Prediction of the Drogue Position in Autonomous Aerial Refueling Based on a Physics-Informed Neural Network

Abstract

Autonomous aerial refueling (AAR) technology is of crucial importance in the aviation field. Accurately predicting the position of the refueling drogue is a core challenge in implementing this technology. An innovative method of a physics-informed neural network (PINN), a fusion of supervised learning and unsupervised learning, integrating physical information with an attention-augmented long short-term memory (AALSTM) neural network is proposed. By constructing a physical model of the refueling drogue, accurate physical constraints are provided for the prediction model. Meanwhile, an AALSTM neural network architecture is designed to predict partial states of the refueling drogue and parameters of the dynamic model. An attention-augmented mechanism is introduced to enhance the ability to capture key information. Simulation experiments verify that introducing an attention-augmented mechanism based on the conventional LSTM can improve prediction accuracy. The PINN significantly outperforms the conventional LSTM method in prediction accuracy, providing strong support for the development of AAR technology.

1. Introduction

In the modern aviation field, aerial refueling technology, as a key factor in enhancing the combat effectiveness of aircraft, is receiving widespread attention [1]. In military scenarios, with the continuous evolution of the warfare, higher requirements are placed on the long-range, persistent and rapid response capabilities of combat aircraft. Aerial refueling technology enables fighter jets to continuously perform missions far from their bases, greatly enhancing the flexibility and strategic deterrence of military operations [2,3]. Given its simplicity, flexibility, and versatility in structure, the hose and drogue aerial refueling method remains one of the main approaches. With the development of aerial refueling technology and the ever increasing requirements of modern flight missions for this technology, there is an urgent need for autonomy in aerial refueling to expand its application scope and ensure its high-precision, high-safety, and high-efficiency implementation [4]. For manned aircraft, autonomous aerial refueling (AAR) technology aims to reduce the risks and complexity of the pilot’s operation and improve the efficiency and safety of refueling docking; for unmanned aerial vehicles, it has become an essential technology for enhancing the combat radius and endurance [5,6].

However, achieving AAR faces numerous challenges. Among them, the accurate prediction of the drogue position is the core challenge for safe and efficient refueling [7]. During the refueling process, the drogue is affected by complex aerodynamics, the relative motion between the tanker and the receiver aircraft, and environmental factors, resulting in a highly uncertain and dynamically changing position [8,9]. Turbulence, gusts, etc., in the air can cause irregular swings and displacements of the drogue, and changes in the speed, acceleration, and attitude of the tanker and receiver aircraft will further exacerbate the complexity of the drogue position [10,11]. These factors make it difficult for the receiver aircraft to accurately predict the future position of the drogue, posing great difficulties for the refueling operation. If the drogue position cannot be accurately predicted, deviations may occur during the docking process between the receiver aircraft and the drogue, and even collision accidents may happen, seriously threatening flight safety [12].

A large number of studies have been carried out on position prediction methods [13,14]. Conventional drogue position prediction methods are mainly based on empirical models or simple physical models [15,16]. These methods may have some accuracy in relatively stable environments, but their limitations gradually emerge in actual complex aerial refueling scenarios. Empirical models often rely on a large amount of historical data and specific flight conditions and lack adaptability to complex dynamic environments; simple physical models have difficulty fully considering the interactions of various complex factors, resulting in large deviations between prediction results and actual situations.

In recent years, with the rapid development of artificial intelligence and machine learning technologies, using data-driven methods for drogue position prediction has become a new research direction [17,18]. These methods can mine the hidden patterns and features in data by learning and analyzing a large amount of flight data, thus establishing more accurate prediction models. However, pure data-driven methods often ignore the constraints of physical principles, which may lead to physically unreasonable prediction results. Moreover, the generalization ability of the models may be affected when data are limited [19].

The emergence of physics-informed neural networks (PINNs) provides a new idea for solving the above problems [20]. PINNs combine physical knowledge with neural networks. While leveraging the powerful nonlinear fitting ability of neural networks, physical equations are introduced as constraint conditions, causing the prediction results of the model to conform to both data characteristics and physical laws [21,22]. This method has unique advantages in dealing with problems with complex physical processes and dynamic changes, and is expected to provide a more accurate and reliable solution for the prediction of the AAR drogue position.

This paper aims to conduct in-depth research on the prediction method of the AAR drogue position based on PINNs. First, a detailed analysis of the motion characteristics and influencing factors of the drogue during the AAR process is carried out. Second, a neural network model integrating physical information is constructed, and the model is trained and optimized using simulated data. Finally, the effectiveness and accuracy of the model are verified through experiments, providing theoretical support and technical guarantee for the development of AAR technology.

The main contributions of this paper are as follows. For the first time, a prediction model that deeply integrates physical information with an attention-augmented long short-term memory (AALSTM) is proposed. It makes full use of the interpretability of physical models and the powerful nonlinear fitting ability of data-driven models, significantly improving the prediction accuracy and stability of the drogue position.

A comprehensive and detailed physical model of the refueling drogue is established. The calculation formulas for gravity, aerodynamic force, and hose tension, as well as the decomposition formulas of each force on the coordinate axes, are presented. The ordinary differential equation of Newton’s second law is established based on the inertial system, making the dynamic model more in line with actual working conditions and providing accurate physical constraints for model integration.

The PINN is designed and comprehensively verified through simulation experiments. The parameter values of the dynamic equations are presented. The data collection and processing methods for AALSTM are detailed. The design of each dimension of the neural network, the division of training data, and the model training method are also discussed. A comparison with conventional methods is made to demonstrate the advantages of the method proposed in this paper in key indicators, providing a strong basis for practical applications.

The rest of the paper is organized as follows. The related work is elaborated in Section 2. The physical model is introduced in Section 3. The design process of PINN based on the AALSTM is presented in detail in Section 4. The simulation test examples and the result analysis are provided in Section 5. Finally, the conclusions are summarized in Section 6.

2. Related Work

In the field of drogue position prediction, previous research has achieved certain results, but there is still room for improvement. Refs. [23,24] conducted an in-depth analysis of the force conditions of the drogue under different flight conditions based on a conventional physical model and predicted the position by establishing accurate kinematic equations. In this study, the analysis of various forces is relatively detailed. For example, corresponding calculation formulas are given for gravity, aerodynamic force, and hose tension. However, the actual aerial refueling environment is extremely complex, and there are many interference factors that are difficult to model accurately, such as unstable airflows and random changes in the atmospheric environment. These factors limit the prediction accuracy of this method in practical applications.

Refs. [25,26,27] used an conventional neural network for drogue position prediction, making full use of the powerful nonlinear fitting ability of the neural network to capture some features in the data. During the data processing stage, the model can model complex nonlinear relationships to a certain extent through learning a large amount of historical data. However, due to the lack of in-depth integration of physical laws, the model cannot reasonably constrain and correct the prediction results according to physical principles when facing complex working conditions, resulting in poor prediction performance. For example, when encountering special flight attitudes or extreme weather conditions, the prediction results often deviate from the actual values.

In the research of AALSTM, refs. [28,29,30] proposed an innovative model that combines an attention mechanism with LSTM for natural language processing tasks. By dynamically adjusting the weights of inputs at different time steps, this method enables the model to focus more on key information, thus effectively improving the accuracy of text classification and sentiment analysis. In text classification tasks, through the attention mechanism, the model can quickly locate keywords and key sentences related to the text category, improving the classification accuracy. This idea provides important reference for applying AALSTM in the prediction of the AAR drogue position. In time-series data processing, the attention mechanism can help the model better capture important features of the data at different time points, which has potential application value for drogue position prediction tasks that need to handle complex time-series data.

In the application of PINNs, refs. [31,32] used PINNs to solve nonlinear dynamic problems. By ingeniously using the constraint conditions of physical models to improve neural network training, they significantly enhanced the generalization ability and prediction accuracy of the model. When solving nonlinear dynamic problems, PINNs integrate conservation laws, boundary conditions, etc. Incorporatingphysical models as constraints into the neural network training process enables the model to not only learn data features but also to follow physical laws.

In the AAR scenario, integrating physical information into neural networks is expected to solve the problem of the lack of physical basis in data-driven methods. Through the constraints of the physical model, the model can reasonably predict the motion state of the drogue according to physical principles under different flight conditions, thereby improving the reliability of drogue position prediction.

Existing prediction methods for the AAR drogue position have their own advantages and disadvantages. The methods based on physical models have a theoretical basis but lack adaptability to complex environments; data-driven methods rely on a large amount of high-quality data and have limited generalization ability. PINNs provide a new direction for solving these problems, but further exploration and improvement are still needed. Therefore, in-depth research on the prediction method of the AAR drogue position based on PINNs has theoretical and practical significance.

3. Physical Modeling

3.1. Modeling Assumptions

Assumption 1. 

Assume that the tanker maintains a uniform linear flight state during the refueling process, and its flight speed and attitude do not change significantly in a short period. This is because the stable flight of the tanker is crucial for maintaining the stable state of the refueling hose and drogue during the AAR process.

Assumption 2. 

Consider the refueling hose as an ideal elastic body with a constant elastic modulus that does not change with the degree of stretching and time.

Assumption 3. 

Assume that the change in the drogue’s attitude is small; the speed change of the drogue caused by disturbances is negligible compared to the speed of the tanker. Therefore, the change in the flow angle is small, and the angle of attack of the drogue is approximately equal to the pitch angle of the drogue.

3.2. Establishment of Coordinate Systems

A schematic diagram of the tanker, receiver, and hose–drogue is introduced in Figure 1 below to better understand their positional relationship.

Geographic coordinate system $\{O-xyz\}$: A coordinate system fixed on the Earth is selected as the global reference frame, based on which the absolute positions and velocities of the tanker and the receiver aircraft can be described.

Refueling pod coordinate system $\{O_t-x_t{y}_t{z}_t\}$: Its origin is located at the connection point of the refueling hose and the pod. The $x_t$-axis points in the flight direction of the aircraft, the $z_t$-axis is vertically downward, and the direction of the $y_t$-axis is determined according to the right-hand rule, pointing towards the wingspan direction. According to the aforementioned assumptions, this coordinate system is also an inertial coordinate system. The position of the drogue is described based on this coordinate system.

Receiver-fixed coordinate system $\{O_r-x_r{y}_r{z}_r\}$: Its origin is fixed at the centroid $O_r$ of the receiver aircraft. The $x_r$-axis points towards the nose of the aircraft, the $y_r$-axis points towards the wingspan direction, and the $z_r$-axis is vertically downward. The relative position between the receiver aircraft and the drogue is described based on this coordinate system.

Drogue-fixed coordinate system $\{O_c-x_c{y}_c{z}_c\}$: The origin of the coordinate system is at the centroid $O_c$ of the drogue. The $x_c$-axis points in the direction of the connection node between the drogue and the hose, the $z_c$-axis is vertically downward, and the $y_c$-axis direction satisfies the right-hand rule. The hose tension acting on the drogue is analyzed in this coordinate system.

Equilibrium towing coordinate system of the drogue $\{O_c-x_b{y}_b{z}_b\}$: The origin of the coordinate system is at the centroid $O_c$ of the drogue. The $x_b$-axis points in the direction of the flight velocity, the $z_b$-axis is vertically downward, and the $y_b$-axis direction satisfies the right-hand rule. According to assumption 3, this coordinate system is also an inertial coordinate system. The aerodynamic force and gravity acting on the drogue are analyzed in this coordinate system.

3.2.1. Force Analysis of the Drogue

During its motion, the drogue is mainly subjected to the action of gravity $\overrightarrow{G}$, aerodynamic force ${\overrightarrow{F}}_a$, and hose tension ${\overrightarrow{F}}_t$.

Given the pitch angle $\theta$ and yaw angle $\psi$ of the drogue, the rotation matrix R for the transformation from the drogue-fixed coordinate system to the drogue equilibrium towing coordinate system is expressed as follows.

$$R=\left[\begin{array}{ccc}cos\theta cos\psi & sin\theta cos\psi & -sin\psi \\ cos\theta sin\psi & sin\theta sin\psi & cos\psi \\ -sin\theta & cos\theta & 0\end{array}\right]$$

(1)

In the equilibrium towing coordinate system, the gravity of drogue is formulated by

$$\overrightarrow{G}={[0,0,-mg]}^T$$

(2)

where m is the mass of the drogue and g is the acceleration due to gravity.

In the drogue-fixed coordinate system, the aerodynamic force is presented as

$${\overrightarrow{F}}_a={[-F_D,F_Y,-F_L]}^T$$

(3)

$${\overrightarrow{F}}_{a_{inertial}}=R·{\overrightarrow{F}}_a$$

(4)

where the aerodynamic lift $F_L$, drag $F_D$, and side force $F_Y$ are, respectively, defined as

$$F_L={\displaystyle \frac{1}{2}}\rho v^{2}S{C}_L$$

(5)

$$F_D={\displaystyle \frac{1}{2}}\rho v^{2}S{C}_D$$

(6)

$$F_Y={\displaystyle \frac{1}{2}}\rho v^{2}S{C}_Y$$

(7)

where $\rho$ is the air density; v is the airspeed of the drogue; S is the reference area of the drogue; $C_L$ is the lift coefficient; $C_D$ is the drag coefficient; and $C_Y$ is the side force coefficient.

The tanker exerts a tension force ${\overrightarrow{F}}_t$ on the drogue through the hose. This tension is the key force for maintaining the stable flight of the drogue and controlling its position. The direction of the tension is along the axial direction of the hose. Let the original length of the hose be $l_0$, the elastic coefficient be k, and the current elongation be $\Delta l$. Then, the tension ${\overrightarrow{F}}_t$ exerted by the hose on the drogue can be calculated by Hooke’s law as follows:

$${\overrightarrow{F}}_t=k·\Delta l·{\overrightarrow{e}}_c$$

(8)

where ${\overrightarrow{e}}_c$ is the unit vector in the axial direction of the end of the refueling hose, which points in the direction of the $x_c$-axis of the drogue-fixed coordinate system. The unit vector is ${\overrightarrow{e}}_c={[1,0,0]}^T$.

The component ${\overrightarrow{F}}_{t_{inertial}}$ of the hose tension in the drogue equilibrium towing coordinate system is expressed as follows:

$${\overrightarrow{F}}_{t_{inertial}}=R·{\overrightarrow{F}}_t$$

(9)

3.2.2. Drogue Dynamic Equations

Based on the inertial system, the drogue dynamic equations are established according to Newton’s second law $\overrightarrow{F}=m\overrightarrow{a}$. The resultant external force ${\overrightarrow{F}}_{total}$ is the vector sum of the gravitational force $\overrightarrow{G}$, the aerodynamic force ${\overrightarrow{F}}_{a_{inertial}}$, and the hose tension ${\overrightarrow{F}}_{t_{inertial}}$. The acceleration $\overrightarrow{a}$ is the second-order derivative of the position vector $\overrightarrow{r}$ with respect to time, expressed as follows:

$$\left\{\begin{array}{c}m{\displaystyle \frac{d^2\overrightarrow{r}}{d{t}^2}}=\overrightarrow{G}+{\overrightarrow{F}}_{a_{inertial}}+{\overrightarrow{F}}_{t_{inertial}}\hfill \\ \overrightarrow{r}\left(0\right)={\overrightarrow{r}}_0\hfill \end{array}\right.$$

(10)

Decompose the differential equation along the three coordinate axes as follows:

$$\left\{\begin{array}{c}m{\displaystyle \frac{d^{2}x}{d{t}^2}}=F_{a_{inertial}x}+F_{t_{inertial}x}\hfill \\ m{\displaystyle \frac{d^{2}y}{d{t}^2}}=F_{a_{inertial}y}+F_{t_{inertial}y}\hfill \\ m{\displaystyle \frac{d^{2}z}{d{t}^2}}=-mg+F_{a_{inertial}z}+F_{t_{inertial}z}\hfill \\ x\left(0\right)=x_0\hfill \\ y\left(0\right)=y_0\hfill \\ z\left(0\right)=z_0\hfill \end{array}\right.$$

(11)

where ${\overrightarrow{r}}_0={[x_0,y_0,z_0]}^T$ represents the initial position at the moment of AAR docking.

4. Design of PINN Based on Attention-Augmented LSTM

4.1. Architecture of PINN

The PINN is a neural network framework that combines deep learning and physical modeling. On the basis of conventional neural networks, it integrates physical laws and prior knowledge, enabling the network’s training and prediction processes to better follow physical laws, thereby improving the model’s generalization ability and prediction accuracy. In the research on the position prediction of the drogue in AAR, this architecture shows unique advantages and adaptability.

The PINN consists of three parts: a data-driven neural network, physical information constraints, and a loss function.

The data-driven neural network usually adopts the structure of a multi-layer perceptron (MLP), which consists of an input layer, several hidden layers, and an output layer. In the scenario of predicting the position of the drogue in AAR, the input layer receives the raw data related to the refueling process. The hidden layers perform feature extraction and representation learning on the input data through a series of nonlinear transformations, mining the complex relationships among these data. The output layer outputs the prediction results, providing an important reference for the aerial refueling process. The neurons in each layer are connected through weights and biases, and these parameters are continuously adjusted during the training process to minimize the loss function.

Physical information constraints are the core part of the PINN, which integrates prior knowledge such as physical laws, conservation laws, and boundary conditions into the network. In this study, the physical model established in Section 3 is used as the physical information constraint and integrated into the network.

The loss function of the PINN consists of two parts: data fitting loss and physical information loss. The data fitting loss measures the difference between the network’s prediction results and the observed data, and usually uses common loss function forms such as mean-squared error (MSE). In the design of the data fitting loss in this study, considering that the output signal contains multiple dimensions, such as the drogue’s position, pitch angle, yaw angle, and tanker’s speed, the mean-squared error is used to calculate the error of each dimension, and the errors of all dimensions are accumulated. This can quantify the prediction error of each dimension and prompt the model to fit the actual data as accurately as possible in each dimension. The physical information loss measures the deviation between the network’s prediction results and the physical information constraints. In this study, the physical constraint loss is designed based on the drogue dynamic equations established according to Newton’s second law. The physical information loss is determined by calculating the sum of the squared residuals between the resultant external force and acceleration obtained from the model’s prediction information at each time step and both sides of the physical model equation. This allows the model to continuously adjust its parameters during the training process to ensure the consistency between the prediction results and the physical model. These two parts of the loss are combined through weighted summation to obtain the final loss function L, which is used to balance the relative importance of the data fitting loss and the physical constraint loss. The value of the weight needs to be determined through experiments to find the optimal balance.

The PINN is a fusion of supervised learning and unsupervised learning. Its architecture is shown in Figure 2.

The present state of the physics-informed system is the three coordinate values of the drogue in the refueling pod coordinate system, $\overrightarrow{s_t}=[x,y,z]$, with auxiliary parameters being the drogue pitch angle $\theta$ and yaw angle $\psi$. The next state is the solution for the state ${\overrightarrow{s}}_{t+\Delta t}$ at time $t+\Delta t$ based on the present state and physical Equation (11). The sizes of the LSTM input layer, hidden layer, and output layer are denoted by n, h, and o, respectively.

4.2. Description of the Attention-Augmented LSTM

In the complex task of predicting the position of the drogue in AAR, the Attention-Augmented LSTM (AALSTM) is introduced. This network architecture skillfully combines the advantages of conventional LSTM with the unique functions of the attention mechanism, demonstrating excellent performance in processing time-series data, especially in adapting to the requirements of drogue position prediction. Compared with the conventional LSTM, the AALSTM algorithm has the following advantages.

Effective capture of long-term dependencies: LSTM effectively solves this problem by introducing key components such as memory cells, input gates, forget gates, and output gates. This advantage is inherited in AALSTM. In the scenario of AAR, the position change of the drogue is comprehensively affected by multiple factors, and its motion trajectory shows complex time-series characteristics. For example, factors such as the adjustment of the tanker’s flight attitude and the continuous action of the air current will affect the drogue’s position at different time scales. AALSTM can store and transmit this long-term influence information through its memory cells, thus accurately capturing the patterns of change in the drogue’s position over a long time span and providing strong support for accurate prediction.

Attention mechanism highlighting key information: The attention mechanism is a key feature of AALSTM. When processing time-series data related to the drogue’s position, the importance of data at different moments for predicting the current position of the drogue is not the same. For example, the data around the moment when the drogue is disturbed by a sudden air current have a higher reference value for understanding and predicting the subsequent position change of the drogue. The attention mechanism can automatically learn and assign different weights to data at different moments, thus highlighting the role of key information. By calculating the attention weights, AALSTM can focus more on the historical data that are most crucial for the current prediction, enhancing the model’s attention to important features and thus improving the prediction accuracy. This ability to accurately capture key information enables the model to quickly screen valuable information and make more reliable predictions when facing the complex and changeable aerial refueling environment.

Dynamic adaptation to complex environment changes: The environmental factors in the process of AAR are complex and changeable. For example, the instability of the air current and the change in the relative motion state between the tanker and the receiver aircraft will lead to dynamic adjustments in the change law of the drogue’s position. The attention mechanism of AALSTM has dynamic adaptability. It can continuously adjust the distribution of attention weights according to the real-time changes in the input data. When the environment changes, the model can quickly refocus on new key information and timely adapt to the changes in the change law of the drogue’s position. This dynamic adaptability enables AALSTM to maintain good prediction performance in different aerial refueling scenarios, enhancing the model’s robustness and generalization ability.

Overall, AALSTM can significantly improve the prediction accuracy in the prediction of the drogue’s position in AAR. By accurately capturing long-term dependencies, highlighting key information, and dynamically adapting to environmental changes, the model can more precisely grasp the change trend of the drogue’s position. At the same time, since the attention mechanism can focus on key information and reduce the processing of irrelevant data, it improves the model’s computational efficiency to a certain extent.

Below is a description of the AALSTM algorithm. Suppose there is an input sequence $\mathbf{X}=[{\mathbf{x}}_1,{\mathbf{x}}_2,\dots ,{\mathbf{x}}_N]$, where ${\mathbf{x}}_i$ represents the input vector at the i-th time step and N is the total number of time steps. Then, calculate the attention weights. First, calculate the score $e_i$ for each time step:

$$e_i={\mathbf{w}}^{T}tanh(\mathbf{W}{\mathbf{x}}_i+\mathbf{b})$$

(12)

where $\mathbf{W}$ is a weight matrix used to map the input vector ${\mathbf{x}}_i$ to a new feature space; $\mathbf{w}$ is a weight vector used to perform a weighted sum of the mapped features; and $\mathbf{b}$ is the bias vector. The attention weight ${\alpha }_i$ is calculated through the softmax function:

$${\alpha }_i={\displaystyle \frac{exp\left(e_i\right)}{{\sum }_{j=1}^{N}exp\left(e_j\right)}}$$

(13)

The attention weight ${\alpha }_i$ represents the relative importance of the input at the i-th time step in the entire input sequence, and ${\sum }_{i=1}^N{\alpha }_i=1$. Perform a weighted sum of the input sequence through the attention weights to obtain the input representation ${\mathbf{h}}_{att}$ with the attention mechanism:

$${\mathbf{h}}_{att}=\sum _{i=1}^N{\alpha }_i{\mathbf{x}}_i$$

Use the input representation ${\mathbf{h}}_{att}$ with the attention mechanism as the input of LSTM, and interact with the hidden state ${\mathbf{h}}_{t-1}$ and cell state ${\mathbf{c}}_{t-1}$ of the conventional LSTM. The description of LSTM is as follows:

Input gate $i_t$:

$$i_t=\sigma ({\mathbf{W}}_i{\mathbf{h}}_{t-1}+{\mathbf{U}}_i{\mathbf{h}}_{att}+{\mathbf{b}}_i)$$

(14)

Forget gate $f_t$:

$$f_t=\sigma ({\mathbf{W}}_f{\mathbf{h}}_{t-1}+{\mathbf{U}}_f{\mathbf{h}}_{att}+{\mathbf{b}}_f)$$

(15)

Output gate $o_t$:

$$o_t=\sigma ({\mathbf{W}}_o{\mathbf{h}}_{t-1}+{\mathbf{U}}_o{\mathbf{h}}_{att}+{\mathbf{b}}_o)$$

(16)

Candidate cell state ${\tilde{\mathbf{c}}}_t$:

$${\tilde{\mathbf{c}}}_t=tanh({\mathbf{W}}_c{\mathbf{h}}_{t-1}+{\mathbf{U}}_c{\mathbf{h}}_{att}+{\mathbf{b}}_c)$$

(17)

Update cell state ${\mathbf{c}}_t$:

$${\mathbf{c}}_t=f_t\odot {\mathbf{c}}_{t-1}+i_t\odot {\tilde{\mathbf{c}}}_t$$

(18)

Update hidden state ${\mathbf{h}}_t$:

$${\mathbf{h}}_t=o_t\odot tanh\left({\mathbf{c}}_t\right)$$

(19)

where $\sigma$ is the sigmoid activation function, ⊙ represents element-wise multiplication, and ${\mathbf{W}}_i,{\mathbf{U}}_i,{\mathbf{b}}_i,{\mathbf{W}}_f,{\mathbf{U}}_f,{\mathbf{b}}_f,{\mathbf{W}}_o,{\mathbf{U}}_o,{\mathbf{b}}_o,{\mathbf{W}}_c,{\mathbf{U}}_c,$ and ${\mathbf{b}}_c$ are the weight matrices and bias vectors of LSTM.

In the entire process, the attention mechanism provides more targeted input for LSTM, enabling LSTM to better process time-series data and improving the performance of the model.

4.3. Input and Output Signals of AALSTM and Sliding Window Design

Feature signals are defined as follows: (1) drogue position $(x,y,z)$, which is defined in the refueling pod coordinate system; (2) drogue pitch angle $\theta$ and yaw angle $\psi$; and (3) tanker speed $v_{tank}$.

Label signals, which are the real values of the model’s output signals corresponding to the inputs, are defined as follows: (1) predicted drogue position $(\widehat{x},\widehat{y},\widehat{z})$; (2) predicted pitch angle $\widehat{\theta }$ and yaw angle $\widehat{\psi }$; (3) and predicted tanker speed ${\widehat{v}}_{tank}$.

Other output signals of the model, which are used for physical information calculation and belong to the type of unsupervised learning, are defined as follows: (1) predicted elongation of the hose $\Delta \widehat{l}$, which reflects the degree of hose stretching and is related to relative motion; and (2) predicted aerodynamic coefficients of the drogue $({\widehat{C}}_D,{\widehat{C}}_Y,{\widehat{C}}_L)$, which are used to predict the aerodynamic forces of the refueling drogue.

The sliding window is designed as follows. The input signal is sampled at a time interval of $\Delta t$. The time-series is converted into the supervised learning form of “input sequence → output label”. Here, the input sequence represents the observations of the past $N_0$ time steps, and the output label represents the real values of the next N time steps.

In the scenario of drogue position prediction, the selection of sliding window parameters needs to be combined with physical characteristics and control decision-making requirements. The choice of the prediction sliding window size $N_0$ is constrained by the physical characteristics of the drogue, and the value of $N_0$ needs to cover the key cycles of drogue movement (such as airflow disturbance cycles and system response delays). According to historical data, the movement cycle of the drogue is generally 1 to 3 s, and the duration of historical data corresponding to $N_0$ is 2 to 4 times the movement cycle of the drogue, which can capture the typical swinging cycle. If $N_0$ is too small, key dynamics (such as airflow fluctuations) will be missed, leading to prediction oscillations; if $N_0$ is too large, historical noise will be introduced, reducing the model’s sensitivity to recent changes.

The prediction step size N needs to match the time scale of control decisions. A too large N may increase the learning difficulty of the model because it needs to capture dependency relationships over a longer time span; a too small N has no obvious effect on control decisions. The prediction results in this study are mainly used in the scenario of AAR docking control, and a 0.5 s advance prediction plays an important role in improving the success rate of AAR docking.

4.4. Loss Function Design

In the AAR drogue position prediction model based on AALSTM, the design of the loss function is crucial for the training and performance improvement of the model. It needs to comprehensively consider the differences between the predicted values and the true values, as well as the constraints of the physical model, to ensure that the model can not only accurately fit the data but also satisfy the physical laws.

4.4.1. Data Fitting Loss

The data fitting loss mainly measures the difference between the model’s predicted values and the true values. Considering that the output signals contain multiple dimensions, such as the drogue position, pitch angle, yaw angle, tanker speed, etc., the mean-squared error (MSE) is used to calculate the error of each dimension, and the errors of all dimensions are accumulated.

Suppose there are a total of M samples. For the k-th sample, the output signal has D dimensions (corresponding to various prediction parameters), the predicted value is ${\widehat{y}}_{k,d}$, and the real value is $y_{k,d}$ ($d=1,2,\dots ,D$). Then, the calculation formula for the data fitting loss $L_{data}$ is given by

$$L_{data}={\displaystyle \frac{1}{M}}\sum _{k=1}^M\sum _{d=1}^D{({\widehat{y}}_{k,d}-y_{k,d})}^2$$

(20)

This design can quantify the prediction error of each dimension and give higher weights to larger errors, prompting the model to fit the actual data as accurately as possible in each dimension.

4.4.2. Physical Constraint Loss

The physical constraint loss is used to ensure that the model’s prediction results conform to the physical model of the refueling drogue established previously. The dynamic equation of the drogue established based on Newton’s second law is the core of the physical model. Therefore, the physical constraint loss can be designed based on this equation.

At each time step, according to the information of the drogue’s position predicted by the model, combined with known physical parameters (such as mass, elastic coefficient, etc.), the resultant external force and acceleration are calculated.

The differential equation of Newton’s second law established based on the inertial system is $m{\displaystyle \frac{d^2\overrightarrow{r}}{d{t}^2}}=\overrightarrow{G}+{\overrightarrow{F}}_{a_{inertial}}+{\overrightarrow{F}}_{t_{inertial}}$. The differential equations expanded in the three axes are expressed as follows:

$$\left\{\begin{array}{c}m{\displaystyle \frac{d^{2}x}{d{t}^2}}=F_{a_{inertial}x}+F_{t_{inertial}x}\hfill \\ m{\displaystyle \frac{d^{2}y}{d{t}^2}}=F_{a_{inertial}y}+F_{t_{inertial}y}\hfill \\ m{\displaystyle \frac{d^{2}z}{d{t}^2}}=-mg+F_{a_{inertial}z}+F_{t_{inertial}z}\hfill \end{array}\right.$$

(21)

Taking the x-axis direction as an example, the residual $r_x$ is calculated as follows:

$$r_x=m{\displaystyle \frac{d^2{x}^{\prime }}{d{t}^2}}-(F_{a_{inertial}x}^{\prime }+F_{t_{inertial}x}^{\prime })$$

(22)

Similarly, the residuals $r_y$ and $r_z$ in the y-axis and z-axis directions can be obtained. The physical constraint loss $L_{physics}$ is given by

$$L_{physics}={\displaystyle \frac{1}{M}}\sum _{i=1}^M(r_{x,i}^2+r_{y,i}^2+r_{z,i}^2)$$

(23)

where M is the total number of time steps. In this way, the model continuously adjusts its parameters during the training process to ensure the consistency between the prediction results and the physical model.

4.4.3. Total Loss Function

The data fitting loss and the physical constraint loss are combined by weighted summation to obtain the total loss function L:

$$L=L_{data}+\lambda L_{physics}$$

(24)

where $\lambda$ is a hyperparameter used to balance the relative importance of the data fitting loss and the physical constraint loss. The value of $\lambda$ needs to be determined through experiments to find the optimal balance between data fitting and physical constraints. If the value of $\lambda$ is too large, the model may focus too much on physical constraints and ignore data fitting; if the value of $\lambda$ is too small, the role of physical constraints may not be obvious, and the model cannot fully utilize physical information to improve the prediction accuracy.

During the model training process, by minimizing the total loss function L, the model can fit the data while following physical laws, thereby improving the accuracy and stability of the AAR drogue position prediction.

5. Simulation Experiments and Result Analysis

5.1. Experimental Setup

5.1.1. Determination of Parameters

Prior to simulation experiments, the parameters in the refueling drogue dynamics equations and other critical calculations were configured as detailed in

Table 1. Simulation parameters.

Parameter	Value	Units
$v_{tank}$	197	m/s
m	30	kg
S	0.2918	m2
g	9.8	m/s2
k	625.7	N/m
$\rho$	0.909	kg/m3
$\Delta t$	0.01	s
$N_s$	30,000	Sample
$N_0$	800	Step
N	50	Step
$N_1$	6	/
$N_2$	6	/
n	9	/
h	128	/
o	13	/

.

5.1.2. Data Collection and Processing

Data Collection: Based on ref. [33], a hose–drogue dynamics model was established. A refueling simulation was carried out from the trail stage of the receiver aircraft (where the receiver aircraft is 21 m behind the drogue) to the successful docking stage to generate numerical simulation data. The data generated by the model were verified for effectiveness by referring to the actual flight test data from ref. [34]. By using this method, time-series data for PINN training were obtained, including the drogue position (the three coordinate values $(x,y,z)$ in the refueling pod coordinate system), attitude (pitch angle $\theta$ and yaw angle $\psi$), and velocity (the tanker velocity ${\overrightarrow{v}}_{tank}$). The projection of the drogue position on the $O_t-y_t{z}_t$ plane is shown in Figure 3.

Data Preprocessing: The collected data were cleaned to remove obviously incorrect and abnormal data points. Then, normalization processing was carried out. Data with different ranges, such as the drogue’s position, were uniformly mapped to the interval [0, 1] to accelerate the convergence speed of model training and improve training stability. For example, for the drogue position $(x,y,z)$, the formula was used for normalization as follows:

$$x_{norm}={\displaystyle \frac{x-x_{min}}{x_{max}-x_{min}}}$$

(25)

where $x_{min}$ and $x_{max}$ were the minimum and maximum values in the x dataset, respectively, and the other signals were processed in a similar way.

5.1.3. Sliding Window Design and Dataset Construction

A multi-variable and multi-step prediction method is adopted to predict the time-series of the refueling drogue state. The dimensions of the input and output data can be expressed as follows:

• Input: [$N_S$, $N_0$, $N_1$];
• Output: [$N_S$, N, $N_2$].

Here, $N_S$ is the number of samples, $N_0$ is the input time steps, N is the output time steps, $N_1$ is the number of input features, and $N_2$ is the number of output labels. The values of these parameters are listed in Table 1.

5.1.4. Data Division

The preprocessed data were divided into a training set, a validation set, and a test set in the ratio of 60%, 20%, and 20%. The training set was used to train the PINN model, enabling the model to learn the relationships between input signals (such as the drogue’s position, attitude, tanker speed, etc.) and output signals (such as the drogue’s position, attitude, tanker speed, hose elongation, aerodynamic coefficients). The validation set was used to adjust the hyperparameters of the model during the training process, such as the learning rate and the number of hidden layer units, to prevent the model from overfitting. The test set was used to finally evaluate the generalization ability and prediction accuracy of the model.

5.2. Model Training

5.2.1. Network Architecture Configuration

The AALSTM network consists of three AALSTM layers and one fully connected layer. The number of hidden units in the first AALSTM layer is set to 128, the second layer has 64 hidden units, and the third layer has 32 hidden units. Each AALSTM layer is equipped with an attention mechanism, which can dynamically allocate weights according to the importance of the input data to better capture the key information related to the output signals. In the fully connected layer, the number of neurons is determined according to the dimension of the output signal to accurately calculate the prediction results such as the drogue position, attitude, tanker speed, hose elongation, and aerodynamic coefficients.

5.2.2. Optimizer and Hyperparameter Selection

The Adam optimizer is selected to train the model. Its characteristic of adaptively adjusting the learning rate helps to improve the training efficiency and stability. The initial learning rate is set to 0.001 and is appropriately adjusted during the training process according to the loss of the validation set. The number of training iterations is set to 5000 to ensure that the model fully learns the data features.

In physics-informed neural networks, the weight parameter $\lambda$ for balancing the data fitting loss and physics constraint loss is a key hyperparameter. The selection of this parameter directly affects the model’s emphasis on data fitting and physical laws. Common weight assignment methods and adjustment strategies include fixed weight methods, dynamic weight methods, validation set performance-based selection, and automatic learning strategies. This study employs a fixed weight method, directly designating $\lambda$ as a constant, and determining its value through experience by starting with smaller values (e.g., 0.1, 1, 5, 10) and gradually increasing them. This approach is simple to implement and computationally efficient. Through simulation comparisons, a $\lambda$ value of 5 was found to meet the prediction requirements.

5.2.3. Training Process

During the training process, a batch of training data (containing 300 samples) is input into the model each time. The model performs forward propagation based on the input data to calculate the predicted values, which include the drogue position, attitude, speed, hose elongation, and aerodynamic coefficients. Then, the weighted sum of the data fitting loss and the physical constraint loss is calculated through the total loss function $L=L_{data}+\lambda L_{physics}$. Next, the backpropagation algorithm is used to update the parameters of the model to minimize the loss function. After a certain number of iterations, the performance of the model is evaluated on the validation set, and the hyperparameters are adjusted according to the validation results.

5.3. Result Analysis

5.3.1. Prediction Accuracy Evaluation

The trained model was evaluated using the test set. Root mean square error (RMSE), mean absolute error (MAE), and mean absolute percentage error (MAPE) were used as evaluation metrics to assess the prediction results of the drogue position, attitude, tanker’s speed, etc., respectively. These metrics indicate that the model has high accuracy in predicting the drogue position and attitude. Figure 4 shows the true values and predicted values of the position and attitude of the drogue, as well as the tanker’s speed based on PINN. The solid lines represent the true values, while the dashed lines represent the predicted values. The evaluation results are shown in

Table 2. Evaluation metrics for state prediction based on PINN.

Drogue State	RMSE (m or deg)	MAE (m or deg)	MAPE (%)
$\widehat{x}$	0.0045	0.0036	0.0158
$\widehat{y}$	0.1219	0.1002	285.5983
$\widehat{z}$	0.1035	0.0808	2.6312
$\widehat{\theta }$	0.1857	0.1445	148.1478
$\widehat{\psi }$	0.3081	0.2545	346.2719
${\widehat{v}}_{tank}$	0.0006	0.0005	0.0002

.

5.3.2. Comparison with Conventional Method

The method proposed in this paper, which uses an AALSTM-based PINN to predict the state of the refueling drogue, is compared with the prediction method based on a conventional LSTM. On the same test set, although the conventional LSTM method can capture certain data patterns, it lacks the integration of physical information. The comparison results show that the method proposed in this paper outperforms the conventional LSTM method in terms of the prediction accuracy of the drogue position. Figure 5 shows the true values and predicted values of the position and attitude of the drogue, as well as the tanker’s speed based on conventional LSTM. The solid lines represent the true values, while the dashed lines represent the predicted values. The accuracy evaluation results are shown in

Table 3. Evaluation metrics for state prediction based on conventional LSTM.

Drogue States	RMSE (m or deg)	MAE (m or deg)	MAPE (%)
$\widehat{x}$	0.0154	0.0126	0.0313
$\widehat{y}$	0.3080	0.2532	1.2633
$\widehat{z}$	0.4223	0.3518	11.1146
$\widehat{\theta }$	1.0204	8615	821.9141
$\widehat{\psi }$	0.6838	0.5548	909.1147
${\widehat{v}}_{tank}$	0.0010	0.0008	0.0004

.

5.3.3. Comparison with AALSTM Method

The method proposed in this paper, which uses an AALSTM to predict the state of the refueling drogue, is compared with the prediction method based on a conventional LSTM and PINN. The comparison results show that the AALSTM method proposed in this paper outperforms the conventional LSTM method, while the prediction results are inferior to the PINN method in terms of the prediction accuracy of the drogue position. Figure 6 shows the true values and predicted values of the position and attitude of the drogue, as well as the tanker’s speed based on conventional LSTM. The solid lines represent the true values, while the dashed lines represent the predicted values. The accuracy evaluation results are shown in

Table 4. Evaluation metrics for state prediction based on AALSTM.

Drogue States	RMSE (m or deg)	MAE (m or deg)	MAPE (%)
$\widehat{x}$	0.0073	0.0057	0.0142
$\widehat{y}$	0.2692	0.2207	1.0983
$\widehat{z}$	0.2090	0.1632	5.2690
$\widehat{\theta }$	0.5098	0.4244	393.4640
$\widehat{\psi }$	0.5885	0.4887	515.6178
${\widehat{v}}_{tank}$	0.0008	0.0006	0.0003

.

5.3.4. Model Stability Analysis

The stability of the model was assessed through multiple repeated trainings and evaluations on different test subsets. The results show that the performance of the model fluctuates slightly during different training processes, and various evaluation indicators are relatively stable. Moreover, when facing different flight conditions and data distributions, the prediction accuracy of the model for the drogue position, attitude, and aerodynamic coefficients can still remain at a certain level, indicating that the model has good stability and robustness. However, under some extreme working conditions, such as when encountering sudden strong air current interference, the prediction accuracy of the model for the aerodynamic coefficients will decline to a certain extent, which also reflects that the model has certain limitations in dealing with extreme situations.

6. Conclusions

6.1. Research Achievements

This paper successfully proposes an AAR drogue position prediction method, PINN, that integrates physical information with an AALSTM-based neural network. By constructing a physical model of the refueling drogue, accurate physical constraints are provided for the prediction model. The designed AALSTM-based neural network architecture effectively enhances the ability to capture key historical information. By comparing the prediction results of the drogue’s position, attitude, and the tanker’s speed, as well as through the comparison of three indicators—RMSE, MAE, and MAPE—it can be shown that introducing an attention-augmented mechanism based on the conventional LSTM can improve prediction accuracy. The PINN significantly outperforms the conventional LSTM method in prediction accuracy, providing strong theoretical and technical support for the practical application of AAR technology.

6.2. Research Limitations and Prospects

Although this study has achieved certain results, there are still some limitations. Under extremely complex meteorological conditions such as turbulence, the current physical model may not accurately describe the changes in aerodynamic forces, leading to a decrease in prediction accuracy. The establishment of the model uses some strong assumptions, and its adaptability to large-scale maneuvers of the tanker needs to be improved. Future research can be carried out in the following aspects: further optimizing the physical model to better adapt to extreme meteorological conditions and large-scale maneuvers of the tanker; and collecting more actual flight data, especially data under extreme working conditions, to enhance the generalization ability of the model.