Introduction to Statistical Orbit Determination 1
Introduction to Statistical Orbit Determination 1 ASEN 5070
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STATISTICAL ORBIT DETERMINATION Observab 1 Prediction Residual R VI ms Exam 2 Review ASEN 5070 LECTURE 26 1101 06 Colorado Center for Asnodynamucs Research The Umveusny of Colorado nm in alum the pusni In islomeda x u at 1 be the m Usmg mm 1 swithr ousmm nomzmm39ulo itv anon I 1 The a n h mldng raqu Mgr iron he stntion I the cm Thus am cstmmte Lhc yummy mu VGIDCIIV I at 2 2 2 x h x P X V pr Zyoc Cololado Camel Io Asuodwamlcs Resealch The Unlvelsny of Cololado Weenability Example 0Q P i p 0p v xv 0p 0p x 7 7 Ti and i i x p p 0x 0x p Hence 0 V x2 l i 1 72 0x 9 9 Next x 0v p 2 1 4L L 1 X L 6X p p p Colorado Center for Astrodynamics Research The University of Colorado Hagenability Example But X XO Vt so 1 t ltIgtttoLt 1 8X00 0 1 and Z 2 H IILltDI tO l 13 L x wtfxft Pl Pl P p as 139 1 2 m H will be rank 2 and the state vector is observable Note that it is H and not H that determines observability Colorado Center for Astrodynamics Research The University of Colorado mbservabilitv Example 06 Same problem except h or That is h is very small If 110 and CM 1 pr 2 and l x ZDO p Then Now Hwill contain a column of zeros and be rank 1 Therefore the state is unobservable Colorado Center for Astrodynamics Research The University of Colorado Alternate Kalman filter time updat An alternate form of the time update for the estimation error covariance matrix can be derived as follows From the conventional time update 13 t lt1 I tk1 PHCDT t tk1 1 Differentiating this expression 1393 t ltigtttk4Pk1IgtT ttk1 lt1 t 1 Pkbe t rm 2 Hum Atlt1gtttk1 3 Pk 1 0 since it is a constant matrix Colorado Center for Astrodynamics Research The University of Colorado rrglgte Kalman lter tlme update Substituting Eq 3 into 2 yields at e At t tAT t This equation may be integrated from rm to 1k With initial conditions Eon pH to determine the time update for 7 This avoids the need for the stale lransillon matrix Slnce the stale devlallon Vector can be oblalned by ally integrating ftAtYt with ic 2HltH Hence the convenllonal Kalman Flller llme update becomes Time update for tH to tk X39 e Fx1 i c x39ou are glven izAzfz lc tH gt2 tAtFtFtAT t The is unchanged Culuradu Canlarlur Amrudynamlus Haaaan 39u rne Unlvaraly nu Culuradu Ext nded Kalman Filter EKF m Algu i i 1 J i L izedns follows A Given Piei Xi e1 and Yi Rh 1 Integrate from 1H 0 it XFXht X39ti1 Xi 1 4730 Mikel AUJ I UJ kelX I mellilcel 1 eeimne eenierim Aerodynemne meagre me unwereiy a Colorado 2 nded Kalman Filter EKF m ompu a Iona 0quot 2 Compute 71P71 I Tinihrl 14731 Vi YL CXtik in UGtXtk8X 3 Compute 197i rFrHEJrRrr i4732 Xi X1 fryr Pr I e minim 4 Replace h with k 1 and return to l orado Center Ior Amrodynamics Research ado Col The Universin oi Color It 39ihrerPrediction Residual 6 474 THE I Rlilth39l lON RESIDUAL It is of interest to examine the Variance ofthc predicted residuals which are sometimes referred to as the innovation or new information which comes from T l quot quot 39 nr immmrinn 39 39 39 quot linl based on the uprim i or predicted state ii at the observation timer til i and is de ned as N in Vi e Hm 14733 As noted previously xr In W IiXi Ii orado Center Ior Amrodynamics Research Col The Universin oi Colorado iPrediction Residual where xi is theter Value of the state deviation Vector and r is the error in Xi Alsoi Elm ll Elllhllli fi and Elit 2 ll Emil Ri Elsi HT i From these conditions it follows that 34 has mean EliA i 2 Elfll x1 ti 7 Elm ElLA 7 Him 0 Colorado Center lor As1rodynami s Hessard1 The Univers ty of Colorado Elm 7 Tami 7 HOT Ella fl EllYL gritNYL 4 ilTl llt39l Fri7705 HHUJTl Pm 7 RA in f 4734 3 Z Hence for a large prediction residual variancecovariance the Kalman gain Ki 7 KEEP 4733 will be small and the observation will have little in uence on the estimate of the state Also large values of the prediction residual relative to the prediction residual standard deviation may be an indication of bad tracking data and hence may be used to edit data from the solution This would be especially important in the case ofthe EKF Colorado Center lor As1rodynami s Hessard1 The Univers ty of Colorado 31 An estimate I m gt n where y HX e and e is N e R An apl39iuri Value ofx is given f x e where e is N 0 F Assume E e 3 0 The estimate is Xi y 7 Hi In What is the bias for this estimator Colorado Centel lor Amrodynamics Research The University 01 Colorado 39s made of x an 77 y 1 Vector based on m observations y x ihly eHi x1 EieHe L x 4K Ii Hr ncc llm hizlx in 3k is h39E Ie roblem31 Em Exo EI39Hxz Hi Colorado Center lor Amrodynamics Research The University 01 Colorado Laramie problem 31 lb What is the vnriMicecovariance associated with the estimation error for x Note that X is biased so use the de nition of P given in the answer Colorado Center lor Astrodynamics Research The University oi Colorado ample problem 31 I1 Tn computo the estimation orror cmrnrinnuo mnlirx mo kiKHxs7HiJ Subtract x from each SidL 57 7 i7 x7KHti7xKe I7KHiti7xHKE 11 Batman is hiasct E 7x KE Hence P E Elrk7x7Eic 7 Eiix7xi7KEii 2 Substituting ELL ti into Eq 3 ErI7KHi x 27 gtlt 1 7KHgt lt7 x Iamp 731T KE 5 7 Ze 7 EV KT 7 I 7 KHifa 7 HIV KHKT Colorado Center lor Astrodynamics Research The University oi Colorado mple problem 31 g to Show that by rede ning the noise vector to be I T L39 and by including in the state vector an unbiased estimator may be formed Assume thman uprinn estimate of with covariance If is available c Let c E e Wham and Then yHxEy Let E bccnmu part 0139 the slate vcctm to be estimated YH E Illlt Colorado Center lor Amrodynamics Research The University of Colorado l gxam Ie roblem 31 Dclinc and Then and the estimate for X is XYmy 7HX w he IL K HFT RJFT R Colorado Center lor Amrodynamics Research The University o1 Colorado ple problem 31 uml P It n gt 0 1 wlwru i r xlli r xlT How we hnvu munrml an a priori value 01 givcn by in E l39 vhcu 6 13 hr cn39ni in hr 0 ll39i7l it liilllllL 01L with l Par We may now slum 1 th unbiuml Eix E X TE xw x K x 7 EX X Colorado Center tor Astrodynamics Research The University at Colorado 35 Consider the linear system de ned in Exercise 21 Assume tu 0 other wise replace l with r 7 m 01 7 mu m on 0 axrto 001 and 400 g 020 001 a Show that the time update for P at the rst measllrenlellt time tlt obtained by integratingthe differential equation for f lEq 4936 with initial conditions P0 is given by L 3 22 4 2n 2 2n 3L 3 7 Pm 2t1 T 2 H 1 a 1 21 1 Colorado Center tor Astrodynamics Research The University at Colorado Fl39um lllt explmsmn given 01 IIH ti C can determine All IPHrnll l39lttn QMquot u U 1 n I t l Heme Flnm Eq l l9wl gt r 4tjt1 Colorado Cenler lor As1rodynamics Research The Universily ol Colorado HI39 1 Ctmlponulil form P11 E12 P13 F21 1 722 F2a 21 13 0 P21 f F23 P31 P32 P33 22 23 0 pm p32 p33 0 0 0 P32 3 0 2 arms 23 7 31 P22 271332 P33 P32 P33 Because is sylnmelliu we Only need to bnlvu lllm39 uppm ll iallgulal39 punlon 1 his vtlllalmll Inmgl nllng his equation tetm by 01m sublth to he initial cundllions PD given in 3910 pmhlt m statement yields 4 21 tf4 2t rE2 rEZ Pm 21 11 Colorado Cenler lor As1rodynamics Research The Universily ol Colorado 39 rlple problem 35 Using Lht conwmmnul 1mm upddtt t uqdllml n 1fu o I T11iu 1 11 W2 1 0 0 1 0 0 0 1 t1 0 2 0 t 1 0 0 0 1 0 0 1 z t 1 yields 1118 same mull l39nl39fm coloran center lor As1m ynzmlns Ramlm me universuyol coloran i ham 2 110806 t stics Re ew 1 Axioms of Probability 2 Conditional Probability 3 Indepen e Density and Distribution Functions 01 u lt0 8 m x 6 Univariale amp Bivariale DenSily FunclionS ProperlieS of Covariance amp CorrelalionS 8 Cenlral Limil Theorem II Statistical Interpretation of Least Squares b5 a ion Error Covari n 2 E5imaion Error Covariance Ill Mi Imum Variance Estimator iv Kalman Sequential Filter v Observah VI Review Homework coloran center lor As1m ynzmlns Ramlm me universuyol coloran