.
Numerical Simulations
We validate the proposed method through extensive simulations. The fixed-wing UAV model includes first-order actuator dynamics with time constants 0.5 s for lateral acceleration and 0.8 s for vertical acceleration (used in altitude hold). Speed controllers and distance controllers employ PID parameters as listed in Table 1.
| Parameter | Value |
|---|---|
| Proportional gain $K_p$ | 1.0 |
| Integral gain $K_i$ | 0.001 |
| Derivative gain $K_d$ | 1.414 |
Initial Condition Robustness Test
We test the method with 10 followers starting from random initial positions, speeds, and headings within certain ranges. The path is a constant-curvature circular arc with radius 500 m. The leader flies at 50 m/s with look‑ahead distance $D=50$ m. The formation configuration is purely lateral: followers 1–5 at $z_{i,d,F}=-50$ m, followers 6–10 at $z_{i,d,F}=+50$ m. Even with initial errors, all fixed-wing UAVs converge to their desired positions within about 10 seconds. The results confirm that the method is stable over a considerable region of initial states, complementing the local theoretical proof.
Comparison on a Complex Path
We simulate a complex path composed of multiple arcs and straight segments with radii ranging from 500 m to 2000 m. Three fixed-wing UAVs (one leader, two followers) are used with lateral offsets of ±100 m. We compare four methods:
- Proposed method (Adaptive Front-Positioned Moving-Target)
- PID method: Position-based PID control for lateral and longitudinal errors.
- Vector-field method: Based on arctan vector fields for heading guidance, with the same inner loops.
- Previous Front-Positioned Moving-Target method (existing method without curvature adaptation).
The simulation results are summarized in Table 2. The leader’s path-following error (RMSE) is 0.081 m for the two front-positioned methods, versus 2.54 m for PID and 0.379 m for vector field. For formation keeping, the adaptive method achieves the smallest RMSE and maximum error in both $X_F$ and $Z_F$ directions. Notably, the $Z_F$ (lateral) maximal error is only 0.096 m for the proposed method, compared to 7.53 m (PID), 15.35 m (vector field), and 3.35 m (previous front-positioned). This dramatic improvement demonstrates the superiority of the adaptive curvature compensation.
| Method | Leader lateral RMSE (m) | Follower $X_F$ RMSE (m) | Follower $Z_F$ RMSE (m) | Max $X_F$ error (m) | Max $Z_F$ error (m) |
|---|---|---|---|---|---|
| Adaptive Front-Positioned Moving-Target | 0.081 | 0.788 | 1.762 | 4.04 | 0.096 |
| PID | 2.540 | 0.964 | 3.109 | 5.49 | 7.525 |
| Vector Field | 0.379 | 1.839 | 3.182 | 9.76 | 15.354 |
| Previous Front-Positioned Moving-Target | 0.081 | 0.789 | 2.232 | 4.04 | 3.346 |
Conclusion
We have presented a formation path-following guidance method for fixed-wing UAVs based on a front-positioned moving-target concept. By carefully designing the virtual target points for each follower and adapting the desired look‑ahead distance according to the path curvature, the method achieves both accurate path-following and tight formation keeping. Lyapunov analysis confirms local asymptotic stability, and simulations demonstrate robust convergence even with large initial errors. Comparative studies on a complex path show that our approach significantly outperforms PID and vector-field methods, and also improves upon an existing front-positioned moving-target method by eliminating steady-state lateral errors during turns. The proposed method is particularly suitable for fixed-wing UAV swarms operating in confined corridors where precise lateral positioning is critical. Future work will incorporate disturbance observers to handle unknown wind perturbations.