Simulation-Based Early Prediction of Rocket, Artillery ...ResearchArticle Simulation-Based Early Prediction of Rocket, Artillery, and Mortar Trajectories and Real-Time Optimization

Research ArticleSimulation-Based Early Prediction of Rocket Artilleryand Mortar Trajectories and Real-Time Optimization forCounter-RAM Systems

Arash Ramezani and Hendrik Rothe

Helmut-Schmidt-UniversityUniversity of the Federal Armed Forces Hamburg Institute of Automation TechnologyChair of Measurement and Information Technology Holstenhofweg 85 22043 Hamburg Germany

Correspondence should be addressed to Arash Ramezani ramezanihsu-hhde

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

Academic Editor Marcello Vasta

Copyright copy 2017 Arash Ramezani and Hendrik Rothe This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

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

1 Introduction

Field camps are military facilities which provide livingand working conditions in out-of-area missions During anextended period of deployment abroad they have to ensuresafety and welfare for soldiers

Current missions in Iraq or Afghanistan have shown thatthe safety of military camps and air bases is not sufficientA growing threat to these military facilities is the use ofunguided rockets artillery projectiles and mortar grenadesDamagewith serious consequences has occurred increasinglyoften in the past few years

This paper focuses on mortars and rockets because theyare more and more used by irregular forces where they haveeasy access to a large amount of these weapons Furtherreasons are the small radar cross-section the short firingdistance and the thick cases made of steel or cast-iron whichmakes mortar projectiles and rockets hard to detect anddestroy

The challenge is to establish an early warning system fordifferent projectiles using analytical and numerical methodsto reduce computing time and improve simulation resultscompared to similar systems An appropriate estimation ofthe ballistic coefficient and the associated calculation ofunknown parameters is the central issue in this field ofresearch

Up to now only a few approaches have been publishedKhalil et al [1] presented a trajectory prediction for thespecial field of fin stabilized artillery rockets Chusilp et al[2] compared 6-DOF trajectory simulations of a short rangerocket using aerodynamic coefficients A very good overviewof modeling and simulation of aerospace vehicle dynamics isgiven by Zipfel [3]

An et al [4] used a fitting coefficient setting methodto modify their point mass trajectory model Chusilp andCharubhun [5] estimated the impact points of an artilleryrocket fitted with a nonstandard fuze Scheuermann et al[6] characterized amicrospoiler system for supersonic finned

HindawiMathematical Problems in EngineeringVolume 2017 Article ID 8157319 8 pageshttpsdoiorg10115520178157319

2 Mathematical Problems in Engineering

projectiles Wang et al [7] established a guidance and controldesign for a class of spin-stabilized projectiles with a two-dimensional 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 impactThe goal is to provide active protection of stationary assetsin todayrsquos crisis regions Therefore a modern counter-RAMsystem with a clear GUI must be developed and will then beemployed for most threats

2 Ballistic Model

The projectile is to be expected as a point mass that isthe entire projectile mass is located in the center of gravityRotation 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 Earth-fixed 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 [11ndash16]

21 Exterior Ballistics Theballisticmodel is principally basedon Newtonrsquos 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 1800ndash4500ms [18]) Anyhow rocketsas well as mortars have ballistic trajectories and the objectis to identify the threat on the basis of different flightcharacteristics

Let 997888rarr119892 denote a reference acceleration (acceleration ofgravity at sea level on Earth) with

10038161003816100381610038161198921003816100381610038161003816 = 980665ms2 (1)

taking effect on the point mass in vertical directionThe air drag 997888rarr119863 can have different values depending on

the design of the projectile that is

(i) muzzle velocity V0

(ii) weight

(iii) aerodynamics

Cd

Ma

Figure 1 Characteristics of the air drag coefficient 119862119889

and the properties of air for example

(i) density(ii) temperature(iii) wind(iv) speed of sound

Considering the general formula

997888rarr119863 = 12 sdot 119862119889 sdot 119860 sdot 120588 sdot |V| sdot

997888rarrV 119862119909 = 119862119889 sdot 119862119860 sdot 119861

(2)

containing all parameters named above with

(i) 119860 cross-section area of the projectile(ii) 120588 air density(iii) V velocity of the projectile(iv) 119862119889 air drag coefficient(v) 119862119860 environmental properties(vi) 119861 ballistic coefficient

it is operative to find an appropriate approximation so thatthe projectile can be specified The parameters 119860 120588 119862119889 119862119860and 119861 are unknown whereas V can be defined precisely fromthe measured radar data

The air drag coefficient 119862119889 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 solu-tions 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 119862119889 that is useful for the subsonic flightregime119872119886 lt 1

Mathematical Problems in Engineering 3Altitud

e

Distance

y

j

ki

z

0

m g

0

x

Figure 2 Mass point model with 3-DOF

(2) 119862119889 inversely proportional to the Mach number that ischaracteristic of the high supersonic flight regime inthis case119872119886 ≫ 1

(3) 119862119889 inversely proportional to the square root of theMach number that is useful in the low-supersonicflight regime119872119886 ge 1

Carlucci and Jacobson [19] give a detailed description of theair drag coefficient

Another coefficient in common use in ballistics is theballistic coefficient 119861 which is defined as

119861 = 1198981198892 (3)

where 119898 and 119889 are the mass and diameter of the projectile[19] Section 32 deals with the problem of estimating theunknown parameters

22 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 Neverthelessthis field of research has a long history and a lot of approachesMore details are discussed in [20ndash24]

In this paper an Earth-bounded coordinate system isused The Earth-bounded coordinate system 119894 119895 119896 is cen-tered in the muzzle with the axes 119894 119895 119896 pointing to fixeddirections in space Axes 119894 is tangent to the Earth 119895 is orthog-onal to 119894 and runs against the gravity and 119896 is orthogonal toboth 119894 and 119895 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

119898119889997888rarrV119889119905 = 997888rarr119892 + 997888rarr119863 (4)

where119898 is the total mass of the projectileFor setting up the system of equations let (119909 119910) denote

the projectile position and (119906 119908) the velocity with 119906 deter-mining the horizontal and 119908 the vertical projection of thevelocity vector

Let 119905 denote the time 0 le 119905 le 119905119891 with 119905 = 0 the initialtime and 119905 = 119905119891 the final time

The system of equations can be written as119889119909119889119905 = 119906119889119910119889119905 = V119889119906119889119905 = minus119862119909V

2 cos120593119889119908119889119905 = minus119892 minus 119862119909V

2 sin120593

(5)

where

V = radic1199062 + 1199082 (6)

is the radial velocity and 120593 is the angle between the thrustvector and the 119909-axis particularly

120593 = 119889119910119889119909 (7)

3 Concept

The purpose of the software is the calculation of trajectoriesIt 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

31 Integration Method There are several integration meth-ods implemented all providing better results compared to theanalytical methods used in [25]

In this paper the equations of motion are basically cal-culated with explicit fixed step-size Runge-Kutta integrationtechniquesThe 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 ℎ119894 = 119905119894+1 minus 119905119894 the algorithm can be programmedon the analogy of [27]

119909119894+1 = 119909119894 + 16 (119905119894+1 minus 119905119894) (1198961 + 4 sdot 1198963 + 1198964) (8)

with1198961 fl 119891 (119909119894) 1198962 fl 119891(119909119894 + ℎ119894

2 1198961)

1198963 fl 119891(119909119894 + ℎ1198944 (1198961 + 1198962))

1198964 fl 119891 (119909119894 minus ℎ119894 (1198962 + 21198963))

(9)


Initial values[a b]

c lt eNo

Yes

Cx =(a + b)

2

Output Cx

Trajectory calculationwith a b

Deviation calculation

Gold Section SearchNew interval a b

Radar data

c = ＜Ｍ(a minus b)

Figure 3 Programming flowchart

With a global discretization error 119874(ℎ4) the algorithm offersa tradeoff between high computing speed and best possibleresults [28]

One may also select the Euler method in the programComparing 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 error119874(ℎ) [29]32 IterativeOptimization In themathematicalmodel whichhas been described in Section 22 there are a number ofparameters missing The other variables are given and canbe easily obtained through the measured trajectory elementsIn order to determine the air drag 997888rarr119863 with the most accurateprecision the following algorithm was developed

The air drag is chosen in away 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

120576 = min119891 (119862119909)

= min119873sum119894=1

radic(119909119898119894 minus 119909119886119894 )2 + (119910119898119894 minus 119910119886119894 )2 + (119911119898119894 minus 119911119886119894 )2(10)

The index119898 refers to the coordinates which are measured bythe radar while the index 119886 belongs to the coordinates whichare calculated by using numerical methodsThe total amountof measurements is called119873

Consequently this is a nonlinear optimizationThe objec-tive function contains parameter 119862119909 In order to find theoptimum one of the fastest methods of one-dimensional

optimization the so-called ldquoGolden Section Searchrdquo 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 theminimumof aunimodal continuous function over an interval without usingderivatives

The method chooses two points 1199061 lt 1199062 on the section[119886 119887] considering Golden Section

1199061 = 119886 + (119887 minus 119886) sdot (3 minus radic5)2 (11)

1199062 = 119886 + (119887 minus 119886) sdot (radic5 minus 1)2 (12)

If the inequality 119891(1199061) lt 119891(1199062) is complied the minimum isin the interval [119886 1199061] In any other case it will be found on thestretch [1199061 119887] When this procedure is repeated the intervalcan be shortened again In case of a new partition [119886 1199062]there are new boundaries 119906lowast1 119906lowast2 with 119906lowast2 = 1199061Therefore 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 aminimal number of function calls arenecessary [31]

Golden Section Search enables an iterative adjustment ofthe trajectory in each step by using the calculated parameter119862119909 for every previous iteration Therefore prediction getsmore precise in the course of time

The programming flowchart is illustrated in Figure 3

Mathematical Problems in Engineering 5

XY-plot

5 6 7 8 94 10

X (m) (103)

100

200

300

400

500

600

700

800

Y(m

)TX-plot

5

6

7

8

9

10

11

X(m

) (10

3)

50 15 20 2510

T (s)

TY-plot

RadarNumeric

0

100

200

300

400

500

600

700

800

Y(m

)

5 10 15 20 250

T (s)

TZ-plot

RadarNumeric

5 10 15 20 250

T (s)

792

794

796

798

800

802

804

806

Z(m

) (10

3)

Figure 4 Forecast with the calculation period 3ndash6 s

Table 1 Rocket Type 63 HE specification [17]

Maximum range 85 kmOverall length 8390mmCaliber 107mmCross-sectional area 772 cm2

Weight 1884 kgLateral moment of inertia 098122 kgm2

Longitudinal moment of inertia 003135 kgm2

Position of center of gravity 3958mmStandard empty weight 8496 kgCombustion duration 06 secImpulse 67 kNsFlight time 215 sec

4 Simulation Results

An 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 aWeibel radar of the typeMFDR-210035 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 1199050 = 3 s be the time at which the radar detects thebullet The final flight time is reached after 119905119891 = 215 s

The duration of calculation is adjustable at will Moretracking points will certainly help to get better resultsbut 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 5669mbetween the calculated and the real trajectory By now itis possible to identify different RAM targets by regardingthe predicted trajectory characteristics This example is illus-trated in Figure 4 After 6 seconds of calculation the meansquare deviation is reduced to 1818mThere are inaccuraciesin all axes of coordinates The estimation of the calculatedparameter 119862119909 needs more iterative calculation steps at thispoint

Finally after 12 seconds of calculation the mean squaredeviation is reduced to 322m and there is still enough


XY-plot

100

200

300

400

500

600

700

800

Y(m

)

5 6 7 84 109

X (m) (103)

TX-plot

5

6

7

8

9

10

11

X(m

) (10

3)

5 10 15 20 250

T (s)

TZ-plot

792

794

796

798

800

802

804

806

Z(m

) (10

3)

5 10 15 20 250

T (s)

RadarNumeric

TY-plot

5 10 15 20 250

T (s)

RadarNumeric

0

100

200

300

400

500

600

700

800

Y(m

)


Table 2 Weibel radar accuracy [17]

Accuracy 20 km Accuracy 40 kmTime (ms) Max range (km) Rg (m) Az (mills) El (mills) Rg (m) Az (mills) El (mills)10 89 020 0491 0491 081 1965 196520 132 009 0220 0220 036 0879 0879

Kalman filter500 236 003 0069 0069 012 0278 0278

time 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 The

more 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 almost3000m the computational error at the point of impact

120576119901 = radic(119909119898119901 minus 119909119886119901)2 + (119910119898119901 minus 119910119886119901)2 + (119911119898119901 minus 119911119886119901)2 (13)

is smaller than 3mHere the index119901 refers to the coordinatesthat are measured and calculated at the point of impact


Details and more examples are discussed by Ramezani et al[10]

5 Summary and Outlook

This 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) projectThe basicmethods 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 tra-jectories 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

Conflicts of Interest

The authors declare that they have no conflicts of interest

References

[1] M Khalil H Abdalla and O Kamal ldquoTrajectory prediction fortypical fin stabilized artillery rocketrdquo in Proceedings of the 13thInternational Conference on Aerospace Sciences and AviationTechnology ASAT-13 2009

[2] P Chusilp W Charubhun and N Nutkumhang ldquoA compar-ative study on 6-DOF trajectory simulation of a short rangerocket using aerodynamic coefficients from experiments andmissile DATCOMrdquo in Proceedings of the Second TSME Interna-tional Conference on Mechanical Engineering Krabi Thailand2011

[3] P H Zipfel Modeling and Simulation of Aerospace VehicleDynamics American Institute of Aeronautics and AstronauticsReston Va USA 2014

[4] S An K B Lee and T H Kang ldquoFitting coefficient settingmethod for the modified point mass trajectory model usingCMA-ESrdquo Journal of the Korea Institute of Military Science andTechnology vol 19 no 1 pp 95ndash104 2016

[5] P Chusilp and W Charubhun ldquoEstimation of impact pointsof an artillery rocket fitted with a non-standard fuzerdquo inProceedings of the 2nd Asian Conference on Defence Technology(ACDT rsquo14) pp 25ndash31 2014

[6] E Scheuermann M Costello S Silton and J Sahu ldquoAerody-namic characterization of a microspoiler system for supersonicfinned projectilesrdquo Journal of Spacecraft and Rockets vol 52 no1 pp 253ndash263 2015

[7] Y Wang W-D Song D Fang and Q-W Guo ldquoGuidance andcontrol design for a class of spin-stabilized projectiles witha two-dimensional trajectory correction fuzerdquo International

Journal of Aerospace Engineering vol 2015 Article ID 90830415 pages 2015

[8] C-H Lee and B-E Jun ldquoGuidance algorithm for projectilewith rotating canards via predictor-corrector approachrdquo inProceedings of the 2014 IEEEConference on Control ApplicationsCCA 2014 pp 2072ndash2077 October 2014

[9] F Fresconi G Cooper andMCostello ldquoPractical assessment ofreal-time impact point estimators for smartweaponsrdquo Journal ofAerospace Engineering vol 24 no 1 pp 1ndash11 2011

[10] A Ramezani J Cors and H Rothe ldquoComparison of methodsfor simulation-based early prediction of rocket artillery andmortar trajectoriesrdquo in Proceedings of the 2012 Autumn Simula-tion Multi-Conference (AutumnSim rsquo12) San Diego Calif USA2012

[11] B Etkin Dynamics of Atmospheric Flight JohnWiley and SonsHoboken NJ USA 1972

[12] O Douglas and C Mark ldquoModel predictive control of a directfire projectile equipped with canardsrdquo Journal of DynamicSystems Measurement and Control Transactions of the ASMEvol 130 no 6 pp 0610101ndash06101011 2008

[13] S A A Ezeddine Military Technical College Cairo Egypt2009

[14] E Gagnon and M Lauzon ldquoCourse correction fuze conceptanalysis for in-service 155mm spin-stabilized gunnery projec-tilesrdquo in Proceedings of the AIAA Guidance Navigation andControl Conference andExhibit HonoluluHawaiiUSAAugust2008

[15] S Jankovic J Gallant and E Celens ldquoDispersion of an artilleryprojectile due to unbalancerdquo in Proceedings of the 18th Interna-tional Symposium on Ballistics San Antonio Tex USA 1999

[16] M S Khalil ldquoTrajectory Predection of Flying Vehiclerdquo TechRep Military Technical College Cairo Egypt 2008

[17] M Knapp P Kossebau A Ramezani and H RotheZunderuntersuchung C-RAM Bundesamt furWehrtechnik undBeschaffung Fachinformationsstelle BWB Koblenz Germany2010

[18] W Wolff Raketen und Raketenballistik Elbe-Dnjepr-VerlagKlitzschen Germany 2006

[19] D Carlucci and S S Jacobson Ballistics Theory and Design ofGuns and Ammunition CRC Press Boca Raton Fla USA 2007

[20] R L McCoy Modern Exterior Ballistics Schiffer MilitaryHistory Atglen Pa USA 1999

[21] C H Murphy Ballistics Research Laboratory Report AberdeenProving Ground Md USA 1963

[22] E J McShane J L Kelley and F V Reno Exterior BallisticsUniversity of Denver Press Denver Colo USA 1953

[23] J D Nicolaides ldquoOn the Free Flight Motion of Missiles HavingSlight Configurational Asymmetriesrdquo Tech Rep AberdeenProving Ground Maryland MD USA 1953

[24] J N Nielsen Missile Aerodynamics American Institute ofAeronautics and Astronautics Reston VA USA 1988

[25] I Shaydurov and H Rothe Flugbahnvoraussage Morsergra-nate Helmut-Schmidt-University Hamburg Germany 2008

[26] A Ramezani Optimale Steuerung einer interplanetaren Flug-bahn zum Mars [Diploma thesis] University of Bremen TheCenter for Industrial Mathematics (ZeTeM) Bremen 2010

[27] WH Press S A TeukolskyW T Vetterling and B P FlanneryNumerical Recipes 3rd Edition The Art of Scientific ComputingCambridge University Press New York NY USA 2007


[28] C Buskens Anleitungen zur Benutzung der Fortran-BibliothekNUDOCCCS Westfalische Wilhelms-University Munster1996

[29] W Dahmen and A Reusken Numerik fur Ingenieure undNaturwissenschaftler Springer-Lehrbuch Aachen Germany2008

[30] H Rothe and S Schrorder Method for Determination of FireGuidance Solution European Patent Office Munchen Ger-many 2006

[31] C F Gerald and P O Wheatley Applied Numerical AnalysisPearson San Luis Obispo Calif USA 2003

Submit your manuscripts athttpswwwhindawicom

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Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Journal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


projectiles Wang et al [7] established a guidance and controldesign for a class of spin-stabilized projectiles with a two-dimensional 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 impactThe goal is to provide active protection of stationary assetsin todayrsquos crisis regions Therefore a modern counter-RAMsystem with a clear GUI must be developed and will then beemployed for most threats

2 Ballistic Model

The projectile is to be expected as a point mass that isthe entire projectile mass is located in the center of gravityRotation 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 Earth-fixed 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 [11ndash16]

21 Exterior Ballistics Theballisticmodel is principally basedon Newtonrsquos 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 1800ndash4500ms [18]) Anyhow rocketsas well as mortars have ballistic trajectories and the objectis to identify the threat on the basis of different flightcharacteristics

Let 997888rarr119892 denote a reference acceleration (acceleration ofgravity at sea level on Earth) with

10038161003816100381610038161198921003816100381610038161003816 = 980665ms2 (1)

taking effect on the point mass in vertical directionThe air drag 997888rarr119863 can have different values depending on

the design of the projectile that is

(i) muzzle velocity V0

(ii) weight

(iii) aerodynamics

Cd

Ma

Figure 1 Characteristics of the air drag coefficient 119862119889

and the properties of air for example

(i) density(ii) temperature(iii) wind(iv) speed of sound

Considering the general formula

997888rarr119863 = 12 sdot 119862119889 sdot 119860 sdot 120588 sdot |V| sdot

997888rarrV 119862119909 = 119862119889 sdot 119862119860 sdot 119861

(2)

containing all parameters named above with

(i) 119860 cross-section area of the projectile(ii) 120588 air density(iii) V velocity of the projectile(iv) 119862119889 air drag coefficient(v) 119862119860 environmental properties(vi) 119861 ballistic coefficient

it is operative to find an appropriate approximation so thatthe projectile can be specified The parameters 119860 120588 119862119889 119862119860and 119861 are unknown whereas V can be defined precisely fromthe measured radar data

The air drag coefficient 119862119889 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 solu-tions 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 119862119889 that is useful for the subsonic flightregime119872119886 lt 1


e

Distance

y

j

ki

z

0

m g

0

x






119861 = 1198981198892 (3)





119898119889997888rarrV119889119905 = 997888rarr119892 + 997888rarr119863 (4)





2 cos120593119889119908119889119905 = minus119892 minus 119862119909V

2 sin120593

(5)

where

V = radic1199062 + 1199082 (6)


120593 = 119889119910119889119909 (7)

3 Concept







119909119894+1 = 119909119894 + 16 (119905119894+1 minus 119905119894) (1198961 + 4 sdot 1198963 + 1198964) (8)

with1198961 fl 119891 (119909119894) 1198962 fl 119891(119909119894 + ℎ119894

2 1198961)

1198963 fl 119891(119909119894 + ℎ1198944 (1198961 + 1198962))

1198964 fl 119891 (119909119894 minus ℎ119894 (1198962 + 21198963))

(9)


Initial values[a b]

c lt eNo

Yes

Cx =(a + b)

2

Output Cx




Radar data






120576 = min119891 (119862119909)

= min119873sum119894=1













XY-plot

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Z(m

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Kalman filter500 236 003 0069 0069 012 0278 0278

















References









































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of





e

Distance

y

j

ki

z

0

m g

0

x






119861 = 1198981198892 (3)





119898119889997888rarrV119889119905 = 997888rarr119892 + 997888rarr119863 (4)





2 cos120593119889119908119889119905 = minus119892 minus 119862119909V

2 sin120593

(5)

where

V = radic1199062 + 1199082 (6)


120593 = 119889119910119889119909 (7)

3 Concept







119909119894+1 = 119909119894 + 16 (119905119894+1 minus 119905119894) (1198961 + 4 sdot 1198963 + 1198964) (8)

with1198961 fl 119891 (119909119894) 1198962 fl 119891(119909119894 + ℎ119894

2 1198961)

1198963 fl 119891(119909119894 + ℎ1198944 (1198961 + 1198962))

1198964 fl 119891 (119909119894 minus ℎ119894 (1198962 + 21198963))

(9)


Initial values[a b]

c lt eNo

Yes

Cx =(a + b)

2

Output Cx




Radar data






120576 = min119891 (119862119909)

= min119873sum119894=1













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Y(m

)TX-plot

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T (s)

TY-plot

RadarNumeric

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Y(m

)

5 10 15 20 250

T (s)

TZ-plot

RadarNumeric

5 10 15 20 250

T (s)

792

794

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800

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804

806

Z(m

) (10

3)














XY-plot

100

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Y(m

)

5 6 7 84 109

X (m) (103)

TX-plot

5

6

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X(m

) (10

3)

5 10 15 20 250

T (s)

TZ-plot

792

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796

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800

802

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806

Z(m

) (10

3)

5 10 15 20 250

T (s)

RadarNumeric

TY-plot

5 10 15 20 250

T (s)

RadarNumeric

0

100

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Y(m

)




Kalman filter500 236 003 0069 0069 012 0278 0278

















References









































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of





Initial values[a b]

c lt eNo

Yes

Cx =(a + b)

2

Output Cx




Radar data






120576 = min119891 (119862119909)

= min119873sum119894=1













XY-plot

5 6 7 8 94 10

X (m) (103)

100

200

300

400

500

600

700

800

Y(m

)TX-plot

5

6

7

8

9

10

11

X(m

) (10

3)

50 15 20 2510

T (s)

TY-plot

RadarNumeric

0

100

200

300

400

500

600

700

800

Y(m

)

5 10 15 20 250

T (s)

TZ-plot

RadarNumeric

5 10 15 20 250

T (s)

792

794

796

798

800

802

804

806

Z(m

) (10

3)














XY-plot

100

200

300

400

500

600

700

800

Y(m

)

5 6 7 84 109

X (m) (103)

TX-plot

5

6

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8

9

10

11

X(m

) (10

3)

5 10 15 20 250

T (s)

TZ-plot

792

794

796

798

800

802

804

806

Z(m

) (10

3)

5 10 15 20 250

T (s)

RadarNumeric

TY-plot

5 10 15 20 250

T (s)

RadarNumeric

0

100

200

300

400

500

600

700

800

Y(m

)




Kalman filter500 236 003 0069 0069 012 0278 0278

















References









































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of





XY-plot

5 6 7 8 94 10

X (m) (103)

100

200

300

400

500

600

700

800

Y(m

)TX-plot

5

6

7

8

9

10

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X(m

) (10

3)

50 15 20 2510

T (s)

TY-plot

RadarNumeric

0

100

200

300

400

500

600

700

800

Y(m

)

5 10 15 20 250

T (s)

TZ-plot

RadarNumeric

5 10 15 20 250

T (s)

792

794

796

798

800

802

804

806

Z(m

) (10

3)














XY-plot

100

200

300

400

500

600

700

800

Y(m

)

5 6 7 84 109

X (m) (103)

TX-plot

5

6

7

8

9

10

11

X(m

) (10

3)

5 10 15 20 250

T (s)

TZ-plot

792

794

796

798

800

802

804

806

Z(m

) (10

3)

5 10 15 20 250

T (s)

RadarNumeric

TY-plot

5 10 15 20 250

T (s)

RadarNumeric

0

100

200

300

400

500

600

700

800

Y(m

)




Kalman filter500 236 003 0069 0069 012 0278 0278

















References









































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of





XY-plot

100

200

300

400

500

600

700

800

Y(m

)

5 6 7 84 109

X (m) (103)

TX-plot

5

6

7

8

9

10

11

X(m

) (10

3)

5 10 15 20 250

T (s)

TZ-plot

792

794

796

798

800

802

804

806

Z(m

) (10

3)

5 10 15 20 250

T (s)

RadarNumeric

TY-plot

5 10 15 20 250

T (s)

RadarNumeric

0

100

200

300

400

500

600

700

800

Y(m

)




Kalman filter500 236 003 0069 0069 012 0278 0278

















References









































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of













References









































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of











Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of




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