An Optimization Method for the Ballistic Simulation of High-Explosive Shells and the Calculation Process of Firing Parameters

Abstract

The present invention relates to the field of anti-aircraft guns, and provides an optimization method for the simulation of anti-aircraft gun trajectories and the calculation process of firing parameters. It aims to solve the problems of low firing accuracy, unsatisfactory hail prevention effect and low operation efficiency caused by the dependence on manual aiming of anti-aircraft guns in manual hail prevention operations. The main solutions include: 1) defining the differential equation of the particle trajectory under non-standard conditions, including the differential expressions of velocity and position; 2) fitting the curve of the air resistance law of the projectile to obtain the expression of the drag coefficient; 3) converting the target coordinates, the initial velocity of the shell and the wind speed into the launch coordinate system; 4) using the Adams predictor-corrector method for ballistic integration to calculate the position and velocity of the shell; 5) predicting the future position of the target based on the ballistic data; 6) correcting the firing angle, including judging whether the target is hit, and if not, correcting the firing angle; 7) controlling the number of iterations to optimize the calculation efficiency. The use of the present invention is to improve the firing accuracy and operation efficiency of anti-aircraft guns in manual hail prevention operations.

Description

Technical Field

[0001] The present invention relates to the field of anti-aircraft guns, and provides an optimization method for high anti-aircraft gun trajectory simulation and shooting parameter calculation process. Background Art

[0002] Hail disaster, as one of the common and highly destructive meteorological disasters globally, often causes serious losses to agriculture, buildings, power facilities and transportation systems. To cope with frequent hail disasters, the technology of artificial weather modification has gradually become an important means of preventing and controlling hail disasters. Among them, anti-aircraft guns, due to their explosion effect, can effectively inhibit the development of hail clouds and become a key tool for hail prevention operations. However, despite the large scale of hail prevention operations, their protection effect is still not ideal. The reasons for this problem are as follows: on the one hand, the aging of technical personnel and the large mobility of personnel have led to insufficient technical levels in hail prevention operations; on the other hand, the current anti-aircraft gun operations mainly rely on manual aiming and shooting, with problems of insufficient accuracy and poor environmental adaptability, directly affecting the hail prevention effect. Although the azimuth and distance of the strong center of the hail cloud can be read through radar, it becomes extremely difficult to visually judge the strong center of the cloud body and related operation information under complex weather conditions. Therefore, improving the accuracy of the high anti-aircraft gun trajectory and providing clear operation information for gun positions has become one of the key issues in improving the efficiency of artificial hail prevention operations.

[0003] Based on the above background, this paper proposes an optimization scheme for high anti-aircraft gun trajectory simulation and shooting parameter calculation process, aiming to improve the efficiency and accuracy of hail prevention operations. First, combined with the actual situation of current artificial hail prevention operations, on the premise of ensuring calculation accuracy and efficiency, a moderately simplified non-standard condition particle trajectory differential equation is used for trajectory simulation. Second, the least squares method is used to re-fit the resistance law curve in 1943 with polynomials to make it more in line with the actual table values. In addition, the fourth-order Runge-Kutta method and the Adams predictor-corrector method are used to solve the trajectory differential equation, and the calculation accuracy of these two methods is verified by comparing with the measured trajectory data. Finally, according to the actual meteorological conditions of artificial hail prevention, the calculation process of shooting parameters is optimized, and considering the influence of wind on the high anti-aircraft gun trajectory, the launch azimuth angle, elevation angle and trajectory flight time of the anti-aircraft gun can be accurately calculated simultaneously. Through the above research, this paper provides a theoretical basis and practical reference for improving the scientificity and effectiveness of artificial hail prevention operations. Summary of the Invention

[0004] The main purpose of this paper is to solve the limitations in the existing artificial hail prevention operations, such as the low shooting accuracy, unsatisfactory hail prevention effect and low operation efficiency caused by the dependence of anti-aircraft guns on manual aiming.

[0005] The present invention provides an optimization method for high anti-aircraft gun trajectory simulation and shooting parameter calculation process, including the following steps:

[0006] Step 1: Define the simplified non-standard differential equation of centroid motion as:

[0007]

[0008]

[0009]

[0010]

[0011] where 、 and are the velocities of the shell in each direction of the coordinate axes respectively, while 、 and are the wind speeds in each direction respectively, is the ballistic coefficient, which depends on the shape, diameter and weight of the projectile, is the air density, is the drag function, represents the neutral acceleration, represents the drag coefficient;

[0012] Step 2: Fit the air drag law curve of the projectile to obtain the expression of the drag coefficient ;

[0013] Step 3: According to the elevation angle and the azimuth angle transform the target coordinates, the initial velocity of the shell and the wind speed in the northeast celestial coordinate system into the launch coordinate system to obtain the coordinates of the target in the launch coordinate system, the velocity component of the initial velocity of the shell in the launch coordinate system , and the components of the wind speed in the launch coordinate system ( );

[0014] Step 4: Ballistic calculation: Use the Adams predictor-corrector method for ballistic integration to obtain the position and velocity of the shell at each moment, and determine the ballistic end point coordinates according to the ballistic data;

[0015] Step 5: According to the ballistic flight time and the motion parameters of the target, predict the position of the target after time;

[0016] Step 6: Shooting angle correction:

[0017] Step 6.1: Calculate the coordinates of the ballistic end point and the predicted target position to check if the distance D is less than a preset threshold :

[0018] If D is less than the preset threshold , it is considered that the target is hit, and the current elevation angle , azimuth angle and the ballistic flight time T are output;

[0019] If D is greater than the preset threshold, based on the coordinates of the ballistic end point and the predicted target position , the current elevation angle and azimuth angle are corrected to obtain the corrected elevation angle and azimuth angle ;

[0020] Step 6.2: Return the corrected elevation angle , azimuth angle , and the predicted target position , and go back to step 4.

[0021] In the above solution, the expression of the drag coefficient in step 2 is as follows, where is the Mach number:

[0022]

[0023]

[0024]

[0025]

[0026]

[0027]

[0028]

[0029]

[0030]

[0031]

[0032]

[0033]

[0034]

[0035] In the above solution, in step 3, according to the azimuth angle and the elevation angle the initial coordinates of the target in the northeast celestial coordinate system are converted into the coordinates in the launch coordinate system by using the transformation matrix ;

[0036] Transformation matrix:

[0037]

[0038] Obtain:

[0039]

[0040]

[0041]

[0042] In the above solution, in step 3, according to the azimuth angle and the elevation angle the initial velocity components of the shell are converted into the components in the launch coordinate system:

[0043]

[0044]

[0045] .

[0046] In the above solution, in step 3, according to the azimuth angle and the elevation angle the conversion matrix for converting the wind speed in the northeast celestial coordinate system into the wind speed in the launch coordinate system is as follows:

[0047]

[0048] Obtain:

[0049] .

[0050] In the above solution, step 4 includes the following steps:

[0051] Step 4.1: Use the Runge-Kutta method to calculate the ballistic data within the time step as the starting value of the Adams method, and calculate the coordinate and velocity values of the first four steps,

[0052] Coordinates of the first four steps:

[0053]

[0054] Velocity values of the first four steps:

[0055]

[0056] Step 4.1.1: According to the initial velocity of the shell Component velocity components in the launch coordinate system , wind speed component ( ), time step , calculate :

[0057]

[0058]

[0059]

[0060] Step 4.1.2: Calculate :

[0061]

[0062]

[0063]

[0064] Step 4.1.3: Calculate :

[0065]

[0066]

[0067]

[0068] Step 4.1.4: Calculate :

[0069]

[0070]

[0071]

[0072] Step 4.1.5: Update the position , , and velocity , , :

[0073]

[0074]

[0075]

[0076]

[0077]

[0078]

[0079] Repeat steps 4.1.2 - 4.1.5 to calculate the coordinate and velocity values of the first four steps:

[0080] Use and as the new initial conditions, repeat steps 4.1.2 - 4.1.5 to calculate the coordinate and velocity values of the second step and ;

[0081] Use and as the new initial conditions, repeat steps 4.1.2 - 4.1.5 to calculate the coordinate and velocity values of the third step and ;

[0082] Use and as the new initial conditions, repeat steps 4.1.2 - 4.1.5 to calculate the coordinate and velocity values of the fourth step and ;

[0083] Step 4.2: Predict the next - step ballistic data according to the Adams prediction formula and ;

[0084] Step 4.2.1: Obtain the known data

[0085] Obtain the coordinate and velocity values of the first four steps:

[0086] Coordinates:

[0087] Velocities:

[0088] Obtain the current time step: ;

[0089] Initialize n = 4;

[0090] Step 4.2.2: Calculate the predicted values

[0091] Use the Adams prediction formula to calculate the predicted coordinates and velocity values for the next step:

[0092] Predicted coordinate values:

[0093]

[0094]

[0095]

[0096] Predicted velocity values:

[0097]

[0098]

[0099]

[0100] Step 4.3: According to the Adams correction formula, correct the predicted values to obtain more accurate ballistic data;

[0101] Step 4.3.1: Substitute into the Adams correction formula:

[0102] Substitute the corrected acceleration value calculated in Step 1 into the Adams correction formula to calculate the corrected coordinate and velocity values:

[0103] Corrected coordinate values:

[0104]

[0105]

[0106]

[0107] Corrected velocity values:

[0108]

[0109]

[0110]

[0111] Step 4.3.2: Repeat Steps 4.2 and 4.3 until the entire ballistic data calculation is completed;

[0112] Step 4.5: Determine the ballistic endpoint coordinates and velocity

[0113] In the above solution, the correction formula in step 6.1 is expressed as:

[0114]

[0115]

[0116] where k is the correction coefficient, is the current firing angle, , is the elevation angle and azimuth angle corresponding to the ballistic endpoint, is the elevation angle and azimuth angle corresponding to the future position of the target.

[0117] In the above solution, in step 6.2, before returning to step 4, the number of iterations is also judged. If the number of iterations is greater than the threshold, it ends; otherwise, the corrected elevation angle , azimuth angle , target prediction position are returned to step 4.

[0118] Because the present invention adopts the above technical means, the following beneficial effects are achieved:

[0119] 1. Simplification of the particle trajectory differential equation under non-standard conditions: Aiming at the differences between the actual conditions and the standard conditions in anti-aircraft gun shooting, by moderately simplifying the particle trajectory differential equation under non-standard conditions, some complex and less influential parameters are eliminated. This simplification method not only improves the calculation efficiency but also ensures the accuracy of the calculation results, thus better adapting to the actual operation environment of artificial hail prevention operations.

[0120] 2. Re-fitting of the drag law curve in 1943: The drag law curve in 1943 was compiled by the Soviet Union and given in the form of table values. To meet the modern calculation requirements, it is crucial to convert the table value law into an analytical form. Through polynomial curve fitting based on the least squares method, the data below Mach 1 was re-fitted, significantly improving the accuracy in the low Mach number region.

[0121] 3. Selection of the numerical solution method for the trajectory: In the trajectory calculation, by combining the fourth-order Runge-Kutta method and the Adams predictor-corrector method, the calculation accuracy and efficiency are balanced. The specific analysis is as follows:

[0122] Problems existing in a single technology:

[0123] Fourth-order Runge-Kutta method: The fourth-order Runge-Kutta method is an improved method based on Taylor series expansion for approximately solving differential equations. It solves differential equations by gradually approaching, and the calculation formula is simple and easy to implement. However, in practical applications, since the right-end function needs to be calculated four times, the calculation efficiency is relatively low.

[0124] Adams Predictor-Corrector Method: Its accuracy is greatly affected by initial conditions, but it has high efficiency.

[0125] Combined effect:

[0126] The ballistic data of the first four steps are calculated by the fourth-order Runge-Kutta method as the starting values of the Adams Predictor-Corrector Method, reducing the amount of calculation and improving the calculation efficiency.

[0127] By combining the prediction and correction steps, the Adams method can ensure a certain calculation accuracy while ensuring the calculation efficiency. Especially at relatively large step sizes (from 0.04 seconds to 0.1 seconds), the error fluctuation of the Adams method is small and its performance is excellent.

[0128] 4. Optimization of the shooting parameter calculation process: In the traditional shooting parameter solving method, the iterative correction method is widely used due to its relatively fast convergence speed. However, this method has deficiencies in the overall algorithm efficiency, and the error in the elevation angle iteration is easily transmitted to the position iteration. To solve this problem, the method is improved by considering the target as always being in a moving state and introducing corrections to both the target elevation angle and coordinates during the iteration process, reducing the number of iteration layers and improving the calculation efficiency.

[0129] 5. Precise control of the iterative calculation process: To ensure that the fire control calculation can be completed within the specified time, an upper limit on the number of iterations is set. This precise control not only avoids the low calculation efficiency caused by excessive iterations but also ensures the accuracy of the calculation results and avoids unnecessary waste of computing resources. Description of the Drawings

[0130] Figure 1 It is a flow diagram of the present invention;

[0131] Figure 2 It is the fitting curve of the drag law in 1943;

[0132] Figure 3 It is the re-fitted curve;

[0133] Figure 4 It is the error graph of the fourth-order Runge-Kutta method;

[0134] Figure 5 It is the error graph of the Adams Predictor-Corrector method;

[0135] Figure 6 It is the comparison of altitude accuracy under different integration step sizes;

[0136] Figure 7 It is the comparison of distance accuracy under different integration step sizes;

[0137] Figure 8 It is the initial parameter table for the solution of antiaircraft gun firing data;

[0138] Figure 9 For the iterative process table

[0139] Figure 10 For the trajectories of the projectile and the target in the last iteration. Specific implementation manners

[0140] The embodiments of the present invention will be described in detail below. Although the present invention will be described and illustrated in conjunction with some specific implementation manners, it should be noted that the present invention is not limited to these implementation manners only. On the contrary, any modifications or equivalent replacements made to the present invention should be covered within the scope of the claims of the present invention.

[0141] In addition, for a better illustration of the present invention, numerous specific details are given in the following specific implementation manners. Those skilled in the art will understand that the present invention can also be implemented without these specific details.

[0142] Fitting optimization of ballistic characteristics and drag law

[0143] Simplifying the differential motion equation of a particle's trajectory under non-standard conditions

[0144] In external ballistics, when studying the motion of the center of mass of a projectile or rocket, the influence of the rotation of the projectile or rocket around its center of mass can be ignored, and the angle of attack is assumed to be zero. In addition, it is assumed that the shape and mass distribution of the projectile or rocket are symmetric about the longitudinal axis, the earth's surface is a plane, the acceleration due to gravity is a constant and acts vertically downward, the Coriolis force is ignored, and the meteorological conditions are assumed to be standard, i.e., no wind and no rain. The equation of motion of the center of mass established under these assumptions can reveal the basic laws of the motion of the center of mass of the projectile or rocket and simplify the calculations. However, in anti-aircraft gun shooting, there are differences between the actual conditions and the standard conditions in the firing tables. To more accurately correct the errors under these non-standard conditions, it is necessary to establish a differential equation of motion of the center of mass of the projectile or rocket applicable to non-standard conditions.

[0145] Since the range of an anti-aircraft gun is relatively short, the earth's surface can be approximated as a plane, and at the same time, the influence of the Coriolis force can be ignored, and the acceleration due to gravity can also be regarded as a constant. Therefore, the differential equation of motion of the center of mass of the projectile or rocket under non-standard conditions can be appropriately simplified, and the simplified differential equation of motion of the center of mass under non-standard conditions is:

[0146] Equation (1):

[0147]

[0148]

[0149]

[0150]

[0151]

[0152]

[0153]

[0154]

[0155] where 、 and are the velocities of the shell in the directions of each coordinate axis respectively, while 、 and are the wind speeds in each direction respectively. is the ballistic coefficient, which depends on the shape, diameter and weight of the projectile. is the air density, which is related to the air pressure and virtual temperature. is the drag function, which is related to the relative velocity of the projectile to the air and the drag coefficient of the standard projectile and can be obtained from the 1943 drag law table. is the virtual temperature related to the altitude and is the virtual temperature related to the altitude.

[0156] Curve Fitting of the Air Drag Law of Projectiles

[0157] In the external ballistics calculation, the form factor of the projectile and the air drag law are usually used to describe the ballistic characteristics. However, the widely used 1943 air drag law of projectiles is given in tabular form. To meet the modern calculation requirements, it is crucial to convert the tabular law into an analytical form, which can be achieved through curve fitting.

[0158] Using the 1stOpt mathematical software, the Levenberg-Marquardt method and the general global optimization method were used to perform curve fitting on the 1943 tabular law. By screening, a set of fitting coefficients with the largest correlation coefficient was obtained, and a rational expression suitable for calculation was obtained. The fitting results are as shown in Figure 1 . The maximum error of this curve appears at , and the absolute value of the error at this point is 3.3%. The variance of the entire curve is . Although the overall fitting effect is good, the fitting accuracy is insufficient in the low Mach number and high Mach number regions. Therefore, we refitted the data below Mach number.

[0159] For the data below Mach 1, polynomial curve fitting based on the least squares method was used to refit the tabular data. The polynomial coefficients are calculated by the following formula:

[0160] Equation (2)

[0161] Among them, is the transpose matrix of a Vandermonde matrix with the number of columns equal to the number of tabulated values and the number of rows equal to the number of polynomial coefficients; is the vector containing the tabulated values, is the obtained polynomial coefficient vector.

[0162] The final fitting result is as follows:

[0163] Equation (3):

[0164]

[0165]

[0166]

[0167]

[0168]

[0169]

[0170]

[0171]

[0172]

[0173]

[0174]

[0175]

[0176] Among them, is the drag coefficient, is the Mach number. The drag law fitting curve in 1943 is as Figure 2 shown, and the curve after refitting is as Figure 3 shown:

[0177] Comparative analysis shows that the accuracy of the new fitting curve is significantly improved in the low Mach number region. The maximum error point of the whole curve appears at , the absolute value of the error at this point is 2.48%, and the variance of the whole fitting curve is . In the range of Mach number from 0.7 to 1, the fitting accuracy has been significantly improved compared with the previous results.

[0178] Ballistic numerical solution method and accuracy analysis

[0179] Overview of ballistic solution methods

[0180] The core objective of the ballistic solution method is to solve the non-linear differential equations that describe the motion of a projectile. Since the equations involve complex factors such as air resistance, wind speed, and gravity, analytical solutions are usually difficult. Therefore, in external ballistic calculations, ballistic table solution methods or numerical solution methods are usually adopted.

[0181] The ballistic table solution method simplifies the calculation process in practical applications by pre-calculating and compiling tables, and is suitable for scenarios with relatively simple calculation conditions. In contrast, due to its high precision and adaptability, the numerical solution method has become the main method in modern external ballistics. The numerical solution method solves differential equations by means of step-by-step approximation, and common algorithms include the Runge-Kutta method and the Adams predictor-corrector method.

[0182] The Runge-Kutta method is an improved method based on Taylor series expansion for approximately solving differential equations. The most commonly used fourth-order Runge-Kutta method is widely used in ballistics, and the calculation formula is as follows:

[0183] Equation (4)

[0184] If the value at the nth position ( is known, then the function value at the position can be obtained by this method. Its truncation error is . Due to its high precision and flexibility, the fourth-order Runge-Kutta method is widely used in ballistic calculations, but the disadvantage is that the right-end function needs to be calculated four times per step.

[0185] The Adams predictor-corrector method only needs to calculate the right-end function once per step, with higher calculation efficiency. However, it depends on the initial conditions and needs to be self-started by the Runge-Kutta method. The calculation process of this method includes two steps: prediction and correction, which are realized by the prediction formula and the correction formula respectively.

[0186] The prediction formula is used to predict the function value of the next step based on the solutions of the previous few steps calculated by the Runge-Kutta method:

[0187] Equation (5)

[0188] The correction formula then uses the latest function value to correct the prediction result, thereby improving the accuracy:

[0189] Equation (6)

[0190] By combining the prediction and correction steps, the Adams predictor-corrector method can ensure a certain calculation accuracy while ensuring the calculation efficiency.

[0191] Accuracy verification of the numerical solution method

[0192] To verify the accuracy of the aforementioned numerical solution method, this paper analyzes the ballistic data of 37-mm anti-aircraft guns from Chongqing 152 Factory provided by Li Zuxian (Li Zuxian, Liu Hongwu, Liao Jun, etc. Safety assessment method for man-made weather modification anti-aircraft gun operations based on external ballistic calculation. Meteorological Science and Technology, 2016, 44(1): 152-156.). The ballistic coefficient of this shell is 1.89. The data includes the measured results of altitude and horizontal distance. The ballistic fuze time is set from 8 seconds to 20 seconds, and the firing angles range from 45° to 85°. A total of 63 groups of altitude data and 63 groups of horizontal distance data are involved.

[0193] During the verification process, the azimuth angle is set to 0, and the integration step size is set to 0.05. , the initial velocity of the shell is 866 m / s, the wind speed is set to 0, and other initial meteorological conditions are set according to the Chinese artillery meteorological standard. A total of 7 fuze durations are considered in this verification, and 9 elevation angles are set for each fuze duration. This application calculates the errors between the coordinates calculated by two numerical solution methods (the fourth-order Runge-Kutta method and the Adams predictor-corrector method) and the measured values. The absolute values of the altitude and horizontal distance errors and their average values for each fuze duration are as Figure 4 and Figure 5 shown:

[0194] The results show that at all firing angles and all fuze durations, the errors of the altitude and horizontal distance calculated by the two methods are less than 0.6%, indicating that under windless conditions, the two numerical solution methods have high accuracy, can accurately simulate the external ballistic trajectory, and meet the requirements of artificial hail suppression operations.

[0195] Further comparison reveals that at different fuze times, the average error of the Adams predictor-corrector method is less than that of the fourth-order Runge-Kutta method, indicating that its accuracy is slightly better. To further verify this conclusion, in this embodiment, the integration step size is adjusted from 0.01 seconds to 0.1 seconds, and the average errors of 63 groups of data at different step sizes are calculated. The absolute value averages of the altitude and horizontal distance errors are as Figure 6 and Figure 7 shown.

[0196] It can be seen from the figure that when the step size is between 0.01 seconds and 0.04 seconds, the errors of the two methods are similar. However, as the step size increases to more than 0.04 seconds, the error of the Adams method fluctuates little, showing better performance than the fourth-order Runge-Kutta method. When the step size increases to 0.1 seconds, the error of the Adams method is lower than that of the Runge-Kutta method.

[0197] In summary, under a relatively small step size (from 0.01 seconds to 0.04 seconds), the accuracies of the two methods are comparable. Under a relatively large step size (from 0.04 seconds to 0.1 seconds), the Adams method performs better and is especially suitable for long-time or real-time calculation scenarios. Therefore, the Adams predictor-corrector method has certain advantages in ballistic calculation and is applicable to the anti-hail AA gun fire control system. By reasonably adjusting the step size, the calculation efficiency can be improved while ensuring the accuracy.

[0198] Anti-aircraft gun firing data solution and iterative process optimization

[0199] Firing data solution

[0200] In fire control solution, the impact point refers to the position where the anti-aircraft gun shell meets the target, which is usually determined by solving the intersection point of the ballistic trajectory and the target trajectory. According to the uniqueness theorem of the existence of the air ballistic, the ballistic coefficient, elevation angle (vertical angle), and muzzle velocity of the shell can completely and uniquely determine a ballistic trajectory. If the target is in a moving state, the firing lead also needs to be considered. At this time, the aiming vector, impact vector, and lead vector form a triangle, which is called the impact triangle.

[0201] The firing data solution aims to determine the elevation angle, azimuth angle of the anti-aircraft gun, and the flight time required for the shell to hit the target. This process usually depends on the differential equation of the ballistic particle and requires the spatial coordinate information of the target. In ballistic calculation, the launch coordinate system

[0202] is usually used for calculation, and the coordinates of the target are usually described in the north-east-up coordinate system . Through appropriate rotation, the launch coordinate system can be converted to the north-east-up coordinate system. Assuming that the azimuth angle of the anti-aircraft gun launch is (the angle between the launch direction and the N axis in the north-east-up coordinate system), the transformation matrix for converting the launch coordinate system to the north-east-up coordinate system is as follows:

[0203] Equation (7)

[0204]

[0205]

[0206] Iterative correction method for firing data

[0207] Among the mainstream methods for solving firing data, the iterative correction method is widely used due to its relatively fast convergence speed. When Li Qiang et al. (Li Qiang, Ouyang Pan. Anti-aircraft fire control algorithm and simulation based on exterior ballistics. Fire Control & Command Control, 2014, 39(5): 131-134.) solved the firing data, they first regarded the target position as a fixed point and calculated through the iteration of the elevation angle. This method first set an elevation angle threshold and calculated the initial elevation angle based on the target position. In the launch coordinate system, when the abscissa of the projectile trajectory calculated by the Runge-Kutta method was consistent with the abscissa of the target, the height difference between the two in the ordinate was calculated. If the height difference was less than the preset threshold, it was considered that the projectile hit the target. Subsequently, the flight time of the projectile was calculated, and the future position of the target was predicted based on this time, and the iteration continued until the projectile-target distance met the requirements, so as to solve the firing data of the moving target.

[0208] However, this method has deficiencies in the overall algorithm efficiency, and the errors in the elevation angle iteration are likely to be transmitted to the position iteration, especially in complex environments, which may lead to divergent results. Therefore, this application improves the method, regards the target as always being in a moving state, and simultaneously introduces the correction of the target elevation angle and coordinates during the iteration process, thereby reducing the number of iteration layers and improving the calculation efficiency.

[0209] Improved solution process

[0210] Compared with the traditional method, the improved algorithm proposed in this application significantly improves the iteration efficiency, reduces the calculation time, and can more accurately synchronously correct the elevation angle and the target position, ensuring that the calculation results of the firing data are more reliable.

[0211] Furthermore, in the operation of anti-hail AA guns, the influence of wind on the ballistic trajectory cannot be ignored. Therefore, it is necessary to improve the above algorithm to retain the advantages of synchronously correcting the elevation angle and the target position, and at the same time introduce the correction of the wind on the ballistic trajectory. This application adopts a simplified non-standard condition particle ballistic differential equation and uses the Adams predictor-corrector method for ballistic solution. The improved solution process is as Figure 1 shown:

[0212] Assume that the initial coordinates of the target in the northeast celestial coordinate system are , the initial velocity of the projectile is , and the initial values of the elevation angle and azimuth angle obtained by iterative solution are and . Since the ballistic differential equation is based on the launch coordinate system, it is necessary to convert the initial velocity components of the projectile into components in the launch coordinate system: , , .

[0213] The wind speed vector is defined based on the northeast - up coordinate system, with the positive direction of each axis being the positive direction of the wind. To consider the influence of the wind on the ballistic trajectory, it is necessary to convert the wind speed based on the northeast - up coordinate system into the wind speed in the launch coordinate system. The conversion matrix for converting the wind speed in the northeast - up coordinate system to the wind speed in the launch coordinate system is as follows:

[0214] Equation (8)

[0215] By solving the system of ballistic differential equations, the ballistic terminal coordinates of the shell can be obtained and its flight time T. Based on the coordinates of this ballistic terminal, a new elevation angle and azimuth angle are calculated relative to the gun point. Using the flight time T of the shell, combined with the flight direction, speed, and acceleration of the target, the position of the target after this period of time can be estimated , and the corresponding new elevation angle and azimuth angle are calculated.

[0216] Next, calculate the distance D between the ballistic terminal coordinates and the target position . If this distance is less than a predetermined threshold , it is determined that the target has been hit, and the current elevation angle, azimuth angle, and ballistic flight time are output.

[0217] If the distance does not meet the threshold, the elevation angle and azimuth angle are corrected, and the corrected values are used to continue the iterative calculation. Assume that the elevation angle and azimuth angle input into the ballistic differential equation are and respectively. Then the corrected elevation angle and azimuth angle .

[0218] In the process of calculating the firing data, it is unrealistic to use only the solution error as the sole criterion for ending the iteration. Any iterative algorithm can achieve high precision when time permits and converges. However, once the iteration time exceeds the allowed limit, even high precision is meaningless. Therefore, the above - mentioned method sets two iteration - ending conditions:

[0219] (1) If the calculated distance D is less than the predetermined threshold , it is determined that the target has been hit, and the iteration ends naturally;

[0220] To avoid excessive iteration times caused by too small precision settings or divergence during the solution process and ensure that the fire - control solution can be completed within the specified time, an upper limit on the number of iterations is set. When the number of iterations reaches the set upper limit, the iterative solution process is terminated.

[0221] Verification and Result Analysis of Firing Data Solution

[0222] To verify the effectiveness of the aforementioned method, a set of initial parameters was set in this paper for calculation, as shown in Figure 8 Table 1 below. This table lists the key parameters for simulating anti-aircraft gun firing, including wind speed, muzzle velocity, target position and velocity, etc. By setting these parameters, detailed iterative calculations were carried out for the firing data in this paper.

[0223] The parameter settings in Table 1 reflect various situations of different wind speeds and target positions, simulating the anti-aircraft gun firing process under different conditions. When the integration step size is set to 0.01 seconds and the threshold is set to 0.5 meters, after 8 iterations, the firing results of elevation angle of 58.824° and azimuth angle of 22.039° were finally obtained, the flight time of the projectile was 15.09 seconds, and the miss distance was only 0.32 meters. The specific iterative data are shown in Figure 9 Table 2 below.

[0224] It can be seen from Table 2 that as the number of iterations increases, the differences in elevation angle and azimuth angle gradually decrease, and the projectile-target distance gradually approaches the target position, indicating that the calculation process gradually converges. After 8 iterations, the finally calculated elevation angle and azimuth angle have reached high precision, and the miss distance is only 0.32 meters, fully meeting the precision requirements for hail prevention operations. Figure 10 The trajectory of the projectile and the target trajectory in the last iteration are shown.

[0225] It should be noted that the number of iterations is significantly affected by the target movement speed and acceleration. As the target speed and acceleration increase, the required number of iterations will also increase. It can be seen from the data in the table that if a looser distance threshold is selected, the number of iterations will be significantly reduced. In addition, the choice of step size has an important impact on the convergence of the calculation. If the step size is too long, the calculation may not converge to the predetermined threshold; if the step size is too small, although the precision is improved, the calculation time will also increase. ] Therefore, the choice of step size must take into account both the convergence and the calculation efficiency.

Claims

1. A method for optimizing the trajectory simulation and firing parameter calculation process of an antiaircraft gun, characterized in that: The following steps are involved: Step 1: Define the simplified nonstandard center-of-mass motion differential equation as: in , and are the speeds of the projectile in each direction of the coordinate axis, and , and are the wind speeds in each direction, is the ballistic coefficient, which depends on the shape, diameter and weight of the projectile. is the air density, is the resistance function, represents the neutral acceleration, represents the drag coefficient; Step 2: Fit the projectile air resistance law curve to obtain the resistance coefficient expression; Step 3: According to the height angle Following the azimuth The target coordinates and the initial velocity of the projectile in the northeast celestial coordinate system are and wind speed into the launch coordinate system to obtain the coordinates of the target in the launch coordinate system , the initial velocity of the projectile The velocity component of the component in the emission coordinate system , the component of wind speed in the launch coordinate system ( ); Step 4: Trajectory calculation: Use the Adams prediction-correction method to perform trajectory integration and obtain the position of the shell at each time. and speed , and determine the coordinates of the trajectory endpoint based on the trajectory data ; Step 5: Based on the ballistic flight time and the target's motion parameters, predicting the target's Position after time ; Step 6: Shooting Angle Correction: Step 6.1: Calculate the coordinates of the trajectory end point Between the predicted target position Is the distance D less than the preset threshold? : If D is less than the preset threshold , it is considered to have hit the target and the current elevation angle is output , azimuth and ballistic flight time T; If D is greater than the preset threshold, then according to the coordinates of the end point of the trajectory and the target predicted position , correct the current elevation angle and azimuth , and get the corrected elevation angle and azimuth ; Step 6.2: Correct the height and low angles , azimuth , target predicted position , return to step 4.

2. The method for optimizing the trajectory simulation and firing parameter calculation process of an antiaircraft gun according to claim 1, characterized in that: Drag coefficient in step 2 The expression is as follows, is the Mach number:

3. The method for optimizing the trajectory simulation and firing parameter calculation process of an antiaircraft gun according to claim 1, characterized in that: In step 3, according to the azimuth and height angle Set the initial coordinates of the target in the northeast celestial coordinate system Use the transformation matrix to transform the target coordinates into the coordinates of the launch coordinate system ; Transformation Matrix: get:

4. The method for optimizing the trajectory simulation and firing parameter calculation process of an antiaircraft gun according to claim 1, characterized in that: In step 3, according to the azimuth and height angle Convert the initial velocity components of the projectile into components in the launch coordinate system: 。 5. The method for optimizing the antiaircraft gun trajectory simulation and firing parameter calculation process according to claim 1, characterized in that: In step 3, according to the azimuth and height angle The conversion matrix for converting the wind speed in the northeast sky coordinate system to the wind speed in the launch coordinate system is as follows: get: 。 6. The method for optimizing the trajectory simulation and firing parameter calculation process of an antiaircraft gun according to claim 1, characterized in that: Step 4 includes the following steps: Step 4.1: Calculate the time step using the Runge-Kutta method The ballistic data in the image is used as the starting value of the Adams method to calculate the coordinates and speed values ​​of the first four steps. Coordinates of the first four steps: The speed values ​​for the first four steps are: Step 4.1.1: Based on the initial velocity of the projectile The velocity component of the component in the emission coordinate system , wind speed component ( ), time step ,calculate : Step 4.1.2: Calculation : Step 4.1.3: Calculation : Step 4.1.4: Calculation : Step 4.1.5: Update location , , and speed , , : Repeat steps 4.1.2-4.1.5 to calculate the coordinate and velocity values ​​of the first four steps: use and As new initial conditions, repeat steps 4.1.2-4.1.5 to calculate the coordinate and velocity values ​​for the second step. and ; use and As new initial conditions, repeat steps 4.1.2-4.1.5 to calculate the coordinate and velocity values ​​for the third step. and ; use and As new initial conditions, repeat steps 4.1.2-4.1.5 to calculate the coordinate and velocity values ​​of the fourth step. and ; Step 4.2: Predict the next trajectory data based on the Adams prediction formula and ; Step 4.2.1: Get known data Get the coordinates and speed values ​​of the first four steps: coordinate: speed: Get the current time step: ; Initialize n=4; Step 4.2.2: Calculate the forecast value Use the Adams forecast formula to calculate the coordinates and velocity forecast values ​​for the next step: Coordinate prediction value: Speed ​​prediction value: Step 4.3: Correct the predicted value according to the Adams correction formula to obtain more accurate trajectory data; Step 4.3.1: Substitute the Adams correction formula: Substitute the corrected acceleration value calculated in step 1 into the Adams correction formula to calculate the corrected coordinate and velocity values: Coordinate correction value: Speed ​​correction value: Step 4.3.2: Repeat steps 4.2 and 4.3 until the entire trajectory data calculation is completed; Step 4.5: Determine the coordinates of the trajectory endpoint based on the trajectory data and speed .

7. The method for optimizing the trajectory simulation and firing parameter calculation process of an antiaircraft gun according to claim 1, characterized in that: The revised formula in step 6.1 is expressed as: Where k is the correction factor, is the current shooting angle, , is the altitude and azimuth corresponding to the end point of the trajectory, are the altitude and azimuth angles corresponding to the target's future position.

8. The method for optimizing the antiaircraft gun trajectory simulation and firing parameter calculation process according to claim 1, characterized in that: In step 6.2, before returning to step 4, the number of iterations is also determined. If the number of iterations is greater than the threshold, the process ends. Otherwise, the corrected high and low angles are , azimuth , target predicted position , return to step 4.

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