1.

INTRODUCTION

At present, in the explosion height test of the shooting range test, the detonator is mainly equipped on the forced bomb, grenade and rocket¹. Because it requires high precision, a high-speed camera is generally used to measure the distance from the explosion point of action in the bomb drop area 1 km. However, in the actual test, assuming that the focal length of the camera is 150mm, the line field of view can cover 133m, and nearly 30% of the projectiles will be out of the field of view². Statistics show that the data acceptance rate is about 70%³. The reason is that these devices use waiting measurement, and the test area is small. However, if the projectiles may fall and have a high prediction accuracy, so that the explosion point measurement device points to the prediction point, the testing capability of the device will be improved. In addition, dynamic test parameters such as the angle of attack, falling speed, and fragmentation speed of the projectile need to be obtained first.

To the relatively accurate impact point of the projectile, it is planned to use high-speed video recording or radar testing to complete the project. Therefore, improving the ability to quickly and accurately predict the impact is crucial to the improvement of other testing capabilities of the range.

At present, through the artillery position reconnaissance calibration radar and ballistic radar, the impact point can be extrapolated according to the data at the end of the ballistic trajectory⁴, and the accuracy is high, but the forecast value cannot be given in advance. However, although the existing shooting table method can obtain the theoretical landing point and spread range of the projectile impact point under standard conditions^5-7, it is impossible to estimate each projectile due to individual differences, such as the influence of factors such as the amount of charge, weather conditions and artillery rifling hit point.

In order to quickly and accurately predict the location of the explosion point, this paper mainly focuses on non-extended range projectiles such as forced bombs and grenades, and takes the drop point test of a certain type of projectile as an example. The test equipment mainly includes initial velocity, ballistic radar, ground weather, and drop point measurement. Fast and accurate prediction method of explosion point is proposed. Based on the real-time test data of the gun position, on the basis of the first indicator projectile, the equivalent ballistic coefficient is obtained by using the external ballistic simulation method, and the ballistic model is established; The initial segment ballistic data provided by the radar in real time is corrected twice to complete the fast and accurate prediction of the impact point of the projectile. The entire forecast time can be controlled within 8 seconds, and the equipment in the bomb drop area can complete corresponding actions according to the forecast data, providing an effective indication method for testing the bomb drop point and some dynamic parameters.

2.

MATHEMATICAL MODEL OF OUTER BALLISTICS

2.1

Motion model of projectile centroid

The nutation angle of the projectile is always small for flight stability, so it can be described by the equation of motion of the particle when studying the projectile motion. The mass point motion equation does not consider the influence of the earth’s rotation and the change of gravitational acceleration with latitude, but consider the change of gravitational acceleration with height and the influence of wind and drift current, and give the mass center momentum equation system with the ground rectangular coordinate system as the independent variable^{8, 9}:

2.2

Air temperature model

In the troposphere,

where τ_οn is the surface air temperature under non-standard meteorological conditions, and h is the plane height. There are different analytical formulas in the substratosphere and stratosphere. Due to the limited elevation of the forced bomb, this paper will not repeat them

2.3

Gravitational acceleration model

In formula (3), g₀ is the surface gravitational acceleration, R₀ is the radius of the earth, and g is the gravitational acceleration at height h.

2.4

Wind speed model

In formula (4), v is the wind speed at the height h from the ground, v₀ is the wind speed at h₀ in the case of real-time measurement¹⁰. It is considered that the wind direction does not change during the ballistic process. And α is the ground roughness, and takes the value of 0.12 on the plain ground.

3.

DESCRIPTION OF THE SIMULATION PROGRAM

• (1) The piecewise function of the air resistance coefficient¹¹ is written by the interpolation method, and the air resistance coefficient at each Mach number can be obtained.

• (2) For the density of air, which changes with temperature, pressure and altitude, the corresponding density function, the corresponding variable acceleration of gravity, speed of sound, and air temperature change with altitude are in section 2. Using the fourth-order Runge-Kutta method with fixed step size, call the corresponding sub-function to solve the differential equation system of the movement of the center of mass of the projectile, and obtain the coordinates in the coordinate system of the shooting range, as shown in Figure 1.

Figure 1.

Ballistic curve diagram in the coordinate system of the shooting range.

4.

SIMULATION OF IMPACT POINT PREDICTION

4.1

Data sources

Table 1 shows the test data of the ground impact point of a certain type of projectile with 18 rounds. The equipment participating in the test mainly included initial velocity radar, ballistic radar, ground meteorology and impact point measurement equipment.

Table 1.

Parameters of initial measurement and true value of impact point measurement.

No.	Initial velocity m/s	Wind direction°	Wind velocity m/s	Air temperature °c	Air pressure MPa	Z(m)	X(m)
1	181.8	332	1.1	35.4	961.0	4684.4	-68
2	181.6	303	2.5	35.8	961.0	4610.2	-71.4
:	:	:	:	:	:	:	:
18	185.9	11	0.9	37	959.7	4782.0	-26.2

The muzzle velocity-impact distance distribution was obtained, as shown in Figure 2.

Figure 2.

Schematic diagram of the distribution of muzzle velocity - impact point distance.

It can be seen from the figure that the distance between the two groups of projectiles was about 300 meters away, and when the initial velocity was basically the same, the impact point may also be quite different. As shown in the figure, when the initial velocity was around 286m/s, the distance also reached about 150m, which was not conducive to the measurement of small-scale testing instruments. Based on the real-time measurement data, this paper discusses the rapid prediction method of the impact point of the projectile.

4.3

Simulation results

According to the above method, the measured data was simulated. Here, the impact point elevation result was the same elevation as the gun position, and the coordinate values of the impact point distance and direction under the shooting range coordinate system were obtained, as shown in Tables 5 and 6.

Table 5.

Prediction results of the first group of projectile impact points.

No.	First error value	Quadratic error value (step size 0.05)	Quadratic error value (step size 0.01)	Remark
Z/m	X/m	Z/m	X/m	Z/m	X/m
1	0	31	Determine the equivalent ballistic coefficient c=1.3989
2	68.8	46.4	5.8	-3.6	5.8	-4.6
3	45	49.6	-49	-0.4	-46	-0.4
4	87.3	34.8	-12.7	0.8	-8.7	-0.2
5	110.9	-15.9	17.9	-5.9	20.9	-5.9
		—
14	86.3	-79.7	24.3	-70.7	27.3	-70.7
15	120.3	-2.6	27.3	-9.6	29.3	-9.6
16	42	-61.6	-16	-40.6	-13	-40.6
17	41	-19.5	-51	-14.5	-49	-15.5	furthest
18	86.1	-97.7	22.1	-63.7	25.1	-63.7

Table 6.

The second group of projectile impact point prediction results.

No.	First error value	Quadratic error value (step size 0.05)	Quadratic error value (step size 0.01)	Remark
Z/m	X/m	Z/m	X/m	Z/m	X/m
1	0	31	Determine the equivalent ballistic coefficient c=1.3989
2	68.8	46.4	5.8	-3.6	5.8	-4.6	nearest
		—
16	94.8	-32.2	-13.2	-8.2	-10.2	-8.2
17	-28.5	-17.3	-50.5	-6.3	-45.5	-6.3
18	57.3	27.2	18.3	23.2	21.3	23.2

Finally, the error between the two simulation results and the actual measured value were obtained, as shown in Figures 3 and 4.

Figure 3.

Prediction error of impact point distance.

Figure 4.

Prediction error of impact point direction.

It can be seen from Figure 4 that after the second correction, the prediction accuracy in distance and direction has been greatly improved. When a fixed step size of 0.05 seconds was selected, the error in distance could be controlled within 51 meters; when the root mean square error was 30.2 m, the direction error could be controlled within 70.7 m, and the root mean square error is 30.1 m. Because there was no time matching problem in the secondary simulation, a fixed step size of 0.01 seconds could be selected to improve the accuracy of the ballistic solution. The error could be controlled within 49 meters, the root mean square error was 29.2 m, and the direction error and root mean square error were the same as when the step size was 0.05 seconds. Individual projectiles would have large errors in the direction, mainly due to the errors brought by the model, and there may also be unpredictable factors such as gusts, jets, and random winds that have a greater impact on the direction. It can be seen from the results that a higher prediction accuracy can be obtained when a smaller calculation step is used. According to the test range that the high-speed camera can cover, the data acceptance rate can reach about 95%.

5.

CONCLUSION

In order to quickly and accurately predict the location of the explosion point, this paper mainly focuses on non-extended range projectiles such as forced bombs and grenades, and takes the drop point test of a certain type of projectile as an example. The test equipment mainly includes initial velocity, ballistic radar, ground weather, and drop point measurement. Based on the real-time test data of the gun position, according to the first indicator projectile, the equivalent ballistic coefficient is obtained by using the external ballistic simulation method, and the ballistic model is established. After the projectile fired, the initial segment ballistic provided in real time by the ballistic radar is used. The speed is corrected twice by the data to complete the fast and accurate prediction of the impact point of the projectile.

The experimental results showed that the entire forecast time can be controlled within 8 seconds, and the equipment in the bomb drop area can complete corresponding actions according to the forecast data, which achieves the purpose of rapid forecasting of the impact point, and solves the problem that the equipment in the drop area is out of the field of view; at the same time, The model is relatively simple, has fast solution speed, and has strong real-time performance. Combined with shooting range communication, it has good application prospects: