Mathematical Problems in Engineering

Volume 2017, Article ID 8157319, 8 pages

https://doi.org/10.1155/2017/8157319

## Simulation-Based Early Prediction of Rocket, Artillery, and Mortar Trajectories and Real-Time Optimization for Counter-RAM Systems

Helmut-Schmidt-University/University of the Federal Armed Forces Hamburg, Institute of Automation Technology, Chair of Measurement and Information Technology, Holstenhofweg 85, 22043 Hamburg, Germany

Correspondence should be addressed to Arash Ramezani; ed.hh-ush@inazemar

Received 30 January 2017; Accepted 2 July 2017; Published 7 August 2017

Academic Editor: Marcello Vasta

Copyright © 2017 Arash Ramezani and Hendrik Rothe. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The threat imposed by terrorist attacks is a major hazard for military installations, for example, in Iraq and Afghanistan. The large amounts of rockets, artillery projectiles, and mortar grenades (RAM) that are available pose serious threats to military forces. An important task for international research and development is to protect military installations and implement an accurate early warning system against RAM threats on conventional computer systems in out-of-area field camps. This work presents a method for determining the trajectory, caliber, and type of a projectile based on the estimation of the ballistic coefficient. A simulation-based optimization process is presented that enables iterative adjustment of predicted trajectories in real time. Analytical and numerical methods are used to reduce computing time for out-of-area missions and low-end computer systems. A GUI is programmed to present the results. It allows for comparison between predicted and actual trajectories. Finally, different aspects and restrictions for measuring the quality of the results are discussed.

#### 1. Introduction

Field camps are military facilities which provide living and working conditions in out-of-area missions. During an extended period of deployment abroad, they have to ensure safety and welfare for soldiers.

Current missions in Iraq or Afghanistan have shown that the safety of military camps and air bases is not sufficient. A growing threat to these military facilities is the use of unguided rockets, artillery projectiles, and mortar grenades. Damage with serious consequences has occurred increasingly often in the past few years.

This paper focuses on mortars and rockets because they are more and more used by irregular forces, where they have easy access to a large amount of these weapons. Further reasons are the small radar cross-section, the short firing distance, and the thick cases made of steel or cast-iron, which makes mortar projectiles and rockets hard to detect and destroy.

The challenge is to establish an early warning system for different projectiles using analytical and numerical methods to reduce computing time and improve simulation results compared to similar systems. An appropriate estimation of the ballistic coefficient and the associated calculation of unknown parameters is the central issue in this field of research.

Up to now, only a few approaches have been published. Khalil et al. [1] presented a trajectory prediction for the special field of fin stabilized artillery rockets. Chusilp et al. [2] compared 6-DOF trajectory simulations of a short range rocket using aerodynamic coefficients. A very good overview of modeling and simulation of aerospace vehicle dynamics is given by Zipfel [3].

An et al. [4] used a fitting coefficient setting method to modify their point mass trajectory model. Chusilp and Charubhun [5] estimated the impact points of an artillery rocket fitted with a nonstandard fuze. Scheuermann et al. [6] characterized a microspoiler system for supersonic finned projectiles. Wang et al. [7] established a guidance and control design for a class of spin-stabilized projectiles with a two-dimensional trajectory correction fuze. Lee and Jun [8] developed guidance algorithm for projectile with rotating canards via predictor-corrector approach. Fresconi et al. [9] developed a practical assessment of real-time impact point estimators for smart weapons.

This paper is based on Ramezani et al. [10]. Real-time prediction of trajectories and continuous optimization is one of the main aims of this work. With the aid of graphical solutions, it is possible to differentiate between several objects and determine firing locations as well as points of impact. The goal is to provide active protection of stationary assets in today’s crisis regions. Therefore, a modern counter-RAM system with a clear GUI must be developed and will then be employed for most threats.

#### 2. Ballistic Model

The projectile is to be expected as a point mass: that is, the entire projectile mass is located in the center of gravity. Rotation is irrelevant in this case, so we regard a ballistic model with 3-DOF.

The Earth can be regarded as a static sphere with infinite radius and represents an inertial system. Based on an Earth-fixed Cartesian coordinate system, the force of inertia is applied in a single direction.

Different projectiles have to be considered in order to set up a mathematical model. While rockets can be regarded as spin-stabilized projectiles, which have a short phase of thrust and are particularly suitable for long distances up to 20 km, mortar grenades are arrow-stabilized and fired on short distances up to approximately 8 km.

Other mathematical models for typical fin stabilized artillery rockets are presented in [11–16].

##### 2.1. Exterior Ballistics

The ballistic model is principally based on Newton’s law and the equations of motion are considered to be under the effect of air drag and the force of gravity only. Additionally, rockets have a thrust vector impelling the projectile for a few seconds (generally, combustion gases have a velocity range of 1800–4500 m/s [18]). Anyhow, rockets as well as mortars have ballistic trajectories and the object is to identify the threat on the basis of different flight characteristics.

Let denote a reference acceleration (acceleration of gravity at sea level on Earth), with taking effect on the point mass in vertical direction.

The air drag can have different values, depending on the design of the projectile, that is,(i)muzzle velocity ,(ii)weight,(iii)aerodynamics, and the properties of air, for example, (i)density,(ii)temperature,(iii)wind,(iv)speed of sound.

Considering the general formulacontaining all parameters named above with (i): cross-section area of the projectile,(ii): air density,(iii): velocity of the projectile,(iv): air drag coefficient,(v): environmental properties,(vi): ballistic coefficient, it is operative to find an appropriate approximation, so that the projectile can be specified. The parameters , , , , and are unknown, whereas can be defined precisely from the measured radar data.

The air drag coefficient for instance depends on the critical velocity ratio, pictured in Figure 1. Since the drag coefficient does not vary in a simple manner with Mach number, this makes the analytic solutions inaccurate and difficult to accomplish.