Risk-Field Visualization and Path Planning for UAV Air Refueling Considering Wake Vortex Effects

Highlights

What are the main findings?

• A wake vortex-aware 3D risk field is established by coupling an HB-P2P (Hallock Burnham-Probabilistic Two-phase Wake Vortex Decay Model) wake evolution model with an RMC-based safety metric tailored to large-span UAVs.
• An improved PSO planner with an RMC-driven multi-objective cost function generates ingress paths that explicitly avoid high-risk wake regions while maintaining path feasibility and smoothness.

What are the implications of the main findings?

• The proposed framework transforms abstract wake vortex disturbances into a quantitative, visualizable risk map, and designs a UAV ingress path planning method that explicitly accounts for the wake vortex risk field in tanker-UAV refueling scenarios.
• It provides a transferable methodology for wake vortex risk mapping and avoidance, supporting safer and more intelligent cooperative airspace operations beyond the specific A330MRTT-RQ-4 scenario.

Abstract

Autonomous aerial refueling is a key technology for enhancing the endurance of unmanned aerial vehicles; however, the wingtip vortices generated by the tanker create a strong three-dimensional wake-vortex flow field, whose downwash and lateral airflow can impose significant rolling moments on the follower Unmanned Aerial Vehicle (UAV), posing a serious threat to flight safety. To address this issue, this study proposes an integrated framework that combines wake-vortex risk-field modeling with optimal path planning. The classical Hallock–Burnham (HB) model is first employed to predict vortex descent and lateral transport, while a two-phase model is used to characterize the temporal decay of vortex circulation. The predicted vortex parameters are then coupled with the UAV’s aerodynamic characteristics, and the rolling-moment coefficient (RMC) is introduced as a risk metric to compute its spatiotemporal distribution in three dimensions, thereby transforming the invisible wake-vortex disturbance into a visualizable and quantifiable dynamic three-dimensional risk map. On this basis, a wake-vortex-aware path-planning algorithm based on particle swarm optimization (PSO) is developed, incorporating adaptive weighting and elitist mutation strategies. A multi-objective cost function considering path length, safety, and smoothness is further constructed to search for an optimal safe path under wake-vortex influence. Simulation results indicate that, compared with the classical A* and Rapidly-Exploring Random Tree (RRT) algorithms, the proposed method reduces cumulative risk exposure by approximately 90% and 75%, respectively, while limiting the increase in path length to about 8% (significantly lower than the increases of 40% for A* and 44% for RRT). In addition, the maximum turning angle is constrained within 10°, and the computation time remains around 0.052 s, satisfying real-time requirements. These results demonstrate that the proposed method can generate safe, efficient, and dynamically feasible paths for UAV aerial refueling and provide a valuable reference for wake-vortex avoidance in similar aerospace missions.

1. Introduction

With the growing demand for long-range reconnaissance and persistent missions, autonomous aerial refueling (AAR) between large tankers and unmanned aerial vehicles (UAVs) has become a key enabler for enhancing unmanned operational effectiveness [1]. However, during the AAR approach phase, the wake vortices generated by the tanker impose significant aerodynamic disturbances on lightweight, high-aspect-ratio, long-endurance UAVs [2,3], potentially inducing abrupt rolling moments and attitude instability. Although CFD and large-eddy simulations have improved the prediction accuracy of wake vortex evolution [4], their computational cost remains prohibitive for real-time planning. Moreover, existing studies lack quantitative descriptions of the aerodynamic influence of tanker wake vortices on large-span UAVs [5], which limits the safety and robustness of autonomous docking.

UAVs typically rely on wireless communication or onboard computers for autonomous control [6]; however, their fuel capacity is constrained by airframe size and payload limitations, resulting in insufficient endurance for long-duration missions [7]. Air-to-air refueling is commonly regarded as a cooperative task in which fuel is transferred from a tanker aircraft to a receiver aircraft during flight [8]. Moreover, it shares certain characteristics with cooperative control problems, such as dynamic event-triggered time-varying formation control in heterogeneous unmanned swarm systems [9]. As UAV applications continue to expand, autonomous UAV aerial refueling has become a research focus, with increasing attention being paid to its necessity and technical challenges [10]. Despite progress in communication control, autonomous guidance, and rendezvous trajectory design, most existing studies assume a disturbance-free or simplified wind environment and do not systematically account for the dominant disturbance source—the tanker wake vortex. Meanwhile, extensive foundational research on aircraft wake vortices has established a comprehensive theoretical framework. Greene proposed an early approximate dissipation model [11]; Sarpkaya revealed the effects of stratification and turbulence intensity on vortex decay and developed the Aircraft Vortex Spacing System (AVOSS) Prediction Algorithm (APA) model [12,13], which later supported NASA’s wake vortex spacing system [14]. Proctor introduced the Terminal Area Simulation System (TASS) Driven Algorithms for Wake Prediction(TDAWP) three-dimensional wake model [15]; Holzäpfel subsequently proposed the Wake Two-Stage Decay Model(D2P)and proposed the Probabilistic Two-phase Wake Vortex Decay Model(P2P)models [16] to incorporate atmospheric and operational effects, supported by integrated wake vortex safety and capacity systems [17], and probabilistic assessment studies [18], forming the basis of the European Wake Vortex Prediction and Monitoring System (WSVBS) wake vortex prediction system [19,20,21]. In the context of wake-encounter dynamics, NASA conducted flight experiments and established a wake-encounter database [22], leading to the development of dynamic response models [23], aerodynamic response models [24], torque-coefficient-based assessment approaches [25], roll-response models [26], and roll-moment-coefficient evaluation methods [27], which together provide essential tools for understanding wake vortex impacts on aircraft.

In research on UAV AAR path planning, Burns employed Dubins geometry and non-linear dynamic inversion to design a rapid rendezvous guidance law [28]; Lu-go-Cardenas proposed a rendezvous-route construction method combining Dubins curves with vector-field theory [29]; and Wilson used the A* algorithm to generate optimal rendezvous trajectories and enhance route-planning automation [30].

From the perspective of existing studies, related research mainly focuses on three aspects. The first aspect is wake-vortex modeling and prediction, including empirical models, semi-empirical models, and high-fidelity numerical simulations for wake evolution analysis. The second aspect is wake-encounter dynamic response, where evaluation methods based on moment coefficients have been established. The third aspect is UAV aerial refueling path planning and autonomous guidance, which are typically designed based on geometric constraints or optimization algorithms. However, these research directions have largely developed independently. In the UAV aerial refueling scenario, most path-planning methods assume a disturbance-free or uniform wind environment and do not incorporate the influence of tanker wake vortices into the planning framework.

Existing studies present two major limitations. First, although wake vortex modeling has become relatively mature, it mainly focuses on the impact of wake vortices on civil transport aircraft, while limited attention has been given to the specific aerodynamic effects on UAVs. Second, current UAV aerial refueling path-planning methods primarily aim to minimize distance or time. They rarely consider aerodynamic safety margins and often neglect the influence of wake vortices on UAV operational safety.

To address these limitations, this study proposes a wake vortex risk field reconstruction method based on the rolling moment coefficient. It also develops an optimal ingress path planning strategy for an RQ-4 UAV refueling with an A330MRTT under wake vortex conditions. The aerodynamic configuration and relevant flight parameters of the RQ-4 UAV and the A330MRTT tanker are introduced in detail in Section 3. The study first compares several representative wake vortex models and selects the HB-P2P model to generate a high-fidelity wake vortex velocity field. The detailed formulation of the HB-P2P wake evolution model is presented in Section 2. The rolling moment coefficient at different UAV positions within the vortex field is then calculated. Based on these results, the study constructs a continuous three-dimensional and time-varying wake vortex risk field.

The study treats this risk field as a dynamic obstacle environment. It then develops a wake vortex aware particle swarm optimization path planning algorithm to generate an ingress path that balances safety, feasibility, and smoothness. Comparative simulations with A star and RRT are conducted to evaluate the method. The results show that the proposed approach produces safer and more efficient ingress trajectories. The method also keeps the peak rolling moment below the required safety threshold.

The three-dimensional wake vortex risk field and the associated path planning algorithm provide real-time safety support for UAV autonomous aerial refueling with large tanker aircraft. The proposed framework can also support wake vortex identification, risk prediction in complex airspace, and intelligent traffic management. These capabilities contribute to safer and more efficient cooperative airspace operations.

The primary contributions of this paper are as follows:

• For autonomous aerial-refueling rendezvous at cruising altitudes, this study develops a spatiotemporal wake vortex evolution prediction method for large tanker aircraft, explicitly accounting for altitude-dependent temperature variations. The model accurately captures essential wake behaviors—including vortex-core descent, lateral transport, and circulation decay—thus providing a high-fidelity wake vortex velocity field that underpins subsequent risk quantification and trajectory-planning processes.
• By coupling the predicted wake characteristics with the UAV’s aerodynamic response, the roll-moment coefficient (RMC) is adopted as a direct risk metric. A continuous, quantifiable, time-evolving three-dimensional risk field is reconstructed, enabling detailed representation of hazardous wake regions.
• A wake-vortex-aware particle swarm optimization–based path-planning algorithm is developed to generate safe ingress trajectories within the dynamic risk environment. The method formulates a multi-objective cost function and incorporates adaptive search mechanisms, thereby enhancing global optimization performance and ensuring efficient online planning under safety constraints.

This study first compares several classical wake-vortex models, selects the one suitable for large tanker aircraft, and constructs a high-fidelity wake-vortex velocity field based on the HB–P2P model, thereby providing an accurate aerodynamic disturbance environment for subsequent risk assessment and path planning [16]. Subsequently, the predicted wake-vortex parameters are coupled with the UAV’s aerodynamic response, and the roll-moment coefficient is introduced as a risk metric. The spatiotemporal wake-vortex risk field is then constructed by calculating the roll-moment coefficient and reconstructing the corresponding three-dimensional risk distribution, thereby explicitly mapping the wake-vortex influence into a visualizable dynamic risk region.

Next, the three-dimensional wake vortex risk field is treated as a dynamic obstacle environment, and a wake-vortex-aware particle swarm optimization-based path-planning algorithm is developed. A multi-objective cost function is designed, and adaptive search and mutation mechanisms are incorporated to generate safe, smooth, dynamically feasible ingress paths within the dynamic risk environment. Finally, the proposed path-planning algorithm is validated through comprehensive simulations and compared with representative methods such as A* and RRT. The effectiveness of the proposed method is quantitatively evaluated in terms of risk exposure, path length, smoothness, and computation time, providing a useful reference for future research and extensions. The overall research process of this study is summarized in Figure 1.

2. Materials and Methods

2.1. Modeling of Wake Vortex Evolution

To characterize the full life cycle evolution of aircraft wake vortices at high altitudes, this study develops an integrated wake vortex modeling approach. First, based on the Kutta–Joukowsky theorem for flow around a cylinder, the relationship between lift and circulation is established to determine key initial wake parameters, including the initial circulation. The effects of high-altitude air density and temperature on wake vortex evolution are also taken into account [23]. During the evolution stage, the two-stage P2P model proposed by Holzäpfel is adopted to describe vortex descent, lateral transport, and circulation decay. This model captures the main wake development characteristics under cruise altitude conditions. Subsequently, the Hallock–Burnham model is introduced to compute the induced velocity distribution and to construct the wake vortex velocity field.

In summary, the Kutta–Joukowsky theorem, the P2P evolution model, and the Hallock–Burnham induced velocity model are combined to establish the HB–P2P wake vortex evolution model. Equations (1)–(9) are formulated with reference to the modeling methodology reported in [23].

According to the Kutta–Joukowsky aerodynamic theorem for flow around a cylinder, the initial circulation of a steadily flying aircraft can be expressed by Equation (1).

$$\left\{\begin{array}{c}{\Gamma }_0={\displaystyle \frac{2n_{y}M}{\rho V{b}_0}}\\ b_0=s·B\end{array}\right.$$

(1)

where $M$ denotes the gravitational force acting on the aircraft (N); $\rho$ is the ambient air density (kg/m³); $V$ is the true airspeed of the aircraft (m/s); $B$ is the wingspan (m); $s$ is the airfoil parameter, taken as π/4; and $n_y$ represents the normal load factor.

In an ideal atmosphere, wake vortices diffuse outward and downward, and their initial descent velocity (characteristic velocity), $w_0$ is given by Equation (2).

$$w_0={\displaystyle \frac{2{\Gamma }_0}{{\pi }^{2}B}}$$

(2)

where ${\Gamma }_0$ is the initial circulation (m²/s) computed from Equation (1).

During the initial descent phase of the wake vortices, the characteristic velocity remains nearly constant. Here, the characteristic time $t_0$ is further introduced to facilitate the subsequent nondimensionalization of the relevant variables. The specific calculation is provided in Equation (3).

$$t_0=s·{\displaystyle \frac{b_0}{w_0}}$$

(3)

The nondimensional atmospheric stratification stability is quantified using the buoyancy frequency, which is computed as given in Equation (4).

$$\left\{\begin{array}{c}{N}^{\mathrm{*}}={\displaystyle \frac{2\pi \sqrt{g}\omega b_0^2}{{\Gamma }_0\sqrt{\mathrm{h}}}}\\ \omega ={\displaystyle \frac{\mathrm{d}\theta }{\mathrm{d}h}}\end{array}\right.$$

(4)

where $\theta$ is the potential temperature and $h$ denotes the altitude.

The nondimensional atmospheric turbulence intensity, ${\mathsf{\epsilon }}^{\mathrm{*}}$ is characterized by the turbulent kinetic energy dissipation rate $\mathsf{\epsilon }$, and is calculated as shown in Equation (5).

$${\mathsf{\epsilon }}^{\mathrm{*}}={\displaystyle \frac{{\left(\mathsf{\epsilon }{\mathrm{b}}_0\right)}^{\frac{1}{3}}}{{\mathrm{w}}_0}}$$

(5)

Within the typical cruise-altitude range of commercial aircraft, atmospheric physical properties exert significant influences on both flight performance and wake vortex behavior. Near the tropopause (approximately 10,500 m altitude), the variation in air density with altitude follows the relationship given in Equation (6).

$$\mathsf{\rho }={{\mathsf{\rho }}_{\mathrm{S}\mathrm{L}}\left(1+{\displaystyle \frac{\mathrm{k}}{{\mathrm{T}}_{\mathrm{S}\mathrm{L}}}}\right)}^{-\left({\displaystyle \frac{\mathrm{g}}{\mathrm{K}\mathrm{R}}}+1\right)}$$

(6)

where ${\mathrm{T}}_{\mathrm{S}\mathrm{L}}$ and ${\mathsf{\rho }}_{\mathrm{S}\mathrm{L}}$ are the temperature and density of the International Standard Atmosphere at sea level, taken as 288.15 K and 1.225 kg/m³, respectively; R′ is the specific gas constant for dry air; and k is the temperature lapse rate (−6.49 K/km).

As shown in Figure 2, after the vortex roll-up stage, the high-altitude evolution of wake vortices is generally divided into a turbulence diffusion stage and a rapid decay stage.

By integrating radar observations from various regions with CFD simulation data, it is evident that the evolution of atmospheric vortex systems results from the interactions of physical processes across multiple temporal and spatial scales. The P2P model provides a more accurate quantification of the circulation decay throughout the high-altitude wake vortex lifecycle.

The calculation procedure for the turbulent diffusion stage is given in Equation (7).

$${\mathsf{\Gamma }}^{\mathrm{*}}\left(t^{\mathrm{*}}\right)=\mathrm{A}-e^{{\displaystyle \frac{-K}{v_1^{\mathrm{*}}\left(t^{\mathrm{*}}-t_1^{\mathrm{*}}\right)}}}$$

(7)

The calculation procedure for the rapid decay stage is given in Equation (8).

$${\mathsf{\Gamma }}^{\mathrm{*}}\left(t^{\mathrm{*}}\right)=\mathrm{A}-e^{{\displaystyle \frac{-K}{v_1^{\mathrm{*}}\left(t^{\mathrm{*}}-t_1^{\mathrm{*}}\right)}}}-e^{{\displaystyle \frac{-K}{v_2^{\mathrm{*}}\left(t^{\mathrm{*}}-t_2^{\mathrm{*}}\right)}}}$$

(8)

where ${\mathsf{\Gamma }}^{*}$ denotes the dimensionless circulation (defined as the mean circulation within a vortex-core radius of 5–15 m); K represents the squared mean vortex-core radius; $v_1^{*}$ is the dimensionless viscosity factor during the diffusion stage; and $t_2^{*}$ and $v_2^{*}$ correspond to the onset time of the rapid decay stage and the effective kinematic viscosity, respectively.

Under different turbulence levels, the calculation of $t_2^{\mathrm{*}}$ is given in Equation (9).

$$\left\{\begin{array}{c}{t}_2^{\mathrm{*}}=0.084{\epsilon }^{\mathrm{*}-0.75}\left({\epsilon }^{\mathrm{*}}\ge 0.2535\right)\\ {\epsilon }^{\mathrm{*}}=\left({t^{\mathrm{*}}}_2^{{\displaystyle \frac{1}{4}}}\right)e^{\left(0.7t_2^{\mathrm{*}}\right)}\left(0.0121\ge {\epsilon }^{\mathrm{*}}\ge 0.2535\right)\\ t_2^{\mathrm{*}}=9.18-180{\epsilon }^{\mathrm{*}}\left({\epsilon }^{\mathrm{*}}\le 0.0121\right)\end{array}\right.$$

(9)

2.2. Induced-Velocity Model for Aircraft Wake Vortices

To assess wake vortex safety, it is necessary to compute the induced velocity generated by the leading aircraft’s wake vortices on the trailing aircraft’s wings. Commonly used tangential velocity models for wake vortices—such as the Rankine, Lamb–Oseen, and Hallock–Burnham models—produce similar results in regions far from the vortex core. As shown in Figure 3, the tangential velocities calculated using the above models exhibit similar behavior at a certain distance from the vortex center. Owing to its concise and unified analytical form, as well as its proven validity supported by extensive measurement data, the Hallock–Burnham model is adopted in this study for simulating wake-induced velocity.

The Hallock–Burnham model has a simple formulation, provides a good fit to wake vortices, and yields relatively smooth velocity variations.

The induced velocity at a point located at a distance r from the vortex-core center is calculated using the Hallock–Burnham model as follows:

$${\mathrm{V}}_T\left(r\right)={\displaystyle \frac{{\mathsf{\Gamma }}_0}{2\pi r}}·{\displaystyle \frac{r^2}{r^2+r_c}}$$

(10)

2.3. Safety Assessment Method

In this study, the rolling moment coefficient (RMC) is selected as the core indicator for UAV safety assessment. The wake vortices generated by large tanker aircraft are dominated by vertical induced velocity. This velocity causes asymmetric lift distribution along the UAV wingspan and produces a rolling response about the longitudinal axis. This roll motion is the most significant dynamic effect when the UAV encounters the wake vortex.

The RMC is obtained by nondimensionalizing the total rolling moment. Compared with other induced force indicators, the RMC more directly reflects the risk of loss of control caused by the wake vortex. It is widely used as a safety criterion in wake encounter studies.

Wake vortices also induce lateral and vertical forces, as well as pitching and yawing moments. However, their contribution to attitude instability is relatively small. In future work, multi-axis aerodynamic loads and a six-degree-of-freedom dynamic model can be introduced to build a more comprehensive risk assessment framework.

The aircraft is simplified into three components: 1. fuselage, 2. wings, and 3. Engine, as shown in Figure 4a. The induced effects of these three components are then computed separately by performing strip-wise integration for each component, as illustrated in Figure 4b.

The aerodynamic response under wake vortex influence is modeled in terms of incremental variations relative to the trimmed steady-flight condition. In this study, the UAV is assumed to operate in a nominal equilibrium state under undisturbed flow, and the corresponding baseline aerodynamic loads are not explicitly recalculated. Instead, the analysis focuses on the additional aerodynamic effects induced by the wake vortex. The induced velocity field modifies the local effective angle of attack, and the resulting incremental lift variation is computed using Equations (12)–(15).

2.3.1. Calculation of Lift Variation

Using the differential-element approach, the lift variation in an individual strip element on the wing is given by:

$$dL(y)={\displaystyle \frac{1}{2}}{{\mathrm{V}}_{\mathrm{f}}}^2\rho cl(y)c(y)dy$$

(11)

where $dL\left(y\right)$ denotes the lift variation in the strip element; $\rho$ is the air density at the aircraft’s flight condition; ${\mathrm{V}}_{\mathrm{f}}$ is the freestream velocity, approximately equal to the approach speed of the following aircraft; $cl\left(y\right)$ represents the variation in the effective lift-line coefficient at spanwise location y; and $c\left(y\right)$ is the local chord length of the wing at the same spanwise position.

By integrating the contributions of all strip elements along the wingspan, the total lift variation in the entire wing can be obtained, as expressed below:

$$\left\{\begin{array}{c}∆L_S={\displaystyle \frac{1}{2}}{\int }_{-{\displaystyle \frac{B}{2}}}^{{\displaystyle \frac{B}{2}}}\rho {{\mathrm{V}}_{\mathrm{f}}}^{2}cl\left(y\right)c\left(y\right)dy\\ cl\left(y\right)=\mu ·C_L^{\alpha }\\ \mu =\arctan {\displaystyle \frac{V_{\tau }\left(y\right)}{{\mathrm{V}}_{\mathrm{f}}}}\approx {\displaystyle \frac{V_{\tau }\left(y\right)}{{\mathrm{V}}_{\mathrm{f}}}}\end{array}\right.$$

(12)

where $V_{\tau }\left(y\right)$ is the induced velocity of the upstream wake on the wing section of the following aircraft; $\mu$ denotes the angle-of-attack perturbation on the wing section caused by the wake vortices of the leading aircraft. Since $\mu$ is small, it can be approximated as ${\displaystyle \frac{V_{\tau }\left(y\right)}{{\mathrm{V}}_{\mathrm{f}}}}$.

$C_L^{\alpha }$ is the lift-curve slope, which can be obtained from the following expression [25]:

$$\left\{\begin{array}{c}{C}_L^{\alpha }={\displaystyle \frac{2\pi \lambda }{2+\sqrt{4+\frac{{\lambda }^2{\beta }^2}{{\eta }_e}\gamma }}}\\ \gamma =1+{\displaystyle \frac{{\left(\tan \chi -\frac{2}{\lambda }·\frac{\eta -1}{\eta +1}\right)}^2}{1-{M\alpha }^2}}\end{array}\right.$$

(13)

where $\lambda$ is the aspect ratio of the aircraft; $\eta$ is the taper ratio; $\chi$ is the sweep angle of the leading edge; ${\eta }_e$ is the airfoil efficiency factor, typically taken as 0.95; and $M\alpha$ is the Mach number corresponding to the aircraft’s angle-of-attack condition.

The fuselage of the UAV can be approximated as a slender circular cylinder at a small angle of attack. According to potential-flow theory, the lift variation acting on the fuselage can be expressed as:

$$\Delta L_b=\rho V_f^2{S}_b\left(\Delta {\alpha }_b\right)\mathit{cos}(\Delta {\alpha }_b)+0.5\rho V_f^2{B}_b\Delta {\alpha }_b^2\mathit{sin}(\Delta {\alpha }_b)$$

(14)

where $\mathsf{\Delta }{L}_b$ is the lift variation in the fuselage, $S_b$ is the projected area of the fuselage, $B_b$ is the projected width of the fuselage, and $\Delta {\alpha }_b$ is the change in the angle of attack of the fuselage relative to the freestream.

According to the panel numerical method, the lift variation in the engine can be calculated as follows [25]:

$$\Delta L_{\mathrm{e}}=\rho V_f\sum _{j=1}^n{V}_j{S}_j$$

(15)

$\Delta L_{\mathrm{e}}$ is the lift variation in the engine, $V_j$ is the wake-induced velocity at strip j, and $S_j$ is the immersed area of the engine corresponding to strip j.

By Equations (13)–(15), the total lift variation acting on the aircraft is given by:

$$\Delta L_{\mathrm{a}}=∆L_S+\Delta L_b+\Delta L_{\mathrm{e}}$$

(16)

2.3.2. Calculation of the Roll-Moment Variation

The calculation of the rolling moment $M_s$ acting on the wing is expressed as follows:

$$M_s={\displaystyle \frac{1}{2}}\rho {{\mathrm{V}}_{\mathrm{f}}}^2{\int }_{{\displaystyle \frac{B}{2}}}^{B}cl\left(y\right)c\left(y\right)·ydy$$

(17)

The calculation of the rolling moment Mb acting on the fuselage is given as follows:

$${\mathrm{M}}_{\mathrm{b}}=\mathsf{\rho }{\int }_{-{\displaystyle \frac{{\mathrm{B}}_{\mathrm{b}}}{2}}}^{{\displaystyle \frac{{\mathrm{B}}_{\mathrm{b}}}{2}}}{\displaystyle \frac{{\mathrm{V}}_{\mathsf{\Gamma }}\left(\mathrm{y}\right)·{\mathrm{V}}_{\mathrm{f}}^2+{\mathrm{V}}_{\mathsf{\Gamma }}^3\left(\mathrm{y}\right)}{\sqrt{{\mathrm{V}}_{\mathrm{f}}^2+{\mathrm{V}}_{\mathsf{\Gamma }}^2\left(\mathrm{y}\right)}}}·\mathrm{l}(\mathrm{y})·\mathrm{y}\mathrm{d}\mathrm{y}$$

(18)

where l(y) denotes the strip length at position y.

The calculation of the rolling moment $M_{\mathrm{e}}$ acting on the engine is given as follows:

$$M_{\mathrm{e}}=\rho V_f\sum _{j=1}^n{V}_j{S}_j{\mathrm{y}}_{\mathrm{j}}$$

(19)

By substituting Equations (16)–(18), the total rolling moment acting on the aircraft can be obtained as follows:

$$M_{\mathrm{a}}=M_{\mathrm{e}}+{\mathrm{M}}_{\mathrm{b}}+M_s$$

(20)

In this study, the rolling moment coefficient is selected as the metric for evaluating the severity of wake vortex encounters, and the RMC is defined as follows:

$$\mathrm{R}\mathrm{M}\mathrm{C}={\displaystyle \frac{2{\mathrm{M}}_{\mathrm{x}}}{\mathsf{\rho }{\mathrm{V}}_{\mathrm{f}}^2{\mathrm{S}}_{\mathrm{f}}{\mathrm{B}}_{\mathrm{f}}}}$$

(21)

where ${\mathrm{M}}_{\mathrm{x}}$ is the rolling moment experienced by the following aircraft during a wake vortex encounter, and $V_f$, ${\mathrm{S}}_{\mathrm{f}}$, and ${\mathrm{B}}_{\mathrm{f}}$ denote the flight speed, wing area, and wingspan of the following aircraft, respectively. A larger RMC indicates a stronger rolling response when the following aircraft penetrates the wake vortex field of the leading aircraft. A roll angle exceeding 40° is considered indicative of a roll-control hazard, and under this condition, the rolling-moment coefficient of the RQ-4 unmanned aircraft is approximately 0.07 [3].

2.4. A PSO-Based Path Planning Algorithm for Wake Vortex Detection

In the particle swarm optimization (PSO) algorithm, each particle represents a candidate solution and is characterized by two essential attributes: velocity and position. During each iteration, a particle updates its velocity and position based on its own best historical position and the best global position found by the entire swarm. The local best corresponds to the optimal solution obtained by an individual particle during the search process, whereas the global best refers to the optimal solution identified by all particles collectively.

The fundamental concept of PSO is that each particle evaluates its current local best value and subsequently updates the global best value, enabling particles to adjust their search behavior according to the global optimum. When updating their velocity, particles typically consider three components: an adaptive inertia weight ${\mathrm{w}}^{\left(\mathrm{t}\right)}$, a cognitive learning factor C1, and a social learning factor C2. These components jointly determine the direction and magnitude of a particle’s movement during the next iteration.

$$\left\{\begin{array}{c}{\mathrm{v}}_{\mathrm{i}}^{\mathrm{k}+1}={\mathrm{w}}^{\left(\mathrm{t}\right)}\times {\mathrm{v}}_{\mathrm{i}}^{\mathrm{k}}+{\mathrm{c}}_1\times {\mathrm{r}}_1\times ({\mathrm{p}}_{\mathrm{b}\mathrm{e}\mathrm{s}\mathrm{t}}-{\mathrm{x}}_{\mathrm{i}}^{\mathrm{k}})+{\mathrm{c}}_2\times {\mathrm{r}}_2\times ({\mathrm{g}}_{\mathrm{b}\mathrm{e}\mathrm{s}\mathrm{t}}-{\mathrm{x}}_{\mathrm{i}}^{\mathrm{k}})\\ {\mathrm{X}}_{\mathrm{i}}^{\mathrm{k}+1}={\mathrm{X}}_{\mathrm{i}}^{\mathrm{k}}+{\mathrm{V}}_{\mathrm{i}}^{\mathrm{k}+1}\end{array}\right.$$

(22)

where ${\mathrm{v}}_{\mathrm{i}}^{\mathrm{k}+1}$ denotes the velocity of the i-th particle at iteration k + 1, while ${\mathrm{v}}_{\mathrm{i}}^{\mathrm{k}}$ represents its velocity at iteration k. The parameter $\mathsf{\omega }$ is the inertia weight; c1 and c2 represent the cognitive and social learning factors, respectively. ${\mathrm{p}}_{\mathrm{b}\mathrm{e}\mathrm{s}\mathrm{t}}$ denotes the local best position of an individual particle, and ${\mathrm{g}}_{\mathrm{b}\mathrm{e}\mathrm{s}\mathrm{t}}$ represents the global best position of the entire swarm. r1 and r2 are random numbers uniformly distributed in the interval [0,1]. Furthermore, ${\mathrm{X}}_{\mathrm{i}}^{\mathrm{k}+1}$ denotes the position of the i-th particle at iteration k + 1, and ${\mathrm{X}}_{\mathrm{i}}^{\mathrm{k}}$ denotes its position at iteration k.

The dynamic adjustment of the inertia weight $\mathsf{\omega }$ and learning factors ${\mathrm{c}}_1$, ${\mathrm{c}}_2$ is essential for balancing global exploration and local exploitation. In this study, a dynamically adjusted learning factor for each particle and a nonlinearly decreasing inertia weight are designed as follows:

$$\left\{\begin{array}{c}{\mathrm{c}}_1={\mathrm{c}}_{\mathrm{m}\mathrm{a}\mathrm{x}}-t·{\displaystyle \frac{\left({\mathrm{c}}_{\mathrm{m}\mathrm{a}\mathrm{x}}-{\mathrm{c}}_{\mathrm{m}\mathrm{i}\mathrm{n}}\right)}{t_{\mathrm{m}\mathrm{a}\mathrm{x}}}}\\ {\mathrm{c}}_2={\mathrm{c}}_{\mathrm{m}\mathrm{i}\mathrm{n}}+t·{\displaystyle \frac{({\mathrm{c}}_{\mathrm{m}\mathrm{a}\mathrm{x}}-{\mathrm{c}}_{\mathrm{m}\mathrm{i}\mathrm{n}})}{t_{\mathrm{m}\mathrm{a}\mathrm{x}}}}\\ {\mathrm{w}}^{\left(\mathrm{t}\right)}={\mathrm{w}}_{\mathrm{m}\mathrm{i}\mathrm{n}}+\left({\mathrm{w}}_{\mathrm{m}\mathrm{a}\mathrm{x}}-{\mathrm{w}}_{\mathrm{m}\mathrm{i}\mathrm{n}}\right)\cdot {\mathrm{e}}^{-\mathsf{\beta }\cdot (\mathrm{t}/{\mathrm{t}}_{\mathrm{m}\mathrm{a}\mathrm{x}}{)}^2}\end{array}\right.$$

(23)

In the equations, ${\mathrm{c}}_{\mathrm{m}\mathrm{a}\mathrm{x}}$ and ${\mathrm{c}}_{\mathrm{m}\mathrm{i}\mathrm{n}}$ denote the maximum and minimum values of the acceleration coefficients, which are set to 0.65 and 0.35, respectively. The improved learning factors ${\mathrm{c}}_1$ and ${\mathrm{c}}_2$ are dynamically adjusted according to the iteration number. Here, $\mathrm{t}$ represents the current iteration, ${\mathrm{t}}_{\mathrm{m}\mathrm{a}\mathrm{x}}$ is the maximum number of iterations, ${\mathrm{w}}_{\mathrm{m}\mathrm{i}\mathrm{n}}$ and ${\mathrm{w}}_{\mathrm{m}\mathrm{a}\mathrm{x}}$ are the lower and upper bounds of the inertia weight, respectively, and $\mathsf{\beta }$ denotes the decay coefficient.

2.4.1. Elite Retention Strategy

At the end of each iteration, particles are ranked according to their fitness values, and the top 10% are designated as elite particles. These elite particles are copied directly into the next generation without updating their velocity or position. This strategy helps preserve high-quality solutions and prevents them from being degraded by random perturbations during the iterative process.

2.4.2. Gaussian Mutation Strategy

For non-elite particles, a Gaussian velocity perturbation is applied to approximately 20% of them (P = 0.2), introducing controlled randomness to enhance exploration capability:

$${\mathit{v}}_i^{(t+1)}\leftarrow {\mathit{v}}_i^{(t+1)}+N(0,{\sigma }^2),\hspace{1em}\sigma =0.1\cdot \left|x_{id}^{\mathrm{m}\mathrm{a}\mathrm{x}}-x_{id}^{\mathrm{m}\mathrm{i}\mathrm{n}}\right|$$

(24)

where $N(0,{\sigma }^2)$ denotes a Gaussian-distributed random variable with zero mean and a standard deviation of $\sigma$, and $\left|x_{id}^{\mathrm{m}\mathrm{a}\mathrm{x}}-x_{id}^{\mathrm{m}\mathrm{i}\mathrm{n}}\right|$ represents the boundary range of the particle’s position in the d-th dimension.

2.4.3. Turning-Angle Constraint

To ensure that the UAV flight trajectory satisfies its dynamic turning-angle limitations, a turning-angle constraint is imposed. If any waypoint violates the turning-angle requirement, an intermediate waypoint is inserted to smooth the trajectory.

The turning angle ${\theta }_i$ is calculated as follows:

$${\theta }_i=\arccos \left({\displaystyle \frac{v_{i-1}\cdot v_i}{\Vert v_{i-1}\Vert \cdot \Vert v_i\Vert }}\right)$$

(25)

where $v_i$ denotes the velocity vector of the UAV at the i-th waypoint along the planned trajectory.

When the turning angle exceeds the allowable threshold, i.e., θ_i > θ_max, an intermediate waypoint is inserted to ensure compliance with the UAV’s dynamic turning limitations.

$$x_{\mathrm{m}\mathrm{i}\mathrm{d}}=x_i+\alpha \cdot {\displaystyle \frac{x_{i+1}-x_i}{\Vert x_{i+1}-x_i\Vert }}$$

(26)

where $x_i$ denotes the position vector of the i-th waypoint along the trajectory, and $x_{\mathrm{m}\mathrm{i}\mathrm{d}}$ denotes the inserted intermediate waypoint.

2.4.4. Definition of the Hazard Zone

Let the hazardous region be defined as a closed set $\mathrm{D}\subset {\mathrm{R}}^3$, whose boundary is represented by an implicit function as follows:

$$\partial \mathrm{D}=\left\{\mathrm{x}\in {\mathrm{R}}^3|\mathsf{\varphi }\left(\mathrm{x}\right)=0\right\}$$

(27)

$$\left\{\begin{array}{c}\mathsf{\varphi }\left({\mathrm{t}}_{\mathrm{i}}\right)>0, \mathrm{the} \mathrm{trajectory} \mathrm{being} \mathrm{outside} \mathrm{the} \mathrm{hazardous} \mathrm{region}.\\ \mathsf{\varphi }\left({\mathrm{t}}_{\mathrm{i}}\right)=0, \mathrm{the} \mathrm{trajectory} \mathrm{lying} \mathrm{on} \mathrm{the} \mathrm{surface} \mathrm{of} \mathrm{the} \mathrm{hazardous} \mathrm{region}.\\ \mathsf{\varphi }\left({\mathrm{t}}_{\mathrm{i}}\right)<0, \mathrm{the} \mathrm{trajectory} \mathrm{being} \mathrm{inside} \mathrm{the} \mathrm{hazardous} \mathrm{region}.\end{array}\right.$$

When $\mathsf{\varphi }\left({\mathrm{t}}_{\mathrm{i}}\right)<$0, the position of the point is updated as follows:

$$p_{\mathrm{n}\mathrm{e}\mathrm{w}}(t)=p\left(t\right)+\lambda {\displaystyle \frac{\nabla \varphi \left(p\left(t\right)\right)}{‖\nabla \varphi \left(p\left(t\right)\right)‖}}$$

(28)

$$\lambda =\max \left(0,\delta -\varphi \left(p\left(t\right)\right)\right)$$

(29)

where ϕ(x) is the implicit function of the hazardous region, $p\left(t\right)$ denotes the position of the point, $\delta$ used to determine whether the step size needs to be adjusted.

In this study, the implicit scalar function ϕ(x) is constructed based on the predicted rolling moment coefficient distribution obtained from the wake vortex model. Specifically, at each spatial point, the rolling moment coefficient is first evaluated according to the previously computed wake-induced velocity field. The scalar field is then defined by comparing these predicted rolling moment values with the allowable safety threshold, thereby implicitly forming a continuous hazardous boundary in space. In this way, the implicit function ϕ(x) is effectively reconstructed from the spatial distribution of the rolling moment coefficient.

The gradient of ϕ(x) is approximated numerically using the variation in the rolling moment coefficient around the particle position. In practical implementation, it is estimated from the difference between the rolling moment coefficient at the particle’s previous position (before update) and the coefficient evaluated at the current position that violates the safety constraint.

2.4.5. Design of the Total Cost Function

A multi-objective cost function $f_{total}$ is designed in this algorithm:

$$f_{total}=w_1\cdot f_{length}+w_2\cdot f_{safety}+w_3\cdot f_{smoothness}$$

(30)

$w_1$, $w_2$, $w_3$ denotes the weight coefficient.

Path length cost $f_{length}$:

$$f_{length}=\sum _{i=1}^{n-1}\Vert x_{i+1}-x_i{\Vert }^2$$

(31)

Safety cost $f_{safety}$:

$$f_{safety}=\sum _{i=1}^n{\max \left(0,R\left(x_i\right)-R_{threshold}\right)}^2$$

(32)

where $R_{threshold}$ denotes a predefined violation-penalty threshold, which is selected within the range 0 < $R_{threshold}$ < 0.07. $R\left(x_i\right)$ denotes the risk value of particle xi. It specifies the allowable upper bound of the risk indicator along the trajectory. For each discretized trajectory point, a safety cost is generated only when the corresponding risk value exceeds $R_{threshold}$; otherwise, the contribution to the safety cost is zero. In other words, the penalty term is activated exclusively for constraint violations, ensuring that safe trajectory segments do not introduce unnecessary cost.

Smoothness cost $f_{smoothness}$:

$$f_{smoothness}=\sum _{i=2}^{n-1}\Vert x_{i+1}-2x_i+x_{i-1}{\Vert }^2$$

(33)

In this study, the “detection” of hazardous areas is not based on a discrete threshold rule or a simple region division. Instead, it is achieved through a continuous three-dimensional wake-vortex risk field. Specifically, the position of each candidate path point is mapped to the corresponding wake-vortex evolution time and spatial coordinates. The induced velocity at this location is computed using the HB-P2P model, and then the roll moment coefficient (RMC) acting on the UAV is calculated. This RMC value is included in the safety cost function and is used in real time in the multi-objective optimization.

Therefore, during particle swarm optimization, any change in a particle’s path in the search space will dynamically change its associated RMC risk value. The cost function automatically penalizes high-risk areas and guides the path toward low-risk regions. In this way, the proposed method enables continuous perception of and dynamic response to wake-vortex hazardous areas, rather than relying on predefined safe corridors or a fixed geometric centerline.

3. Simulation Results and Analysis

3.1. Data Provenance

The data used in this paper are all obtained from publicly available online sources. The parameters involved include wingspan, wing area, fuselage height, takeoff weight, zero-fuel weight, indicated airspeed, true airspeed, ground speed, maximum Mach number, service ceiling, and others.

Table 1. A330 MRTT Aircraft Parameters.

Category	Parameter	Value
General Dimensions	Length	58.80 m
	Wing Span	60.30 m
	Height	17.40 m
	Wing Area	362 m2
Maximum Weights	Maximum Take-off Weight (MTOW)	233,000 kg
	Maximum Landing Weight (MLW)	182,000 kg
	Maximum Fuel Weight (MFW)	111,000 kg
	Maximum AAR Mission Altitude	10.500 km
	Maximum Cruise Speed	1053.5 km/h
	Typical Cruise Speed	980–1005 km/h

presents the parameters of the A330 MRTT aircraft required for the simulation, and

Table 2. RQ-4 Unmanned Aircraft Parameters.

Category	Parameter	Value
Aircraft Geometric Data	Length	14.5 m
	Wing Span	39.9 m
	Height	4.7 m
	Wing Area	50.3 m2
Aircraft Performance Data	Maximum Level Flight Speed	575 km/h
	Service Ceiling	18,288 m
	Aspect Ratio	31.6
	Range	14,001 km
Aircraft Weight	Empty Weight	6781 kg
	Normal Takeoff Weight	14,628 kg

presents the parameters of the RQ-4 UAV used in the simulation.

3.2. Evolutionary Analysis of Induced Velocity Distribution

Based on Equations (1)–(9) and the characteristic data of the A330 MRTT, the distributions of the wake vortex-induced velocity of the A330 MRTT at different time instants are obtained.

Figure 5 illustrates the spatiotemporal evolution of the A330 MRTT wake vortex induced velocity field calculated using the Hallock–Burnham model. The computational domain spans ±125 m laterally and +50 to −150 m vertically, with the initial wingtip-vortex position set as the origin. The induced velocity is visualized using a color scale, and the vortex cores are marked by black dots. The results reveal a typical dual-vortex structure: at t = 0 s, the circulation is Γ = 908.972 m²/s and the vortex cores are located at (±23.7 m, 0.0 m), exhibiting distinct high-velocity regions that decay radially outward. As time progresses, these high-velocity regions contract and the velocity field gradually weakens; by t = 219.2 s, the circulation has decreased to 90.133 m²/s, indicating substantial attenuation of the wake vortex. Figure 6 shows the relationship between wake vortex circulation and time, as well as the variation of vortex core descent position with time. Over the full 253.5 s duration, the circulation exhibits nonlinear decay: from 0 to 92.7 s, it decreases slowly—corresponding to the turbulent diffusion phase in the P2P model—after which a rapid-decay phase begins, with circulation dropping from 753.47 m²/s to nearly zero, consistent with the fast-decay mechanism described by Equation (8). The vertical motion of the vortex cores shows a similar two-stage behavior: in the initial phase (0–92.7 s), the cores descend nearly linearly by about 84.3 m, whereas in the later phase the descent rate diminishes due to reduced mutual induction and increasing atmospheric stratification effects, eventually stabilizing around −120 m after t ≈ 150 s. These evolution characteristics provide important insights for assessing wake vortex behavior and ensuring flight-safety separation.

3.3. Evolutionary Analysis of RMC Distribution

In the present study, the UAV is assumed to enter the wake vortex field vertically with its nose aligned toward the vortex center. The induced lift acting on each wing segment is evaluated using Equations (10)–(12), after which the total rolling moment is obtained from Equation (13). Subsequently, Equation (14) is applied to nondimensionalize the rolling moment, yielding the spatial distribution of the RMC.

Figure 7 illustrates the spatiotemporal evolution of the hazardous region encountered by the RQ-4 UAV when flying into the A330 MRTT wake vortex at an altitude of 10,500 m. Different contour colors correspond to various RMC levels: red indicates high-risk regions with RMC > 0.07, yellow corresponds to medium-risk regions with RMC = 0.05–0.07, green denotes low-risk regions with RMC = 0.03–0.05, and blue represents safe regions with RMC < 0.03. According to the safety criterion given in Equation (15), an RMC exceeding 0.07 signifies a severe risk of roll instability. In the initial stage, the left and right vortex cores each form a distinct high-risk center, with hazardous zones concentrated within approximately ±10 m of the cores where the induced velocity reaches its maximum (Equation (10)), resulting in significant lift asymmetry and roll moments exceeding the safety threshold. As time progresses, although the circulation decays during 0–92.7 s, the increasing vortex-core radius (Equation (7)) causes the medium- and low-risk regions to expand laterally, enlarging the overall affected area. After 92.7 s, the wake enters a rapid-decay phase, during which the circulation decreases sharply and the high-risk regions vanish accordingly. Because the RMC is proportional to circulation (Equation (14)), the hazard level decreases simultaneously. Meanwhile, the center of the hazardous region shifts downward following the descent of the vortex core, consistent with the vertical evolution predicted by Equation (2). The irregular distribution of the rolling-moment coefficient (RMC) is mainly caused by the non-uniform evolution of the wake vortex velocity field and the characteristics of the RMC calculation. The wake vortex flow field itself is highly non-uniform. Near the vortex core, the velocity gradient is large. As the UAV penetrates the wake region, the vortex spacing increases and the circulation decays, which reduces the vortex strength. At the same time, the vortex core expands to a larger area, leading to significant variations in induced velocity at different spatial locations.

This non-uniformity causes the iso-risk boundaries to appear irregular in shape. In particular, near the vortex core, the rapid change in induced velocity results in strong local variations in the risk envelope, producing noticeable fluctuations and irregular contours. When the vortex spacing increases and the circulation decays, the overall wake intensity decreases. However, the local risk boundary is still influenced by the velocity gradient distribution. As a result, the RMC field continues to exhibit irregular boundaries.

Using a time step of 1 s, the three-dimensional distribution of the hazardous region is obtained by applying an interpolation-based smoothing method, as shown in Figure 8, which presents the detailed three-dimensional distribution of the hazardous region.

The color variations in the risk-field visualization are introduced solely to illustrate the spatial distribution of the rolling moment coefficient (RMC) and to facilitate intuitive interpretation of the wake vortex risk. In the path-planning algorithm, however, a single predefined RMC threshold is adopted as the safety constraint. The color intervals, therefore, do not represent multiple constraint boundaries.

3.4. Experimental Results of the Path Planning Algorithm

To strengthen the comparative analysis and clarify the technical contribution of the proposed method, this study evaluates its performance against representative path planning algorithms that are widely used in UAV applications. Graph-based methods, such as A star, and sampling-based methods, such as RRT, have been extensively adopted in autonomous flight and aerial refueling studies. Therefore, they provide suitable baseline approaches for performance comparison.

To partially account for the dynamic characteristics of the UAV control system, a strict constraint on the maximum turning angle is imposed during the simulation. This constraint reflects the maneuverability limits and response capability of the onboard flight control system.

Using the same five initial points, the paths generated by different planning algorithms are compared to demonstrate their respective performance. In the experiments, five identical starting points (marked by green circles) are selected, and each algorithm plans a safe trajectory to the target location (marked by a red circle). The z-axis coordinate represents the relative height in meters (m), with the initial vortex core height specifically defined as 0 m in this study. The x-axis coordinate denotes the horizontal distance in meters (m), calculated by multiplying the leading aircraft’s velocity by the wake vortex evolution time, with the origin set at the initial vortex generation point. The y-axis coordinate indicates the lateral distance in meters (m), with the origin positioned at the midpoint between the two vortex cores.

Compared with the A* and RRT algorithms, the improved PSO algorithm demonstrates superior path quality and planning efficiency in wake vortex avoidance. As shown in Figure 9, the A* algorithm employs a grid-based search strategy that guarantees feasible paths but incurs high computational cost and produces noticeable “zigzag” trajectories due to spatial discretization, requiring additional smoothing. As shown in Figure 10, the RRT algorithm generates more natural, curved paths through random sampling; however, its solutions exhibit substantial randomness, local sharp turns, and limited optimality. For example, the red trajectory generated by the RRT algorithm shows many unnecessary avoidance maneuvers, which are mainly related to its algorithmic mechanism. RRT is an incremental search method based on random sampling, and its primary objective is to rapidly construct a feasible path rather than to ensure global optimality or smoothness. Therefore, in a continuously distributed wake vortex risk field, the algorithm may produce repeated local detours due to randomly expanded directions. In addition, RRT mainly relies on local feasibility checks during tree expansion and lacks global guidance from the risk-field gradient, which can lead to frequent direction adjustments near the hazard boundary. Furthermore, a maximum turning-angle constraint is imposed on the UAV in this study, which further increases the polyline characteristics of the generated path. In contrast, the improved PSO algorithm leverages continuous-space optimization and smoothness constraints to produce highly smooth paths that fully satisfy the UAV’s dynamic constraints. As shown in Figure 11, it enables the trajectory to closely follow the boundary of the hazardous region while maintaining safety, thereby reducing unnecessary detours and shortening the overall flight distance.

It should be noted that flying along the geometric midpoint between the two wake vortices may approximate a low-risk region only under ideal symmetric and steady conditions. In realistic environments, however, wake vortices undergo descent, lateral transport, and asymmetric decay, leading to a spatiotemporally non-uniform risk distribution. Under different altitudes and time separations, the midpoint is not always the safest region.

3.5. Comparison of Loss Function Results

To objectively evaluate the performance differences among the three path planning algorithms, this section conducts a quantitative comparative analysis in terms of hazard-loss metrics, path length, and computational efficiency. In the experimental setup, 50 test points are uniformly selected on the wake vortex field plane at t = 250 s and used as the initial positions, while the target point is uniformly set to the refueling location. For each initial point, path planning is performed using the PSO, A*, and RRT algorithms, respectively.

Based on the 50 path planning comparison trials and an extended set of 200 stability tests, the improved PSO algorithm demonstrates significant advantages in hazard loss, path length, path smoothness, computational efficiency, and planning success rate. The performance metrics of different path planning algorithms are summarized in

Table 3. Performance comparison of different path-planning algorithms (results are presented as mean ± standard deviation).

Algorithm	Hazard Loss (−)	Path Length (km)	Computation Time (s)	Max Turning Angle (deg)
A*	0.1452 ± 0.0246	7.448 ± 0.4022	0.127 ± 0.054	26.41 ± 8.7
RRT	0.0711 ± 0.0193	5.945 ± 0.988	0.0257 ± 0.038	22.73 ± 21.8
IPSO	0.0126 ± 0.0043	5.297 ± 0.087	0.0278 ± 0.005	4.72 ± 4.5

. As shown in Figure 12, The hazard loss, defined as the cumulative RMC values along the path, indicates that PSO consistently maintains the risk level below 0.02, with an average of only 0.015 and a standard deviation of 0.003, indicating excellent risk control and stability. In contrast, the hazard loss of the A* algorithm fluctuates substantially between 0.12 and 0.18, while that of RRT averages around 0.06 due to randomness in sampling. In terms of path length, the PSO-generated trajectories remain stable at approximately 5.3 km, increasing by only about 8% relative to the straight-line distance; the A* paths are noticeably longer at around 7.4 km, and the RRT paths vary widely between 5 and 6.8 km, In Figure 13, the fact that the path length becomes stable over time does not mean that the UAV stops during flight. It should be clarified that the path length in the figure refers to the total length of the optimal path obtained by the optimization algorithm in each planning cycle, rather than the real-time cumulative flight distance of the UAV. In the simulation, the planning start points differ only in lateral distance and altitude, while the horizontal distance remains the same. Since the horizontal distance is much larger than the lateral distance and altitude, after the algorithm converges, the path length tends to remain stable around an optimal value. As shown in Figure 14, With respect to computational efficiency, the improved PSO achieves an average planning time of just 0.052 s with a standard deviation of 0.008 s, far faster and more predictable than A* (0.20 s) and RRT (0.13 s), and offering better scalability. As shown in Figure 15, Regarding path smoothness, PSO achieves an average turning angle of only 6.8°, inherently satisfying UAV dynamic constraints without requiring post-processing; in contrast, A* produces jagged trajectories with an average turning angle of 30.2°, while RRT yields moderately smooth but unstable paths with an average of about 17.5°. As shown in

Table 4. Comparison of Planning Failure Counts for the Three Algorithms.

Number of Path Planning Trials	A* Algorithm	RRT Algorithm 2	IPSO Algorithm
50	0	0	0
100	2	2	0
150	2	4	0
200	4	4	0

, in 200 path planning experiments, the improved PSO algorithm successfully found feasible paths in all cases, resulting in zero failures and a 100% success rate, whereas the reliability of A* and RRT degrades significantly as the number of experiments increases. Overall, the improved PSO algorithm exhibits superior global search capability, environmental adaptability, and task reliability in complex dynamic environments, enabling stable, rapid, and safe path planning.

4. Discussion

The proposed framework integrates high-fidelity wake vortex modeling, quantitative risk field reconstruction, and intelligent optimization-based path planning to address the autonomous aerial refueling ingress problem of an RQ-4 UAV in the wake vortex environment generated by an A330MRTT tanker. Traditional UAV path planning methods mainly consider geometric obstacles or simplified threat regions and typically optimize distance or time. In contrast, the present study explicitly incorporates the three-dimensional and time-varying wake vortex hazard into the planning process through a rolling moment coefficient-based risk metric. This approach shifts the perspective from treating wake vortices as implicit or bounded disturbances to using a physically grounded risk field as a primary basis for trajectory design.

Previous studies on aerial refueling and wake vortices have largely focused on predicting wake vortex evolution or assessing encounter risk under specific traffic configurations. Other studies have concentrated on designing refueling envelopes and control strategies under assumed safety margins. These approaches are mainly evaluation-oriented or constraint-oriented. They do not directly convert wake vortex information into constructive guidance for path generation. In contrast, this study couples the Hallock Burnham P2P wake model with a rolling moment coefficient-based hazard metric to construct a continuous spatiotemporal wake vortex risk field. The study then embeds this field into a multi-objective cost function within an improved particle swarm optimization planner. Simulation results show up to 90 percent and 75 percent reductions in cumulative risk exposure compared with A star and RRT, respectively. The path length increases by about 8 percent, and the maximum turning angle remains below 10 degrees. These results support the hypothesis that risk field-aware optimization can significantly enhance safety while maintaining path feasibility and smoothness.

The proposed wake vortex aware particle swarm optimization uses adaptive learning factors and a nonlinearly decreasing inertia weight. This design helps balance safety, feasibility, and smoothness in the complex cost landscape created by the wake vortex risk field. Graph-based planners such as A star and RRT can find shortest or collision-free paths. However, they do not have a mechanism to account for graded and spatially distributed hazards. As a result, they may guide the UAV through regions with high induced rolling moments. In contrast, the proposed method continuously steers candidate trajectories away from hazardous areas. The method keeps the peak rolling moment below the required safety threshold and ensures that the generated paths remain dynamically feasible for the aerial refueling maneuver.

Beyond the specific A330MRTT and RQ-4 scenario, the results indicate that transforming wake vortex disturbances into a quantitative and visualizable risk map provides a transferable approach for wake vortex avoidance in cooperative airspace operations. The same framework can be adapted to other tanker and receiver combinations. It can also support applications such as formation flight, rapid wake vortex identification, and risk prediction in complex airspace. At the same time, several limitations should be acknowledged. The current risk field relies on deterministic wake models and nominal atmospheric conditions. The simulations also assume accurate tracking of the planned path. Stochastic atmospheric effects, modeling uncertainties, and control errors are not fully considered. In addition, validation is limited to simulation, and broader benchmarking under different aircraft types and environmental conditions is still needed.

Future research will focus on developing real-time wake vortex risk field updating methods based on onboard sensor data. The study will also integrate the proposed planning algorithm with high-fidelity flight dynamics and closed-loop control systems. Extending the framework to multi-UAV and multi-tanker scenarios and combining it with intelligent air traffic management concepts are promising directions for enabling safer and more efficient cooperative airspace operations.

In practical implementations, the wake vortex risk field cannot be assumed to be perfectly known. Sensor noise, atmospheric disturbances, and onboard estimation errors may introduce uncertainty into the reconstructed rolling moment coefficient distribution. Therefore, incorporating sensor uncertainty modeling and closed-loop state estimation mechanisms into the framework will be essential for enhancing robustness under imperfect information conditions. The integration of onboard flow sensing, wake detection systems, and real-time filtering techniques would enable adaptive risk field updating and improve operational reliability.

In addition, for HALE (High Altitude Long Endurance) UAV platforms with large aspect ratios and flexible wings, wake-induced rolling moments may couple with aeroelastic effects and structural load constraints. Future extensions should therefore incorporate aeroelastic considerations and structural safety limits into the planning framework, so as to better reflect the operational characteristics of high altitude long endurance vehicles.

Furthermore, in multi-UAV or multi-tanker cooperative scenarios, the interaction and superposition of multiple wake systems may generate highly nonlinear and time-varying risk distributions. Extending the present single-wake framework to multi-agent environments will be an important step toward supporting large-scale cooperative aerial operations and intelligent airspace management.