A Trajectory Prediction Method for Reentry Glide Vehicles via Adaptive Cost Function

Abstract

This paper proposes a trajectory prediction method via the adaptive cost function to address the difficulties in inferring the attack intention and maneuver mode, as well as the accumulation of prediction error during the trajectory prediction of reentry glide vehicles. Firstly, the vehicle guidance task is divided into two distinct categories: conventional guidance and no-fly zone avoidance guidance. A task-matched time-varying parameter prediction model set is then constructed. Secondly, taking into account the maneuverability, guidance intent, and battlefield situation of the vehicle, an adaptive intent cost function adapted to the guidance task is proposed, which avoids the estimation failure problem caused by manually setting cost coefficients in traditional methods. Finally, long-term trajectory prediction of vehicles is achieved using Bayesian theory to infer the attack intent and parametric model with the maximum a posteriori probability. The results of the simulations demonstrate that the proposed prediction method is capable of accurately inferring the vehicle’s attack intention and parameter model, and of effectively reducing the accumulation of prediction errors and the time required for the algorithmic process compared to existing methods.

1. Introduction

The gliding of reentry glide vehicles (RGVs) in near space has the characteristics of fast flight speed and strong maneuverability, which is a hot research topic among military powers [1,2,3]. As a result of the weaponization of RGVs, their rapid strike characteristics pose new challenges to the defense’s early warning, detection, tracking, and interception [4,5,6]. Consequently, it is of paramount importance to investigate the trajectory prediction of the RGV.

In terms of prediction mechanisms, trajectory prediction techniques for reentry glide unmanned aerial vehicles are mainly divided into two categories of trajectory prediction methods: data analysis-based and model-based. The first method predicts the vehicle trajectory by analyzing the historical tracking data, which has the characteristics of high short-term prediction accuracy and simple method implementation. Li et al. [7] used the prediction idea of decomposition and then integration, achieving a high accuracy short-term prediction of the trajectory sequence by predicting the trend term, the period term, and the random term, respectively. Hu et al. [8] conducted a statistical analysis of the vehicle’s motion trajectory through Gaussian process regression, which demonstrated superior performance in predicting trajectories with similar data distributions. However, this approach lacked the support of a motion model and exhibited inadequate trajectory constraints. With the development of artificial intelligence [9,10,11], deep learning based on batch data has also been applied to the trajectory prediction field of RGVs. Yang et al. [12] implemented a sequence-to-sequence trajectory prediction method with the help of the attention mechanism and the long short-term memory (LSTM) structure, taking the vehicle’s state information and angle of attack as inputs. It should be pointed out that the aforementioned methods directly process the superficial information of the RGV and lack the deep model constraints, which makes it challenging to meet the demand for high precision medium- and long-term trajectory prediction.

In addition, the second method is classified into three types in terms of prediction mechanism: parameter estimation, mode recognition, and intention inference. Among these, parameter estimation encompasses the reentry glide unmanned aerial vehicle as a non-cooperative target, integrates the unknown information, such as target mass, force area, aerodynamic coefficients, etc., into one parameter, and realizes the iterative prediction of the vehicle trajectory with the help of the dynamics model [13,14]. In the absence of a priori information, determining suitable and easily predictable parameters is a crucial aspect of the method. Wang et al. [15] achieved trajectory prediction by estimating the lift-to-drag ratio of the vehicle. The prediction effect was found to be better when the lift-to-drag ratio varied linearly. Hu et al. [16] used a set of aerodynamic parameters that are approximately linear with the attitude of the vehicle, combined with a vector autoregressive (VAR) model, to achieve a relatively accurate description of the trajectory tracking segment, laying the foundation for medium- and long-term trajectory prediction of the RGVs. Li et al. [17] constructed a set of suitable control variables, which enabled the prediction of trajectories under different maneuver modes of the vehicle. Li et al. [18] combined the control parameters and aerodynamic parameters with the LSTM network, and the simulation demonstrated that the control parameters exhibited superior performance in the trajectory prediction of the RGV. The aforementioned method is applicable to the case of simple parameter change rules and is subject to significant limitations. However, during the reentry of the vehicle, a specific maneuver mode is employed to conduct a surprise defense or change the movement pattern in real time according to the pre-determined attack intention. Consequently, the parameter estimation method is no longer applicable. Based on this, numerous scholars have explored trajectory prediction methods based on the mode recognition and intention inference of the RGVs.

In terms of vehicle maneuver mode recognition, the construction of pre-built corresponding prediction models based on different maneuver modes, and then the matching of these models through recognition techniques, have been demonstrated to be effective methodologies [19,20]. Chen et al. [21] employed historical tracking data to ascertain the parameters of the autoregressive integrated moving average (ARIMA) model, which was then combined with the unscented Kalman filter (UKF) algorithm to achieve the prediction of the vehicle trajectory. Wei et al. [22] utilized a double the sinusoidal summation function to align the reentry vehicle trajectory with the periodic jump characteristics. Cheng et al. [23] generated a set of maneuver modes based on the definition of maneuver modes to train the SVM, thereby achieving mode recognition and the prediction of vehicle maneuver trajectories. Sun et al. [24] employed linear attenuation terms and amplitude attenuation sinusoidal terms to derive analytical expressions for variables, such as height and speed, for a longitudinal jump-gliding trajectory of the vehicle under simplified conditions. They then implemented the calculation of the trajectory prediction pipeline with the assistance of an LSTM network. In terms of vehicle intention inference, since the reentry guidance always possesses a certain purpose, the trajectory prediction accuracy can be further improved by introducing the attack intention to correct the prediction result of the maneuver mode recognition method [25,26]. Zhang et al. [27] derived a recursive formula for the maneuver mode and motion state by constructing an intention cost function to achieve trajectory prediction under the uncertainty of vehicle maneuver. In order to address the issue of abrupt changes in maneuvering modes, Hu et al. [28] proposed a solution based on reference [27], transforming the trajectory prediction problem into the probability of the target passing through the reachable area by utilizing the intention cost function and the Bayesian principle. The algorithm did not rely on the dynamic process of the vehicle and demonstrated high short-term prediction accuracy; however, it was time-consuming and, therefore, not conducive to online prediction. In a subsequent study, Li et al. [29] enhanced the intention cost function proposed in reference [28] and developed a set of longitudinal and lateral time-varying prediction models, thereby enabling the multi-model and multi-intention fusion trajectory prediction of the RGV. However, the cost coefficient needed to be set manually, and the algorithm was not universal. Xu et al. [30] established a relationship between the initial state of the vehicle, the control variables and the flight area through a deep neural network, which improved the computational efficiency of attack intention inference and trajectory prediction. However, the algorithm did not consider the effect of the existence of a no-fly zone on the trajectory of the vehicle.

In conclusion, the ability to rapidly and accurately infer the vehicle’s intended attack and maneuver mode is crucial for accurately predicting the trajectory of the RGV over an extended period of time. In order to achieve this objective, we propose a reentry glide unmanned vehicle trajectory prediction method via adaptive cost function. The method divides the target task into no-fly zone evasive guidance and conventional guidance and constructs a set of time-varying parameter prediction models for task matching and coefficient-adaptive intention cost functions. Then, the attack intention and parameter models are inferred based on Bayesian theory and maximum a posteriori probability, and accurate prediction of RGV trajectories is achieved. The main contributions are summarized as follows:

• Classify the guidance tasks of vehicles and provide judgment criteria, which solves the problem of method failure caused by traditional trajectory prediction methods not distinguishing between vehicle guidance tasks.
• The intention cost functions with adaptive cost coefficients have been proposed. Unlike the existing approaches in [27,28,29], which use fixed cost coefficients, the proposed intention cost function, tailored to guidance tasks, can comprehensively consider the vehicle’s maneuverability and battlefield situations, thereby enhancing the universal applicability of the cost function.
• Based on Bayesian theory, the maximum posterior probability attack intention and parameter model are inferred, effectively reducing error accumulation during the prediction process and significantly improving the algorithm’s runtime. This meets the defense side’s requirements for medium- to long-term high-precision trajectory prediction.

The remainder of this paper is organized as follows. In Section 2, the RGV motion model is presented. Predictive model sets for task matching are introduced in Section 3. In Section 4, the RGV trajectory prediction method is proposed. Numerical simulation and analysis in Section 5. The conclusions are provided in Section 6.

2. RGV Motion Model

With considering the Earth’s rotation, the three degree of freedom motion model for RGV in the velocity–turn–climb (VTC) coordinate system is shown in the following equation [1,13,16]:

$$\left\{\begin{array}{l}\dot{r}=v\sin \theta \\ \dot{\varphi }={\displaystyle \frac{v\cos \theta \sin \Psi }{r\cos \phi }}\\ \dot{\phi }={\displaystyle \frac{v\cos \theta \cos \Psi }{r}}\\ \dot{v}=-{\displaystyle \frac{D}{m}}-g\sin \theta -{\omega }_e^{2}r\left(\cos \phi \sin \phi \cos \Psi \cos \theta -{\cos }^2\phi \sin \theta \right)\\ \dot{\theta }={\displaystyle \frac{L\cos \sigma }{mv}}+{\displaystyle \frac{\cos \theta }{v}}\left({\displaystyle \frac{v^2}{r}}-g\right)-2{\omega }_e\cos \phi \sin \Psi +{\displaystyle \frac{{\omega }_e^{2}r}{v}}\left(\cos \phi \sin \phi \cos \Psi \sin \theta +{\cos }^2\phi \cos \theta \right)\\ \dot{\Psi }={\displaystyle \frac{L\sin \sigma }{mv\cos \theta }}+{\displaystyle \frac{v\cos \theta \sin \Psi \tan \phi }{r}}+{\omega }_e^{2}r{\displaystyle \frac{\cos \phi \sin \phi \sin \Psi }{v\cos \theta }}+{\displaystyle \frac{2{\omega }_e}{\cos \theta }}\left(\cos \phi \cos \Psi \sin \theta -\sin \phi \cos \theta \right)\end{array}\right.$$

(1)

where $r$ is the radial distance from the vehicle to the Earth’s center, $\varphi$ and $\phi$ are the latitude and longitude of the vehicle, respectively, $m$ denotes the vehicle’s mass, $v$ denotes the vehicle’s speed, $\theta$ and $\Psi$ are the flight-path angle and velocity heading angle, respectively, $\sigma$ is the bank angle, ${\omega }_e$ is the self-rotation angular rate of the Earth, and $L$ and $D$ denote the aerodynamic lift and drag, respectively. Equation (2) is as follows:

$$\left\{\begin{array}{l}L=0.5C_{L}S\rho v^2\\ D=0.5C_{D}S\rho v^2\end{array}\right.$$

(2)

where $C_L$ and $C_D$ denote the lift coefficient and drag coefficient of the vehicle, respectively, $S$ is the reference area of the vehicle, and $\rho$ is the atmospheric density corresponding to the flight altitude.

When the RGV is a cooperative vehicle, the lift coefficient and drag coefficient under different operating conditions can be obtained through wind tunnel tests, CFD software, and engineering calculation methods. Based on Equations (1) and (2), trajectory planning for the vehicle under multiple constraints can be conducted. However, when the RGV is a non-cooperative vehicle, for the defender, the mass, reference area, lift coefficient, and drag coefficient of the vehicle are generally unknown. In such cases, when the defender uses Equation (1) to predict the RGV’s trajectory, it is necessary to integrate the unknowns into variables that can be obtained through measurement equipment and tracking filtering algorithms.

The unknown states of a non-cooperative target are represented by parameters determined by lift and drag coefficients, as in the following Equation (3):

$$\left\{\begin{array}{l}{K}_D={\displaystyle \frac{C_{D}S}{2m}}\\ K_L={\displaystyle \frac{C_{L}S}{2m}}\end{array}\right.$$

(3)

Due to the shorter duration of the tracking and prediction segments compared to the entire gliding process of the vehicle, the influence of the Earth’s rotation is ignored when using these dynamic equations. Combined with Equation (3), the last three items of the RGV kinematic model can be expressed as follows:

$$\left\{\begin{array}{l}\dot{v}=-K_D\rho v^2-g\sin \theta \\ \dot{\theta }=K_L\rho v\cos \sigma +{\displaystyle \frac{\cos \theta }{v}}\left({\displaystyle \frac{v^2}{r}}-g\right)\\ \dot{\Psi }={\displaystyle \frac{K_L\rho v\sin \sigma }{\cos \theta }}+{\displaystyle \frac{v\cos \theta \sin \Psi \tan \phi }{r}}\end{array}\right.$$

(4)

At this juncture, the vehicle’s movement mode is determined by three unknown control variables: $K_D$, $\sigma$, and $K_L$. The defense side can obtain the acceleration of incoming vehicles through filtering algorithms using deployed warning radar or other devices, which can accurately estimate unknown parameters. In this paper, the “Current” statistical (CS) model and the UKF algorithm are employed in the vehicle tracking section to achieve the estimation of the control variables. In addition, it is pointed out here that the tracking algorithm uses a coordinate-coupled motion model with an acceleration expansion dimension. The position and velocity variables of the vehicle are described in the East–North–Up (ENU) coordinate system, while the acceleration variable is described in the VTC coordinate system. Detailed definitions can be found in reference [31]. The conversion relationship between the control variables and non -gravitational acceleration in the VTC coordinate system is as follows:

$$\left\{\begin{array}{l}{K}_D={\displaystyle \frac{-a_{\mathrm{V}}}{\rho v^2}}\\ K_L\sin \sigma ={\displaystyle \frac{a_{\mathrm{T}}}{\rho v^2}}-{\displaystyle \frac{\cos \theta \sin \Psi \tan \phi }{\rho r}}\\ K_L\cos \sigma ={\displaystyle \frac{a_{\mathrm{C}}}{\rho v^2}}-{\displaystyle \frac{\cos \theta }{\rho r}}\end{array}\right.$$

(5)

where $a_{\mathrm{V}}$, $a_{\mathrm{T}}$, and $a_{\mathrm{C}}$ denote non-gravitational accelerations in the VTC coordinate system. As the focus of this paper is on analyzing the trajectory prediction problem of vehicle, the measurement model and filtering process in the tracking calculation are detailed in references [31,32], and the measurement error will be presented in the experimental simulation section.

3. Predictive Model Sets for Task Matching

Establishing a predictive model set is an effective way to solve the trajectory prediction problem of RGVs. Unlike reference [29], this paper constructs different sets of prediction models based on task judgment, considering the different motion characteristics brought about by different guidance tasks of RGVs.

3.1. The Judgement of the Vehicle’s Guidance Task

The guidance tasks of RGVs are primarily divided into three categories: no-fly zone avoidance, conventional guidance, and maneuver to break through a defense. When the vehicle performs a conventional guidance mission towards a predefined target point, it encounters a no-fly zone that is highly unfavorable due to the defense force of the defending party, as well as the geographical environment and other factors. When the vehicle possesses sufficient maneuvering capability, it will typically a perform C-type maneuver in the lateral direction in order to circumvent the no-fly zone. Conversely, when the maneuvering capability is insufficient to fully evade the no-fly zone, the vehicle will employ an S-type maneuver in order to breach the defense.

This paper assumes that the vehicle has sufficient maneuverability and analyzes two tasks: avoiding a no-fly zone and conventional guidance. These are denoted as ${\mathrm{T}}_b$ and ${\mathrm{T}}_f$, respectively. The judgement logic is based on two conditions: the angle and the distance. To facilitate the determination of the guidance task of the vehicle, a horizontal plane dynamic coordinate system is established with the current position of the vehicle as the origin. The schematic diagram of the task determination logic is shown in Figure 1.

When the following equation is satisfied, the guidance task of the vehicle is ${\mathrm{T}}_b$:

$$\Psi \in \left({\Psi }_{b\min },{\Psi }_{b\max }\right)\&\left\{{\mathrm{\Gamma }}_{\left|\sigma \right|},{\mathrm{\Gamma }}_{-\left|\sigma \right|}\right\}\cap {\mathrm{\Gamma }}_b\ne \varnothing$$

(6)

Otherwise, the guidance task is ${\mathrm{T}}_f$, where $\Psi \in \left({\Psi }_{b\min },{\Psi }_{b\max }\right)$ is the angle condition indicating that the vehicle’s speed is pointing towards the no-fly zone at this time, ${\Psi }_{b\min }$ denotes the minimum heading angle from the vehicle to the no-fly zone boundary, and ${\Psi }_{b\max }$ denotes the maximum heading angle. The distance condition, denoted by $\left\{{\mathrm{\Gamma }}_{\left|\sigma \right|},{\mathrm{\Gamma }}_{-\left|\sigma \right|}\right\}\cap {\mathrm{\Gamma }}_b\ne \varnothing$, indicates that the vehicle will intersect with the no-fly zone if it continues to fly at the current bank angle or at the changed bank angle. This implies that the no-fly zone affects the normal trajectory of the vehicle. ${\mathrm{\Gamma }}_{\left|\sigma \right|}$ and ${\mathrm{\Gamma }}_{-\left|\sigma \right|}$ denote the edges of the instantaneous turning circle when the sign of the bank angle is positive and negative, respectively, and ${\mathrm{\Gamma }}_b$ denotes the edges of the no-fly zone. The expressions for ${\Psi }_{b\min }$ and ${\Psi }_{b\max }$ are as follows:

$$\left\{\begin{array}{l}{\Psi }_{b\max }={\Psi }_b+\arcsin \left(R_b/S_b\right)\\ {\Psi }_{b\min }={\Psi }_b-\arcsin \left(R_b/S_b\right)\end{array}\right.$$

(7)

where ${\Psi }_b$ is the heading angle from the vehicle to the center of the no-fly zone, $R_b$ is the radius of the no-fly zone, and $S_b$ is the range-to-go from the vehicle to the center of the no-fly zone. For ease of description, we convert the distance variables mentioned in the paper into radians corresponding to the corresponding distances on Earth. The range-to-go is expressed as follows:

$$S_b=\arccos \left(\cos {\phi }_b\cos \phi \cos ({\varphi }_b-\varphi )+\sin {\phi }_b\sin \phi \right)$$

(8)

where $\left({\varphi }_b,{\phi }_b\right)$ are the latitude and longitude of the center of the no-fly zone, and ${\Psi }_b$ is calculated by the following Equation (9):

$${\Psi }_b=\arcsin \left({\displaystyle \frac{\sin \left({\varphi }_b-\varphi \right)\cos {\phi }_b}{\sin S_b}}\right)$$

(9)

When the instantaneous turning circle of the vehicle intersects the no-fly zone, the following characteristic exists:

$$S_{tb}<R_{turn}+R_b$$

(10)

where $R_{turn}$ is the instantaneous turning radius, $R_b$ is the radius of the no-fly zone, and $S_{tb}$ denotes the center distance between the instantaneous turning circle and the no-fly zone. The instantaneous turning radius is calculated by the following Equation (11):

$$R_{turn}={\displaystyle \frac{m{v}^2{\cos }^2\theta }{L\sin \left|\sigma \right|R_e}}\approx {\displaystyle \frac{2}{K_L{R}_e\rho \sin \left|\sigma \right|}}$$

(11)

$S_{tb}$ can be determined by the following equation:

$$S_{tb}=\sqrt{{\left({\varphi }_b-{\varphi }_{turn}\right)}^2+{\left({\phi }_b-{\phi }_{turn}\right)}^2}$$

(12)

where $\left({\varphi }_{turn},{\phi }_{turn}\right)$ are the latitude and longitude of the center of the instantaneous turning circle of the vehicle, respectively, calculated as follows:

$$\left\{\begin{array}{l}{\varphi }_{turn}=\varphi +sign\left(\sigma \right)R_{turn}\cos \Psi \\ {\phi }_{turn}=\phi -sign\left(\sigma \right)R_{turn}\sin \Psi \end{array}\right.$$

(13)

After determining the vehicle’s guidance mission according to Equation (6), different sets of prediction models are constructed for the requirements of different guidance missions.

3.2. Predictive Model Set Construction

In order to achieve the different guidance tasks, it is necessary for the vehicle to undergo continuous adjustments to both the angle of attack and the bank angle, in accordance with the battlefield situation. Among these variables, the angle of attack exerts a predominant influence on the vehicle’s longitudinal maneuverability, while the bank angle exerts a predominant influence on the lateral maneuverability. In light of environmental constraints, such as heat flow density, dynamic pressure, and overload in the near space, the angle of attack is generally set to a segmented profile that varies less over the tracking and prediction period. In contrast, the bank angle is varied by changing the sign and magnitude for different lateral maneuvers. When the sign of the bank angle remains constant, the vehicle performs a C-type maneuver in the lateral direction. Conversely, when the sign of the bank angle is repeatedly flipped, it performs an S-type maneuver in the lateral direction. Furthermore, the magnitude of the bank angle will also have different values for different guidance tasks. Based on this analysis, the prediction model sets, which include the vertical and lateral models, are constructed for different guidance tasks in this paper. The vertical prediction model set consists of $K_D$, $K_L$, and $\left|\sigma \right|$. The lateral prediction model set consists of bank angle signs. In order to enhance the long-term trajectory prediction capability for RGVs, the model set is updated in real time in accordance with the prevailing circumstances on the battlefield and the specific guidance mission.

3.2.1. Predictive Model Set for Conventional Guidance

(1)

Longitudinal parameters

In Section 2, the CS-UKF filtering algorithm can be used to estimate the non-gravity acceleration of RGVs, and the estimated control parameters $K_D^e, \left|{\sigma }^e\right|, K_L^e$ can be derived based on Equation (5). Afterwards, for the longitudinal prediction of the RGV, the least squares method is used to fit the control parameters of the tracking segment and extract the long-term maneuvering characteristics of the vehicle, which can obtain the fitted parameters $K_D^f, \left|{\sigma }^f\right|, K_L^f$ and the predictive parameters $K_D^p, \left|{\sigma }^p\right|, K_L^p$ at a future time. For ease of description, let ${\rm M}=\left[K_D,\left|\sigma \right|,K_L\right]$, $M^e=\left[K_D^e,\left|{\sigma }^e\right|,K_L^e\right]$, $M^f=\left[K_D^f,\left|{\sigma }^f\right|,K_L^f\right]$, and $M^p=\left[K_D^p,\left|{\sigma }^p\right|,K_L^p\right]$. It is assumed that the parameter filtered values satisfy a Gaussian distribution with the true values as the mean, and that the parameter predicted values satisfy a Gaussian distribution with the predicted results as the mean. The parameter distributions are shown in the following Equation (14):

$$\left\{\begin{array}{l}{M}^e\sim \mathcal{N}\left(M,{\left({\sigma }^e\right)}^2\right)\\ M^m\sim \mathcal{N}\left(M^p,{\left({\sigma }^p\right)}^2\right)\end{array}\right.$$

(14)

where $M^m=\left[K_D^m,\left|{\sigma }^m\right|,K_L^m\right]$ represents the possible values of the predicted values of the parameters at future moments, while ${\sigma }^e=\left[{\sigma }_{K_D}^e,{\sigma }_{\left|\sigma \right|}^e,{\sigma }_{K_L}^e\right]$ and ${\sigma }^p=\left[{\sigma }_{K_D}^p,{\sigma }_{\left|\sigma \right|}^p,{\sigma }_{K_L}^p\right]$ are the standard deviations of $M^e$ and $M^m$, respectively.

When the longitudinal parameter prediction model set is constructed for the vehicle performing a guidance mission ${\mathrm{T}}_f$ as determined by Equation (6), the $N$ parameters are extracted from each of the a distributions to form the $M^m\sim \mathcal{N}\left(M^p,{\left({\sigma }^p\right)}^2\right)$ $N^3$ model set. In order to enhance the accuracy of trajectory prediction, it is essential to ensure that the range of the model set encompasses the actual values as closely as possible. Concurrently, it is crucial to minimize the number of models to reduce the computational time required by the algorithm. Given that the distribution of parameters in the prediction section is unknown, it is not possible to determine the value directly. In this paper, therefore, we deflate the standard deviation ${\sigma }^e$ of the parameter filtering value $M^e$, thereby obtaining the following:

$${\sigma }^p=\lambda {\sigma }^e$$

(15)

where $\lambda$ is the scale factor. In the absence of data regarding the true value of the parameter $M$ during parameter tracking, it is reasonable to assume that the result of the function fitting is proximate to the true value. Consequently, the filtered parameter values are also expected to satisfy a Gaussian distribution with the fitted result as the mean value, as follows:

$$M^e\sim \mathcal{N}\left(M^f,{\left({\sigma }^f\right)}^2\right)$$

(16)

$$\left\{\begin{array}{l}{M}^f\approx M\\ {\sigma }^e\approx {\sigma }^f\end{array}\right.$$

(17)

where ${\sigma }^f=\left[{\sigma }_{K_D}^f,{\sigma }_{\left|\sigma \right|}^f,{\sigma }_{K_L}^f\right]$ is the standard deviation of $M^f$.

It should be noted that in the actual tracking process, the parameter models obtained through the measurement equipment are discrete. For ease of distinction, $M^e$ and $M^f$ with subscript $k$ are used to represent the parameter models actually obtained during the tracking phase. Assuming that $M_k^e$ values at different moments obey the same variance of the Gaussian distribution, the definition of the standard deviation shows the following:

$${\sigma }^f={\displaystyle \sum _{k=1}^{n_e}\left|M_k^e-M_k^f\right|}$$

(18)

where $n_e$ is the total step length of the tracking segment and $k$ is the time instant.

When sampling the parameters from Equation (14), $M^m$ is expressed as follows:

$$\left\{\begin{array}{l}{K}_D^m=K_D^p+{\sigma }_{K_D}^p{\delta }_{K_D}^m\\ {\left|\sigma \right|}^m={\left|\sigma \right|}^p+{\sigma }_{\left|\sigma \right|}^p{\delta }_{\left|\sigma \right|}^m\\ K_L^m=K_L^p+{\sigma }_{K_L}^p{\delta }_{K_L}^m\end{array}\right.$$

(19)

where ${\delta }_{K_D}^m$, ${\delta }_{\left|\sigma \right|}^m$. and ${\delta }_{K_L}^m$ are random numbers satisfying the standard normal distribution.

(2)

Lateral parameters

When the vehicle is conventionally guided, the heading angle corridor is employed to ascertain the sign of the vehicle’s bank angle at the present moment, thereby establishing a set of lateral parameter prediction models. In contrast to the conventional heading angle corridor, the no-fly zone may encroach upon the width of the corridor, resulting in repeated deflections of the bank angle sign. To circumvent this phenomenon, some compensation must be incorporated when constructing the corridor [33]. The schematic diagram of the heading angle corridor with compensation is shown in Figure 2.

In order to compensate for the heading angle corridor, the upper and lower boundaries of the heading angle are determined based on the relative position of the vehicle to the no-fly zone. When $\Psi \ge {\Psi }_{b\max }$, it can be obtained as follows:

$$\left\{\begin{array}{l}{\Psi }_{\min }=\max \left(\Psi -\Delta {\Psi }_{fed},{\Psi }_{b\max }\right)\\ {\Psi }_{\max }=\Psi +\Delta {\Psi }_{fed}+\max \left({\Psi }_{b\max }-\left(\Psi -\Delta {\Psi }_{fed}\right),0\right)\end{array}\right.$$

(20)

When $\Psi \le {\Psi }_{b\min }$, it can be obtained as follows:

$$\left\{\begin{array}{l}{\Psi }_{\max }=\min \left(\Psi +\Delta {\Psi }_{fed},{\Psi }_{b\min }\right)\\ {\Psi }_{\max }=\Psi -\Delta {\Psi }_{fed}-\max \left({\Psi }_{b\min }-\left(\Psi +\Delta {\Psi }_{fed}\right),0\right)\end{array}\right.$$

(21)

where $\Delta {\Psi }_{fed}$ denotes the range of the vehicle’s original heading angle corridor.

The vehicle’s bank angle sign can be determined from the compensated heading angle corridor as follows:

$$sign\left({\sigma }_{k+1}\right)=\left\{\begin{array}{ll}-1,& \Psi -{\Psi }_f\ge {\Psi }_{\max }-\Psi \&\Psi >{\Psi }_f\\ 1,& \Psi -{\Psi }_f\le {\Psi }_{\min }-\Psi \&\Psi <{\Psi }_f\\ sign\left({\sigma }_k\right),& {\Psi }_{\min }-\Psi <\Psi -{\Psi }_f<{\Psi }_{\max }-\Psi \end{array}\right.$$

(22)

where ${\Psi }_f$ indicates the heading angle from the vehicle to the intention to attack.

The final set of $N^3$ predictive models, $\Lambda =\left\{{{\rm M}}_1,M_2,\cdots ,M_n,\cdots ,M_{N^3}\right\}$, is determined by combining the vehicle longitudinal parameter model sets, where $M_n=\left\{K_{D,n_1}^p,sign\left({\sigma }_{k+1}\right)\left|{\sigma }_{n_2}^p\right|,K_{L,n_3}^p\right\}$, $n_1,n_2,n_3\in \left\{1,2\cdots ,N\right\}$.

3.2.2. Predictive Model Set for Avoiding a No-Fly Zone

(1)

Longitudinal parameters

In the event that the vehicle is undertaking a guidance mission with the objective of avoiding the no-fly zone, there are two scenarios in which the current turning capability is insufficient to achieve it. In the first scenario, when both $\Psi$ and ${\Psi }_f$ are situated on either side of ${\Psi }_b$, there is an overlap between the vehicle’s instantaneous turning circle and the no-fly zone. This necessitates a maneuver to circumvent the no-fly zone in order to achieve evasion. In the second scenario, when $\Psi$ and ${\Psi }_f$ are on one side of ${\Psi }_b$, the instantaneous turning circle and the no-fly zone at $sign\left(\sigma \right)=sign\left(\Psi -{\Psi }_b\right)$ intersect. This results in a small portion of the no-fly zone blocking the vehicle’s guidance trajectory. The schematic diagrams of the two cases are shown in Figure 3.

In this instance, it is essential to enhance the magnitude of the bank angle in order to optimize the turning capability. Based on the longitudinal parameter prediction model sets in Section 3.2.1, the values of the bank angle amplitude $\left|{\sigma }^p\right|$ are selected to encompass its definition domain $\left[0,\pi /2\right]$ to the greatest extent possible.

Sample $N$ points equidistantly over the range $\left[0,\pi /2\right]$, and the following Equation (14) is used to obtain the weight of each sampling point:

$${\vartheta }_{\left|\sigma \right|}^m={\displaystyle \frac{1}{\sqrt{2\pi }{\sigma }_{\left|\sigma \right|}^p}}{e}^{-{\displaystyle \frac{{\left({\sigma }^m-{\sigma }^p\right)}^2}{2{\sigma }_{\left|\sigma \right|}^p}}}$$

(23)

The weights of the normalized sampling points can be expressed as follows:

$${\vartheta }_{\left|\sigma \right|}^{m_n}={\displaystyle \frac{{\vartheta }_{\left|\sigma \right|}^{m_n}}{{\displaystyle \sum _{n=1}^N{\vartheta }_{\left|\sigma \right|}^{m_n}}}}$$

(24)

The actual sampling is conducted by generating a random number ${\delta }^{\prime }$ within $\left[0,1\right]$, comparing it with the weights ${\vartheta }_{\left|\sigma \right|}^{m_n}$ of the all points, and interpolating to obtain a new sampling point $\left|{\sigma }^m\right|$.

(2)

Lateral parameters

During lateral prediction, the vehicle must consider the potential scenarios in its relative positional relationship with the no-fly zone and the intent to attack. This requires the vehicle to combine its own maneuvering capability, energy loss, and mission accomplishment in order to select the least costly lateral maneuver mode. Thus, the lateral parameter prediction model set considers both cases of bank angle sign $\left\{1,-1\right\}$, with the final lateral maneuver mode being inferred with the aid of the intentional cost function presented in the next section of the paper.

Combined with the longitudinal parameter model set, a final set of predictive models is identified, i.e., $\Lambda =\left\{M_1,M_2,\cdots ,M_{2n-1},M_{2n},\cdots ,M_{2N^3}\right\}$, where $M_{2n}=\left\{K_{D,n_1}^p,\left|{\sigma }_{n_2}^p\right|,K_{L,n_3}^p\right\}$, $M_{2n-1}=\left\{K_{D,n_1}^p,-\left|{\sigma }_{n_2}^p\right|,K_{L,n_3}^p\right\}$, and $n_1,n_2,n_3\in \left\{1,2\cdots ,N\right\}$.

4. RGV Trajectory Prediction

In the trajectory prediction method for RGVs, different cost functions need to be constructed based on different guidance tasks. On this basis, combined with Bayesian theory, the parameter model and attack intention of the vehicle can be inferred, which is beneficial for improving the final trajectory prediction accuracy.

4.1. Intention Cost Function Construction

When oriented towards trajectory prediction for RGVs, the intention cost function can provide a basis for inferring the attack intent and predicting the model. The literature [28,29] comprehensively considered the factors of range, angle, and no-fly zone when constructing the intention cost function. However, there are significant differences in the estimation of each factor, and the coefficients of the different factors are not given a clear determination method. This makes it challenging to popularize and apply the prediction method. Consequently, this paper proposes the construction of distinct adaptive cost functions according to the objective of the vehicle guidance task. Furthermore, it unifies the expression of different factors and provides specific weight expression methods.

4.1.1. Intention Cost Function for Conventional Guidance

In the case of the ${\mathrm{T}}_f$ mission, the design of the intention cost function primarily considers the angle cost and energy cost. For the conventional guidance of RGVs, the gliding process of the vehicle has two characteristics: (1) the heading angle should point towards the attack intention or within a small range near it; (2) under the influence of the parameter model, the vehicle’s own energy should reach a suitable planned speed, generally around 5 Ma, when it reaches the actual attack intention. Therefore, when constructing the cost function, intentions or models that satisfy these two characteristics should be assigned a smaller cost, while those that do not should be assigned a larger cost. The specific form is as follows:

$$I_f=\max \left(\left|\Delta {\Psi }_f\right|-\Delta {\Psi }_{fed},0\right)e^{\left|{\displaystyle \frac{S_{ref}-S_f}{S_{ref}}}\right|}$$

(25)

where $\max \left(\left|\Delta {\Psi }_f\right|-\Delta {\Psi }_{fed},0\right)$ is the angle cost, while $\left|\Delta {\Psi }_f\right|=\left|\Psi -{\Psi }_f\right|$ denotes the absolute value of the difference between the vehicle’s heading angle and the heading angle to the attack intention. The angle cost is uniformly 0 when $\left|\Delta {\Psi }_f\right|\le \Delta {\Psi }_{fed}$ and $\left|\Delta {\Psi }_f\right|-\Delta {\Psi }_{fed}$ when $\left|\Delta {\Psi }_f\right|>\Delta {\Psi }_{fed}$. That means the intention cost function tends to select the parameter prediction model $M$ which results in a decrease in $\left|\Delta {\Psi }_f\right|$. $\left|S_{ref}-S_f\right|$ is energy cost, and $S_{ref}$ denotes the range-to-go of the vehicle when its speed is reduced to the planning speed $v_{ref}$ under the current parameter prediction model, while $S_f$ denotes the range-to-go of the vehicle to the intent to attack. When $S_f=S_{ref}$, the energy cost is 0. When $S_f\ne S_{ref}$, the larger the difference between $S_f$ and $S_{ref}$, the larger the energy cost, and the intention cost function tends to select the parameter prediction model that minimizes the proximity.

The range-to-go $S_f$ is calculated as follows:

$$S_f=\arccos \left(\cos {\phi }_f\cos \phi \cos ({\varphi }_f-\varphi )+\sin {\phi }_f\sin \phi \right)$$

(26)

where $\left({\varphi }_f,{\phi }_f\right)$ are the latitude and longitude of the intent to attack.

The vehicle’s calculation of $S_{ref}$ by integrating according to the parameter prediction model $M$ is too time-consuming for rapid model inference by the defense. In order to improve the prediction speed and facilitate the estimation of the intentional cost of different parameter prediction models, the range-to-go $S_{ref}^{\prime }$ of the vehicle under equilibrium gliding condition is compared instead of the actual $S_{ref}$, ignoring the range deviation caused by the actual longitudinal motion of the vehicle, and $S_{ref}^{\prime }$ is expressed as follows:

$$S_{ref}^{\prime }={\displaystyle \frac{k\cos \sigma }{2}}\ln \left({\displaystyle \frac{gr-v_{ref}^2}{gr-v_0^2}}\right)$$

(27)

where $v_0$ is the current speed of the vehicle, and $k=K_L/K_D$ denotes the lift–drag ratio.

4.1.2. Intention Cost Function for Avoiding the No-Fly Zone

When the objective of the vehicle is to avoid the no-fly zone, while simultaneously minimizing the impact on the final guidance task, it is not necessary to consider the energy cost when designing the intention cost function. Instead, the cost of the avoidance angle and the cost of the guidance angle should be the primary focus. The formula is as follows:

$$I_b={\epsilon }_1\Delta {\Psi }_b+{\epsilon }_2\Delta {\Psi }_{bf}$$

(28)

where ${\epsilon }_1\Delta {\Psi }_b$ is the avoidance angle cost, ${\epsilon }_1$ is the avoidance cost coefficient, ${\epsilon }_2\Delta {\Psi }_{bf}$ is the guidance angle cost, and ${\epsilon }_2$ is the guidance cost coefficient.

(1)

Avoidance angle cost

We can see that the larger $\Delta {\Psi }_b$ is, the more costly the avoidance angle is. Due to the limited turning capability of the vehicle, it is necessary to quantitatively describe the avoidance of the no-fly zone under bank angle control in the parameter prediction model, and to select the least cost. $\Delta {\Psi }_b$ is expressed as follows:

$$\Delta {\Psi }_b=\left\{\begin{array}{l}{\Psi }_{b\max }-\Psi , M=M_{2n}\\ \Psi -{\Psi }_{b\min }, M=M_{2n-1}\end{array}\right.$$

(29)

In calculating the avoidance cost coefficient, as outlined in Section 3.2.2, it is essential to consider the relationship between $\Psi$, ${\Psi }_b$, and ${\Psi }_f$. In the set of bank angle amplitudes for which the vehicle can avoid the no-fly zone, the least energy is lost when the trajectory is at a tangent to the no-fly zone. When Equation (30) or Equation (31) are satisfied, the avoidance cost coefficient requires the use of tangent bank angle.

$$\left(\Psi -{\Psi }_b\right)\left({\Psi }_f-{\Psi }_b\right)\le 0$$

(30)

$$\left(\Psi -{\Psi }_b\right)\left({\Psi }_f-{\Psi }_b\right)>0\&{\mathrm{\Gamma }}_{sign\left(\sigma \right)=sign\left(\Psi -{\Psi }_b\right)}\cap {\mathrm{\Gamma }}_b\ne \varnothing$$

(31)

At this point, the coefficient ${\epsilon }_1$ is determined by the following equation:

$${\epsilon }_1=e^{\left|{\sigma }_{\tan }-\left|\sigma \right|\right|/{\sigma }_{\tan }}$$

(32)

When both Equations (30) and (31) are not satisfied, the avoidance cost coefficient is determined by the following equation:

$${\epsilon }_1=e^{\left|{\sigma }^{\prime }-\left|\sigma \right|\right|/{\sigma }^{\prime }}$$

(33)

where ${\sigma }_{\tan }$ is the bank angle required for the vehicle’s instantaneous turning circle and no-fly zone tangent, and ${\sigma }^{\prime }$ is the bank angle amplitude at the previous moment. In the case where $\left|\sigma \right|={\sigma }_{\tan }$, the cost of the avoidance coefficient is 1. Conversely, when $\left|\sigma \right|\ne {\sigma }_{\tan }$, the greater the deviation, the greater the avoidance cost coefficient. The exact procedure for calculating ${\sigma }_{\tan }$ is as follows.

From the relationship of externally tangent between the vehicle’s instantaneous turning circle and the no-fly zone, we can derive the following equation:

$$S_{tb}=R_{turn}+R_b$$

(34)

where $S_{tb}$ denotes the center distance between the instantaneous turning circle and the no fly zone. From Equations (12) and (13), we can obtain the following:

$$\begin{array}{l}{\left({\varphi }_b-{\varphi }_{turn}\right)}^2+{\left({\phi }_b-{\phi }_{turn}\right)}^2\\ ={\left({\varphi }_b-\varphi -sign\left(\sigma \right)R_{turn}\cos \Psi \right)}^2+{\left({\phi }_b-\phi +sign\left(\sigma \right)R_{turn}\sin \Psi \right)}^2\\ ={\left(R_{turn}+R_b\right)}^2\end{array}$$

(35)

Expanding the above equation gives the following:

$$\begin{array}{l}2R_{turn}\left(sign\left(\sigma \right)\sin \Psi \left({\phi }_b-\phi \right)-sign\left(\sigma \right)\cos \Psi \left({\varphi }_b-\varphi \right)\right)+{\left({\varphi }_b-\varphi \right)}^2+{\left({\phi }_b-\phi \right)}^2\\ =R_b^2+2R_b{R}_{turn}\end{array}$$

(36)

Collapsed into an expression for instantaneous turning radius, Equation (37) is as follows:

$$R_{turn}={\displaystyle \frac{{\left({\varphi }_b-\varphi \right)}^2+{\left({\phi }_b-\phi \right)}^2-R_b^2}{2\left(R_b-sign\left(\sigma \right)\sin \Psi \left({\phi }_b-\phi \right)+sign\left(\sigma \right)\cos \Psi \left({\varphi }_b-\varphi \right)\right)}}$$

(37)

Associative Equations (11) and (37) lead to the following:

$$\left|{\sigma }_{\tan }\right|=\arcsin {\displaystyle \frac{4\left(R_b-sign\left(\sigma \right)\sin \Psi \left({\phi }_b-\phi \right)+sign\left(\sigma \right)\cos \Psi \left({\varphi }_b-\varphi \right)\right)}{\left({\left({\varphi }_b-\varphi \right)}^2+{\left({\phi }_b-\phi \right)}^2-R_b^2\right)K_L{R}_e\rho }}$$

(38)

(2)

Guidance angle cost

$\Delta {\Psi }_{bf}$ is defined as follows:

$$\Delta {\Psi }_{bf}=\left\{\begin{array}{l}{\Psi }_{b\max }-{\Psi }_f, sign\left(\sigma \right)=1\\ {\Psi }_f-{\Psi }_{b\min }, sign\left(\sigma \right)=-1\end{array}\right.$$

(39)

When ${\Psi }_{b\min }\le {\Psi }_f\le {\Psi }_{b\max }$, the intent to attack is at the back of the no-fly zone relative to the vehicle, and the cost of guidance becomes greater no matter which direction the vehicle goes to avoid it. When ${\Psi }_f>{\Psi }_{b\max }$ or ${\Psi }_f<{\Psi }_{b\min }$, the intention to attack is on the side of the no-fly zone relative to the vehicle, and avoiding the no-fly zone in the direction of the intention to attack will result in a lower guidance cost; otherwise, a higher guidance cost will be incurred.

The guidance cost coefficient ${\epsilon }_2$ is calculated as follows:

$${\epsilon }_2=e^{-{\displaystyle \frac{S_f}{S_f+S_b}}}$$

(40)

When $S_b<S_f/2$, ${\epsilon }_2<0.5$ indicates that the vehicle is far from the attack site. For avoiding the no-fly zone, the impact of the guidance task is a smaller proportion; thus, the guidance cost is smaller. When $S_b\ge S_f/2$, ${\epsilon }_2\ge 0.5$ indicates that the vehicle is closer to the attack site, and avoiding the no-fly zone will have a direct impact on the completion of the final guidance task of the vehicle; therefore, the guidance cost is larger.

If there are multiple no-fly zones affecting the vehicle’s trajectory, the closest no-fly zone is selected as the primary no-fly zone to characterize the integrated intention cost of avoiding the no-fly zone. Equation (41) is as follows:

$$I_{b,M_j}=\arg \min \left(S_{b_i}\right), \exists I_{b_i}, i=1,2,\cdots n_b$$

(41)

where $n_b$ is the number of no-fly zones affecting the vehicle trajectory.

4.2. Inference of Vehicle Intention and Parameter Model via the Adaptive Cost Function

In Section 2, we presented the motion model of reentry glide RGVs under continuous time conditions. It should be noted that since the vehicle is a non-cooperative vehicle, its state variables are mainly obtained through measurement equipment and filtering algorithms. At this point, the motion model used to infer intent and parameter models is actually discrete. The measurement cycle of the measuring equipment is the discrete time interval of the motion model. At the tracking time instant $k$, the state of the vehicle obtained by the defense through the detection device and filtering algorithm is denoted as $x_k$ and the specific form is as follows:

$$x_k=[r_k, {\varphi }_k, {\phi }_k, v_k, {\theta }_k, {\Psi }_k]$$

(42)

where $r_k, {\varphi }_k, {\phi }_k, v_k, {\theta }_k$, and ${\Psi }_k$ are the estimated values of the discrete $r, \varphi , \phi , v, \theta$, and $\Psi$. When using Bayes’ theorem to infer a parameter, it usually refers to calculating the posterior probability by estimating the likelihood probability with known prior probabilities. For non-cooperative RGVs, the estimation of likelihood probability relies on the intention cost function to quantify the cost of different attack intentions or parameter prediction models. Next, this section will provide a detailed derivation process for inferring the attack intent and parameter prediction model of the vehicle based on Bayes’ theorem.

4.2.1. Inference of Parameter Model

When the reentry glide RGV is a non-cooperative vehicle, the true control parameter model and guidance intention are unknown to the defense side. But, fortunately, we have the estimated values of the vehicle’s state during the tracking phase. In Section 3.2.1, we constructed a set of predictive parameter models for the future period through appropriate simplification and derivation. The process of the vehicle from state $x_{k-1}$ to state $x_k$ can be seen as a motion process achieved through the control parameter model ${\mathit{M}}_n$ with the aim of attacking a certain target ${\eta }_T$ in the intended space $\Theta$ which includes all the important places that might be attacked. When in the subsequent state $x_k$, the intent cost function can quantify the cost associated with different parameter models completing this state transition, thereby aiding the defense in inferring which parameter model the vehicle might utilize. Normalizing the costs of these different parameter models yields their likelihood probabilities, as follows:

$$\mathrm{P}\left(x_k|x_{k-1},{\eta }_T,M_n\right)\triangleq {\displaystyle \frac{e^{-I\left({\eta }_T,M_n,x_k\right)}}{{\displaystyle \sum _{M_n\in \Lambda }{e}^{-I\left({\eta }_T,M_n,x_k\right)}}}}$$

(43)

where ${\displaystyle \sum _{M_n\in \Lambda }\mathrm{P}\left(x_k|x_{k-1},{\eta }_T,M_n\right)=1}$.

According to Bayes’ theorem, the posterior probability of the parameter model $M_n$ is as follows:

$$\mathrm{P}\left(M_n|x_{1:k},{\eta }_T\right)={\displaystyle \frac{\mathrm{P}\left(M_n,x_{1:k},{\eta }_T\right)}{\mathrm{P}\left(x_{1:k},{\eta }_T\right)}}$$

(44)

When working with the partition to the right of the equal sign, the state quantity is split into two parts and converted into the form of the likelihood probability multiplied by the prior probability.

$$\mathrm{P}\left(M_n|x_{1:k},{\eta }_T\right)={\displaystyle \frac{\mathrm{P}\left(x_k|x_{1:k-1},M_n,{\eta }_T\right)\mathrm{P}\left(x_{1:k-1},M_n,{\eta }_T\right)}{\mathrm{P}\left(x_k|x_{1:k-1},{\eta }_T\right)\mathrm{P}\left(x_{1:k-1},{\eta }_T\right)}}$$

(45)

The two prior probabilities on the numerator and denominator combine to form the posterior probability of the parameter model $M_n$ at the previous point in time.

$$\mathrm{P}\left(M_n|x_{1:k},{\eta }_T\right)={\displaystyle \frac{\mathrm{P}\left(x_k|x_{1:k-1},M_n,{\eta }_T\right)\mathrm{P}\left(M_n|x_{1:k-1},{\eta }_T\right)}{\mathrm{P}\left(x_k|x_{1:k-1},{\eta }_T\right)}}$$

(46)

Due to the Markovian nature of the state transfer process, a simplification of the state transfer under the parameter model $M_n$ yields the following:

$$\mathrm{P}\left(M_n|x_{1:k},{\eta }_T\right)={\displaystyle \frac{\mathrm{P}\left(x_k|x_{k-1},M_n,{\eta }_T\right)\mathrm{P}\left(M_n|x_{1:k-1},{\eta }_T\right)}{\mathrm{P}\left(x_k|x_{1:k-1},{\eta }_T\right)}}$$

(47)

where $\mathrm{P}\left(x_k|x_{k-1},M_n,{\eta }_T\right)$ is the likelihood probability of the parameter model $M_n$, which is denoted by the intention cost function, $\mathrm{P}\left(M_n|x_{1:k-1},{\eta }_T\right)$ is the posterior probability of the parameter model $M_n$ at the previous moment in time, and $\mathrm{P}\left(x_k|x_{1:k-1},{\eta }_T\right)$ denotes the likelihood probability of the intention ${\eta }_T$. $\mathrm{P}\left(x_k|x_{1:k-1},{\eta }_T\right)$ is calculated as follows:

$$\mathrm{P}\left(x_k|x_{1:k-1},{\eta }_T\right)={\displaystyle \sum _{M_n\in \Lambda }\mathrm{P}\left(x_k,M_n|x_{1:k-1},{\eta }_T\right)}$$

(48)

From Bayes’ theorem, the posterior probability of ${\displaystyle \sum _{M_n\in \Lambda }\left(x_k,M_n\right)}$ is expressed as follows:

$$\begin{array}{l}{\displaystyle \sum _{M_n\in \Lambda }\mathrm{P}\left(x_k,M_n|x_{1:k-1},{\eta }_T\right)}={\displaystyle \sum _{M_n\in \Lambda }\frac{\mathrm{P}\left(x_{1:k-1},x_k,M_n,{\eta }_T\right)}{\mathrm{P}\left(x_{1:k-1},{\eta }_T\right)}}\\ ={\displaystyle \sum _{M_n\in \Lambda }\frac{\mathrm{P}\left(x_k|x_{1:k-1},M_n,{\eta }_T\right)\mathrm{P}\left(x_{1:k-1},M_n,{\eta }_T\right)}{\mathrm{P}\left(x_{1:k-1},{\eta }_T\right)}}\\ ={\displaystyle \sum _{M_n\in \Lambda }\mathrm{P}\left(x_k|x_{1:k-1},M_n,{\eta }_T\right)\mathrm{P}\left(M_n|x_{1:k-1},{\eta }_T\right)}\end{array}$$

(49)

Therefore, the posterior probabilities of the parametric model $M_n$ is as follows:

$$\mathrm{P}\left(M_n|x_{1:k},{\eta }_T\right)={\displaystyle \frac{\mathrm{P}\left(x_k|x_{k-1},M_n,{\eta }_T\right)\mathrm{P}\left(M_n|x_{1:k-1},{\eta }_T\right)}{{\displaystyle \sum _{M_n\in \Lambda }\mathrm{P}\left(x_k|x_{k-1},M_n,{\eta }_T\right)\mathrm{P}\left(M_n|x_{1:k-1},{\eta }_T\right)}}}$$

(50)

According to Equation (50), the parameter prediction model for the vehicle with maximum a posteriori probability under the attack intent ${\eta }_T$ is as follows:

$$M_n=\underset{M_n\in \Lambda }{\arg \max }\left(\mathrm{P}\left(M_n|x_{1:k},{\eta }_T\right)\right)$$

(51)

4.2.2. Inference of Attack Intention

Next, the posterior probability of attack intention ${\eta }_T$ is derived as follows:

$$\mathrm{P}\left({\eta }_T|x_{1:k}\right)={\displaystyle \frac{\mathrm{P}\left(x_{1:k},{\eta }_T\right)}{\mathrm{P}\left(x_{1:k}\right)}}$$

(52)

Similarly, splitting the state quantity into two parts yields the following:

$$\begin{array}{l}\mathrm{P}\left({\eta }_T|x_{1:k}\right)={\displaystyle \frac{\mathrm{P}\left(x_k,x_{1:k-1},{\eta }_T\right)}{\mathrm{P}\left(x_k,x_{1:k-1}\right)}}\\ ={\displaystyle \frac{\mathrm{P}\left(x_k|x_{1:k-1},{\eta }_T\right)\mathrm{P}\left(x_{1:k-1},{\eta }_T\right)}{\mathrm{P}\left(x_k|x_{1:k-1}\right)\mathrm{P}\left(x_{1:k-1}\right)}}\\ ={\displaystyle \frac{\mathrm{P}\left(x_k|x_{1:k-1},{\eta }_T\right)\mathrm{P}\left({\eta }_T|x_{1:k-1}\right)}{\mathrm{P}\left(x_k|x_{1:k-1}\right)}}\end{array}$$

(53)

where $\mathrm{P}\left(x_k|x_{1:k-1},{\eta }_T\right)$ is the likelihood probability of the intention ${\eta }_T$ sought in Equation (48) above, $\mathrm{P}\left({\eta }_T|x_{1:k-1}\right)$ is the posterior probability of the intention ${\eta }_T$ at the previous moment, and $\mathrm{P}\left(x_k|x_{1:k-1}\right)$ denotes conditional probability of one-step prediction for the vehicle. $\mathrm{P}\left(x_k|x_{1:k-1}\right)$ is expressed as follows:

$$\mathrm{P}\left(x_k|x_{1:k-1}\right)={\displaystyle \sum _{{\eta }_T\in \Theta }\mathrm{P}\left(x_k,{\eta }_T|x_{1:k-1}\right)}$$

(54)

where $\Theta$ denotes the collection of all intention. From Bayes’ theorem, the posterior probability of ${\displaystyle \sum _{{\eta }_T\in \Theta }\mathrm{P}\left(x_k,{\eta }_T\right)}$ is expressed as follows:

$$\begin{array}{l}{\displaystyle \sum _{{\eta }_T\in \Theta }\mathrm{P}\left(x_k,{\eta }_T|x_{1:k-1}\right)}={\displaystyle \sum _{{\eta }_T\in \Theta }\frac{\mathrm{P}\left(x_k,{\eta }_T,x_{1:k-1}\right)}{\mathrm{P}\left(x_{1:k-1}\right)}}\\ ={\displaystyle \sum _{{\eta }_T\in \Theta }\frac{\mathrm{P}\left(x_k|{\eta }_T,x_{1:k-1}\right)\mathrm{P}\left({\eta }_T,x_{1:k-1}\right)}{\mathrm{P}\left(x_{1:k-1}\right)}}\\ ={\displaystyle \sum _{{\eta }_T\in \Theta }\mathrm{P}\left(x_k|{\eta }_T,x_{1:k-1}\right)\mathrm{P}\left({\eta }_T|x_{1:k-1}\right)}\end{array}$$

(55)

Therefore, the posterior probability of the intention ${\eta }_T$ is calculated as follows:

$$\mathrm{P}\left({\eta }_T|x_{1:k}\right)={\displaystyle \frac{\mathrm{P}\left(x_k|x_{1:k-1},{\eta }_T\right)\mathrm{P}\left({\eta }_T|x_{1:k-1}\right)}{={\displaystyle \sum _{{\eta }_T\in \Theta }\mathrm{P}\left(x_k|x_{1:k-1},{\eta }_T\right)\mathrm{P}\left({\eta }_T|x_{1:k-1}\right)}}}$$

(56)

According to Equation (56), the attack intention of the vehicle with maximum a posteriori probability is as follows:

$${\eta }_T=\underset{{\eta }_T\in \Theta }{\arg \max }\left(\mathrm{P}\left({\eta }_T|x_{1:k}\right)\right)$$

(57)

4.3. RGV Trajectory Prediction Process

As illustrated in Section 4.2, inferring the attack intention of the vehicle necessitates the calculation of the intention cost of the entire parameter prediction model for all potential attack intents, which can result in a significant computational burden in practice. Given that the attack intention is typically not altered frequently during the actual guidance, in order to reduce the prediction time, a single attack intention inference is conducted at the outset of the trajectory prediction, with the attack intent not being iteratively updated subsequently. The prediction block diagram of the RGV trajectory prediction algorithm via the adaptive cost function is depicted in Figure 4.

The prediction steps are outlined below.

Step1: Initialize information about radar parameters, prediction parameters, no-fly zones, and the intended position;

Step2: The vehicle is tracked according to the CS-UKF filtering algorithm to obtain the control parameters $K_D^f, \left|{\sigma }^f\right|, K_L^f$, and the predicted $K_D^p, \left|{\sigma }^p\right|, K_L^p$ is obtained using least squares fitting;

Step3: Calculate the vehicle footprint, determine the attack intention set $\Theta$, and execute Step4~Step6 for each possible attack intention;

Step4: Evaluate the vehicle guidance task according to Section 3.1; if the vehicle task is ${\mathrm{T}}_f$, go to step5; if ${\mathrm{T}}_b$, go to step6;

Step5: Construct the parameter prediction model set $\Lambda$ for conventional guidance according to Section 3.2.1, and construct the intention cost function $I_f$ according to Section 4.1.1 to compute the corresponding cost;

Step6: Construct the parameter prediction model set $\Lambda$ for avoiding the no-fly zone guidance according to Section 3.2.2, and construct the intention cost function $I_b$ according to Section 4.1.2 to compute the corresponding cost;

Step7: Infer the attack intention ${\eta }_T$ with maximum a posteriori probability according to Section 4.2.2;

Step8: Infer the parameter prediction model $M_n$ with maximum a posteriori probability according to Section 4.2.1, and iteratively integrate according to the reconstructed set of equations of motion of the vehicle to obtain the predicted target position;

Step9: When the prediction time reaches the planning time, finish the prediction; otherwise, loop to execute Step4~Step6 and Step8.

5. Simulations

In order to ascertain the efficacy of the algorithm proposed in this paper, a series of simulation experiments have been conducted on the reentry glide segment of the CAV-H [34]. According to publicly available information, the mass of the vehicle is 907 kg, the area is 0.4839 m², and the initial state of the reentry glide $\left({\varphi }_0,{\phi }_0,h_0,v_0,{\theta }_0,{\Psi }_0\right)$ is $\left(0°,0°,70 \mathrm{km},6000 \mathrm{m}/\mathrm{s},-0.1°,65°\right)$. The guidance is carried out using the predictor–corrector algorithm [35,36,37], with a guidance period of 1 s. The early warning radar deployment position is set as ${[23°, 3.6°, 100 \mathrm{m}]}^{\mathrm{T}}$, the detection error is set as ${[30 \mathrm{m}, 0.05°, 0.05°]}^{\mathrm{T}}$, and the detection period and guidance period are the same. Given that the location of different no-fly zones will result in significant differences in the vehicle’s guidance trajectory, it is necessary to conduct a comprehensive evaluation of the algorithm’s effectiveness. To this end, simulations are set up with two cases of no-fly zones, in addition to three intended targets, as shown in

Table 1. Intended targets parameters and no-fly zones parameters for different cases.

Causality	Case 1		Case 2		Radius/(°)
	Longitude/(°)	Latitude/(°)	Longitude/(°)	Latitude/(°)
Target 1	15	18	15	18	-
Target 2	30	25	30	25	-
Target 3	35	15	24	20	-
No-fly zone 1	15	11	15	11	2
No-fly zone 2	25	14	11	17	2

. The fundamental parameters involved in the algorithm are the number of longitudinal parameter model sets, where $N$ is 5, the deflation coefficient, where $\lambda$ is 0.5, the heading angle corridor, where $\Delta {\Psi }_{fed}$ is 5°, and the planning speed, where $v_{ref}$ is 1500 m/s.

The footprint of the RGV is solved using the constant value of the bank angle method [38], where the range of values of the bank angle is $\left[-70°,70°\right]$. In order to verify the superiority of the proposed method in this paper, the predictive methods from the literature [16,28,29] are used for simulation comparison. For the sake of clarity, the aforementioned methods will be referred to as Method 1, Method 2 and Method 3, respectively. The proposed method in this paper will be designated as Method 4. The cost coefficients in Method 2 and Method 3 will be set to the same values as those presented in literature [29].

5.1. Case 1

The trajectory of the RGV in case 1 is depicted in Figure 5. Figure 5a illustrates that the vehicle crosses between the two no-fly zones during the reentry gliding process and reaches the second intended target. Figure 5b depicts that the vehicle’s longitudinal altitude early change is relatively smooth, and the altitude jumps at the latter stage due to the imbalance of the longitudinal force. The entire gliding section has a flight time of approximately 1200 s and a range of approximately 4300 km. In order to predict the trajectory of case 1, 200 s and 400 s are used as the starting point for tracking for 200 s, and the vehicle’s position after 150 s is predicted.

Table 2. The posteriori probability of the intended target of case 1.

Method	${\mathit{t}}_0=200 \mathbf{s}$			${\mathit{t}}_0=400 \mathbf{s}$
	Target 1	Target 2	Target 3	Target 1	Target 2	Target 3
Method 2	0	50.24%	49.76%	0	100.00%	0
Method 3	0	6.15%	93.85%	0	100.00%	0
Method 4	0	99.99%	0.01%	0	100.00%	0

presents the inferred probability of the target attack intention in the footprint. Figure 6 and Figure 7 illustrate the simulation results of the prediction methods.

Table 3. Predicted effect of case 1.

Predicted Start Time	Method	Longitudinal Error/(km)	Latitudinal Error/(km)	Skyward Error/(km)	Overall Error/(km)	Prediction Time/(s)
$t_0=200 \mathrm{s}$	Method 1	6.89	−12.44	1.86	14.34	2.37
	Method 2	−26.36	37.69	5.45	46.32	8.56
	Method 3	46.23	−58.89	0.55	74.87	0.90
	Method 4	4.65	−2.71	0.98	5.47	0.28
$t_0=400 \mathrm{s}$	Method 1	−18.23	15.22	−1.65	23.80	2.37
	Method 2	33.40	5.33	7.88	34.73	5.98
	Method 3	15.50	−4.29	−5.25	16.92	0.61
	Method 4	7.98	9.87	−4.78	13.56	0.35

displays the prediction effect.

Figure 6a illustrates that at that time, intended targets 2 and 3 are within the footprint, whereas target 1 is not, contingent on the vehicle’s attack intent and maneuverability. Table 2 demonstrates that the three methods equipped with intention inference provide disparate a posteriori probability for intended targets within the footprint. Since the intention cost coefficients for Method 2 and 3 are artificially given in case 1, Method 2 barely determines the true intent to attack, while Method 3 determines that the posterior probability of target 3 being the intention to attack is as high as 93.85%. The adaptive intention cost function proposed in this paper, on the other hand, better integrates the objectives of avoiding no-fly zones and achieving the guidance mission. Under the influence of adaptive cost coefficients, it significantly improves the accuracy of attack intention inference. As can be seen from Figure 6b, in Method 2, as the predicted position gradually approaches the heading between the current position and the intended position, the exponential component in the intention cost calculation becomes excessively small, causing the final cost to converge towards a constant value of 1. In Method 3, due to a misjudgment of the vehicle’s attack intention, the proportion of the no-fly zone in the cost calculation increases, resulting in a fluctuating and decreasing trend in the positional cost. In Method 4, a sudden change occurs in the predicted guidance task of the aircraft at 464s, leading to a significant mutation in the calculated cost value before and after this point. However, within the same guidance task, the cost values generally appear smooth and exhibit a monotonically decreasing trend.

From Figure 6c, it can be seen that the prediction errors of Methods 1 and 4 are generally lower than the other two prediction methods. Method 2 has a high accuracy of prediction error in the short term because it converts the prediction of the vehicle control parameters into the prediction of the position within the one-step footprint, but the error gradually increases as the prediction time increases. In Method 3, the prediction error continues to increase due to errors in inferring the vehicle’s attack intention. Combined with Table 3, it can be seen from Figure 6d,e that the prediction errors in the longitude and latitudinal directions for Methods 2 and 3 are the main reason for the large final overall prediction errors. Method 1 resulted in significant longitudinal errors due to the accumulation of errors caused by least squares fitting. Due to the inability to accurately infer the inclination within the positional inference algorithm of Method 2, the prediction error for the skyward direction is also the largest of the four prediction methods. Method 3 resulted in a significant deviation of the trajectory in the lateral plane from the true trajectory due to misjudgment of intent. However, Method 4 constructs a set of parameter prediction model and an intention cost function based on the accurate judgement that the vehicle mission is to avoid the no-fly zone and be guided towards intended target 2. The accumulation of the prediction error is reduced by choosing the parameter prediction model that makes the intention cost as small as possible, which ultimately reduces the prediction error of Method 4 to less than half that of Method 1.

From Figure 7a, it can be seen that at time $t_0=400 \mathrm{s}$, only intended target 2 is within the vehicle footprint, so the difference in prediction error caused by different prediction methods is not related to the attack intention inference. As can be seen from Figure 7b, the position predicted by Method 2 is near the heading between the current position and the intended position. In Method 3, since there is no influence from the no-fly zone at this time, the cost value gradually increases as the remaining voyage decreases. In Method 4, the cost values within the track deviation angle corridor are all zero. However, at the predicted vehicle heading angle at 676 s, it exceeds the corridor range, and a larger cost is beneficial for correcting the predicted position of the vehicle.

Figure 7c illustrates that the prediction errors of Methods 1, 3, and 4 are all subject to fluctuations. Method 4 exhibits the smallest final prediction error of 13.56 km, whereas Method 2 continues to demonstrate an upward trend, with the final prediction error at the 150 s mark reaching as high as 34.73 km. When considered alongside Table 3, Figure 7d demonstrates that the predicted trajectories in the lateral plane for all four prediction methods are situated around the true trajectory and do not undergo any significant shifts. Method 2 significantly suffers from longitudinal prediction errors owing to neglecting the vehicle control variable’s variations in trajectory prediction, while Method 1 exhibits a notable error stemming from challenges in accurately fitting the bank angle overturning law using vector autoregressive model. Method 3 represents an improvement upon Methods 1 and 2. It employs a combination of the intention cost function with the estimation of multiple lateral maneuver modes based on the fitting of control parameters and the fusion of multi-model trajectories. This approach has been shown to yield more accurate predictions than Methods 1 and 2. However, the fusion of trajectories in erroneous lateral maneuver modes has been found to reduce the final prediction accuracy. In contrast, this paper constructs a task-matched parameter prediction model set to reduce the number of lateral parameter models based on the judgement that the vehicle is conventionally guided. The parameter prediction model with the maximum a posteriori probability is selected by the intention cost function. This removes the effect of the incorrect model on the prediction error, thereby improving the prediction accuracy of Method 3 by approximately twofold. From Figure 7e, it can be seen that the longitudinal trajectory of the vehicle jumps significantly in the prediction section, and the prediction trajectories of Methods 1, 3, and 4 are smoother, so that the prediction error is larger in the altitude descent section, but the overall error is kept at a low level.

In terms of prediction time, it can be seen from Table 3 that Method 2 takes the most time to predict owing to it using a Monte Carlo algorithm to solve the posteriori probability of the discrete region after discretizing the vehicle’s footprint, which results in it consuming more time. When Method 1 uses the vector autoregressive model to fit parameters, the coefficients between multiple variables increase significantly in terms of time cost. The redundancy in the set of lateral parameter prediction model for Method 3 has the consequence of increasing the time required for prediction. Although the Method 4 proposed in this paper also constructs the set of parameter prediction model, the lateral parameter prediction models are reduced to between one-quarter and one-half of the original one by dividing the vehicle guidance task and analyzing the motion characteristics. Furthermore, the algorithm’s prediction time is reduced to less than 0.5 s. Consequently, the proposed methodology presented in this paper addresses the need for rapid prediction with a minimal computational burden, while simultaneously enhancing the accuracy of the prediction.

5.2. Case 2

Figure 8 illustrates the guided trajectory of the RGV in case 2. From Figure 8a, it can be seen that the RGV successfully evaded the two no-fly zones on one side during the reentry gliding process, reaching the second intended target. Figure 8b depicts a relatively smooth longitudinal altitude change, with a comparable flight time and range to that of case 1. In order to predict the trajectory of case 2, 300 s and 500 s are used as the starting point for tracking for 200 s, and the vehicle position after 150 s is predicted.

Table 4. The posteriori probability of the intended target of case 2.

Method	${\mathit{t}}_0=300 \mathbf{s}$			${\mathit{t}}_0=500 \mathbf{s}$
	Target 1	Target 2	Target 3	Target 1	Target 2	Target 3
Method 2	0	50.35%	49.65%	0	100.00%	0
Method 3	0	0.01%	99.99%	0	100.00%	0
Method 4	0	99.99%	0.01%	0	100.00%	0

presents the inferred probability of an attack target within the footprint. Figure 9 and Figure 10 illustrate the simulation results of the prediction methods.

Table 5. Predicted effect of case 2.

Predicted Start Time	Method	Longitudinal Error/(km)	Latitudinal Error/(km)	Skyward Error/(km)	Overall Error/(km)	Prediction Time/(s)
$t_0=300 \mathrm{s}$	Method 1	−14.51	12.90	1.71	19.49	2.33
	Method 2	−79.02	67.54	−0.69	103.96	10.20
	Method 3	−21.35	20.10	−0.25	29.32	1.05
	Method 4	0.52	2.12	−0.25	2.27	0.33
$t_0=500 \mathrm{s}$	Method 1	33.84	−26.45	3.62	43.10	2.43
	Method 2	−64.75	42.42	6.14	77.65	14.97
	Method 3	27.56	−23.36	1.11	36.14	0.54
	Method 4	4.19	−5.31	1.35	6.90	0.29

outlines the prediction effect.

As illustrated in Figure 9a, the situation is analogous to that of case 1, with intended targets 2 and 3 located within the footprint and target 1 located outside. As can be seen from Table 4, Method 4 accurately deduces the true attack intent of the vehicle, Method 2 still just barely arrives at a correct judgement, and the probability of Method 3’s intention judgement being wrong rises to 99.99%. From Figure 9b, it can be seen that the predicted position of Method 2 becomes closer to the no-fly zone while approaching the heading of the current position and the intended position, resulting in the computed cost value gradually converging to 0. The predicted position of Method 3 is at the capability boundary at a tangent to the no-fly zone, and the reduction in the remaining range in the later stages of prediction due to the smaller influence of the no-fly zone results in the rapid increase in the cost value. Method 4 predicts a sudden change in the cost value at 572 s due to the judgment of the guidance task, but the overall smooth change in the generation value is still due to the heading angle being outside the corridor.

As illustrated in Figure 9c, Method 4 exhibits the lowest prediction error. Method 1 is better than Method 3 in terms of accuracy. Method 2’s prediction error is more accurate in the short term but rapidly diverges in the long term. Combined with Table 5, it can be seen from Figure 9d that although Method 2 correctly determines the vehicle’s attack intention, its intention cost function, with fixed cost coefficients, does not achieve the purpose of avoiding the no-fly zone when predicting the trajectory, but instead goes straight towards target 2, which is inconsistent with the real trajectory, resulting in a large prediction error. Method 3, with the intention cost function, is more accurate in parameter model inference, although it incorrectly infers the intention of the attack, and reasonably selects the parameters to circumvent the no-fly zone, which substantially improves the prediction accuracy compared to Method 2. Since the vehicle in the prediction segment mainly conducts C-type maneuver to achieve no-fly zone avoidance, Method 1 obtains a high prediction accuracy by modeling the dynamic relationship between control parameters through vector autoregression model. The prediction error of Method 4 is still the smallest. From Figure 9e, it can be seen that Methods 3 and 4 predict the vehicle height change more accurately, while the remaining methods have large errors, but the maximum error is within 2 km. This indicates that the prediction accuracy is generally higher in the case of relatively smooth longitudinal height changes by the vehicle.

Figure 10a illustrates that only intended target 2 is within the vehicle footprint at $t_0=500 \mathrm{s}$. From Figure 10b, it can be seen that the predicted position cost calculation results of Method 2 and Method 3 are similar to those of case 1, but the cost of Method 4 is relatively high most of the time because the heading angle exceeds the corridor range under conventional guidance tasks. Combined with Table 5, Figure 10c,d demonstrate that Method 1 results in a significant prediction error due to its inability to accurately predict the change pattern of the lateral control volume change. Method 2, however, fails to consider the constraints imposed by the vehicle in the discrete prediction of the footprint. This results in a predicted trajectory that is directed towards target 2 and differs significantly from the true trajectory. Method 3 also has a large prediction error due to the redundancy of the lateral parameter prediction models, but the prediction accuracy is substantially improved compared to Methods 1 and 2. Method 4 reaches a prediction error of 6.90 km after determining that the vehicle mission is conventionally guided and within the constraints of the heading angle corridor with compensation. As illustrated in Figure 10e, the predicted vertical height jumps resulting from the application of Methods 1, 2, 3, and 4 exhibited considerable discrepancies, with Method 2 exhibiting the greatest error, reaching 6.14 km. However, Methods 3 and 4 predict more gradual longitudinal height changes and exhibited the highest degree of accuracy. In terms of the time required for prediction, it is consistent with the findings of case 1.

6. Conclusions

This paper proposed a trajectory prediction method for the RGV via the adaptive cost function. On the basis of distinguishing the guidance tasks of RGVs, the method constructs corresponding prediction model sets and intention cost functions for different guidance tasks and combines Bayesian theory to derive the attack intention and prediction model with the maximum a posteriori probability. Combined with the dynamic equation system, it realizes the long-term trajectory prediction of the vehicle. The main conclusions of this paper are as follows:

(1)

The vehicle guidance tasks were divided, and the task-matched set of time-varying parameter prediction models was constructed. This reduced the redundancy of the lateral parameter model and ensured the fast implementation of the prediction algorithm.

(2)

In consideration of the vehicle’s maneuvering capability, guidance intention, and battlefield situation, the proposed intention cost functions with adaptive cost coefficients improved the accuracy of guidance intention cost estimation.

(3)

The attack intent and parameter prediction model of the vehicle were inferred based on Bayesian theory and the maximum a posteriori probability, which reduced the error accumulation in the prediction process.

The simulation results demonstrated that the proposed prediction method was capable of accurately inferring the attack intention and accurately predicting the position of the RGV in a variety of scenarios. Compared with the methods in the literature [28,29], the method proposed in this paper reduces the computation time by 97.19% and 83.16%, respectively, and improves the accuracy by 88.27% and 92.79%, respectively.