ADVANCES in MATHEMATICAL and COMPUTATIONAL METHODS

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1 ADVANCES in MATHEMATICAL and COMPUTATIONAL METHODS Requirements on the accuracy of determination of target position and movement parameters LUDEK JEDLICKA Department of Weapons and Ammunition University of Defence Brno Kounicova Str CZECH REPUBLIC ludek.jedlicka@unob.cz Abstract: The article is focused on the exterior ballistic analysis of the small calibre experimental weapon system equipped with a passive target tracking unit. There are briefly described the structure of the weapon system the structure of the used fire control system target position and movement parameters in the article. The effect of the accuracy of measurement of the target position and movement parameters on the position of the mean point of impact is investigated. On basis of the carried out sensitivity analysis the minimal from the exterior ballistic point of view requirements on the accuracy of the target position and movement parameters determination are set for the given experimental weapon system. Key-Words: Exterior ballistics trajectory modelling fire control systems fire data ballistic solution aiming angles 1 Introduction The fire accuracy of guns is largely affected by the accuracy of the determination of the relative position between the weapon and the target and also the accuracy of determination of the target movement parameters. In case of the direct fire the most important characteristics of the target is the slant range the angle of site and the target angular velocity as the basic input factors for the fire data computation [6 9]. Contemporary direct fire weapon systems are usually equipped with the laser range finder for measurement of the slant range and manually can be measured the angle of site and the angular velocity of the target. Often it is used also the operator s estimation of the target velocity and direction of movement for the prediction of future target position. Disadvantage of the use of laser rangefinder is the disclosure of own position to the enemy due to the emission of electromagnetic radiation [6]. The newly developed weapon system utilises the passive target tracking unit that is based on the battlefield scene image processing [8] and does not emit any electromagnetic radiation. From the image processing the slant range of the target is obtained and from the resolvers of the servo drives propelling the passive target tracking unit the angle of site and the target angular velocities are obtained. The question for designers of such weapon system is still how accurately the position and movement parameters must be determined to hit the target with the required probability. The accuracy requirements on these parameters can be analysed from at least the exterior ballistic and target tracking point of views. In the next chapters the exterior ballistic analysis of the requirements on the accuracy of measurement of the target position and movement parameters will be carried out. 2 Experimental weapon system 2.1 Structure of the experimental weapon system The experimental weapon system can be described as a static distributed weapon system consisting of three main parts: small calibre automatic weapon mounted on a stationary gun carriage passive target tracking unit and controlling computer. The schematic arrangement and mutual connection among individual parts can be seen in Fig. 1. Weapon of calibre 7.62x54 mm Controlling computer Passive target tracking unit Fig. 1 Structure of the experimental distributed weapon system ISSN: ISBN:

2 ADVANCES in MATHEMATICAL and COMPUTATIONAL METHODS The experimental weapon system is based on the small calibre automatic weapon of calibre 7.62x54 mm with projectile s initial velocity of 855 m.s -1 and maximum range of fire 39 m. The automatic weapon is mounted on the stationary ground gun carriage. The gun carriage is equipped with two servo drives (elevation and bearing) with resolvers for aiming the weapon. The passive target tracking unit purchases the image of the scene for the controlling computer and also provides information about state of its two servo drives with resolvers (elevation and bearing) for the controlling computer. Controlling computer provides all functions for the image processing controlling of the passive target tracking unit drives the controlling of the weapon drives and also for the fire control system. 2.2 Structure and tasks of the experimental fire control system Important part of each sophisticated direct fire weapon system is the fire control system. In the heart of such fire control system a ballistic computer connected with number of sensors and also with the weapon control system is placed. His essential task is to calculate the aiming angles (elevation and azimuth) for the particular firing task (firing against stationary or moving target). The simplified structure of the experimental fire control system is shown in Fig. 2. Ballistic data Meteo sensors (temperature pressure wind speed and direction) Ballistic computer Passive target tracking unit (elevation bearing) (slant range angle of site angular velocities) Controlling computer (slant range angle of site angular velocities) Weapon drives Fig. 2 Structure of the experimental fire control system The ballistic computer calculates the projectile trajectories with use of ballistic data meteorological information and position of the target. Further it iterates the aiming angles for the given target position (real for static targets or predicted for moving target). For the calculation of the projectile trajectories the point mass model is utilised. This simple exterior ballistic model described e.g. in [4] and [5] is sufficiently accurate for solving the direct fire firing tasks. This model is also fully comparable with point mass trajectory models presented in [1] [2] and [3]. Use of more sophisticated exterior ballistic model (e.g. modified point mass trajectory model six degrees of freedom model see [1] [3]) for the direct fire weapon systems would bring a lower computation speed that is not balanced with the advantage of more accurate trajectory calculation. The point mass trajectory model consists of six first order differential equations. These equations describe projectile motion as a motion of a point mass in the space. The model is based on following equations: dv p c v w G dt dx vx dt x N N 43 x x 43 pn N dvy p N N c43 vy wy G43 g dt pn N dy vy dt dv z p N N c43 vz wz G43 dt pn N dz vz dt where: c 43 ballistic coefficient (drag law 1943) p N standard atmospheric pressure p N standard atmospheric pressure at sea level w wind speed τ N standard virtual air temperature τ N standard virtual air temperature at sea level G 43 drag function (drag law 1943). The system of equations (1) is solved by the means of standard Runge-Kutta method with variable time step. The basic function of any ballistic computer is to calculate the aiming angles (angle of elevation and bearing) for the particular firing task. To successfully fulfil this task it is necessary to know the following: trajectory model (in this case the point mass trajectory model) characteristics of the weapon system (initial velocity ballistic coefficient) atmospheric conditions (air pressure air temperature air humidity direction and speed of wind) position of the target (slant range of the target azimuth and angle of site) (1) ISSN: ISBN:

3 Height of mean point of impact [m] ADVANCES in MATHEMATICAL and COMPUTATIONAL METHODS angular velocity of the target (vertical and horizontal) in case of moving target. The angle of elevation can be calculated for the particular firing task (i.e. position of the target) iteratively. In this case one of the most efficient methods the secant method is used. The method can be described by the following relation: i i 1 y i 1 where: i1 i 2 1 y i 2 y i θ angle of elevation y height of point of impact on the target. The iteration process is ended when the height of point of impact y differs from (centre of the target) by less than 1 calibre i.e m. All calculations are carried out under the standard atmospheric conditions. In case of moving target it is further necessary to predict the future position of the target with the use of target angular velocities and the time of projectile flight from the muzzle of barrel to the target. The aiming angles (elevation and bearing) can be obtained again iteratively. The passive target tracking unit determines the slant range of the target. The angle of site and angular velocities in horizontal and vertical directions are obtained from the servos that move the passive target tracking unit in horizontal and vertical directions. Atmospheric characteristics air pressure air temperature wind speed and direction are obtained from the corresponding meteorological sensors. Use of these sensors allows calculation of projectile trajectories for the non-standard conditions. 3 Effect of target position and movement parameters accuracy on the position of the mean point of impact 3.1 Slant range The range of the target is one of the most important characteristics of the target for the calculation of the aiming angles. The accuracy of this characteristic directly affects the accuracy of fire and consequently also the target hit probability. The effect of the accuracy of the target range measurement on the position of the mean point of impact is expressed by means of change of height of point of impact on the target. Calculation of the aiming angles is realised for the measured target range (with error) and the projectile trajectory is calculated for the real (correct) (2) target range and the height of mean point of impact is recorded. Based on the [7 8] it can be estimated that the range of errors of the slant target range measurement will not exceed ± 15 % of the target range up to 35 m. The change of the height of mean point of impact is presented in dependency on the relative range of the target. The relative range of the target is defined as a ratio of the measured and the real range of the target. Four real ranges of the target (4 6 8 and 1 m) were chosen for the calculations. The effect of the inaccurate target range measurement on the position of the height of mean point of impact is shown in Fig. 3 and in Table m 6 m 8 m 1 m Relative range of target [1] Fig. 3 Effect of inaccuracy of target range measurement Table 1 Change of height of mean point of impact [m] due to target range measurement inaccuracy Error of target range measurement [%] Range [m] Angle of site Another characteristic of the target position is the angle of site. The angle of site is the vertical angle that is formed by the line of site and the horizontal. The effect of inaccurate measurement of the angle of site will be expressed in a similar way as the effect of the target range measurement. For the simulations were ISSN: ISBN:

4 Side of mean point of impact [m] Height of mean point of impact [m] ADVANCES in MATHEMATICAL and COMPUTATIONAL METHODS chosen on basis of the technical capability of the weapon system following values of the angle of site ±1 ±5 and ±1. The angle of site measurement is usually more accurate than the slant target range measurement and therefore the expected errors of the angle of site measurement were chosen from the range of ±1 % and the slant range of the target was set on 6 m which is the maximum effective range of fire for the used weapon system. The effect of the inaccurate angle of site measurement on the position of the height of mean point of impact is shown in Fig. 4 and in Table Fig. 4 Effect of inaccuracy of angle of site measurement Table 2 Change of height of mean point of impact [m] due to angle of site measurement inaccuracy Error of angle of site measurement [%] Angle of site [ ] Relative target angle of site [1] calculated with use of the trajectory model (1) for the chosen target ranges are shown in Table 3. Table 3 Change of time of flight with range of target Range [m] ToF [s] The ground targets e.g. military vehicles tanks armour fighting vehicles etc. move with velocity from up to 7 km.h -1 (2 m.s -1 ) in the terrain. The angular velocity can reach in dependency on the target range values in the order of magnitudes.1 rad.s -1. The values of both horizontal and vertical angular velocities are obtained from the resolvers of the servo drives propelling the passive target tracking unit. The effect of the inaccurate angular velocities determination on the position of the mean point of impact will be investigated on the target in the range of 6 m and moving in the muzzle level. The expected errors of the angular velocities are chosen from the range of ±1 % and the angular velocities were chosen and.1 rad.s -1 for the horizontal and.1.5 and.25 rad.s -1 for the vertical directions Horizontal angular velocity The inaccurate measurement of the horizontal angular velocity causes the inaccurate prediction of the future position of the target which means that the point of impact moves in the horizontal. The results of calculations for the target in range of 6 m are shown in Fig. 4 and summarised in Table rad.s rad.s -1.5 rad.s rad.s -1.5 rad.s -1.1 rad.s Angular velocity Knowledge of the target angular velocity with respect to the weapon and the time of projectile flight to the target (ToF) plays important role during prediction of the target position on the projectiles arrival. The values of ToF Relative horizontal angular position [1] Fig. 4 Effect of inaccuracy of horizontal angular velocity measurement ISSN: ISBN:

5 Height of point of impact [m] ADVANCES in MATHEMATICAL and COMPUTATIONAL METHODS Table 4 Change of side of mean point of impact [m] due to horizontal angular velocity measurement inaccuracy Error of horizontal angular velocity measurement [%] Angular velocity [rad.s -1 ] Vertical angular velocity The vertical angular velocity reaches lower values in case of ground targets than the horizontal angular velocities due to the nature of the motion. The effect of the inaccurate measurement of the vertical angular velocity is similar to the effect of inaccurate measurement of the angle of site. The mean point of impact moves in the vertical. The results of calculations for the target in range of 6 m are shown in Fig. 5 and summarised in Table rad.s -1.5 rad.s -1.1 rad.s Relative vertical angular position [1] Fig. 5 Effect of inaccuracy of vertical angular velocity measurement Table 5 Change of height of mean point of impact [m] due to vertical angular velocity measurement inaccuracy Error of vertical angular velocity measurement [%] Angular velocity [rad.s -1 ] Discussion It can be seen pro the previously carried out analysis that the accuracy of measurement can significantly affect the position of the mean point of impact. For further consideration about the position of the mean point of impact it is assumed use of the NATO standard target (2.3 x 2.3 m). The position of the target is given by the slant range and the angle of site. From the Table 1 it can be seen that for the chosen standard target is the minimal accuracy of the slant range measurement from the ballistic point of view ±15 % for the target in the range of 6 m see Fig. 3 and Table 1. The required accuracy of measurement increases with increasing target range to the ±5 % for the target range 1 m. The angles of site for the target at the range of 6 m should be measured with better than ±1 % accuracy to keep the mean point of impact in the target see Fig. 4 and Table 2. In case of angle of site +1 or -1 must be noted that with ±1 % accuracy measurement will be the mean point of impact on the edge of the target and the hit probability will be about 5 % (one half of the dispersion pattern will lie outside the target). For the measurement of the horizontal angular velocity of the target at range of 6 m the ±1 % accuracy is sufficient see Fig. 4 and Table 4. Obviously with the increasing range of the target and increasing angular velocity the requirements on the accuracy of measurement will increase e.g. for the same conditions and the target range of 8 m the mean point of impact lies on the edge of the target. The vertical angular velocities are in practice lower than the horizontal ones. The required accuracy of their measurement should be kept under ±9%. The calculations carried out for a particular target ranges can be realised for any target range from to the maximum range of fire and the obtained results can be used in the same way. Also the properties of the weapon ISSN: ISBN:

6 ADVANCES in MATHEMATICAL and COMPUTATIONAL METHODS system can be changed and the results can be utilised in the same way. 5 Conclusion It can be concluded that the carried out analysis can be used for the setting the accuracy requirements on the target position and movement parameters measurements. The results of the analysis can be used as a basis for the setting of the minimal accuracy requirements on the target position and movement parameters. For the particular weapon system and the expected range of fire up to the 8 m it holds true that the slant range must be measured with better than ±7 % accuracy. The angle of site should be measured with better than ±.75 % accuracy. The angular velocities must be measured with better than ±1 % in case of horizontal and ±8 % in case of vertical directions. To set the requirements on the accuracy of the target position and movement parameters it is necessary to finish a similar analysis from the target tracking and image processing point of view. From the results of both analyses the final requirements on accuracy of target position and movement parameters for the experimental weapon system can be set. [9] CECH V. Armament of tank T-72 (in Czech). Brno: VA AZ p. Acknowledgement: The work presented in this paper has been supported by the Ministry of Defence of the Czech Republic (research project No. MOFVT42). References: [1] McCoy Robert L. Modern Exterior Ballistics: The Launch and Flight Dynamics of Symmetric Projectiles. New York: Schiffer Aviation History ISBN p. [2] MOSS G. M. LEEMING D. W. FARRAR C. L. Military ballistics. London: Brassey s ISBN p. [3] Textbook of ballistics and gunnery. London: Her Majesty s Stationery Office Vol. I. 86 p. [4] JIRSAK C. and KODYM P. Exterior ballistics and theory of fire (in Czech). Prague: Naše vojsko p. [5] HAUCK G. Aussere Ballistik (in German). Berlin: Militarverlag der DDR [6] MIL-HDBK-799(AR) Fire Control Systems- General Washington: USA Department of Defence April p. [7] JEDLICKA Ludek KOMENDA Jan BEER Stanislav. Analysis of ballistic systems up to caliber of 2 mm suitable for firing against ground targets at distances to 1 m (in Czech). [Research report VZ MOFVT42]. Brno: University of Defence p. [8] VITEK R. Mathematical model of the target position determination from the image information (in Czech). [Research Report VZ MOFVT42]. Brno: University of Defence p. ISSN: ISBN: