Factor analysis of interceptor capture probability

Journal of Systems Engineering and Electronics Vol. 22, No.6, December 2011, pp.940- -947Available online at www.jseepub.comFactor analysis of interceptor capture probabilityLei Chen*, Zhijun Gu, Xianzong Bai, and Kebo LiCollege of Aerospace and Material Engineering, National University of Defense Technology, Changsha 410073, P. R. ChinaAbstract: The capture probability of interceptors has been deeplymaneuverable target, the precision of presetting attitudestudied. Firsty, the definition of capture probability is analyzed.angles is determined by the frequency that the ground sen-It is transformed into calculating the probability that the relativesors transfer the target position information to the kineticposition vector between the target and the interceptor locates ininterceptor [5- -7]. The problem could be solved efficientlya certain cone. The relative position vector and associated CO-if some search mechanisms are actualized. .variance matrix are projected in line-of-sight coordinates, and the3-dimensional integral of a probability function in a cone is cal-The maximum capture distance of an interceptor is de-culated to obtain the capture probability. The integral equationtermined by the capability of the interceptor seeker. Itis a complicated expression of probability, and it is simlified towould be determined by the magnitude of the seeker fieldan explicit approximate expression according to some assump-angle to capture the target if the target moved into the ef-tions based on the characteristics of the engineering problems.fective capturing range of the intercepto [8]. It is assumedThe approximation precision is analyzed by comparative simula-tion difference, which indicates that approximate assumptions arethat the optical axis of interceptor points from the center ofreasonable. Utilzing the explicit expression, the characteristicsan interceptor seeker to the probability distribution centerof capure probablity are analyzed respectively with the factors, of taret position, thus the interceptor altitude is controlledsuch as the distance between the interceptor and the target, theto approximate the presetting attitude angle exactly. Theprecision of relative position vector, the maximum capture distanceposition distribution of targets is stochastic because the er-and the maximum field angle of interceptor seeker.ror of targets trajectory propagation is stochastic. ThusKeywords: intereptor, capture probablity, maximum capture dis- the observe angle is stochastic. The capure probabilitytance, maximum field angle, precision.is defined as the probability that the distance between anDOI: 10.3969./issn.1004 4132 2011.06.010interceptor and a target is smaller than the maximum cap-ture distance of interceptor seeker, and the observe angle issmaller than the maximum field angle. It is significant for1. Introductioninterceptor design to obtain the capture probability.Kinetic energy weapons play an important role in thdomains of bllistic missile defense and air defense [1]. as the navigation precision of interceptors, the detectionThe typical representative is ground-based interceptorcharacteristics of seekers, the prediction precision of tar-(GBI) of ground-based midcourse defense (GMD) systemgets. The interceptor could not capture the target unlessin United States. The precondition of handing over be-the following two conditions are met [4,9,10]: () the dis-tween midcourse guidance and terminal guidance is thattance between interceptor and target L is smaller than thethe interceptor seeker captures the target in the kinetic in-maximum capture distance of seeker Lmax (L≤Lmax);terception process [2]. In the midcourse guidance, preset-(i) the observe angle E (the angle between the optical axisting attitude angles are calculated based on the intercep-of interceptor and line-of -sight (LOS)) is smaller than thetor navigation position and the target position transferredmaximum field angle of interceptor seekera(e≤a).by a battle management, command, control and commu-The calculation methods of capture probability are pri-nications (BM/C3) system on the ground [3,4]. The tar-marily analyzed in this paper, and the influence of inter-get acquisition is infuenced directly by the precision of ceptor navigation precision and target tracking precision ispresetting attitude angles. For a non-maneuverable tar-analyzed emphatically.get, the precision of presetting attitude angles is compara-2. Problem definition of capture probabilitybly accurate, the target should be captured easily. For theManuscript reeved April 27, 2010.The position distribution of kinetic interceptors is normal*Corresponding author.distribution in earth中国煤化工THCNMH G .Lei Chen et al: Factor analysis of interceptor capture probability .941tem, so it is the distribution of targets. The position vec-Thus the probability density function of the relative posi-tor of the interceptor distribution centre is marked as rv0, tion vector could be expressed asthe covariance matrix of position errors is marked as Po,the position vector of the target distribution centre is rtof(r) =(V2x)°|P|/2°and the covariance matrix of the position error the is Pas shown in Fig.1. The direction of LOS is defined topoint from the centre of an interceptor to the centre of tar-exp[如-n)"P-'(-则.(5)get position distribution, and the direction vector of LOSisro= rto - ro. It would be confirmned that the intercep-The capture probability Pc could be expressed astor captures the target if such conditions are met that thereal observe angle Eε between the real relative position vec-Po= f[(√2n)°pP1/2"torr = rt - r。and the LOS direction is smaller than thegiven field angle a, and the real relative distance L = |川|is smaller than the maximum capture distance of seekerexp|-2(r-ro)T P-1(r-ro)| ds2.6)Lmax.TargetAs shown in Fig. 2, a joint error elipsoid is obtained bysynthesizing the position errors of interceptor and target.一PThe calculation of capture probability is transformed into amathematical problem: calculating the prprobabilityroangle between ro and the 3-dimensional normal stochas-tic vector r (the covariance matrix of vectorr is P and一Tthe mean vector is ro) is smaller than ax, and the length of- Jor is shorter than Lmax. The set of r forms a cone whichInterceptormeets the conditions above, the axis of the cone is ro andFig. 1 Sketch of relative positions of interceptor and targetthe semi-vertex angle of the cone is a. Capture probabilityCapture probability Pc is defined as the probability thatPc is the probability that the stochastic vector r locates indistance between interceptor and target L is smaller thanthis cone.Lmax and the observe angle E is smaller than the field an-gle a. It could be expressed asPc= P{(ε≤a)∩(L < Lmax)}(1Pc= P {(rro≥|r |ro| cosax)∩(r|≤Lmax)}. (2)The probability density functions of interceptor positionvector r”and target position vector rt could be expressedrespectively asfo1r)= (V2n"pP1/2:Fig. 2 Joint error llipsoid and vector cone3. Capture probability equations in LOScoordinatesfr(r)= (Va"P172'The origin 0 is located in the centre of the distribution of5[-(m-ro)"P"'(r-ro)]. (3)an interceptor. The axis Ozs points from the distributioncentre of the interceptor to the distribution centre of tar-mutual independent, the mean vector and covariance ma-gets, which is the direction of LOS ro. The axis Oxs isvertical in the plane which is formed by three points, thetrix of a relative position vector r are expressed asearth center, the interceptor distribution centre and the tar-E(r)= E(rt-ro)=rto-rvo=roget distribution centre [11,12]. The LOS coordinate systemCov(r)=Cov(rt-ru)= P+P= P. (4) is aright handed co中国煤化工Fig.3.YHCNMH G .942Journal of Systems Engineering and Electronics Vol. 22, No. 6, December 2011(x2+y2≤z2tan2an:x2+y2+z2≤Lmax .(12)Interceptor/zTargetz>0The calculation of capture probability is transformed intoa 3-dimensional probability integral because P。could notbe a diagonal matrix in practical situation. The expressionFig 3 Definition of an LOS coordinate systemof capture probability would be complicated. It is very dif-ficult to obtain the analytic solution of the 3-dimensionalThe unit axes vectors of LOS coordinate system areintegral. The numerical integral costs plenty of time, andro_a kind of simplified method must be found to perform therol =rto - rvo|factor analysis.y:= rtoXro04. Calculation method of probability integral|rto X roo|x=ySxz9.7)4.1 Transform covariance matrix into diagonal matrixThe coordinate transformation matrix from ECI to theThe LOS distance is much longer than the xs, Ys directionLOS coordinate system isdistances in practical situation. Thus the position error ofMEcI- .s=[x% x z9]T.8)axis Oz% in LOS coordinate system is nearly independentWith the help of coordinate transformation matrix, vectorto the position errors of axis Oxs and axis Oys. The jointerror covariance matrix P: is a block diagonal matrix asro and joint covariance matrix P in ECI could be expressedfollows:in the LOS coordinate system asoρTxsTys 0TOs = roMEcI- +SP。=| ρσxsOys(13)P。= MEcI _sPMEc1- +s.9)2。」According to the definition of LOS coordinate systems, thevector rOs could be expressed asIn order to simplify the calculation, the LOS coordinatero。=(0 0 zo)T(10)system is revolved with an angle θ around the axis Ozsand its axes would point to the main axes of error ellip-wherezo= |ros| = |ro| = |rto - ro|l. It is the distancesoid. P; could be transformed into a diagonal matrix in thebetween interceptor distribution centre and target distribu-new coordinate system (see Fig. 4). Some relative coefi-tion centre, which is namely the mean distance betweencients in (13) could be deleted. The transformation matrixinterceptor and target [13,14]. Equation (6) is expressedaround the axis Ozs is M (0), and the covariance matrix isin LOS coordinate system, and the capture probability Pctransformed as P'.could be expressed asPc=[[[raTOn2lrl|"ox|coea(√2x)"|pP[|/2exp[-(-ropPp:(0-ro)]da.、. (1)Letrg= (x y) , the integral regionSl : rsrOs ≥|rsl|ros| cosa, |rx|≤Lmax could be expressed asFig 4 Revolving method of the LOS coordinate systemP'= M(6)P。MT(6)= diag(o2叼2)=。一吃2 cos2θ + ρσxzsOys sin 20 +呢sin20. sin 20 + ρσzsOys cos2002y。-σ2。(14)2工sin 20 + ρOxsoOys cos202 sin2θ- ρσxsOyssin20 + oy cos200中国煤化工MHCNMH G .Lei Chen et al: Factor analysis of interceptor capture probability43P' is a diagonal matrix, so(oo?φ+ ksin2中)2三sin20 + ρσxoσys cos20 = 0. (15)2元J。2σzσydq(21)The expression of 0 could be obtained as云cos2中+ksin2φtan20 =2pσxsOys(16)Assume2-啦更(φ)=去cos2φ+ksin2φ=And the following equation could be obtained(啦=o2。cos20+pOzoOyssin20+2sin2θ1+(后-1)cos2φ+(k- 1)sin2φ. (22)吼= 2。sin2θ- ρσxsσys sin20 + oy cos2 0.Equation (19) could be simplifed asσ2=σ2。R2更(φ)1_ 1(17)P=2元J。1-exp2σxσ)(d)d中. (23)The capture probability could be expressed asThe 2-dimensional integral is transformed into th11-dimensional integral. It is essential to integrate the ex-(√2m)3 σxσyσzpression numerically in order to obtain the high precisionresult of capture probability. Some approximation theories| dxdydz =should be utilized to obtain analytical solutions.=[+(备+号+气。p")]It is useful to compress the space. Let the coefficientk= σx/σy(18)x'=x，y' = ky.(24)The integral in (19) could be transformed into the follow-.2 Approximating method of 2-dimensional integraling expressionThe integral of anisotropic probability density function inp=1[ exp(--中!"dx'dy'.= 2rσ2202a circle is expressed as最+敞≤1(25)dxdy.2πσxσy[-(备+号)x2+y2≤R2tion in a circle is transformed into an integral of theisotropic probability density function in an ellipse, asThis kind of 2-dimensional integral is very important inshown in Fig. 5(b). Equation (25) could be approximatedthe calculations of collision probabilities of space objects.to the following expression, if the ellipse is replaced withThe calculation of clisinin probability could usually be the circle which ocupies the same area.reduced to the problem of a 2-dimensional integral in aso-called encounter plane which is vertical to the relativep__ 1fexp (_x"+y2\P= 2x02(-- 2σ2 )dr'dy. (26)velocity. The calculation method of collision probabilityr12+y'2 ≤kR2has been researched in [15- -29]. With comprehensive con-sideration on the precision and speed of the calculation, they|ymethods in [17] and [28] are used to analyze this problem.With the help of polar coordinate system transforma-tion,出= pcosφ, y= ρsinφ.(20)Letk = σx/oy, the integral above could be transformed as(a) Anisotropic pro (b) Isotropic probabi- (c) Elipsc is replacedP= 2πoxσybability densitylity density funcwith the circlefunction in a circletion in an elipsewhich ocupies thesame areaxp |(co?o+ ksin4川]pdpdφ=牌gs EIpe pprpp中国煤化工apressionTYHCNMHG944Journal of Systems Engineering and Electronics Vol.22, No.6, December 201 1Thus, the probability integral is transformed into an inte-gral of the isotropic probability density function in a circle[o(Drmx一5万)+(品)一小as shown in Fig. 5(c). And there is an analytical solutionof this integral.exp_z tan2 a(31)2ησxσy .kR2'r2In practical interception situation, the magnitude of dis-P=1- exp .、20=1- exp(27) .tance zo is hundreds of kilometers, the magnitude ofσ is .several kilometers, and the magnitude of a is several de-4.3 Calculation of triple integralgrees. Thus, z0/σz》1,0 13°.The capture probability Pc will increase while the distancezo increases firstly. And Pe drops to 0 rapidly after it ismaximized. There is a curve of Pc versus z0 in Fig. 7,.8-whena = 3°, Lmax = 200 km, andσx = σy= σz.6--10 km. As shown in Fig. 7, Pc increases from 0 to 0.35while distance z0 increases from 0 to 181.5 km. Eventhough z0 increases continuously, Pc drops to 0 rapidly.The reason is that the distance z0 exceeds the maximumcapture distance Lmax of the interceptor.015a/(°)0.4Fig.9 Curve of P versus a0.5.4 Analysis of position errors0.25。0.2-Capture probability Pc will decrease while the position er-0.15rors increase. And decreasing speeds in different directionsare different.0.05There are curves of Pc versus different position errors50100150250in Fig.10 and Fig.11, whena = 3°, Lmax = 200 km,Z0 = 150 km. The curve of Pc versus σx is shown in而/ kmFig. 10, whenσz = 10km,σx = σy. Pc is almost ap-Fig.7 Curve of P versus zoproximate to 0 whenσx > 50 km. The curve of Pc versus2 Analysis of maximum capture distanceσz is shown in Fig. 11, whenσx = σy = 10km. Pc isPe will increase while Lmax increases if the distance z0approximate to 0 onlyifσx > 500 km.is certain. There is a curve of Pc versus Lmax in Fig.8,whena=3°,z0= 150 km, andσx=σy=σz = 10 km..8--As shown in Fig. 8, the capture probability is approximate.6-to 0 when Lmax < 120 km, and the capture probabilityis not infuenced by the maximum capture distance prac-tically when Lmax > 180 km. It is determined by the 3σ.2--principle, and z should be located in the region of zo士3σz.20 30x/ kmFig 10 Curve of Pc versus σn0.3r0.25 F0.2a 0.10.1F-05F100 110 120130140150160 170 180 190 20000200 300 400 500ILmx/kmFig. 8 Curve of P versus Lmax中国煤化工YHCNMH G .946Jourmal of Systems Engineering and Electronics Vol. 22, No. 6, December 2011The phenomenon iluminates that the infuence caused[2] E. J. Ohlmeyer, T. R. Pepitone. In-flight alignment techniquesfor navv theaterwidemissiles. ProcoftheAlAAGuidance.by the position error of the LOS direction is smaller, andNavigation, andControlConference and Exhibit, 2001-4401.the infuence caused by the position error of the vertical di-[3] C. Phillips, s. Malyevac. Midcourse motor and KKV divertrection to the LOS is much larger. In practical interception,propellant allocations for an exoatmospheric interceptor. Proc.it should be considered that the LOS direction is the direc-of the AIAA Guidance, Navigation, and Control Conferenceand Exhibi, 1998 4307.tion whose position error is the largest in order to lower the[4] H. B. Hablani, D. W. Pearson. Miss distance error analysisinfuence caused by the position error. The LOS directionof exoatmospheric interccptors. Journal of Guidance, Controlshould be accorded with the direction of velocity because[5] T. L. Vincent, R. G. Cotrell, R. W. Morgan. Minimizingand Dynamics, 2004, 27(2): 283. -289.the target position error on the direction of velocity is themaneuver advantage requirements for a hit-to-kill interceplor.largest. Considering the requirement of impact velocity,Proc. of the AIAA Guidance, Navigation, and Control Confer-the inverse-orbit interception is applied commonly, and in-ence and Exhibit, 2001- 4276.terceptor hits the target head-on.[6] L. Chen, B. Zhang. Novel TPN control algorithm for exoatmo-spheric intercepl. Journal of Systems Engineering and Elec-6. Conclusiontronics, 2009, 20(6): 1290- 1295.[7] J. Waldmann. Linc-of-sight rate estimation and linearizingThe integral of a 3-dimensional probability function is sim-control of an imaging seeker in a tactical missile guided byplifed to calculate the capture probability of an interceptorproportional navigation. IEEE Trans. on Control SystemsTechnology, 2002, 10(4): 556- -567.based on the practical interception characteristics. The ex-[8] H. B. Hablani. Pulsed guidance of exoatmospheric interceptorplicit expression of capture probability is obtained, whichwith image processing delays in angle measurements. Proc. ofis testified to be a high- precision equation by the simula-the AIAA Guidance, Navigation, and Control Conference andExhibit, 2000.tion results.9] J. Akpert. Miss dsiatance anlysis for command guided mis-Based on the explicit expression of capture probability,siles. Joumal of Guidence, Control and Dynamics, 1988,the factors are discussed respectively, such as the distance11(6); 481- 487.between an interceptor and a target, the maximum capture[10] R. w. Nesline, P. Zarchan. Miss distance dynamics in homingmissiles. Proc. of ihe ALAA Guidance and Control Conference,distance of interceptor, the maximum field angle and the1984: 8498.precision of position errors.[11] F. Tyan. Unified approach to missile guidance laws: a 3D ex-(i) The capture probability will increase while the dis-tension. IEEE Trans. on Aerospace and Eleclronic Systems,2005, 41(4): 1771199tance between an interceptor and a target increases firstly,[12] W. Grossman. Bearings- only tracking: a hybrid coordinateand the capture probability drops to 0 rapidly after it issystem approach. Proc. of the Conference on Decision andControl, 1991: 2032- 2037.maximized.(i) The capture probability will increase while the max-[13] C. D. Yang, C. C. Yang. A unified approach to proportionalnavigation. IEEE Trans. on Aerospace and Electronic Sys-imum capture distance increases. The capture probabil-tems, 1997, 33(2): 557- 567.ity is approximate to 0 if the maximum capture distance is[14] E. Dufios, P. Penel, P. Vanheeghe. 3D guidance law model-ing. IEEE Trans. on Aerospace and Electronics Systems, 1999,smaller than zo(1 - 3σz), and the capture probability is not35(1): 72- 83.infuenced by the maximum capture distance practically if[15] F K. Chan. Spacecrafi cllision probabiliy. Los Angeles:maximum capture distance is larger than zo(1 + 3σz).Aerospace Press, 2008.(ii) The capture probability will increase while the max-[16] R. P. Patera. General method for calculating stelite colli-sion probability. Joumal of Guidance, Control and Dynamics,imum field angle increases. The capture probability will beapproximate to 1 if the field angle is larger than a certain2001, 24(4): 716-722.esearch on calulatioal method of col-[17] X. Z. Bai, L. Chen. Research on calculational methodvalue, when the maximum capture distance is bigger thanlision pobability between space objects. Joumal of Astronauzo(1 + 3σz).R. P. Patera. Method for calculating collision probability be-(iv) The capture probability will decrease while the po-tweena satellite anda snace lether Lourmal ofGuidance Con-twrana SD ynur as 202 25: 9045406Cudiuanc.on.R. P. Patera. Space vehicle confict probability for llipsoidaldirections are different. The influence caused by the po-confict volumes. Journal of Guidance, Control and Dynam-sition ertor of LOS direction is smaller, and the influenceics, 2007, 30(6); 1818-1821caused by the position error of the vertical direction toLOS[20] M. R. Akella, K. T. Alfriend. The probability of ollision be-is larger.tween space objects. Journal of Guidance, Control and Dy-namics, 2000, 23(5): 769 -772.[21] S. Alfano. A numerical implementation of spherical objectReferencescollision probability. Journal of the Astronautical Sciences,[1] L. L. Croix, s. Kurzius. Peeling the onion: an heuristic2005, 53(1): 103- -10922] S. Allfano. Satelite ollision probability enhancements.overview of hit-to-kill missile defense in the 21st century.Proc. of SPIE, 2005, 5732: 225- 249.Jourmal of Gui中国煤化工2006, 29(3):TYHCNMH G .Lei Chen et al: Factor analysis of interceptor capture probability947588- 592.Zhijun Gu was borm in 1979. He received his Ph.D.K. T. Alfriend, M. R. Akella, J. Frisbee. Probability of cllifrom College of Aerospace and Materials Engineer-ing in National University of Defense Technology.[24]I日KChanCollisionnrobabilitvanalvsesforeathorhitineHis research interests are fight vehicle dynamics,guidance and control.E-mail: gzjnudt@ 263.com25] F K. Chan. Improved analytical expressions for computingspacecraft collision probabilities. Proc. of the AAS/AIAASpace Flight Mechanics Meeting, 2003.[261 H. Klinkrad. Collision avoidance for operational ESA satel-Xianzong Bai was borm in 1983. He is a doctoriallites. Proc. of the 4th European Conference on Space Debris,candidate in College of Aerospace and Material En-2005.gineering in National University of Defense Tech-[27] N. Berend. Estimation of the probability of collision betweennology. His research interests are space objects col-two catalogued orbiting objects. Advances in Space Research,lision avoidance and space traffic management.1999, 23(1): 243- 247.E-mail: baixianzong@ 163.com28] X. Z. Bai, L. Chen. A rapid algorithm of space debris colli-sion probability bascd on space compression and infinite se-ries. Acta Mathematicae Applicatae Sinica, 2009, 32(2): 336-353. (in Chinese)Kebo Li was born in 1986. He is a postgraduate29] H. Klinkrad. Space debris- models and risk analysis. Chich-in College of Aerospace and Material Enginceringester: Praxis Publishing Ltd, 2006.in National University of Defense Technology. Hisresearch interests are fight vehicle dynamics, guid-Biographiesance and control.E-mail: likeboreal@qq.comand Ph.D. degrees in flight vehicle design from Na-tional University of Defense Technology in 1997andMaterials Engineering in NationalUniversitvofDefense Technology now, His research interests areinterceptor dynarmics and guidance, spacecraft col-lision probability.E-mail: clwhl@ 263.net中国煤化工MHCNMH G