Close formation flight of fixed-wing drones offers significant advantages in fuel savings and mission capability. However, the wingman experiences severe aerodynamic disturbances from the leader’s wake vortex, complicating controller design. In this work, we address these challenges by first developing an efficient aerodynamic model for swept-wing fixed-wing drones using a multi-lifting-line method. We then propose a prescribed-time incremental nonlinear dynamic inversion (PT-INDI) controller that ensures rapid convergence and strong robustness. To obtain accurate angular acceleration signals despite sensor noise, we design a modified tracking differentiator (MTD) based on predefined-time convergence. Simulation results demonstrate that the proposed approach achieves superior tracking performance under aerodynamic coupling, parameter uncertainties, and measurement noise.
1. Aerodynamic Modeling of Close Formation Flight for Fixed-Wing Drones
1.1 UAV Model
We consider a flying-wing fixed-wing drone XQ7B with eight control surfaces. The geometric parameters are listed in Table 1.
| Parameter | Value | Symbol | Unit |
|---|---|---|---|
| Wing area | 1.546 | $S_{ref}$ | m$^2$ |
| Wingspan | 2.808 | $b$ | m |
| Mean aerodynamic chord | 0.78 | $c$ | m |
| Mass | 15 | $m$ | kg |
| Roll inertia | 2.369 | $I_x$ | kg$\cdot$m$^2$ |
| Pitch inertia | 1.211 | $I_y$ | kg$\cdot$m$^2$ |
| Yaw inertia | 3.522 | $I_z$ | kg$\cdot$m$^2$ |
| Product of inertia | 0.022 | $I_{xz}$ | kg$\cdot$m$^2$ |
The dynamics of fixed-wing drones are described by the standard rigid-body equations involving angular rates $p,q,r$, airflow angles $\mu,\alpha,\beta$, and kinematic variables. Due to the time-scale separation, the inner loops (angular rate and airflow angle) are controlled separately.
1.2 Modeling the Leader’s Wake Vortex Using a Multi-Lifting-Line Method
For swept-wing fixed-wing drones, the classical Prandtl lifting-line theory is inadequate because of three-dimensional cross-flow effects. We adopt a multi-lifting-line (MLL) approach. The leader’s wing is divided into $2N$ trapezoidal panels along the span. Each panel carries a horseshoe vortex with the bound vortex at the quarter-chord line and free vortices trailing downstream. The control points are located at the three-quarter-chord midpoints. Applying the flow-tangency condition at each control point yields a linear system for the unknown vortex strengths $\Gamma_i$:
$$
(\mathbf{V}_\infty + \sum_{i=1}^{2N} \mathbf{V}_{j,i}) \cdot \mathbf{n}_j = 0, \quad j=1,\dots,2N
$$
Here $\mathbf{V}_{j,i}$ is the induced velocity at control point $j$ from horseshoe vortex $i$, computed via the Biot–Savart law. The Kurylowich vortex model is used to describe the velocity profile of the trailing vortices:
$$
V_\theta(r) = \frac{\Gamma}{2\pi r} \left(1 – e^{-1.26 (r/r_c)^2}\right)
$$
with core radius $r_c = \sqrt{36.2 \nu t / \cos(\Lambda)}$, where $\Lambda$ is the quarter-chord sweep angle, $\nu$ the kinematic viscosity, and $t$ the vortex age.
1.3 Aerodynamic Coupling in Close Formation
The induced velocity field from the leader perturbs the flow over the wingman. We compute the effective translational and rotational components at the wingman’s center of mass using averaging methods. The resulting increments in force and moment coefficients are:
$$
\begin{aligned}
\Delta C_D &= \frac{\partial C_D}{\partial \alpha} \Delta \alpha + \frac{\partial C_D}{\partial \beta} \Delta \beta \\
\Delta C_Y &= \frac{\partial C_Y}{\partial \beta} \Delta \beta + \frac{b}{2V} \left( C_{Y_p} \Delta p + C_{Y_r} \Delta r \right) \\
\Delta C_L &= \frac{\partial C_L}{\partial \alpha} \Delta \alpha + \frac{c}{2V} C_{L_q} \Delta q
\end{aligned}
$$
$$
\begin{aligned}
\Delta C_l &= C_{l_\beta} \Delta \beta + \frac{b}{2V} \left( C_{l_p} \Delta p + C_{l_r} \Delta r \right) \\
\Delta C_m &= C_{m_\alpha} \Delta \alpha + \frac{c}{2V} C_{m_q} \Delta q \\
\Delta C_n &= C_{n_\beta} \Delta \beta + \frac{b}{2V} \left( C_{n_p} \Delta p + C_{n_r} \Delta r \right)
\end{aligned}
$$
These increments depend on the relative position between the leader and wingman fixed-wing drones.
1.4 Validation
The MLL model was validated against CFD simulations (XFlow) for various lateral, longitudinal, and vertical offsets. The results show excellent agreement in predicting the coefficient increments for the wingman fixed-wing drone. Notably, the lift increase is maximized when the wingtips overlap by 1/8 span, which also coincides with the largest roll moment disturbance.
2. Prescribed-Time Incremental Nonlinear Dynamic Inversion (PT-INDI)
2.1 Controller Design
Consider a nonlinear system perturbed by uncertainties and disturbances:
$$
\dot{\mathbf{x}} = \mathbf{f}(\mathbf{x}) + \mathbf{g}(\mathbf{x})\mathbf{u} + \mathbf{d}
$$
Under fast sampling, the incremental form is:
$$
\dot{\mathbf{x}} = \dot{\mathbf{x}}_0 + \mathbf{g}(\mathbf{x}_0) \Delta \mathbf{u} + \mathbf{d}
$$
We propose the PT-INDI control law:
$$
\mathbf{u} = \mathbf{u}_0 + \mathbf{g}^{-1}(\mathbf{x}_0) \left[ -\rho(t)(\kappa_0 \mathbf{e} + \kappa_1 \mathbf{e}^{p/q}) + \dot{\mathbf{x}}_c \right]
$$
where $\mathbf{e} = \mathbf{x} – \mathbf{x}_c$, $\rho(t)$ is the prescribed-time modulation function:
$$
\rho(t) = \begin{cases}
\frac{T_p^m}{(T_p – t)^m}, & 0 \le t < T_p \\
1, & t \ge T_p
\end{cases}
$$
$T_p$ is the prescribed convergence time, $m$ a positive integer, and $\kappa_0, \kappa_1 > 0$.
2.2 Stability Proof
Lemma 1: For a system with Lyapunov function $V$ satisfying $\dot{V} \le -\rho(t)(k_0 V + k_1 V^{1-\epsilon}) + d_0$, the error converges to a residual set within $T_p$.
Theorem 1: Under the PT-INDI law, the closed-loop system is globally prescribed-time stable, and the tracking error $\mathbf{e}$ converges to a bounded neighborhood of zero by $t = T_p$.
Proof: Choose $V = \frac{1}{2} \mathbf{e}^T \mathbf{e}$. Its derivative yields:
$$
\dot{V} \le -\rho(t) \left( \kappa_0 \|\mathbf{e}\|^2 + \kappa_1 \|\mathbf{e}\|^{(p+q)/q} \right) + \|\mathbf{e}\| \|\mathbf{d}\|
$$
Using Young’s inequality and Lemma 1, we conclude prescribed-time stability.
3. Modified Tracking Differentiator (MTD) Based on Predefined-Time Control
3.1 Preliminaries
Classical tracking differentiators (TDs) suffer from tradeoffs between speed and noise sensitivity. We introduce a predefined-time sliding-mode TD to achieve finite-time convergence and noise attenuation simultaneously.
3.2 Design of MTD
Theorem 2: Consider the system:
$$
\begin{cases}
\dot{x}_1 = x_2 \\
\dot{x}_2 = -\frac{1}{T_s} \phi_1(s) – \frac{1}{T_c} \phi_2(x_1)
\end{cases}
$$
with $s = x_2 + \frac{1}{T_c} \phi_1(x_1)$, $\phi_1(s) = \exp(\lambda_1 |s|^{\lambda_1}) \text{sign}(s)$, $\phi_2(x_1) = \exp(\lambda_2 |x_1|^{\lambda_2}) \text{sign}(x_1)$, $0 < \lambda_1, \lambda_2 < 1$, $T_s, T_c > 0$. Then the derivative system:
$$
\begin{cases}
\dot{z}_1 = z_2 \\
\dot{z}_2 = -\frac{1}{T_s} \phi_1(z_2 + \frac{1}{T_c} \phi_1(z_1 – u)) – \frac{1}{T_c} \phi_2(z_1 – u(t))
\end{cases}
$$
is a tracking differentiator such that $z_1 \rightarrow u(t)$ and $z_2 \rightarrow \dot{u}(t)$ within $T_s + T_c$.
3.3 Convergence Analysis
The sliding surface $s=0$ is reached in predefined time $T_s$, and then $x_1$ converges to zero in predefined time $T_c$. This ensures overall finite-time convergence regardless of initial conditions.
3.4 Simulation Results of MTD
We tested the MTD on a step input with and without Gaussian noise (variance 0.004). Compared to classic TD with $R=500$, the MTD converged faster (within 0.4 s) and provided smoother derivative estimates with reduced noise amplification.
4. Wingman Inner-Loop Controller Design in Close Formation
4.1 Airflow Angle Control
The airflow angle dynamics for the wingman fixed-wing drone are:
$$
\dot{\boldsymbol{\Phi}} = \dot{\boldsymbol{\Phi}}_0 + \mathbf{G}_{\Phi} \Delta \boldsymbol{\Omega} + \mathbf{d}_{\Phi}
$$
where $\boldsymbol{\Phi}=[\mu,\alpha,\beta]^T$. The PT-INDI control law is:
$$
\boldsymbol{\Omega}_c = \boldsymbol{\Omega}_0 + \mathbf{G}_{\Phi}^{-1} \left[ -\rho(t)(\kappa_{\Phi0} \mathbf{e}_{\Phi} + \kappa_{\Phi1} \mathbf{e}_{\Phi}^{p_{\Phi}/q_{\Phi}}) + \dot{\boldsymbol{\Phi}}_c \right]
$$
with $\mathbf{e}_{\Phi} = \boldsymbol{\Phi} – \boldsymbol{\Phi}_c$.
4.2 Angular Rate Control
The angular rate dynamics are:
$$
\dot{\boldsymbol{\Omega}} = \dot{\boldsymbol{\Omega}}_0 + \mathbf{G}_{\Omega} \Delta \mathbf{u} + \mathbf{d}_{\Omega}
$$
The control surface deflection command is:
$$
\mathbf{u} = \mathbf{u}_0 + \mathbf{G}_{\Omega}^{-1} \left[ -\rho(t)(\kappa_{\Omega0} \mathbf{e}_{\Omega} + \kappa_{\Omega1} \mathbf{e}_{\Omega}^{p_{\Omega}/q_{\Omega}}) + \dot{\boldsymbol{\Omega}}_c \right]
$$
where $\mathbf{e}_{\Omega} = \boldsymbol{\Omega} – \boldsymbol{\Omega}_c$, and $\dot{\boldsymbol{\Omega}}_0$ is obtained from the MTD.
5. Digital Simulation Verification
5.1 Simulation Setup
The controller parameters are listed below. Disturbances include aerodynamic coupling, sensor noise, model uncertainties (30% perturbation in $\mathbf{F}$ and $\mathbf{G}$ matrices), and time delays (Table 2). We compare three schemes: (1) standard INDI, (2) predefined-time backstepping, and (3) the proposed PT-INDI+MTD.
| Signal | Bias | Noise variance | Delay (s) | Sampling time (s) |
|---|---|---|---|---|
| $p,q,r$ (rad/s) | 0.00003 | 0.000004 | 0.128 | 0.0192 |
| $V$ (m/s) | 2.5 | 0.00085 | 0.1 | 0.0625 |
| Control surfaces (rad/s) | 0.0045 | 0.00000055 | 0.0397 | 0.01 |
5.2 Results
Ideal case (no disturbances): All three schemes track the commanded $\mu,\alpha,\beta$ accurately. The proposed PT-INDI achieves convergence within the prescribed time $T_p = 1.2$ s.
With disturbances and noise: Standard INDI and predefined-time backstepping exhibit large oscillations and fail to track the reference. In contrast, the PT-INDI with MTD provides smooth tracking with small steady-state error. The control surface deflections are smooth without high-frequency chattering, thanks to the noise-filtering property of the MTD.
6. Conclusion
This paper presented a comprehensive approach for close formation control of fixed-wing drones. A multi-lifting-line aerodynamic model accurately captures the wake vortex effects for swept-wing configurations. The proposed PT-INDI controller combines prescribed-time convergence with incremental robustness, while the MTD supplies clean angular acceleration estimates. Simulation results demonstrate that the method effectively handles aerodynamic coupling, model uncertainties, and sensor noise, making it suitable for practical close formation flight of fixed-wing drones.