Comparison of methods for simulation-based early prediction of rocket, artillery and mortar trajectories PDF
Preview Comparison of methods for simulation-based early prediction of rocket, artillery and mortar trajectories
Comparison of Methods for Simulation-Based Early Prediction of Rocket, Artillery and Mortar Trajectories ArashRamezani,JostCors,HendrikRothe Helmut-Schmidt-University/UniversityoftheFederalArmedForcesHamburg InstituteofAutomationTechnology ChairofMeasurementandInformationTechnology Holstenhofweg85,D-22043Hamburg [email protected] Keywords:EarlyWarningSystem,RAM(Rocket,Artillery, moreandmoreusedbyirregularforcesinAfghanistan,where and Mortar) Threats, Ballistic Coefficient, Iterative Opti- they have easy access to a large amount of these weapons. mization,TrajectoryPrediction. Further reasons are the small radar cross section, the short firingdistance,andthethickcasesmadeofsteelorcast-iron, Abstract whichmakesmortarprojectilesandrocketshardtodetectand Thethreatimposedbyterroristattacksisamajorhazardfor destroy. militaryinstallations,e.g.inIraqandAfghanistan.Thelarge Thechallengeistoestablishanearlywarningsystemfordif- amountsofrockets,artilleryprojectiles,andmortargrenades ferent projectiles using analytical and numerical methods to (RAM) that are available, pose serious threats to military reducecomputingtimeandimprovesimulationresultscom- forces.Animportanttaskforinternationalresearchanddevel- paredtosimilarsystems[1].Anappropriateestimationofthe opment is to protect military installations and implement an ballisticcoefficientandtheassociatedcalculationofunknown accurate early warning system against RAM threats on con- parametersisthecentralissueinthisfieldofresearch.Real- ventionalcomputersystemsinout-of-areafieldcamps. time prediction of trajectories and continuous optimization This paper presents the most commonly used mathematical aretwoofthemainaimsofthispaper. methods for determining the trajectory, caliber and type of With the aid of graphical solutions it is possible to differ- a projectile based on the estimation of the ballistic coeffi- entiatebetweenseveralobjectsanddeterminefiringlocations cient.Thealgorithmsareimplementedonlow-endcomputer aswellaspointsofimpact. systems used in military camps. A comparison between the The goal is to provide active protection of stationary assets methodsgivesinformationabouterrorpropagationandrelia- in today’s crisis regions. Therefore, a modern counter-RAM bility of the system. In addition, simulation-based optimiza- systemwithaclearGUImustbedevelopedandwillthenbe tion processes are presented that enable iterative adjustment employedformostthreats. of predicted trajectories in real time. Combinations of these methodswillbecomparedtoincreasetheaccuracyofsimu- 2. BALLISTICMODEL lation. Theprojectileistobeexpectedasapointmass,i.e.theen- A graphical user interface (GUI) is programmed to present tireprojectilemassislocatedinthecenterofgravity.Rotation the results. It allows for comparison between predicted and is irrelevant in this case, so a ballistic model with 3 degrees actual trajectories. Finally, different aspects and restrictions offreedom(3-DOF)isassumed. formeasuringthequalityoftheresultsarediscussed. TheEarthcanberegardedasastaticspherewithaninfinite radius and represents an inertial system. Based on an earth- 1. INTRODUCTION boundCartesiancoordinatesystem,theforceofinertiaisap- Field camps are military facilities which provide living pliedinasingledirection. and working conditions in out-of-area missions. During an Differentprojectileshavetobeconsidered:Whilerocketscan extended period of deployment abroad, they have to ensure beregardedasspin-orfin-stabilizedprojectiles,whichhave safetyandwelfareforsoldiers. a short phase of thrust and are particularly suitable for long Current missions in Iraq or Afghanistan have shown that distancesupto20km,mortargrenadesarefin-stabilizedand the safety of military camps and air bases is not sufficient. firedonshortdistancesuptoapproximately8km[2]. A growing threat to these military facilities is the use of unguided rockets, artillery projectiles and mortar grenades. 2.1. ExteriorBallistics Damages with serious consequences have occurred increas- inglyofteninthepastfewyears. Thissectiondealswiththemeasurementanddescriptionof This paper focuses on mortars and rockets because they are movementwhichabodyexperiencesafterfiringwithanan- gleθ andtheinitialvelocityv .Theballisticmodelisprin- 0 0 cipally based on Newton’s law and the equations of motion areconsideredtobeundertheeffectofairdragandtheforce of gravity only, amounting 98% of the forces acting on the projectile [3]. Therefore, no additional parameters are con- sidered. Let(cid:126)gdenoteareferenceacceleration(accelerationofgravity atsealevelonEarth),with |g|=9.80665m/s2, (1) takingeffectonthepointmassinverticaldirection. The air drag (cid:126)D can have different values, depending on the designandotherparametersoftheprojectile,i.e. • muzzlevelocityv , 0 Figure1. CharacteristicsoftheairdragcoefficientC • massm, D • aerodynamics, andthepropertiesofair,e.g. • densityρ, • temperature, • wind, • speedofsound. Consideringthegeneralformula Figure2. Masspointmodelwith3-DOF 1 (cid:126)D=− ·ρ·(cid:126)v2·A·C , (2) D 2 Wecansetupthesystemofequationsasfollows: it is necessary to find an appropriate approximation for the parameters ρ, A,C , so that the projectile can be specified. D x˙ = vcos(θ), (4) TheairdragcoefficientC forinstancedependsonthecritical D y˙ = vsin(θ), (5) velocityratio,picturedinFig.1.Thevelocityvcanbedefined D preciselybythemeasuredradardata. v˙ = − −g·sin(θ), (6) Section 2.4. deals with the problem of estimating the un- m g knownparameters. θ˙ = − ·cos(θ), (7) v 2.2. EquationsofMotion withtheinitialconditions The earth-bounded coordinate system {x,y,z} is centered x(t=0) = 0, (8) inthemuzzle,withtheaxesx,y,zpointingtofixeddirections in space. Axis x is tangent to the earth, y is orthogonal to x y(t=0) = 0, (9) andrunsagainstthegravity,andzisorthogonaltobothxand v(t=0) = v , (10) 0 y,settinguparight-handedtrihedron.Themodelisillustrated θ(t=0) = θ . (11) 0 inFig.2. With the aforementioned parameters the equilibrium of 2.3. NumericalIntegration forcesinthiscasecanbedescribedwiththeformula There are basically two approaches to predict trajectories 0 of projectiles with the help of radar data. On the one hand, d2(cid:126)x m =m−g+(cid:126)D. (3) the underlying equations can be solved analytically. This is dt2 0 realized by Siacci’s approximation method, which was well known for calculating bullet trajectories of small departure Consequently,thisisanon-linearoptimization.Theobjective anglesduringWorldWarI[4]. functioncontainsparameterC.Inordertofindtheoptimum, Ontheotherhand,numericalapproximationisworthyofcon- one of the fastest methods of one-dimensional optimization, sideration for solving the initial value problem. It is impor- the so-called ”Golden Section Search” (GSS), is applied. It tanttodistinguishbetweenone-stepandmulti-stepmethods. onlyneedsonevalueoftheobjectivefunctionforeachstepof Thereareseveralnumericalmethodsimplemented,bothone- thecalculation.Thesecondvalueistakenfromthepreceding step and multi-step, all providing better results compared to iterationstep. theanalyticalmethodsusedin[5]: Thismethodpossessesarobustandlinearconvergencespeed tofindtheminimumofaunimodalcontinuousfunctionover • The Euler method as an explicit, one-step method for anintervalwithoutusingderivatives[7]. numericalintegrationofordinarydifferentialequations. Themethodchoosestwopointsu <u onthesection[a,b] 1 2 Itisafirst-ordernumericalprocedure. consideringGoldenSection: • AmodificationoftheEulermethodwithahigherorder √ andmoreaccuracy. (b−a)·(3− 5) u = a+ (14) 1 2 • The fourth-order Runge-Kutta method is commonly √ (b−a)·( 5−1) usedandoffersatradeoffbetweenhighcomputingspeed u = a+ (15) 2 2 andbestpossibleresults[6]. • Adams-Bashforthmethodsareusedaslinearmulti-step Iftheinequality f(u )< f(u )iscomplied,theminimumis 1 2 methodsforthenumericalsolutionofordinarydifferen- in the interval [a,u ]. In any other case, it will be found on 1 tialequations. thestretch[u ,b].Whenthisprocedureisrepeated,theinter- 1 val can be shortened again. In case of a new partition [a,u ] 2 • The Kalman filter functions as a recursive estimator to therearenewboundariesu∗,u∗withu∗=u .Therefore,only 1 2 2 1 approximate system values using a series of measure- twovaluesofthegoalfunctionalareneededtobemeasured mentsobservedovertime. duringthefirststepofthecalculation[8].Thegoalisanop- timalreductionfactorforthesearchinterval.Additionally,a 2.4. IterativeOptimization minimalnumberoffunctioncallsisnecessary[9]. Inthedifferentialequationswhichhavebeendescribedin GoldenSectionSearchenablesaniterativeadjustmentofthe section 2.2., there are a number of parameters missing. The trajectory in each step by using the calculated parameter C othervariablesaregivenandcanbeeasilyobtainedthrough for every previous iteration. Therefore, prediction gets more the measured trajectory elements. The unknown parameters precise over time. The programming flowchart is illustrated canbeclassifiedunder inFig.3. ρAC D C= . (12) 2m In order to determine the so-called ”ballistic coefficient” C withthemostaccurateprecision,thefollowingalgorithmwas developed. Theparametersarechoseninawaysothattheexteriorballis- ticmodelcompliestothemeasuredtrajectoryoftheprojectile inthebestpossibleway.Thisimpliesthatthesumofthede- viations between the calculated and the measured positions shouldbeminimal: ε = minf(C) (13) N (cid:113) = min∑ (xr−xc)2+(yr−yc)2+(zr−zc)2. i i i i i i i=1 Figure3. GSSflowchart Theindexr referstothecoordinateswhicharemeasuredby theradar,whiletheindexcbelongstothecoordinateswhich arecalculatedbyusingnumericalmethods.Thetotalamount In addition, Newton’s method and Simpson’s rule are im- ofmeasurementsiscalledN. plementedandcanbechosenintheGUIaswell. 3. SIMULATIONSOFTWARE • Prognosezeit: Choose the beginning time and the end The purpose of the software is the calculation of trajecto- timeoftheprediction. ries. It receives the measured position of the projectile from • Optimierung: Choose an optimization method and cal- thetrackingradarandreturnsthepredictedtrajectory.AGUI culate the parameter C. This function is optional. The isnecessarytofacilitatethehandling.Ithastobeeasytolearn valuecanalsobeenteredmanually. andenablesoldiersinmilitaryinstallationstooperatewithin a short space of time. The programming language used for • Dateiausgabe: Use the checkbox to create a text file thissoftwareisC.TheselectionwindowisshowninFig.4. showingthecalculatedcoordinatesoftheprojectile. Fig.5illustratestheprocessoftheprogramwithaflowchart. Figure5. Programflowchart 4. RESULTS Thesimulationrunswiththreedifferentdatarecordscalled A1, A2 and B. Two of them are radar data of an unspeci- fiedmortargrenademeasuredattheTechnicalCenterofthe Federal Armed Forces in Meppen/Germany. With a logging of data fromt =22s tot =54s and an interruption between Figure4. Selectionwindow t =37s and t =41s, this data record is used to analyze tra- jectoriesincludingmeasuringmistakes.Thegapcanbefilled withalinear(A1)orratherquadraticinterpolation(A2). • Berechnungsmethode:Chooseaintegrationmethod. Thethirddatarecordisanartificiallygeneratedtrajectoryof arocketTypeM105withthesimulationsoftwarePRODAS1. • Analysezeit: Determine the first and the last radar data Itcalculatesthetrajectoryaccordingtotherequirementsfrom usedfortheoptimizationoftheunknownparameterC. t=0stot=19s(B). • Abbruchkriterium: Choose a criterion for stopping the algorithm, either computation time or any y-coordinate (above/belowsealevel). • Schrittweite:Choosetheincrement. 1ProjectileRocketOrdananceDesign&AnalysisSystem Thispaperanalyzesthecomputationalerroratthepointof impact (cid:113) ∆ = (xr −xc)2+(yr −yc)2+(zr −zc)2 (16) r A A A A A A andtheaveragedeviationofthepredictedtrajectory 1 N (cid:113) ∆ = ∑ (xr−xc)2+(yr−yc)2+(zr−zc)2. (17) r,m N i i i i i i i=1 The index A refers to the coordinates that are measured and calculatedatthepointofimpact. Themoreradardataavailablefortheanalysis,thecloseristhe prediction to the measured trajectory. More tracking points will certainly help to get better results, but sometimes a fast interceptionoftheRAMthreatisindispensable.Thisfacthas been proven for all methods. Only Siacci’s algorithm shows irregularities for the data records of the mortar grenade, be- causethismethodisinsteadintendedtobullettrajectoriesof smalldepartureangles. Withanincrementof0,5secondstheaveragedeviationcanbe calculatedforthedifferentintegrationmethodsatthepointof impact.TheresultsareshowninTab.1.Theinfluenceofthe Table1. Averagedeviationatthepointofimpact Method ∆ A1 ∆ A2 ∆ B r r r Euler 78,54m 43,08m 5,93m Mod.Euler 87,89m 31,21m 7,40m Runge-Kutta 67,88m 28,87m 7,33m Adams-Bashforth 42,35m 26,33m 17,61m Kalmanfilter 93,24m 47,66m 35,90m Siacci 39,67m 49,20m 27,80m different data records on the results is very high. Quadratic interpolation should be the solution for measuring mistakes. Ofcourse,theartificiallygeneratedtrajectoryhasthesmall- estdeviation. Theeffectoftheincrementvariesforeachmethod.Itcanbe insignificantly small or decisive with regard to the calcula- Figure6. Variationofthecomputationalerroratthepointof tion.AnexampleisshowninFig.6. impactwithdifferentincrements Regarding the average deviation of the predicted trajectory throughtheentirecalculation,similarresultscanbeproduced withanincrementof0,5secondsshowninTab.2. Table2. Averagedeviationofthepredictedtrajectory Thegoldensectionsearchisthebestchoiceforiterativeop- Method ∆ A1 ∆ A2 ∆ B timization. It can be used for every data record. Simpson’s r,m r,m r,m Euler 25,33m 11,53m 1,81m rule has a lower accuracy for some integration methods and Mod.Euler 24,85m 6,57m 1,92m shouldbeusedinexceptionalcases.Newton’smethodisun- Runge-Kutta 19,83m 6,73m 1,77m suited for this purpose, because it can’t be applied on some Adams-Bashforth 9,66m 6,03m 5,06m records. The use of derivatives creates difficulties in finding a local minimum point. Fig. 7 shows a typical course of the Kalmanfilter 21,65m 21,76m 6,96m error function. For this reason, optimization methods using Siacci 11,36m 10,38m 13,13m derivativessuchasNewton’salgorithmshouldbeavoided. [3] Kuhrt, A.: Ein analytisches Verfahren zur Lo¨sung des Hauptproblemsdera¨ußerenBallistikgestreckterFlug- bahnen fu¨r Echtzeitfeuerleitsysteme. Shaker Verlag. Aachen,2008. [4] Germershausen, R.: Waffentechnisches Taschenbuch. RheinmetallGmbH.Du¨sseldorf,1977. [5] Shaydurov, I.; Rothe, H.: Flugbahnvoraussage Mo¨rsergranate. Internal Report. Helmut-Schmidt- University.Hamburg,2008. [6] Hairer, E. et al.: The Numerical Solution of Differential-Algebraic Systems by Runge-Kutta Meth- ods.Westfa¨lischeWilhelms-UniversityMu¨nster.1996. [7] Press, W. H. et al.: Numerical Recipes in C - The Art Figure7. Errorfunction of Scientific Computing. Cambridge University Press. Cambridge,1992. 5. SUMMARYANDOUTLOOK [8] Rothe,H.;Schro¨der,S.:MethodforDeterminationof Thispaperintroducesanalgorithmforearlywarningsys- Fire Guidance Solution. European Patent Office, DE- tems used for command and control applications in out-of- C1-3407035.Mu¨nchen,2006. area missions and is based on the MONARC2 project [10]. [9] Gerald, C. F.; Wheatley, P. O.: Applied Numerical Thebasicmethodshavebeentestedsuccessfullyandtheyare Analysis.Pearson.SanLuisObispo,2003. used in fire guidance solutions for German frigates of type 124and125. [10] Rothe, H.; Kuhrt, A.: Feuerleitalgorithmus und Soft- Themostimportantaspectisthatonecandistinguishbetween warefu¨rdasMONARC-ProjektderdeutschenMarine. differentprojectilesinordertopredictthetrajectoriesandhit Uniforschung,Wiedemeier&Martin,Bd.15.2005,S. pointsmoreaccurately.Tocalculatetheirtrajectories,differ- 36-39.Du¨sseldorf,2005. entflightphasesareanalyzedindetailandthedesignsofthe projectiles are estimated by the use of iterative optimization Biography methodsforapproximatingenvironmentalandballisticprop- Arash Ramezani currently works as a research assistant erties. at the mechanical engineering department of the Helmut- The most commonly used algorithms are compared to im- Schmidt-University in Hamburg. He has studied Applied proveexistingtrajectorypredictionalgorithmsandtofindthe MathematicsattheUniversityofBremenandtheUniversity bestpossiblesolutionformilitaryinstallationsinout-of-area ofQueenslandinAustraliaandreceivedhisDiplomadegree missions. in2010.Hisresearchinterestsincludemodelling,simulation Future work will focus on giving the user specific informa- andvisualizationofballisticproblems. tion of the projectile data. Furthermore, work also has to be Jost Cors currently studies mechanical engineering at the doneona3-dimensionalsimulation. Helmut-Schmidt-University in Hamburg. He received his Attheend,asophisticatedsimulationsoftwarewillbeestab- BachelorofSciencein2011andisnowworkingonhismas- lishedthroughwhichitwillbepossibletoshowandevaluate ter’s thesis ”Investigation of Methods for Trajectory Predic- areal-timebattlefieldscenario. tion”attheChairofMeasurementandInformationTechnol- ogy.HeisacommissionedofficeroftheGermanArmy. REFERENCES HendrikRotheisaprofessoroftheChairofMeasurement [1] Isaacson, J. A.; Vaughan, D. R.: Estimation and Pre- and Information Technology at the mechanical engineering dictionofBallisticMissileTrajectories.RandPublish- department of the Helmut-Schmidt-University in Hamburg. ing.SantaMonica,CA,1996. HehasstudiedelectricalengineeringattheTechnicalUniver- sityofIlmenau,Thuringia,fromwhichhereceivedaDiploma [2] Wolff,W.:RaketenundRaketenballistik.Elbe-Dnjepr- and his doctoral grades in 1973 and 1980 respectively. His Verlag.Klitzschen,2006. research interests include ballistics, optics, optronics, and nanometrology. 2Modularnavalartilleryconcept
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