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Mathematical Problems in Engineeringprojectiles. Wang et al. [7] established a guidance and controldesign for a class of spin-stabilized projectiles with a twodimensional trajectory correction fuze. Lee and Jun [8]developed guidance algorithm for projectile with rotatingcanards via predictor-corrector approach. Fresconi et al. [9]developed a practical assessment of real-time impact pointestimators for smart weapons.This paper is based on Ramezani et al. [10]. Real-timeprediction of trajectories and continuous optimization is oneof the main aims of this work. With the aid of graphicalsolutions, it is possible to differentiate between several objectsand determine firing locations as well as points of impact.The goal is to provide active protection of stationary assetsin todayโs crisis regions. Therefore, a modern counter-RAMsystem with a clear GUI must be developed and will then beemployed for most threats.Cd2MaFigure 1: Characteristics of the air drag coefficient ๐ถ๐ .2. Ballistic ModelThe 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 ballisticmodel with 3-DOF.The Earth can be regarded as a static sphere with infiniteradius and represents an inertial system. Based on an Earthfixed Cartesian coordinate system, the force of inertia isapplied in a single direction.Different projectiles have to be considered in order toset up a mathematical model. While rockets can be regardedas spin-stabilized projectiles, which have a short phase ofthrust and are particularly suitable for long distances up to20 km, mortar grenades are arrow-stabilized and fired onshort distances up to approximately 8 km.Other mathematical models for typical fin stabilizedartillery rockets are presented in [11โ16].2.1. Exterior Ballistics. The ballistic model is principally basedon Newtonโs law and the equations of motion are consideredto be under the effect of air drag and the force of gravityonly. Additionally, rockets have a thrust vector impelling theprojectile for a few seconds (generally, combustion gases havea velocity range of 1800โ4500 m/s [18]). Anyhow, rocketsas well as mortars have ballistic trajectories and the objectis to identify the threat on the basis of different flightcharacteristics. Let ๐ denote a reference acceleration (acceleration ofgravity at sea level on Earth), with 2 ๐ 9.80665 m/s ,(1)taking effect on the point mass in vertical direction. The air drag ๐ท can have different values, depending onthe design of the projectile, that is,(i) muzzle velocity V0 ,(ii) weight,(iii) aerodynamics,and the properties of air, for example,(i) density,(ii) temperature,(iii) wind,(iv) speed of sound.Considering the general formula 1 V ,๐ท ๐ถ๐ ๐ด ๐ V 2(2)๐ถ๐ฅ ๐ถ๐ ๐ถ๐ด ๐ตcontaining all parameters named above with(i) ๐ด: cross-section area of the projectile,(ii) ๐: air density,(iii) V: velocity of the projectile,(iv) ๐ถ๐ : air drag coefficient,(v) ๐ถ๐ด : environmental properties,(vi) ๐ต: ballistic coefficient,it is operative to find an appropriate approximation, so thatthe projectile can be specified. The parameters ๐ด, ๐, ๐ถ๐ , ๐ถ๐ด ,and ๐ต are unknown, whereas V can be defined precisely fromthe measured radar data.The air drag coefficient ๐ถ๐ for instance depends on thecritical velocity ratio, pictured in Figure 1. Since the dragcoefficient does not vary in a simple manner with Machnumber, this makes the analytic solutions inaccurate anddifficult to accomplish.One can see from this figure that there is no simpleanalytic solution to this variation. With computer powernowadays, we usually solve or approximate the exact solutions numerically, doing the quadratures by breaking the areaunder the curve into quadrilaterals and summing the areas. Ingeneral, there are three forms of the drag coefficient:(1) Constant ๐ถ๐ that is useful for the subsonic flightregime: ๐๐ 1

Mathematical Problems in Engineering3Let ๐ก denote the time, 0 ๐ก ๐ก๐ , with ๐ก 0 the initialtime and ๐ก ๐ก๐ the final time.The system of equations can be written asy 0โ๐๐ฅ ๐ข,๐๐กAltitude โmg โ 0jโkโiโ๐๐ฆ V,๐๐ก ๐๐ข ๐ถ๐ฅ V2 cos ๐,๐๐กxDistance(5)๐๐ค ๐ ๐ถ๐ฅ V2 sin ๐,๐๐กzwhereFigure 2: Mass point model with 3-DOF.V ๐ข2 ๐ค2(2) ๐ถ๐ inversely proportional to the Mach number that ischaracteristic of the high supersonic flight regime: inthis case, ๐๐ 1(3) ๐ถ๐ inversely proportional to the square root of theMach number that is useful in the low-supersonicflight regime: ๐๐ 1Carlucci and Jacobson [19] give a detailed description of theair drag coefficient.Another coefficient in common use in ballistics is theballistic coefficient ๐ต, which is defined as๐ต ๐,๐2(3)where ๐ and ๐ are the mass and diameter of the projectile[19]. Section 3.2 deals with the problem of estimating theunknown parameters.2.2. Equations of Motion. The aerodynamics and ballisticsliterature are quite diverse and terminology is far fromconsistent. This has particular significance in the coordinatesystems used to define the equations of motion. Nevertheless,this field of research has a long history and a lot of approaches.More details are discussed in [20โ24].In this paper, an Earth-bounded coordinate system isused. The Earth-bounded coordinate system {๐, ๐, ๐} is centered in the muzzle, with the axes ๐, ๐, ๐ pointing to fixeddirections in space. Axes ๐ is tangent to the Earth, ๐ is orthogonal to ๐ and runs against the gravity, and ๐ is orthogonal toboth ๐ and ๐, setting up a right-handed trihedron. The modelis illustrated in Figure 2.With the aforementioned parameters, the equilibrium offorces in this case can be described with the formula V ๐ ๐๐ ๐ท,๐๐ก(4)where ๐ is the total mass of the projectile.For setting up the system of equations, let (๐ฅ, ๐ฆ) denotethe projectile position and (๐ข, ๐ค) the velocity, with ๐ข determining the horizontal and ๐ค the vertical projection of thevelocity vector.(6)is the radial velocity and ๐ is the angle between the thrustvector and the ๐ฅ-axis: particularly๐ ๐๐ฆ.๐๐ฅ(7)3. ConceptThe purpose of the software is the calculation of trajectories.It receives the measured position of the projectile from thetracking radar and returns the predicted trajectory.A C code was written for the simulation and a GUI easesthe handling of the results. Radar data can be read in and willbe plotted for comparison.This chapter gives an overview of the methods used inthis paper. An integration method for differential equationsis introduced, which is used to solve the equations of motionin the previous section.3.1. Integration Method. There are several integration methods implemented, all providing better results compared to theanalytical methods used in [25].In this paper, the equations of motion are basically calculated with explicit, fixed step-size Runge-Kutta integrationtechniques. The advantage of this scheme over other schemesis that the approximating problems that result can be solvedvery efficiently and accurately. More details are discussed byRamezani [26].Knowing โ๐ ๐ก๐ 1 ๐ก๐ the algorithm can be programmedon the analogy of [27]๐ฅ๐ 1 ๐ฅ๐ 1(๐ก ๐ก ) (๐ 4 ๐3 ๐4 )6 ๐ 1 ๐ 1(8)with๐1 fl ๐ (๐ฅ๐ ) ,๐2 fl ๐ (๐ฅ๐ โ๐๐ ),2 1โ๐3 fl ๐ (๐ฅ๐ ๐ (๐1 ๐2 )) ,4๐4 fl ๐ (๐ฅ๐ โ๐ (๐2 2๐3 )) .(9)

4Mathematical Problems in EngineeringInitial values[a, b]c ๏ผ ๏ผญ(a b)Noc eTrajectory calculationwith a, bYesCx (a b)2Deviation calculationRadar dataOutput CxGold Section SearchNew interval a, bFigure 3: Programming flowchart.With a global discretization error ๐(โ4 ), the algorithm offersa tradeoff between high computing speed and best possibleresults [28].One may also select the Euler method in the program.Comparing to Runge-Kutta, the results are less precise dueto a lower order of consistency. Anyhow, the Euler methodachieves a significant improvement of computing time in themajority of cases with a global discretization error ๐(โ) [29].3.2. Iterative Optimization. In the mathematical model whichhas been described in Section 2.2, there are a number ofparameters missing. The other variables are given and canbe easily obtained through the measured trajectory elements. In order to determine the air drag ๐ท with the most accurateprecision, the following algorithm was developed.The air drag is chosen in a way so that the exterior ballisticmodel complies to the measured trajectory of the projectilein the best possible way. This implies that the sum of thedeviations between the calculated and the measured mortarpositions should be minimal:๐ min ๐ (๐ถ๐ฅ )๐222 min (๐ฅ๐๐ ๐ฅ๐๐ ) (๐ฆ๐๐ ๐ฆ๐๐ ) (๐ง๐๐ ๐ง๐๐ ) .(10)๐ 1The index ๐ refers to the coordinates which are measured bythe radar, while the index ๐ belongs to the coordinates whichare calculated by using numerical methods. The total amountof measurements is called ๐.Consequently, this is a nonlinear optimization. The objective function contains parameter ๐ถ๐ฅ . In order to find theoptimum, one of the fastest methods of one-dimensionaloptimization, the so-called โGolden Section Search,โ isapplied. It only needs one value of the objective functionfor each step of the calculation. The second value is takenfrom the preceding iteration step. This method possesses arobust and linear convergence speed to find the minimum of aunimodal continuous function over an interval without usingderivatives.The method chooses two points ๐ข1 ๐ข2 on the section[๐, ๐] considering Golden Section:๐ข1 ๐ ๐ข2 ๐ (๐ ๐) (3 5)2(๐ ๐) ( 5 1)2,(11).(12)If the inequality ๐(๐ข1 ) ๐(๐ข2 ) is complied, the minimum isin the interval [๐, ๐ข1 ]. In any other case, it will be found on thestretch [๐ข1 , ๐]. When this procedure is repeated, the intervalcan be shortened again. In case of a new partition [๐, ๐ข2 ],there are new boundaries ๐ข1 , ๐ข2 with ๐ข2 ๐ข1 . Therefore, onlytwo values of the goal functional are needed to be measuredduring the first step of the calculation [30].The goal is an optimal reduction factor for the searchinterval. Additionally, a minimal number of function calls arenecessary [31].Golden Section Search enables an iterative adjustment ofthe trajectory in each step by using the calculated parameter๐ถ๐ฅ for every previous iteration. Therefore, prediction getsmore precise in the course of time.The programming flowchart is illustrated in Figure 3.

Mathematical Problems in Engineering5TX-plotXY-plot80011700106009X (m) (103 )Y (m)500400300876200510045678X (m) (103 0.2Z (m) (103 )500Y (m)15T T (s)T (s)RadarNumericRadarNumericFigure 4: Forecast with the calculation period 3โ6 s.Table 1: Rocket Type 63 HE specification [17].Maximum rangeOverall lengthCaliberCross-sectional areaWeightLateral moment of inertiaLongitudinal moment of inertiaPosition of center of gravityStandard empty weightCombustion durationImpulseFlight time8.5 km839.0 mm107 mm7.72 cm218.84 kg0.98122 kgm20.03135 kgm2395.8 mm8.496 kg0.6 sec6.7 kNs21.5 sec4. Simulation ResultsAn example is calculated for a rocket Type 63 HE on acommon Intel x86. The data specification of the rocket islisted in Table 1.The trajectory was recorded by a Weibel radar of the typeMFDR-2100/35. It detects the RAM target at high accuracy. Itis designated to the RAM target with information receivedfrom a search radar. The accuracy is listed in Table 2. TheKalman filter illustrates the track error over time.Let ๐ก0 3 s be the time at which the radar detects thebullet. The final flight time is reached after ๐ก๐ 21.5 s.The duration of calculation is adjustable at will. Moretracking points will certainly help to get better results,but sometimes a fast interception of the RAM threat isindispensable. Starting the forecast with a 3-second period ofcalculation, there will be a mean square deviation of 566.9 mbetween the calculated and the real trajectory. By now itis possible to identify different RAM targets by regardingthe predicted trajectory characteristics. This example is illustrated in Figure 4. After 6 seconds of calculation, the meansquare deviation is reduced to 181.8 m. There are inaccuraciesin all axes of coordinates. The estimation of the calculatedparameter ๐ถ๐ฅ needs more iterative calculation steps at thispoint.Finally, after 12 seconds of calculation, the mean squaredeviation is reduced to 32.2 m and there is still enough

6Mathematical Problems in EngineeringTX-plotXY-plot80011700106009X (m) (103 )Y (m)500400300876200510045678X (m) (103 0.2Z (m) (103 )500Y (m)15T T (s)T (s)RadarNumericRadarNumericFigure 5: Forecast with the calculation period 3โ15 s.Table 2: Weibel radar accuracy [17].Time (ms)1020500Max. range (km)8.913.2Rg (m)0.200.0923.60.03Accuracy 2.0 kmAz (mills)El (mills)0.4910.4910.2200.220Kalman filter0.0690.069time for the command and control system to initiate allnecessary steps, for example, warning and defending. Theresults are shown in Figure 5. It is quite obvious that thesimulated altitude is overestimated. Mortar grenades have astrong change in altitude on their trajectory, a challenge forsimulation-based early prediction systems.The prediction of the trajectory allows the calculation ofthe point of impact. The future position of the projectile iscalculated through extrapolation of the measured values.It is clear that the prediction gets significantly better witheach iteration. Thus, certain areas in the field camp can bewarned partially and a counterattack can be initiated. TheRg (m)0.810.360.12Accuracy 4.0 kmAz (mills)1.9650.879El (mills)1.9650.8790.2780.278more the radar data available for the analysis, the closer theprediction to the measured trajectory. More tracking pointswill certainly help to get better results, but sometimes a fastinterception of the RAM threat is indispensable.With a prediction of 3 seconds into the future, forexample, which corresponds to an intercept range of almost3000 m, the computational error at the point of impact222๐๐ (๐ฅ๐๐ ๐ฅ๐๐ ) (๐ฆ๐๐ ๐ฆ๐๐ ) (๐ง๐๐ ๐ง๐๐ )(13)is smaller than 3 m. Here, the index ๐ refers to the coordinatesthat are measured and calculated at the point of impact.

Mathematical Problems in Engineering7Details and more examples are discussed by Ramezani et al.[10].[8]5. Summary and OutlookThis paper introduces an algorithm for early warning systemsused for command and control applications in out-of-areamissions and is based on the MONARC (modular navalartillery concept) project. The basic methods have been testedsuccessfully and they are used in fire guidance solutions forGerman frigates of type 124 and 125.The most important aspect is that one can distinguishbetween different projectiles in order to predict the trajectories and hit points more accurately. To calculate theirtrajectories, different flight phases are analyzed and thedesigns of the projectiles are estimated by the use of iterativeoptimization methods for approximating environmental andballistic properties.Future work concentrates on giving the user specificinformation of the projectile data. Further work has also tobe done on a 3-dimensional simulation.At the end, sophisticated simulation software will beestablished through which it will be possible to show andevaluate a real-time battlefield scenario.[9][10][11][12][13][14]Conflicts of InterestThe authors declare that they have no conflicts of interest.[15]References[16][1] M. Khalil, H. Abdalla, and O. Kamal, โTrajectory prediction fortypical fin stabilized artillery rocket,โ in Proceedings of the 13thInternational Conference on Aerospace Sciences and AviationTechnology, ASAT-13, 2009.[2] P. Chusilp, W. Charubhun, and N. Nutkumhang, โA comparative study on 6-DOF trajectory simulation of a short rangerocket using aerodynamic coefficients from experiments andmissile DATCOM,โ in Proceedings of the Second TSME International Conference on Mechanical Engineering, Krabi, Thailand,2011.[3] P. H. Zipfel, Modeling and Simulation of Aerospace VehicleDynamics, American Institute of Aeronautics and Astronautics,Reston, Va, USA, 2014.[4] S. An, K. B. Lee, and T. H. Kang, โFitting coefficient settingmethod for the modified point mass trajectory model usingCMA-ES,โ Journal of the Korea Institute of Military Science andTechnology, vol. 19, no. 1, pp. 95โ104, 2016.[5] P. Chusilp and W. Charubhun, โEstimation of impact pointsof an artillery rocket fitted with a non-standard fuze,โ inProceedings of the 2nd Asian Conference on Defence Technology(ACDT โ14), pp. 25โ31, 2014.[6] E. Scheuermann, M. Costello, S. Silton, and J. Sahu, โAerodynamic characterization of a microspoiler system for supersonicfinned projectiles,โ Journal of Spacecraft and Rockets, vol. 52, no.1, pp. 253โ263, 2015.[7] Y. Wang, W.-D. Song, D. Fang, and Q.-W. Guo, โGuidance andcontrol design for a class of spin-stabilized projectiles witha two-dimensional trajectory correction fuze,โ 26][27]Journal of Aerospace Engineering, vol. 2015, Article ID 908304,15 pages, 2015.C.-H. Lee and B.-E. Jun, โGuidance algorithm for projectilewith rotating canards via predictor-corrector approach,โ inProceedings of the 2014 IEEE Conference on Control Applications,CCA 2014, pp. 2072โ2077, October 2014.F. Fresconi, G. Cooper, and M. Costello, โPractical assessment ofreal-time impact point estimators for smart weapons,โ Journal ofAerospace Engineering, vol. 24, no. 1, pp. 1โ11, 2011.A. Ramezani, J. Cors, and H. Rothe, โComparison of methodsfor simulation-based early prediction of rocket, artillery andmortar trajectories,โ in Proceedings of the 2012 Autumn Simulation Multi-Conference (AutumnSim โ12), San Diego, Calif, USA,2012.B. Etkin, Dynamics of Atmospheric Flight, John Wiley and Sons,Hoboken, NJ, USA, 1972.O. Douglas and C. Mark, โModel predictive control of a directfire projectile equipped with canards,โ Journal of DynamicSystems, Measurement and Control, Transactions of the ASME,vol. 130, no. 6, pp. 0610101โ06101011, 2008.S. A. A. Ezeddine, Military Technical College, Cairo, Egypt,2009.E. Gagnon and M. Lauzon, โCourse correction fuze conceptanalysis for in-service 155 mm spin-stabilized gunnery projectiles,โ in Proceedings of the AIAA Guidance, Navigation andControl Conference and Exhibit, Honolulu, Hawaii, USA, August2008.S. Jankovic, J. Gallant, and E. Celens, โDispersion of an artilleryprojectile due to unbalance,โ in Proceedings of the 18th International Symposium on Ballistics, San Antonio, Tex, USA, 1999.M. S. Khalil, โTrajectory Predection of Flying Vehicle,โ Tech.Rep., Military Technical College, Cairo, Egypt, 2008.M. Knapp, P. Kossebau, A. Ramezani, and H. Rothe,Zuฬnderuntersuchung C-RAM, Bundesamt fuฬr Wehrtechnik undBeschaffung, Fachinformationsstelle BWB, Koblenz, Germany,2010.W. Wolff, Raketen und Raketenballistik, Elbe-Dnjepr-Verlag,Klitzschen, Germany, 2006.D. Carlucci and S. S. Jacobson, Ballistics: Theory and Design ofGuns and Ammunition, CRC Press, Boca Raton, Fla, USA, 2007.R. L. McCoy, Modern Exterior Ballistics, Schiffer MilitaryHistory, Atglen, Pa, USA, 1999.C. H. Murphy Ballistics Research Laboratory Report, AberdeenProving Ground, Md, USA, 1963.E. J. McShane, J. L. Kelley, and F. V. Reno, Exterior Ballistics,University of Denver Press, Denver, Colo, USA, 1953.J. D. Nicolaides, โOn the Free Flight Motion of Missiles HavingSlight Configurational Asymmetries,โ Tech. Rep., AberdeenProving Ground, Maryland, MD, USA, 1953.J. N. Nielsen, Missile Aerodynamics, American Institute ofAeronautics and Astronautics, Reston, VA, USA, 1988.I. Shaydurov and H. 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8[28] C. Buฬskens, Anleitungen zur Benutzung der Fortran-BibliothekNUDOCCCS, Westfaฬlische Wilhelms-University Muฬnster,1996.[29] W. Dahmen and A. Reusken, Numerik fuฬr Ingenieure undNaturwissenschaftler, Springer-Lehrbuch, Aachen, Germany,2008.[30] H. Rothe and S. Schroฬrder, Method for Determination of FireGuidance Solution, European Patent Office, Muฬnchen, Germany, 2006.[31] C. F. Gerald and P. O. Wheatley, Applied Numerical Analysis,Pearson, San Luis Obispo, Calif, USA, 2003.Mathematical Problems in Engineering

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ResearchArticle Simulation-Based Early Prediction of Rocket, Artillery, and Mortar Trajectories and Real-T