Optimization of Multistage Coilgun Based on Neural Network and Intelligent Algorithm

Abstract

The parameter optimization of a multistage synchronous induction coilgun (SICG) is a time-consuming task. Traditional machine learning methods can accelerate the process by building predictive models, but they require separate modeling for an SICG with different stages, which requires numerous datasets and is a cumbersome process. This paper proposes a method for building a predictive model for an SICG with different stages based on a recurrent neural network (RNN). In this method, the feed time of a 2- to 10-stage SICG is selected from the standard orthogonal design table as the training and test datasets, and the current filament method (CFM) is used to calculate the dataset label. The gate recurrent unit (GRU) neural network is used to study the training dataset, and the predictive model has good accuracy with respect to the test dataset, with an average error of 0.022. The predictive model and a particle swarm optimization (PSO) algorithm are applied to optimize the feed time of the SICG with different stages. The results show that the three-stage SICG can achieve a muzzle velocity of 50 m/s for a projectile, while the maximum muzzle velocity of the three-stage SICG in all datasets is 46.87 m/s.

1. Introduction

Electromagnetic launchers are currently under the spotlight of military studies as they surpass mechanical and chemical launchers because of their high initial speed, long range, good controllability, low cost, fast response speed, and silent, light- and smoke-free emission [1,2,3]. A synchronous induction coilgun (SICG) is a popular type of electromagnetic launcher. It is mainly composed of a driving coil, a driving circuit, a pulse power source, and a projectile. It uses the magnetic traveling wave generated by the pulse to accelerate a projectile. By simply increasing the number of drive coils, an SICG can accelerate large-mass loads to very high speeds, which makes it a hot topic of research [4,5,6].

A multi-stage SICG (MSSICG) usually has the same parameters for components such as the driving coil and a projectile due to processing and cost constraints. Therefore, the feed time of each driving coil of an MSSICG becomes the crucial factor to determine the final muzzle velocity of the projectile [7,8,9], for which the number of coils can be increased to achieve the desired muzzle velocity. An inappropriate feed time often leads to undesirable outcomes.

The current methods for optimizing the feed time of MSSICGs mainly include the orthogonal experimental method and intelligent algorithms. The orthogonal experimental method can obtain reliable results with fewer tests, but its optimization results are limited by the range of the selected level, and its convergence speed is slow, so it is difficult to determine data trends [10]. On the other hand, intelligent algorithms, such as genetic algorithms (GAs) [11], ant colony algorithms (ACOs) [12], simulated annealing (SA) algorithms [13], etc., can more easily determine the global optimal solution, so they are more favored. However, obtaining the global optimal solution usually requires many calculations executed via iteratively calling the circuit model, which is very time consuming. Therefore, to save a great deal of calculation time, it is recommended to replace the current filament model with the predictive model in the optimization process. For example, Liu et al. [14] used three typical machine learning methods, namely, LSSVM, KELM, and ESN, to perform non-parametric modeling of a five-level SICG. The results showed that the error between the predictive model and the current filament model was only 2.9%. However, these machine learning methods can only handle fixed-length inputs. Additionally, they can only learn the input information at the current moment and will not refer to the input information at other moments. However, when the number of coils of an MSSICG is used as the optimization target, the number of feeds will change, which means that the length of the input will also change. At the same time, there is a strong sequential relationship between each feed time, so it is necessary to develop a machine learning method that is not constrained by the fixed length input space and can store the state of the previous input and combine it with the current input.

This paper uses a particle swarm optimization (PSO) [15] algorithm based on the prediction model of the gated recurrent unit (GRU) neural network [16] to optimize the feed time of an MSSICG. By optimizing the feed time, we aim to achieve a final projectile muzzle velocity of 50 m/s with the minimum number of coil stages, thereby reducing costs and increasing efficiency.

2. Simulation Model

2.1. Basic Principle and Structure

A 3D schematic diagram of an MSSICG is shown in Figure 1. According to Faraday’s law of electromagnetic induction, when a changing current flows through a coil, the induced current will act on the projectile. In addition, according to Lenz’s law, when the current on the coil continues to increase, the current in the opposite direction will act on the projectile. If the projectile’s center is located on the right side of the coil center at this time, according to the left-hand rule, the projectile will be accelerated by the force to the right.

A typical external circuit and current waveform of an MSSICG are shown in Figure 2. The capacitor and the coil form the RLC oscillation circuit. The trigger switch is closed to discharge the capacitor instantaneously. In addition, the addition of a diode can prevent the reverse charging of the capacitor, thus ensuring that there is only forward current in the drive coil, which decays exponentially according to the RL circuit formula. The structural parameters of an MSSICG are shown in

Table 1. Structural parameters of MSSICG.

Rc (mm)	Lc (mm)	D (mm)	N	S (mm)	Rp (mm)
4	35	1	6	5	3.25
Lp (mm)	V (m/s)	M (g)	U (V)	C (mF)
25	0	6.37	400	1

, where R_c is the inner radius of the coil, L_c is the axial length of the coil, D is the diameter of the coil wire, N is the number of layers of the coil wire, S is the distance between the coils, R_p is the radius of the projectile, L_p is the axial length of the projectile, V is the initial velocity of the projectile, M is the mass of the projectile, U is the initial voltage of the capacitor, and C is the capacitance of the capacitor.

2.2. Current Filament Model

In order to build a predictive model, one must first obtain a valid model input and output dataset. Compared with costly prototype experiments, the mainstream method of obtaining data is through numerical simulation, which currently mainly includes the finite element method (FEM) and the current filament method (CFM) [17]. The FEM uses different discretization types to construct numerical model equations that are similar to partial differential equations. The solution obtained is the approximate solution of the partial differential equation. The solution accuracy can be adjusted by adjusting the mash division and the solution time step, but the disadvantage of this procedure is that the solution process is very lengthy. In order to improve the efficiency of obtaining datasets, we will use the CFM for modeling and simulation.

As its name suggests, the CFM entails dividing a projectile with a certain length and thickness into a circle of current filaments, which can solve the problem of an uneven distribution of induced current on the projectile caused by the skin effect. The basic idea is similar to the finite element method, both of which discretize the projectile and divide it into m and n layers along the axial and radial directions of the projectile. The calculation accuracy of the CFM can be improved by adjusting the values of m and n. The equivalent circuit model of the CFM is shown in Figure 3.

After the projectile is divided, each ring can be regarded as a single-turn coil. According to Kirchhoff’s law, the circuit equations of each turn of the driving coil and each current filament loop of the projectile can be expressed as follows:

$$\left\{\begin{array}{l}{R}_d{I}_d+L_d\frac{d{I}_d}{dt}+M\frac{d(M{I}_p)}{dt}=U_d\\ R_p{I}_p+L_p\frac{d{I}_p}{dt}+M\frac{d(M{I}_d)}{dt}=0\end{array}\right.$$

(1)

Since the model is symmetrical along the Z-axis, when the projectile moves toward the Z-axis, the relationship between velocity and displacement is as follows:

$$\frac{dz}{dt}=v$$

(2)

Substitute Equation (2) into Equation (1) and expand it; then, organize it into a matrix form:

$$\left[\begin{array}{cc}{R}_d& 0\\ 0& R_a\end{array}\right]\left[\begin{array}{c}{I}_d\\ I_a\end{array}\right]+\left[\begin{array}{cc}{L}_d& 0\\ 0& L_p\end{array}\right]\left[\begin{array}{c}\frac{d{I}_d}{dt}\\ \frac{d{I}_p}{dt}\end{array}\right]+\left[\begin{array}{cc}0& M\\ M& 0\end{array}\right]\left[\begin{array}{c}\frac{d{I}_d}{dt}\\ \frac{d{I}_p}{dt}\end{array}\right]+v\left[\begin{array}{cc}0& \frac{dM}{dx}\\ \frac{dM}{dx}& 0\end{array}\right]\left[\begin{array}{c}{I}_d\\ I_p\end{array}\right]=\left[\begin{array}{c}{U}_d\\ 0\end{array}\right]$$

(3)

Perform a force analysis on the projectile. When the projectile moves a distance of d_z along the axis, according to the law of conservation of energy, the work executed by the electromagnetic force should be equal to the change in electromagnetic energy. The force F_z acting on the projectile in the axial direction is as follows:

$$F_z=\frac{dM}{dz}{I}_d{I}_p$$

(4)

If the mass of the projectile is m, then according to Newton’s second law, it can be obtained:

$$\frac{dv}{dt}=\frac{F}{m}=\frac{1}{m}\frac{dM}{dz}{I}_d{I}_p$$

(5)

Since the external circuit capacitor voltage U_d satisfies the following conditions when time t is greater than time t₀, it follows that

$$\frac{d{U}_d}{dt}=-\frac{1}{C}{I}_d$$

(6)

By combining Equations (2), (3), (5), and (6), the differential equation group of a single-turn driving coil and the projectile current filament loop can be obtained and then generalized to multi-stage driving coils and expressed in matrix form as follows:

$$\left\{\begin{array}{l}\frac{dI}{dt}=(L+M{)}^{-1}(U-v\frac{dM}{dx}I-RI)\\ \frac{d{U}_d}{dt}=-\frac{I_d}{c}\\ \frac{dv}{dt}=\frac{1}{m}{\displaystyle \sum _{i=1}^m{\displaystyle \sum _{j=1}^n{I}_{di}^{}}}{I}_{ai}^{}\frac{d{M}_{da,ij}}{dx}\\ \frac{dx}{dt}=v\end{array}\right.$$

(7)

Based on Equation (7), it is possible to calculate the electromagnetic and motion characteristics of any MSSICG, where R, L, and I are the resistance matrix, inductance matrix, and current column vector of the driving coil d₁~d_m and current filament a₁~a_n, respectively; U is the initial voltage column vector; U_d is the sub matrix of U; M is the mutual inductance matrix; and c, m, v, and x represent capacitance, mass, velocity, and displacement, respectively.

2.3. Experimental Verification

A prototype of a single-stage SICG is shown in Figure 4. Its parameters are the same as those in Table 1. We will use it to verify the accuracy of the current wire method.

First, we must verify the accuracy of the external circuit simulation. We connect a 0.1 Ω resistor in series with the external circuit, charge the capacitor fully, use a switch to trigger the capacitor to discharge, and measure the voltage on the resistor. A comparison between the results of the simulation and the experiment is shown in Figure 5. Then, we apply ANOVA to compare the results. The analysis of variance result for the simulated data is 3.8561, while the analysis of variance result for the experimental data is 4.0255. This demonstrates that our external circuit simulation is valid.

Next, we verify the accuracy of the model’s simulation. We place a photoelectric sensor near the muzzle of the barrel and connect it to the single-chip microcomputer. The photoelectric sensor can convert the optical signal into an electrical signal. When an obstruction passes through the sensor, the output of the sensor changes from a low level to a high level, and the duration of the high level is recorded by a single-chip microcomputer. Assuming that the speed of the projectile remains unchanged during the high level period the speed of the projectile is equal to the length of the projectile divided by the duration of the high level period. The single-chip microcomputer completes the calculation and outputs the result to the LCD screen.

The measurement results are shown in Figure 6a, in which it is shown that the resulting muzzle velocity is 20.718 m/s. The simulation results of the current wire model are shown in Figure 6b. It can be seen from the figure that the muzzle velocity is 22.424 m/s. It can be seen that the difference between the two results is less than 10%. The difference should stem from the fact that the actual hand-wound coil is not as ideal as it is in the simulation model. In addition, there are other pertinent factors such as friction and air resistance. The above experiment was repeated three times, and the difference between the results was less than 10% each time, which shows that it is feasible to replace the results of real experiments with the results calculated using the CFM.

3. Recurrent Neural Network

A neural network can be regarded as a black box that can fit any function. Approximation via superposition of a sigmoidal function [18] has proved this point. As long as the corresponding network is fully trained, the desired output can be obtained through the input. However, the no free lunch theorem also points out that there is no machine learning algorithm that can fit all problems in a limited search space [19]. Ordinary neural networks can only process independent and fixed-length inputs. The previous input and the current input are completely irrelevant with respect to a neural network. However, the number of feeds of an MSSICG varies with the number of coils, so the length of the input will also change. At the same time, the output is not only affected by the current feeding time but also by the previous feeding time. A recurrent neural network (RNN) can solve the problems mentioned above [20].

The network structure of an RNN is shown in Figure 7. It can be seen from the figure that an RNN is essentially a fully connected neural network, in which the role of the embedding layer is to convert the input into a low-dimensional and dense vector, and the role of the linear layer is to convert the output dimension of the hidden layer into the required output dimension. However, the difference is that an RNN splices multiple fully connected neural networks together. The activation value of the current RNN Cell is calculated from the current input and the activation value of the previous RNN Cell, so the RNN can remember each moment’s information. The structure of an RNN Cell is shown in Figure 8.

In addition, the activation function usually chooses tanh because the value range of the tanh function is between a negative and positive value, which is more conducive to the calculation of RNN. We can conclude that the formula modelling the behavior of an RNN is as follows:

h_t = tanh (W_hhh_{t − 1} + b_hh + W_ihx_t + b_ih)

(8)

where h_t is the output of the hidden layer at the current moment, tanh is the activation function, W_hh and W_ih are the weights of the hidden layer, b_hh and b_ih are the biases of the hidden layer, and x_t is the input at the current moment.

However, RNNs also suffer from the problem of gradient descent during the training process. When the input sequence is very long, the advantage of RNN that can save each piece of input information becomes a disadvantage, so we used an improved GRU on the basis of an RNN. The structure of the GRU is shown in Figure 9.

A GRU can save the important information in the input and ignore the unimportant information. This is because a GRU introduces an update gate Z_t and a reset gate r_t. The reset gate r_t can control whether the calculation of the candidate state ĥ_t depends on the state h_{t − 1} at the previous moment. The update gate Z_t controls how much information the current state h_t needs to keep from h_{t− 1} and how much new information needs to be received from ĥ_t. The formula for a GRU can be summarized as follows:

$$\left\{\begin{array}{l}{r}_t=\sigma (W_r\cdot [h_{t-1},x_t])\\ {\overline{h}}_t=\tanh (W\cdot [r_t\ast h_{t-1},x_t])\\ z_t=\sigma (W_z\cdot [h_{t-1},x_t])\\ h_t=(1-z_t)\ast h_{t-1}+z_t\ast {\overline{h}}_t\end{array}\right.$$

(9)

It can be gleaned from the formula that there is a linear relationship between the current state h_t and the previous state h_{t − 1}, and there is also a non-linear relationship, which can alleviate the problem of gradient disappearance to a certain extent. At the same time, in the problem we want to solve, we only need to transform the final hidden layer output ${\mathrm{h}}_{\mathrm{n}}$ into the desired output through a linear layer, so the output layer of the network and the corresponding linear layer are unnecessary. The simplified network structure is shown in Figure 10.

4. Predictive Model

Before building a predictive model, we first need to obtain a sufficient dataset. In order to make the data representative, we must select uniformly dispersed and neatly comparable data from the standard orthogonal design table for training. Then, we set the parameters of the MSSICG’s external circuit, coils, projectiles, etc., to the same value; set the number of coils in the range from 2 to 10; and use the feed time of each coil as a training factor. The number of level values selected by each factor is 8. The selected factors and levels are shown in

Table 2. Factors and levels.

Levels	Factors
	t2 (ms)	t3 (ms)	t4 (ms)	t5 (ms)	t6 (ms)
1	1.20	2.10	2.90	3.55	4.10
2	1.30	2.20	2.98	3.62	4.17
3	1.40	2.30	3.06	3.69	4.24
4	1.50	2.40	3.14	3.76	4.31
5	1.60	2.50	3.22	3.83	4.38
6	1.70	2.60	3.30	3.90	4.45
7	1.80	2.70	3.38	3.97	4.52
8	1.90	2.80	3.46	4.04	4.59
Levels	Factors
	t7 (ms)	t8 (ms)	t9 (ms)	t10 (ms)
1	4.65	5.20	5.65	6.02
2	4.72	5.26	5.70	6.07
3	4.79	5.32	5.75	6.12
4	4.86	5.38	5.80	6.17
5	4.93	5.44	5.85	6.22
6	5.00	5.50	5.90	6.27
7	5.07	5.56	5.95	6.32
8	5.14	5.62	6.00	6.37

.

We used the standard orthogonal design table L₆₄(8⁹) to select data for numerical simulation because the standard orthogonal design table still has orthogonality even after deleting a given column; thus, to address the situation in which the number of coils will change, we only needed to delete the data of the corresponding column from the table. Finally, we obtained a total of 520 sets of data through numerical simulation, some of which are shown in

Table 3. Dataset.

	1	2	3	…	385	…	520
t2 (ms)	1.20	1.20	1.20	…	1.20	…	1.90
t3 (ms)	2.10	2.20	2.30	…	2.10	…
t4 (ms)	2.90	3.06	3.22	…	2.90	…
t5 (ms)	3.56	3.76	3.97	…		…
t6 (ms)	4.10	4.38	4.45	…		…
t7 (ms)	4.66	5.00	5.14	…		…
t8 (ms)	5.20	5.56	5.26	…		…
t9 (ms)	5.66	6.00	5.80	…		…
t10 (ms)	6.02	6.07	6.12	…		…
vout (m/s)	90.30	80.80	81.80	…	53.63	…	35.84

. After randomly sorting the data in Table 3, we removed the first 400 data and assigned them as the training dataset and removed the last 120 data and assigned them as the test dataset.

The prediction result of the prediction model when applied to the test dataset following the completion of training is shown in Figure 11, and the average error of the calculated test dataset is only 0.022. This shows that it is feasible to replace the current filament model with the GRU-based predictive model. Next, we combine the prediction model with the optimization algorithm to optimize the feed time of the drive coil of the MSSICG to further verify the feasibility and stability of the prediction model.

5. Parameter Optimization

In this section, PSO will be used to optimize the feed time of the drive coil of the MSSICG. PSO is an evolutionary computing technology. It uses a particle to simulate the foraging process of one bird in a flock of birds. Each bird has its own position and speed. They will find the location with the most food, share the food information of their location, and then all the birds will approach the location with the most food. However, the moving speed of each bird is different. They will constantly look to see whether there is a location with more food during their movement. Eventually, all the birds will be close to the place with the most food on the map. We can write the following formula based on the above description:

$$\left\{\begin{array}{l}{a}_1=c_1\times rand\times (p_{best}-x_i)\\ a_2=c_2\times rand\times (g_{best}-x_i)\\ v_{i+1}=\omega \times v_i+a_1+a_2\\ x_{i+1}=x_i+v_i\end{array}\right.$$

(10)

where i = 1, 2, ..., N, N is the number of iterations; a₁ and a₂ denote the acceleration of the particles, while v_i is the velocity of the particles; ω is the inertia factor; c₁ and c₂ are the learning factors; rand is a random number from zero to one; p_best is the optimal position of the particle; g_best is the global optimal position; and x_i is the position of a particle. The flowchart of the PSO algorithm is shown in Figure 12.

According to the flowchart of PSO, the corresponding pseudocode is shown in Algorithm 1.

Algorithm 1. The corresponding pseudocode

1.procedure PSO 2.   for each particle i 3.      Initialize velocity Vi and position Xi for particle i 4.       Evaluate particle i and set pBesti = Xi 5.   end for 6.   gBest = max(pBesti) 7.   while not stop 8.      for i = 1 to N 9.         Update the velocity and position of particle i 10.         Evaluate particle i 11.         if fit(Xi) > fit(pBesti) 12.            pBesti = Xi 13.         if fit(pBesti) > fit(gBest) 14.            gBest = pBesti 15.      end for 16.   end while 17.   print gBest 18.end procedure

We set the target velocity to 50 m/s and used PSO to determine the minimum number of coils that can reach the target velocity. The optimization results are shown in

Table 4. Optimization results.

Number of Coils	Parameters		Velocity (m/s)
	t2 (ms)	t3 (ms)
3	1.59	2.44	50.12

.

It can be gleaned from the table that only three coils are required to reach the target velocity. We imported the same parameters into the CFM model for calculation. The muzzle velocity obtained via our calculations is 49.07 m/s, and the velocity curve is shown in Figure 13. It can be calculated that there is a 2.1% error between the prediction model and the CFM model. Although the CFM simulation result does not reach the target value of 50 m/s, under the same conditions, when the number of coils is three, the maximum muzzle velocity obtained from the orthogonal design table is 46.87 m/s. In addition, the optimization process of iteratively calling the predictive model through PSO only took about 0.0156 seconds, while it took several days to calculate the parameters with the orthogonal design table using CFM.

6. Conclusions

This manuscript delineates a methodology for the optimization of feed time and coil quantity in a 2- to 10-stage SICG utilizing GRU and PSO algorithms. The entire experimental process comprised three steps:

Step 1: An CFM model was established to simulate an MSSICG and calculate its electromagnetic and kinematic characteristics. The simulation data were validated against external circuit experimental data, with the error maintained within 10%. This indicated that it was feasible to use the simulation model’s calculations in lieu of the actual experimental results. Step 1 provides a new method for acquiring large volumes of data for MSSICGs.

Step 2: GRU neural networks were trained using the simulation data in order to establish a prediction model. The predicted root mean square error was only 0.022, indicating that the GRU-based prediction model can replace the current filament model. This model can further save computational resources and significantly reduce computation time.

Step 3: The predicted output value of the prediction model was used as the objective function, with the target muzzle velocity of the projectile set at 50 m/s. A PSO algorithm was used to find the minimum number of coils that could achieve the target speed, thereby reducing component costs. The optimized design results were compared with the optimal results obtained from the orthogonal test design under the same coil level. When the minimum number of coils was three, the projectile could reach the target speed, with a corresponding initial muzzle velocity of 50.12 m/s. In contrast, the maximum initial muzzle velocity determined using the orthogonal design table was only 46.87 m/s. The PSO algorithm iteration optimization process only takes about 0.0156 s, while calculating parameters in the orthogonal design table using the CFM takes several days.

In summary, this paper provides a simulation method for acquiring large volumes of data for MSSICGs and proposes a prediction model that combines GRU neural networks and PSO algorithms to optimize an MSSICG’s feed time and coil quantity. The accuracy and optimization efficiency of the model have been verified, which can help to improve MSSICGs’ service performance, save design time, and reduce maintenance costs.

This method can be improved further. We did not consider the distance between the coils of the MSSICG as a factor when building the model. In fact, the distance between the coils also has a sequence relationship. When adding a coil, the best distance from the previous coil needs to be judged according to the distance between all the previous coils. Therefore, if the distance between the coils is added as a factor when constructing the dataset, the prediction model will be further improved. In addition, when training the prediction model, we ignored the output of the GRU but used the output of the hidden layer to transform it into the result we desired. In fact, if we can build a dataset concerning the optimal feed time, the optimal feed time can be predicted through the GRU, even without using an optimization algorithm.