Solving the Inverse Problem of Determining the Firing Point Based on a Fixed Section of the Trajectory Using Deep Learning Neural Networks

The article presents mathematical models and numerical algorithms for solving direct and inverse problems of external ballistics in order to reconstruct the trajectory from a fixed section and determine the firing point. When solving inverse problems of external ballistics, two approaches are considered. The first approach is based on numerical methods for solving systems of differential equations of projectile motion and algorithms for solving inverse problems. The second approach is based on processing trajectory measurement data using deep learning neural networks. The algorithm for solving the inverse problem using numerical methods consists of several stages. Since the projectile coordinate measurement data are obtained with some error depending on the characteristics of the locator, a procedure for approximating the data with fourth-degree polynomials is used to obtain smooth curves. To determine the type of projectile, its ballistic coefficient is calculated. The projectile with the closest ballistic coefficient and its main characteristics are determined from the projectile database. The trajectory from the beginning of the fixed section to the firing point is calculated by numerically integrating the equations of motion with a negative time step. To clarify the trajectory parameters and coordinates of the firing point, the problem of minimizing the deviations of the calculated trajectory from the points recorded by the locator is solved. The general algorithm for solving the inverse problem using neural networks includes the following stages: preprocessing of the initial data; estimation of the directional firing angle; classification of the projectile type; reconstruction of the projectile trajectory; determination of the coordinates of the firing point from the solution of the optimization problem. To solve the problems of classifying the projectile type and reconstructing the trajectory, convolutional neural networks of deep learning were built. The data for training the neural networks were obtained as a result of modeling the trajectories of 20 types of projectiles with different initial firing angles, wind direction and speed. The results of a study of the dependence of the error in determining the coordinates of the firing point on the root-mean-square error of measuring the projectile coordinates by a radar are presented.