Projectile trajectory prediction method based on BiLSTM-KAN model under small sample

Abstract

The invention discloses a projectile trajectory prediction method based on a BiLSTM-KAN model under a small sample. The method is used for filling up missing information in a projectile trajectory. The method comprises the following steps: 1) establishing a six-degree-of-freedom projectile motion model for numerical simulation, and solving by using a fourth-order Runge-Kutta method, so as to construct a projectile trajectory simulation data set; 2) preprocessing the obtained projectile trajectory data set, constructing the preprocessed projectile trajectory data set into a supervision data set containing samples and labels, and dividing the data set into a training set and a test set; (3) a BiLSTM-KAN model is built; 4) carrying out training convergence on the BiLSTM-KAN model on the training set; and 5) finally, performing a projectile trajectory prediction experiment on the converged model on a test set.

Description

Technical Field

[0001] The present invention belongs to the technical field of trajectory prediction of uncontrolled projectiles for artillery, and particularly relates to a method for predicting the ballistic trajectory of projectiles based on the BiLSTM-KAN model under small samples. Background Art

[0002] In modern warfare, as the main firepower source of the army, it is extremely important to improve the shooting accuracy of artillery at targets. The feedback and correction technology of artillery is an important link to ensure the shooting accuracy of artillery. This link monitors the flight process of the projectile and feeds back the ballistic trajectory data to ensure that the artillery can self-correct after each launch and achieve accurate strikes on targets in subsequent shootings. However, when measuring equipment obtains ballistic trajectory data, due to the complexity of the battlefield environment and the artificial interference caused by the enemy, the measuring equipment may not be able to effectively monitor the flight process of the projectile, resulting in the situation of missing trajectory data. Therefore, it is necessary to predict the missing data of the projectile ballistic trajectory in this case to complete the supplement, so as to ensure the shooting accuracy of the artillery.

[0003] The traditional methods for predicting the ballistic trajectory of projectiles by the fire control system mainly include firing table function approximation and establishing a projectile ballistic model. Among them, firing table function approximation is to virtually extend the firing table to obtain a virtual ballistic outside the firing table, so that virtual closed-loop calibration can be carried out in advance, extending the calibration time, reducing the projectile-target deviation, and more accurately grasping the fighter opportunity. However, since the ballistic and correction amounts of each weapon and each type of ammunition it is equipped with need to be calculated separately, the versatility of this method is insufficient. The method of establishing a projectile ballistic model is based on the measuring equipment to obtain the initial velocity and ballistic parameters of the artillery projectile, and through the projectile motion differential equation, a ballistic model is established, and the method of numerical integration is used for iterative solution to complete the prediction of the ballistic trajectory. This method has the characteristics of strong real-time calculation ability and less computer resource occupation, and is widely used to solve the problem of projectile trajectory prediction. However, it does not consider the influence of meteorological factors and needs to perform multiple iterations to obtain data with higher accuracy, resulting in weak real-time performance. Therefore, it cannot meet the requirements of projectile trajectory prediction in different environments.

[0004] In addition to the above-mentioned projectile ballistic trajectory prediction methods, deep learning-based methods can also be used to complete the prediction of ballistic trajectories. This is because the projectile ballistic trajectory data belongs to time series data, and the Recurrent Neural Networks (RNNs) in deep learning can be used for the prediction of time series data. However, RNNs can only effectively process short time series data, and when faced with large-scale data sets such as projectile ballistic trajectory data, gradient explosion and gradient disappearance will occur. To overcome the above problems in RNN neural networks, the Long Short-Term Memory (LSTM) neural network and GRU neural network based on gating mechanisms have been successively proposed. More and more researchers obtain a large number of sample data sets through simulation and establish projectile ballistic trajectory prediction models based on LSTM or GRU neural networks. However, in practical applications, due to the influence of various environmental factors, geographical factors, human factors, etc., the data obtained by simulation is different from the real data. Therefore, it is necessary to obtain real projectile trajectory data as the training set. However, the sample data used in the existing model training is large in scale, which will cause large time costs and cost costs in practical applications. While the model trained with small sample data can reduce the cost of obtaining the data set and can significantly reduce the training time. However, traditional neural networks have disadvantages such as high parameter consumption and low interpretability, resulting in inaccurate prediction results of the model when the number of training samples is small. Summary of the Invention

[0005] The purpose of the present invention is to provide a projectile ballistic trajectory prediction method based on the BiLSTM-KAN model under small samples to overcome the deficiencies mentioned in the background art.

[0006] The technical solution to achieve the purpose of the present invention is as follows:

[0007] A projectile ballistic trajectory prediction method based on the BiLSTM-KAN model under small samples includes:

[0008] Step 1, establish a 6-degree-of-freedom artillery projectile motion model, use the fourth-order Runge-Kutta method for numerical integration to solve, and take the projectile range, projectile cross deviation, and projectile height as the characteristic attributes of the ballistic trajectory data set, so as to construct a ballistic trajectory simulation data set;

[0009] Step 2, perform data preprocessing on the data set obtained by simulation and construct a training set and a test set. First, perform first-order difference processing to increase the stationarity of the data, and then perform normalization processing to eliminate the dimension and order of magnitude of the data; divide the data into sample features and sample labels, construct a supervised data set, and finally divide the data set into a training set and a test set;

[0010] Step 3: Build the BiLSTM-KAN model, which consists of two layers of BiLSTM networks and KAN. Add a dropout layer between the two BiLSTM layers, and select the mean squared error function as the loss function of the model during training. The BiLSTM network is composed of two LSTM group neural network units. One processes the input sequence forward, and the other processes the input sequence backward. The hidden states output by the two LSTMs are concatenated to obtain the final hidden state of the BiLSTM network, and then this hidden state is input into KAN, and finally the output is completed by KAN.

[0011] Step 4: First, input the sample feature data in the training set constructed in Step 2 into the BiLSTM-KAN model constructed in Step 3, and output the predicted trajectory data. Secondly, calculate the mean squared error between the predicted trajectory data and the sample label data in the training set constructed in Step 2. Then use the backpropagation algorithm to transmit the mean squared error to each LSTM neuron, aiming to minimize the mean squared error, and continuously update the weight matrix and coefficients in each LSTM neuron. Finally, when the mean squared error begins to change stably, it indicates that the model has converged, and at this time, stop training.

[0012] Step 5: Input the input trajectory data in the test set constructed in Step 2 into the BiLSTM-KAN model after training in Step 4, and output the predicted trajectory data.

[0013] Compared with the prior art, the present invention has the following remarkable advantages:

[0014] (1) By using the first-order difference and normalization methods, the present invention greatly improves the smoothness of the projectile trajectory data and eliminates the influence of the dimension and order of magnitude of different features in the projectile trajectory simulation data set.

[0015] (2) By introducing bidirectional long short-term memory network units, compared with LSTM, the output of the BiLSTM network combines the outputs of the forward LSTM network and the backward LSTM network, and can effectively utilize the forward and backward dependencies in the ballistic trajectory data.

[0016] (3) By introducing KAN, compared with the traditional neural network which has the disadvantages of high parameter consumption and low interpretability, KAN can show better performance with fewer parameters in tasks with small data volume and high precision requirements.

[0017] The present invention will be further described in detail below with reference to the accompanying drawings. BRIEF DESCRIPTION OF THE DRAWINGS

[0018] Figure 1 It is a flowchart of a specific implementation method for projectile trajectory prediction based on the BiLSTM-KAN model.

[0019] Figure 2 It is a schematic structural diagram of the LSTM neural network unit in the present invention.

[0020] Figure 3 It is a schematic structural diagram of KAN in the present invention.

[0021] Figure 4 It is a process diagram of the projectile trajectory prediction of the BiLSTM-KAN model in the present invention. Specific Embodiments

[0022] The following describes the specific embodiments of the present invention in conjunction with the accompanying drawings, so that those skilled in the art can better understand the present invention. It should be particularly noted that in the following description, when the detailed descriptions of known functions and designs may dilute the main content of the present invention, these descriptions will be ignored here.

[0023] Reference Figure 1 , Figure 2 , Figure 3 and Figure 4 , the specific implementation scheme of the present invention is elaborated. The projectile trajectory prediction method based on the BiLSTM-KAN model under small samples includes the following steps:

[0024] Step 1, construct a projectile trajectory simulation data set under different conditions, specifically as follows:

[0025] (1) First, establish a six-degree-of-freedom artillery projectile motion model, and the specific expression is as follows:

[0026]

[0027] In the formula, m is the projectile mass, v is the projectile velocity, and t represents time; (F x2 , F y2 , F z2 ) are the projection components of the resultant force acting on the projectile in the ballistic coordinate system; (x, y, z) are the projectile position components on the (X g , Y g , Z g ) axes in the ground coordinate system; θ is the ballistic inclination angle of the ideal trajectory; ψ 1 is the component of the angle between the projectile velocity vector and the ideal trajectory in the vertical plane, and ψ 2 is the lateral component of this angle; J and I are the polar moment of inertia and the equatorial moment of inertia of the projectile respectively; (ω x1 , ω y1 , ω z1 ) are the projectile rotation angular velocity components in the projectile axis coordinate system; (M x1 , M y1 , M z1)They are the projection components of the aerodynamic resultant moment of the projectile in the ideal ballistic coordinate system; is the angle between the Y-axis of the ideal ballistic coordinate system and the projectile axis coordinate system, is the angle between the Z-axis of the ideal ballistic coordinate system and the projectile axis coordinate system; γ is the spin angle of the projectile; δ 1 is the elevation angle of attack, δ 2 is the azimuth angle of attack.

[0028] (2) Secondly, the fourth-order Runge-Kutta method is used to numerically integrate the six-degree-of-freedom projectile motion model to obtain a large number of ballistic trajectory sequences. The specific expression of the fourth-order Runge-Kutta method is as follows:

[0029]

[0030] In the formula, t n represents the current time, y n represents the current value, y n+1 represents the value at the next time, k 1 represents the slope at the start of the time period, k 2 represents the slope of the time period, k 3 represents the slope of the time period, k 4 represents the slope at the end of the time period, f represents the six-degree-of-freedom projectile motion differential equation system, h represents the integration step size, with a value of 0.001, indicating that a trajectory point is obtained every 0.001 s.

[0031] (3) The projectile trajectory simulation data sets under different conditions can be constructed as three-dimensional vectors {D 1 , D 2 ,..., D k}, where D k ={(x a , y a , z a )|m = 1, 2,..., q} is the k-th ballistic trajectory data, q represents the total number of trajectory points,

[0032] (x a , y a , z a ) are the range, lateral deviation, and altitude of the a-th trajectory point, respectively, which are three trajectory characteristic quantities.

[0033] Step 2: Preprocess the projectile trajectory simulation data set, divide the trajectory data into sample features and sample labels, construct a supervised data set, and divide the data set into a training set and a test set.

[0034] (1) First, perform data preprocessing on the projectile trajectory simulation dataset. The first-order difference method is used to increase the stationarity of the data. The specific form of the first-order difference method is as follows:

[0035] Δm t =m t+1 -m t

[0036] where t represents the time, m t represents the eigenvalue at the t-th moment, m t+1 represents the eigenvalue at the (t + 1)-th moment, and Δm t represents the difference between the (t + 1)-th moment and the t-th moment;

[0037] (2) Secondly, use the normalization method to eliminate the influence of different feature dimensions and magnitudes in the projectile trajectory simulation dataset:

[0038]

[0039] where: P x is the normalized trajectory data; P is the original trajectory data; P MAX is the maximum value of the trajectory features in the simulation data; P MIN is the minimum value of the trajectory features in the simulation data.

[0040] (3) Finally, starting from the initial trajectory point of a single ballistic trajectory, continuously select M trajectory points as sample features, then select the subsequent N trajectory points as sample labels, and then determine the subsequent sample features and sample labels in the same way, and so on, so as to construct a supervised dataset. Finally, divide the dataset into a training set and a test set.

[0041] Step 3, build the BiLSTM-KAN model. Among them, BiLSTM is a bidirectional long short-term memory network, which is composed of two LSTMs. One processes the input sequence forward; the other processes the input sequence backward. After the processing is completed, the hidden states output by the two LSTMs are concatenated to obtain the final hidden state of the BiLSTM network, and then this hidden state is input into KAN for dimensionality reduction, and finally KAN completes the final output. The mean square error function is used as the model loss function to construct the BiLSTM-KAN model, as follows:

[0042] (1) The structure of the LSTM neural network unit consists of a forget gate, an input gate, and an output gate. Among them, the role of the forget gate is to selectively forget a part of the information at the previous moment. The specific operation is to combine the hidden state h t-1 at the previous moment with the network input x t at the current moment as the input, and then obtain the proportion f t of the retained information through the sigmoid layer:

[0043] f t = σ(W f ·[h t-1 , x t + b f )

[0044] In the formula, W f and b f respectively represent the weight matrix and bias vector of the forget gate, and their function is to perform a linear transformation on the matrix concatenated by h t-1 and x t . σ represents the sigmoid activation function, and its function is to limit the output value within the range of 0 to 1.

[0045] The function of the input gate is to selectively store new information in the cell state. The specific operation mainly includes three parts. First, determine the proportion i t :

[0046] i t = σ(W i [h t-1 , x t + b i )

[0047] In the formula, W i and b i respectively represent the weight matrix and bias vector of the input gate, and their function is the same as that of the forget gate.

[0048] Then determine the new information to be stored, that is, the candidate information state

[0049]

[0050] In the formula, W C and b C respectively represent 's weight matrix and bias vector.

[0051] Finally, combine i t , and f t obtained through the forget gate, so as to obtain the new cell state C t :

[0052]

[0053] The function of the output gate is to determine which parts of the cell state C t at the current moment to output. First, the output proportion o t needs to be obtained through the sigmoid layer:

[0054] o t = σ(W o ·[h t-1 , x t + b o )

[0055] In the formula, W o and b o are respectively the weight matrix and bias vector of the output gate. Under the action of the sigmoid activation function, a value between 0 and 1 will also be output.

[0056] After obtaining o t , combining it with h t-1 can obtain the hidden state h at the current moment t :

[0057] h t = o t × tanh(C t )

[0058] In the formula: tanh represents the tanh activation function, which outputs a value between -1 and 1.

[0059] (2) Add a dropout layer between two BiLSTM layers. During the model training process, a certain proportion of neurons are randomly selected and deleted to improve the robustness of the model and reduce the risk of model overfitting. The dropout rate of the dropout layer is set to 0.1.

[0060] (3) KAN is a new type of neural network architecture. Its principle is based on the Kolmogorov - Arnold theorem, that is, any multivariate continuous function can be expressed as a combination of a finite number of univariate continuous functions.

[0061] Each weight parameter in KAN can be replaced by a univariate function, and these functions are parameterized in the form of spline functions, thus providing extremely high flexibility and being able to simulate complex functions with fewer parameters, enhancing the interpretability of the model.

[0062] KAN uses the B - spline function S(x) as the univariate function for each node, and its expression is:

[0063]

[0064] In the formula, k i represents the i - th coefficient that needs to be continuously optimized during training, B i(x) represents the i-th B-spline basis function defined on the grid, and n represents the number of basis functions. More grids mean more control and higher precision. The flexibility of the spline function enables it to adaptively model complex relationships in the data by adjusting its shape, thereby minimizing the approximation error and enhancing the ability of the KAN network to learn subtle patterns from high-dimensional datasets.

[0065] (4) The mean square error function is the loss function of the BiLSTM-KAN model during the training phase, and its specific expression is as follows:

[0066]

[0067] In the formula, MSE is the mean square error, N is the length of the ballistic trajectory data input into the model, and y j represents the true ballistic trajectory data of the j-th trajectory point, represents the ballistic trajectory data predicted by the model for the j-th trajectory point.

[0068] Step 4: Input the sample feature data in the training set constructed in Step 2 into the BiLSTM-KAN model constructed in Step 3, and output the predicted ballistic trajectory data; secondly, calculate the mean square error between the obtained predicted data and the sample label data in the training set constructed in Step 2; then use the backpropagation algorithm to transfer the mean square error to each neuron, and aim to minimize the mean square error, continuously update the weights between neurons and the parameters inside the neurons; finally, when the mean square error starts to change stably, it indicates that the model has converged, and stop the model training process.

[0069] Step 5: Use the BiLSTM-KAN model trained in Step 4 as the test model on the test set. First, input the ballistic trajectory data in the test set constructed in Step 2 into the BiLSTM-KAN model trained in Step 4, and output the predicted trajectory data; secondly, calculate the root mean square error and mean absolute error between the predicted sequence and the label data in the training set constructed in Step 2, and evaluate the quality of the model according to the root mean square error and mean absolute error. The specific calculation methods of the root mean square error and mean absolute error are as follows:

[0070]

[0071] In the formula, RMSE and MAE respectively represent the root mean square error and mean absolute error, N is the length of the ballistic trajectory data input into the model, and y j represents the true ballistic trajectory data of the j-th trajectory point, represents the ballistic trajectory data predicted by the model for the j-th trajectory point.

[0072] Embodiment

[0073] The specific implementation of the present invention will be described below in conjunction with the accompanying drawings, so that those skilled in the art can better understand the present invention. It should be noted that in the following description, when the detailed description of known functions and designs may dilute the main content of the present invention, these descriptions will be ignored here. Refer to Figure 1 , Figure 2 , Figure 3 and Figure 4 , the method for predicting the projectile trajectory based on the BiLSTM-KAN model under small samples, the specific steps are as follows:

[0074] Step 1, solve the 6-degree-of-freedom projectile motion model using the fourth-order Runge-Kutta method, and consider three influencing factors: ground wind speed, initial muzzle velocity of the gun, and elevation angle of the gun. A total of 154 projectile trajectory data are obtained, and the projectile range, lateral deviation of the projectile, and altitude of the projectile are used as the characteristic attributes of the projectile trajectory database to construct the ballistic trajectory simulation data set under different conditions.

[0075] (1) In order to simulate real ballistic data, the present invention sets the initial velocity, launch angle, and ground wind speed parameters of the external ballistic simulation, and adds errors to the projectile initial velocity, weapon elevation angle, and projectile mass. The final obtained data set is shown in Table 1:

[0076] Table 1 Composition of the simulation data set

[0077]

[0078] (2) The present invention establishes a 6-degree-of-freedom projectile motion model, and the specific expression is as follows:

[0079]

[0080] In the formula, m is the projectile mass, v is the projectile velocity, and t represents time; (F x2 , F y2 , F z2 ) are the projection components of the resultant force acting on the projectile in the ballistic coordinate system; (x, y, z) are the projectile position components on the (X g , Y g , Z g ) axes in the ground coordinate system; θ is the ballistic inclination angle of the ideal trajectory; ψ 1 is the component of the angle between the projectile velocity vector and the ideal trajectory in the vertical plane, and ψ 2 is the lateral component of this angle; J and I are the polar moment of inertia and equatorial moment of inertia of the projectile respectively; (ω x1 , ω y1 , ω z1 ) are the projectile rotational angular velocity components in the projectile axis coordinate system; (M x1 , M y1,M z1 are the projection components of the aerodynamic resultant moment of the projectile in the ideal ballistic coordinate system; is the angle between the Y-axis of the ideal ballistic coordinate system and the projectile axis coordinate system, is the angle between the Z-axis of the ideal ballistic coordinate system and the projectile axis coordinate system; γ is the spin angle of the projectile; δ 1 is the elevation angle of attack, δ 2 is the azimuth angle of attack.

[0081] In the present invention, it is assumed that the initial velocity of a 155 mm artillery projectile is 800 m / s, the ground wind speed is a constant wind force of 2 - 10 m / s, the elevation angle of the weapon is set to 20° - 50°, the rifling pitch η = 20, the angular velocity of the Earth's rotation w D = 7.2922×10 -5 rad / s, the effective Earth radius in China R = 6358.299 km, the standard value of the gravitational acceleration on the ground G = 9.80 m / s 2 , the specific heat ratio of air is 1.404, the length of the projectile is 0.9 m, the projectile diameter is 0.155 m, the mass of the projectile is 45.5 kg, the initial coordinate values of the artillery launch are all 0, and the external ballistic aerodynamic parameters are obtained by parameter identification of the projectile flight experimental data.

[0082] During the launch of a 155 mm artillery, the ballistic trajectory of the projectile will be affected by many uncertain factors such as artillery launch disturbances, gun barrel conditions, and meteorological conditions. In order to more realistically simulate the complexity of the battlefield environment, a random error of -40 to 40 m / s is added to the initial velocity of the projectile to simulate the impact of the strong shock during artillery launch on the initial velocity of the projectile, and a random error of -1° to 1° is added to the elevation angle of the artillery to simulate the jump angle error caused by the weapon line jitter during artillery launch. At the same time, there will be a certain deviation in the mass of the projectile, and the present invention assumes that this deviation follows a normal distribution of N(45.5, 0.1).

[0083] (3) Secondly, the fourth-order Runge-Kutta method is used to numerically integrate the six-degree-of-freedom projectile motion model to obtain a large number of ballistic trajectory sequences. The specific expression of the fourth-order Runge-Kutta method is as follows:

[0084]

[0085] In the formula, t n represents the current time, y n represents the current value, y n+1 represents the value at the next time, k 1 represents the slope at the beginning of the time period, k 2 represents the slope of the time period, k 3 represents the slope of the time period, k4 represents the slope at the end of the time period, f represents the differential equations of motion of the six-degree-of-freedom projectile, h represents the integration step size, with a value of 0.001, indicating that a trajectory point is obtained every 0.001 s.

[0086] Step 2: Preprocess the projectile trajectory simulation data set, divide the trajectory data into two parts: sample features and sample labels, form a supervised data set, and divide the data set into a training set and a test set.

[0087] Among them, preprocess the projectile trajectory simulation data set and divide the trajectory data into sample features and sample labels, specifically as follows:

[0088] (1) First, preprocess the projectile trajectory simulation data set, and use the first-order difference method to increase the stationarity of the data. The specific form of the first-order difference method is:

[0089] Δm t = m t+1 - m t

[0090] In the formula, t represents the time, m t represents the eigenvalue at the t-th moment, m t+1 represents the eigenvalue at the (t + 1)-th moment, Δm t represents the difference between the (t + 1)-th moment and the t-th moment;

[0091] (2) Secondly, use the normalization method to eliminate the influence of different feature dimensions and magnitudes of the projectile trajectory simulation data set:

[0092]

[0093] In the formula: P x is the normalized trajectory data; P is the original trajectory data; P MAX is the maximum value of the trajectory features in the simulation data; P MIN is the minimum value of the trajectory features in the simulation data.

[0094] (3) Starting from the ballistic starting position, continuously select 30 trajectory points as sample features, and then continuously select the next 1 trajectory point as the sample label. Determine the next set of sample features and sample labels in the same way, and so on to obtain a supervised data set. The supervised data set is divided into a training set and a test set. Among them, the training set contains 104 ballistic trajectories, and the test set contains 50 ballistic trajectories.

[0095] Step 3: Build the BiLSTM-KAN model. BiLSTM is a bidirectional long short-term memory network, which is composed of two LSTMs. One processes the input sequence forward, and the other processes the input sequence backward. After processing, the hidden states output by the two LSTMs are concatenated to obtain the final hidden state of the BiLSTM network. Then, this hidden state is input into KAN for dimensionality reduction, and finally, KAN completes the output. The mean square error function is used as the model loss function to construct the BiLSTM-KAN model, as follows:

[0096] (1) The structure of the LSTM neural network unit consists of an input gate, a forget gate, and an output gate. Among them, the role of the forget gate is to selectively forget part of the information at the previous moment. The specific operation is to combine the hidden state h t-1 at the previous moment with the network input x t at the current moment as the input, and then obtain the proportion f t of the retained information through the sigmoid layer:

[0097] f t =σ(W f ·[h t-1 ,x t +b f )

[0098] In the formula, W f and b f represent the weight matrix and bias vector of the forget gate respectively. Their role is to perform a linear transformation on the matrix concatenated by h t-1 and x t . σ represents the sigmoid activation function, which is used to limit the output value within the range of 0 to 1.

[0099] The role of the input gate is to selectively store new information in the cell state. The specific operation mainly includes three parts. First, determine the proportion i t of the new information to be stored:

[0100] i t =σ(W i [h t-1 ,x t +b i )

[0101] In the formula, W i and b i represent the weight matrix and bias vector of the input gate respectively, and their role is the same as that of the forget gate.

[0102] Then determine the new information to be stored, that is, the candidate information state

[0103]

[0104] Where W C and b C respectively represent the weight matrix and the bias vector of

[0105] Finally, combining the i obtained in the above two steps t , and f obtained through the forget gate t , thus obtaining a new cell state C t :

[0106]

[0107] The role of the output gate is to determine which parts of the cell state C at the current moment are output. First, the output ratio o needs to be obtained through the sigmoid layer t : t :

[0108] o t = σ(W o ·[h t-1 , x t + b o )

[0109] Where W o and b o are respectively the weight matrix and the bias vector of the output gate. Under the action of the sigmoid activation function, a value between 0 and 1 will also be output.

[0110] After obtaining o t , combining it with h t-1 can obtain the hidden state h at the current moment t :

[0111] h t = o t × tanh(C t )

[0112] Where tanh represents the tanh activation function, which will output a value between -1 and 1.

[0113] (2) Add a dropout layer between two BiLSTM layers. During the model training process, a certain proportion of neurons are randomly selected and deleted to improve the robustness of the model and reduce the risk of model overfitting. The dropout rate of the dropout layer is set to 0.1.

[0114] (3) KAN is a new type of neural network architecture, and its principle is based on the Kolmogorov - Arnold theorem, that is, any multivariate continuous function can be expressed as a combination of a finite number of univariate continuous functions.

[0115] In KAN, each weight parameter can be replaced by a univariate function, and these functions are parameterized in the form of spline functions, thus providing extremely high flexibility and being able to simulate complex functions with fewer parameters, enhancing the interpretability of the model.

[0116] KAN uses the B - spline function S(x) as the univariate function for each node, and its expression is:

[0117]

[0118] In the formula, k i represents the i - th coefficient that needs to be continuously optimized during training, B i (x) represents the i - th B - spline basis function defined on the grid, n represents the number of basis functions, and more grids mean more control and higher precision. The flexibility of the spline function enables it to adaptively model complex relationships in the data by adjusting its shape, thereby minimizing the approximation error and enhancing the ability of the KAN network to learn subtle patterns from high - dimensional datasets.

[0119] (4) The mean - square error function is the loss function of the BiLSTM - KAN model during the training phase, and its specific expression is as follows:

[0120]

[0121] In the formula, MSE is the mean - square error, N is the length of the ballistic trajectory data input into the model, y j represents the true ballistic trajectory data of the j - th trajectory point, represents the ballistic trajectory data predicted by the model for the j - th trajectory point.

[0122] Step 4: Input the sample feature data in the training set constructed in Step 2 into the BiLSTM - KAN model constructed in Step 3, and output the predicted ballistic trajectory data; secondly, solve the mean - square error between the obtained ballistic trajectory prediction data and the sample label data in the training set constructed in Step 2; then use the backpropagation algorithm to transfer the mean - square error to each neuron, and with the aim of minimizing the mean - square error, continuously update the weights between neurons and the parameters inside the neurons; finally, when the mean - square error starts to change stably, it indicates that the model has converged, and stop the model training process.

[0123] Step 5: Use the BiLSTM-KAN model obtained in Step 4 to predict the data on the test set. Input the input trajectory sequence in the test set constructed in Step 2 into the test model and output the predicted trajectory sequence. Secondly, calculate the errors in range, cross deviation, and altitude between the predicted trajectory data and the sample labels in the test set constructed in Step 2, and then find the average of the root mean square error and the mean absolute error of the three features as the final evaluation result. The LSTM model is set as the comparison model in the experiment. When the sampling time of the trajectory points is 0.01 s, input 30 trajectory points and predict the next 300 and 500 trajectory points respectively, that is, when the input time is 0.3 s, predict the ballistic trajectories in the next 3 s and 5 s respectively. The prediction results of the two models are shown in Table 2. The number of training rounds of the model of the present invention is selected as 200, the learning rate is selected as 0.001, and the Adamw optimizer is selected as the optimizer.

[0124] Table 2 Comparison table of model prediction results

[0125]

[0126] It can be seen from the experimental results that when the input time of the model is fixed, when the output time of the model is 3 s and 5 s, the prediction effect of the BiLSTM-KAN model is significantly improved compared with the LSTM model.

Claims

1. A projectile trajectory prediction method based on BiLSTM-KAN model under small sample conditions, characterized in that: include: Step 1: Establish a 6-DOF artillery projectile motion model, use the 4th-order Runge-Kutta method for numerical integration, and use the projectile range, projectile lateral deviation, and projectile shooting height as characteristic attributes of the ballistic trajectory data set to construct a ballistic trajectory simulation data set; Step 2: Preprocess the data set obtained by simulation and construct the training set and test set. First, perform first-order difference processing to increase the stability of the data, and then perform normalization processing to eliminate the dimension and order of magnitude of the data; divide the data into sample features and sample labels, construct a supervised data set, and finally divide the data set into a training set and a test set; Step 3: Build a BiLSTM-KAN model, which consists of two layers of BiLSTM networks and KAN. Add a dropout layer between the two BiLSTM layers, and select the mean square error function as the loss function of the model during training. The BiLSTM network is composed of two LSTM neural network units, one for forward processing of the input sequence and the other for reverse processing of the input sequence. The hidden states of the two LSTM outputs are concatenated to obtain the final hidden state of the BiLSTM network, which is then input into the KAN, which then completes the final output. Step 4: First, input the sample feature data in the training set constructed in step 2 into the BiLSTM-KAN model constructed in step 3, and output the predicted trajectory data; second, calculate the mean square error between the predicted trajectory data and the sample label data in the training set constructed in step 2; then use the back propagation algorithm to transfer the mean square error to each LSTM neuron, with the purpose of minimizing the mean square error, to continuously update the weight matrix and coefficients in each LSTM neuron; finally, when the mean square error begins to change steadily, it indicates that the model has converged, and the training is stopped at this time; Step 5: Input the input trajectory data in the test set constructed in step 2 into the BiLSTM-KAN model trained in step 4, and output the predicted trajectory data.

2. The projectile trajectory prediction method based on the BiLSTM-KAN model under small sample conditions according to claim 1 is characterized in that: The 6-DOF projectile motion model is: Where m is the mass of the projectile, v is the velocity of the projectile, and t is the time; (F x2 ,F y2 ,F z2 ) are the projected components of the resultant force acting on the projectile in the ballistic coordinate system; (x, y, z) are the projected components of the resultant force acting on the projectile in the ballistic coordinate system; g ,Y g ,Z g ) axis; θ is the ballistic inclination angle of the ideal trajectory; ψ1 is the component of the angle between the projectile velocity vector and the ideal trajectory in the vertical plane, and ψ2 is the lateral component of the angle; J and I are the polar moment of inertia and equatorial moment of inertia of the projectile, respectively; (ω x1 ,ω y1 ,ω z1 ) are the angular velocity components of the projectile in the projectile axis coordinate system; (M x1 ,M y1 ,M z1 ) are the projection components of the aerodynamic torque of the projectile in the ideal ballistic coordinate system; is the angle between the Y axis of the ideal ballistic coordinate system and the missile axis coordinate system. It is the angle between the Z axis of the ideal ballistic coordinate system and the projectile axis coordinate system; γ is the rotation angle of the projectile; δ1 is the elevation angle of attack, and δ2 is the directional angle of attack.

3. The projectile trajectory prediction method based on the BiLSTM-KAN model under small sample conditions according to claim 1 is characterized in that: The 4th-order Runge-Kutta method is expressed as follows: Where, t n represents the current time, y n Indicates the current value, y n+1 Indicates the value at the next moment, k1 indicates the slope at the beginning of the time period, and k2 indicates The slope of the time period, k3 represents The slope of the time period, k4 represents the slope at the end of the time period, f represents the 6-DOF projectile motion differential equations, and h represents the integration step size, which is 0.001, indicating that a trajectory point is obtained every 0.001s.

4. The projectile trajectory prediction method based on the BiLSTM-KAN model under small sample conditions according to claim 1 is characterized in that: The constructed projectile trajectory simulation data set is: {D1,D2,...,D k } Where D k ={(x a ,y a ,z a )|m=1,2,...,q} is the kth ballistic trajectory data, q represents the total number of trajectory points, (x a ,y a ,z a ) are the three trajectory feature quantities of the a-th trajectory point, namely, the range, lateral deviation and shooting height.

5. The projectile trajectory prediction method based on the BiLSTM-KAN model under small sample conditions according to claim 1 is characterized in that: Step 2 specifically includes the following steps: First, the projectile trajectory simulation data set is preprocessed, and the first-order difference method is used to increase the stability of the data. The specific form of the first-order difference method is: Δm t =m t+1 -m t Where t represents the time, m t represents the eigenvalue at time t, m t+1 represents the eigenvalue at time t+1, Δm t Represents the difference between time t+1 and time t; Secondly, the normalization method is used to eliminate the influence of different feature dimensions and magnitudes of the projectile trajectory simulation data set: Where: P x is the normalized trajectory data; P is the original trajectory data; P MAX is the maximum value of the trajectory feature in the simulation data; P MIN is the minimum value of the trajectory feature in the simulation data; Finally, starting from the starting point of a single ballistic trajectory, M trajectory points are continuously selected as sample features, and then the next N trajectory points are selected as sample labels. Then, the same method is used to determine the subsequent sample features and sample labels, and so on, so as to construct a supervised data set, and finally divide the data set into a training set and a test set.

6. The projectile trajectory prediction method based on the BiLSTM-KAN model under small sample conditions according to claim 1 is characterized in that: The structure of the LSTM neural network unit consists of a forget gate, an input gate, and an output gate: The function of the forget gate is to selectively forget part of the information of the previous moment. The specific operation is to t-1 With the current network input x t Combined as input, and then through the sigmoid layer to get the proportion of retained information f t : f t =σ(W f ·[h t-1 ,x t ]+b f ) Where W f With b f They represent the weight matrix and bias vector of the forget gate respectively, and σ represents the sigmoid activation function; The function of the input gate is to selectively store new information in the cell state. The specific operation mainly includes three parts. First, determine the proportion of new information that needs to be stored. t : i t =σ(W i [h t-1 ,x t ]+b i ) Where W i With b i Represent the weight matrix and bias vector of the input gate respectively; Then determine the new information that needs to be stored, that is, the candidate information state Where W C With b C Respectively The weight matrix and bias vector of ; Finally, combine the i obtained in the previous two steps t , And f obtained through the forget gate t , thus obtaining a new cell state C t : The function of the output gate is to determine the output of the current cell state C t Which parts of the output need to be passed through the sigmoid layer first to get the output ratio o t : the t =σ(W o ·[h t-1 ,x t ]+b o ) Where W o With b o They are the weight matrix and bias vector of the output gate respectively; Get o t After that, with h t-1 Combined, we can get the hidden state h at the current moment t : h t =o t ×tanh(C t ) Where: tanh represents the tanh activation function.

7. The projectile trajectory prediction method based on the BiLSTM-KAN model under small sample conditions according to claim 1 is characterized in that: KAN uses the B-spline function S(x) as the univariate function of each node, and its expression is: Where k i represents the i-th coefficient that needs to be continuously optimized during training, B i (x) represents the i-th B-spline basis function defined on the grid, and n represents the number of basis functions.

8. The projectile trajectory prediction method based on the BiLSTM-KAN model under small sample conditions according to claim 1 is characterized in that: The mean square error function is specifically expressed as follows: Where MSE is the mean square error, N is the length of the ballistic trajectory data input to the model, and y j represents the actual ballistic trajectory data of the jth trajectory point, Represents the model-predicted ballistic trajectory data for the jth trajectory point.

9. The projectile trajectory prediction method based on the BiLSTM-KAN model under small sample conditions according to claim 1 is characterized in that: It also includes calculating the root mean square error and mean absolute error between the predicted sequence and the labeled data in the training set constructed in step 2, and evaluating the quality of the model based on the root mean square error and the mean absolute error.

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