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# Introduction to Statistical Orbit Determination 1 ASEN 5070

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STATISTICAL ORBIT DETERMINATION Smoothing Monte Carlo Simulation ASEN 5070 LECTURE 37 120806 Colorado Center lor Amrodynamice Research The University ol Colorado Smoothing 415 SMOOTHING r c t lter in this case we are searching for the best estimate of the state at some time ta based on all observations through time t wheret gt i For the case where there is no random component to the dynamical equation ofstatkfol39 example H Hbl atc r predic tion equation Eqs 4419 and 4422 will give the smoothed solution How e v r as noted the batch estimation approach has dif culty including the effects of process noise The smoothing aigoritirms have been deveio ed to overcome this dif culty Following Jazwins i 1970 the smootiring algorithm can be de rived using a Bayesian approach of maximizing the ensi 39flinction of the state conditioned on knowledge of the observations through time tr Our system is described in Section 49 tsee Eq 4946 Kit1 I fir1 fit1X4 Fflv39139fr ll k yr Hrrxr er min was Smoothing 9 We will use the notation to indicate the best estimate of x at t based on obser ations through h where in general t gt k Following the Maximum Likelihood philosophy we wish to nd a rccursit expression for xl in terms of which maximizes the conditional density function IlXXI1Yi Wllt l39t Yly1ygytyr t45t alter some algebra JtY Ple KM YLl lel thwH 1 le V lm1 J39xx11Xt Yil cgmn 2on5 Smoothing 9 Finally x 2 5 54 7 lt1nutxtit 1415 Wll ere 5i Pyli I llt vWNWHimBld lk n lrlll391lLlQiFl wimp 1751 thmum 141510 Eq 41 t is the smoothing algorithm Computation goes backward in index k with he lter solution as initial conditions Note that the lter solutions for xi Pkl DUAHJU and F HI 1 are required and should be stored in the filtering process The time update ofthe covariance matrix Fig may be stored or rccomputetl Cumin 2on5 Smoothing 0 The equation for propagating the smoothed covariance is derived nextUnzwin ski 1970 Rausch 6 IL 1965 It can easily be shown from Eq 14159 that is unbiased hence the smoothed covariance is de ned by P E xj ux a WT 41511 The equation for the smoothed covariance is given by P Pi Slsipi1 Filly91F 439153924 I Copyright 2006 Smoothing computational algorithm Given from the ltering algorithm 25 Pf l 135511 lt1gttn1 set Iv 7 1 5H Pg Fm tH Pf lYl 41529 iii71xii 5mm 7 lt1gt m m xiii Copyright 2006 0 Smoothing computational algorithm Given from the ltering algorithm and the previous step of the smoothing algo rithm 442 r 2 I42 4 X2 Pl1 Pl2 XPJ 077122 set A 1 2 and compute 2 ij ltIgtT rm m2 JDfrI 41530 V Ar 4 2 2 X2 Xzig bi 2Xi 1 qlfl li 2 2 xiigi J and so on 39 V Copyright 2006 Filtering Exam le Problem 41 of Ch 4 Exercises 9 41 Generate 1000 equally spaced observations of one cycle of a sine wave with amplitude 1 and period 10 Add Gaussian random noise with zero mean and variance 2 025 Set up a sequential estimation procedure to estimate the amplitude of the sine wave as a function of time using the noisy raw data Model the sine wave as a GaussMarkov process as given by Eq 4960 77x41 mi177i Filui where mi1 e UHi ti U2 Fi1 ll m3 1 x G 8 K 739 Copyright 2006 Filtering Exam 1e Problem 41 of Ch 4 Exercises and 7 is the time constant The sequential algorithm is given by 1 77 thtialmisl 1 2 1000 tit1 m e t7 ti 1gt p1 10513ti 1Pr 1 ITti7tr 1 FiQi l FiT Note that P I Q and 1quot are scalars 777 thus 1 assume R7 1 Q7 1 g 2 00 1 P 1 77KYii im iKiYii i Y is the observation data P I Kirim K Nexti Plot your observations the truth data and 77 versus time You will need to guess initial values for a and B 4 Copyright 2006 9 Filtering Exam 1e Problem 41 of Ch 4 Exercises 9 25 W 2 Raw l I l Tth l Filtered RMS 0178m Time Figure 410 Process noisesine wave recoverquot showing truth raw dam truth plus noise and the ltered solution 1 0 r7 21511 U 13 Where N T 0 12 7 i i lli w I 1iis 5 Copyright 2006 10 Computational Algorithm for Smoother Given from the ltering algorithm Pzil P351 D U 1771 set C F7 1 SH Plijf Trtt1Pf 1 1 441529 gt254 23 541022 7 I a mi lib V Copyngii 2006 11 Computational Algorithm for Smoother Given from the ltering algorithm and the previous step of the smoothing algo rithm kt 1353 13415 x MM m2 set I I39 2 and compute SH sz T mi mg 171 1 41530 542 X 572 38171 qflil Tri xtg niitl so on Copyngii 2006 12 Solve for the smoothed history of ll 1 using the computational algorithm for smoother Plot noised single cycle of a sine Wave The sequential algorithm is given by l v i39IHI 1 L l mum 71mm 1 wvt y 7 m T ampI rvlgtT 104 lmc mm 139 ll 1qu l are wulzu39x 7 W1th I171Lltumcl71137 1 n 7 III U 71 Iquot 771771l7TJNT ILquot 7 dam Where I39rl7ll 1 lut Copyright 2006 the true Values of I the lter Values determined by the algorithm below and the smoothed Values Compute the RMS of the smoothed t data is 5 Time Figure M01 Process noisesine wove momy mom me num me ltered and me smoomed solmion W n a 7 219 a 7 015 In x Copyright 2006 Filter Smoother Consistency Check 9 The smoother covariance at stage k is given by Pf P7 51 Pin 7 sz1539f 41524 Where 57 Pf DTULH flll l tfk s JBf DTUker 1 Flflcly fkQTT1 Li 1 I llNil BjIDTn1 ttBf41 1 41510 c BC is the lter covariance at stage k and k 11 is the lter covariance at stage k 1 based on k observations at 39 Copyright 2006 15 Filter Smoother Consistency Check 9 Calculate the n x n difference matrix difference between the lter and smoother covariance at stage k 2 k 2 5P Pk Pk forkE 0 l 2 Z I 6 B should be non negative de nite ie have no negative eigenvalues I Denote the square root ofthe ith main diagonal element of 63 as 041 I Calculate the n x 1 difference vector 5 x between the ltered and smoothed state estimate 2 quotk quot2 5 xk xk xk n Copyright 2006 16 Filter Smoother Consistency Check Calculate the ratio 139 Z 139 RM 6xk him for eachiE0 12 n andkE0 l 21 If for eachi and k we have R Z s3 Then the ltersmoother solutions are said to be globally consistent If for eachi and k we have i RkZ l gt 3 The ltersmoother solutions have globally failed the consistency test 1 Copyright 2006 17 Monte Carlo Simulations 9 Assume We have a state Vector X with associated error covariance x is a Vector of zero mean error realizations of the state Vector X Hence 0 factor P into Where S is upper triangular and can be computed Via Cholesky decomposition or orthogonal transformations Note that S is not unique Copyrigat 2006 19 Monte Carlo Simulations 9 Premultiply Pby 0 using soEeeTI u Copyrig1t2006 20 Monte Carlo Simulations Copyright 2006 Therefore e can be realized as an calculated from Therefore x is a realization of errors of the Vector X for Which P is the error covariance 9 Copyright 2006 Monte Carlo Simulations Implementation procedure using Matlab Given an nVector X and P compute S Generate an n Vector of Gaussian mndom numbers with If desired an nVector of Gaussian mndom number b With mean M and Variance Monte Carlo Simulations 9 0 Compute a realization of error in x from 0 Generate a new realization of X 0 P as its error covariance Copyrig1L2006 Z3 Monte Carlo Simulations 9 Another realization may be computed by genemting a new Vector of random numbers e Unless you specify the seed Matlab will genemte a different mndorn Vector each 39 e w Copyrig1L2006 Z4 0 We could also use S 0 Let 0 Note that Copyright 2006 Monte Carlo Simulations 9 0 Hence this will be a diITerent realization of x given the same mndorn vector 0 However it can be shown that Where Q is an orthogonal transformation matrix 0 Therefore w Copyright 2006 Monte Carlo Simulations e Monte Carlo Simulations Proofthat Because S is not unique ie there is more than one Value of S for which We must choose Q so that S is upper triangular IfWe do not require S to be triangular there are an in nite number of solutions for S ie Q may be any Copyrigat 2006 orthogonal Iquot am 27 STATISTICAL ORBIT DETERMINATION Givens Operational Procedure Kalman Filter with Process Noise ASEN 5070 LECTURE 29 111006 Colorado Center or As1rodynamics Research The Urrirrersiiy at Colorado In summary given a prim39i infonna ou F and E imd observations yHXE r1m the matrix we wish to reduce to Ilpper Irinngular form is n 1 p vs A F B I H H1 3 H2 ya m H m ym Where P is upper rimiglllar Colorado Center lor As1rodynamics Research The University of Colorado a4l A PRIOR INFORMATION AND INITIALIZATION 5438I dition of New Data After application of this algorithm the n m k n 1 matrix will appear as n 1 A A FBIquot R h n 5441 Q 39 th 0 e m which is the required form for solution of the least squares estimation problem as Colorado Cenler for Ash39odynamics Research The University of Colorado Once the away has been reduced to the form given by Eq 5441 subsequent observations can be included by considering the following array R I Ru R12 quot39 R1 11 0 R R b H 11 1 22 2quot 2 3442 m quot2 0 0 quot39 R31 b3 0 0 R7117 bu Hm11 Hm12 quot39Hm1n lm1 0 5 2 where Colorado Cenler for Ash39odynamics Research The University of Colorado Addition of New Data Tlieu by application of n Givens rerniion 0 rows 1 nnd 17 1 Herein can be ulled Successive opplicnrions moving down the rnniu diagonal can be used to u 1 15 r up p mangulal form R b39 0 E m1 0 2 2 is replaced by e2 721 and the procedure is repeated with ilie next observation and so on It is also obvious that n group of m observations could be included by replacing the array Hm duel with an nnny in which Hm has diineusion m gt n and ym1 has dimension 777 The Givens rotation would be used as before lo reduce be augmented nrrny to upper triangular form Note coiorauo cenier lor As1rodynamlcs Research The universiiy ol coiorauo Estimation in the Presence of E Process Noise State Noise Compensation Algorithm section 49 The state dynamics of a linear system underthe in uence ofprocess noise are described by X F Xt B tut 491 The state deviation vector is propagated according to XtAtxtB tut WhereAtandBtare known functions and ut is a white ie uncorrelated in time random noise process with E ut 0 Eutu71Qt6t71 coiorauo cenier lor As1rodynamlcs Research an The unruersriy 0 color o I E xnatbh in the Presence of E Process N0se 5 tir is the Dirac Delta function which is zero everywhere except at t 1 where it is in nite in such a way that the integral ofthe function 5 71 across the singularity is unity 392 The algorithm that results from the assumption that fl is white noise with known covariance is known as Slum Nul xp Viupenh39lninll SNO The use of more sophisticated models such as the process to compensate for state midor meat 39 39 referred to as Dwimliir Mmiul Conlpmlsm iml i It the case ofDMC plDCCSS noise parameth are often iliCiiltiCd in the state vector to be estimated Appendix F pim itles more details on SN39 and DMC Colorado centerior Asirodynsrnios aesesroh The University at Colorado i I E xnatbh in the Presence of Process N0se 392 The solution of Eq 491 can be obtained by the method of Variation of parameters The homogeneous equation is given y ska Atxt 493 which as previously noted has a solution of the form xt m inlcn 494l The method of variation of parameters selects 0 as a function of time so that xt ltIgtttn7t 495l Colorado centerior Asirodynsrnios aesesroh The University at Colorado i nation in the Presence of Process NOISS it follows then that m thit tttJC u Mu UJCU 49ut Substituting Eq 496 into Eq 491 yields MtfoK UH I lltWC AlfJXKfH Bmlllf 4m Recuiltllut Mr 1 4 ltlrr to 498 Colorado Center lor mrodynamlcs Research The Unlverslty at Colorado wation in the Presence of Process NOISS ltIt moo Biut Hence 71 ltIr1rtuBtum Integrating Eq 4910 yields t cm co 1TtuBrgturdT tit Substituting Eq 4911 into Eq 493l results in xf ttuln Mt tgltIquot739 taB7uTdT t it Substituting Eq 498 into Eq 497 and using Eq 495 reduces Eq 497l to 499 4910l 4911 4912l Colorado Center lor mrodynamlcs Research The Unlverslty at Colorado 4 E Lmation in the Presence of g rocess NOISe Using the properties of the trausltlor matrlx we may wrlte ltIgtrtgt1r1rtg pmuyng m r 4913 At t to xtu x hence Eq 4912 can be used to detennine that CD x0 With these results Eq 4912 can be written as l 42 14120er t7BTuTdT 4914 in Culuradu Seniarlur Amrudynamlca Research rm unmrmty a Colorado 1 Ebul lauull in the r39leaellue of Process Noise Eq 4 9 14 Illustrates how the true value of 4 IS propagated In the presence of process nolse We wlsh to understand how the estlmate of the state propagates In the presence of process nolse Note that Eq 4 9 14 IS a stochastlc Integral and cannot be valu ted In a determlnlstlc sense Now 339Extlyk for tat 2 E proxy 1 1 4tzxz3zuzlyk but E uzyH o Culuradu Seniarlur Amrudynamlca Research rna Unluarally ul Culuradu Estimation in the Presence of g Process Noise 8 tAtExtyH Atit itAtit 4917 Eq 4917 has the solution it 131 tkilfiki1 4919 The equation for propagating the estimation error covariance Matrix is given by 13tEit xtit xtT 4921 Colorado Center for Astrodynamics Research The University of Colorado Estimation in the Presence of Process Noise The solution for13t is developed in section 49 of the text It involves the solution of the Differential Eq 130 At13t13tAT tBtQtBT t 4935 with LC PtkPr We may express the solution of Eq 4935 in integral form by using the Method of Variation of Parameters see notes The homogeneous Eq is 130 At13t13tAT with C Pt0P0 Colorado Center for Astrodynamics Research The University of Colorado 4 gation in the Presence of E Process NOIse which has he selnllon Fa Muml n wa 4937 here on convenience H has been replaced by m Letting Pn become a func llnn nrllme Eq 4937 becomes r i ffuPo I TttoJ ltIgtttnPgltigtTtfn kI ttoPntIJTttn 4938 qumting Eqs 4935 and 4938 Air l4Tl 4 BLtQtBT1 rigtttnPn Ttm IgttiuPoigtTtm awrmannxbthnl 4939 Colorado Gamer lor Aalrodynamlca Research The Unlverany o1 Colorado ggation in the Presence of g Process NOISe Howm er from Eqs 4936 and 4937 A1071Ilrfl11gtffano I rltol ltI39tfll13niquotrrn 49401 Using Eq 4940 E 493 reduces to 1411 39ltIgtTlln BllQnBTrl or I1 IVIUJUBIQBTINIFTUJ Ul 4041 Colorado Gamer lor Aalrodynamlca Research The Unlverany o4 Colorado istiination in the Presence of Process N0ise Substituting Eq 4942 into 4937 ifrlt1gtotoPoltIaia TBTQrBT7qundeiszijug fu gttoPultDTitU 141rDJltbnTBTQrBTT a gtlt lt1gtTtu7rIgtTrrodT 4943 Ifwe use q wmno r 1417 and NoomWrmn MtwmrnmiiT Wt r then after letting r0 1 Eq 4943 becomes PoMri71gtPt71 Tlt mm I itTB739QTBTTI TTTLIT 4944 vfiri Colorado Center lor A rodynamics Research The University of Colorado Estimation in the Presence of Process Noise 13 tt lt1 arkl PHQT MM f tIB I Q 137 119 m a r 4944 IVI Eq 4919 and 4944 are the equations for propagating the state and estimation error covariance for a continuous sy em It is more convenient to replace f a continuous random process with a white random sequence Thus fis considered to be a piecewise constant function between observations L9 Colorado Center lor A rodynamics Research The University at Colorado Estimation in the Presence of g Process Noise with covariance E u uT t1 sty where 6y is the Kronecker delta and 1 i j y 0 i j Becauseur is constant between observations Eq 4914 may be written as XZk1 101qu xk rtktgtlk uk 4946 1m Where 1 tk1tk fqgttklTBTdT 4947 k I tk1tk is called the process noise transition matrix The equation for propagating the estimation error covariance matrix is Colorado Center for Astrodynamics Research The University of Colorado Estimation in the Presence of Process Noise T k Xk1 k1 1 k1 39 39 P 1 E x x x 49 48 Where it ltIgt Imam 4919 Xm Dlktgttk Xk rtk1gttkuk 4946 Substituting these equations into Eq 4948 results in 13m E Ik1atk5 k Xk FIk19tkuk Earner where E 16 Xk u 0 Becauseuk can only effect the state for z gt 2k Colorado Center for Astrodynamics Research The University of Colorado 4 Estimation in the Presence of Q Process Noise Hence 131m DE ik xk1 k1 Xk1T 17 FE lltug FT D tk1 tk PkCDT tk1 tk rtk19 tkriTtk19 tic Hence the Kalman Filter with process noise also known as state noise compensation SNC is as follows Time update Yew D 51 tic 5 tic Ftk1 I 1qu quf 11m tie Ftk1 tkQk1 T 11m tie Colorado Center for Astrodynamics Research The University of Colorado Estimation in the Presence of Process Noise Measurement update 1 T T Kk1 BC1Hk1 HkanHm Rk1 Xk1 Xk1 Kk1 yk1 T Hklxk1 Btu I Kk1Hk1 Em Q is generally determined by trial and error and is usually a diagonal matrix This formulation ofthe Kalman Filter in known as state noise compensation SNC See appendix F and web handout Colorado Center for Astrodynamics Research The University of Colorado g IE STATISTICAL ORBIT DETERMINATION Exam 2 key Satellite Tracking Examgle of SNC and M ASEN 5070 LECTURE 32 112706 The University of Colorado ASEN 5070 Exam 2 NaveInner o 2006 o l 30 Given the joint probability density function fXy cx 13 y2 O lt X lt 2 0 lt y lt 1 F 1 The value of c 2 gx the marginal density function ofx 3 F112 4 Are X and y independent Why or why not 5 p0ltylt120ltxlt1 Colorado Center for Astrodynamics Research The University of Colorado g Exam 2 Key 0ltxlt2 0ltylt1 mow 2 Find F1 FL p x515ys fjjfxydxdy i xo3y2w 44 fxycx13y2 1 Findc Colorado Center for Aslrodynamics Research The Universily of Colorado Exam 2 Key 3 Are xand y independent 1 2 1102 0 x13y2dx A gxhyfxay 13y2 Hence X and y are independent gxfxl3y2 dy Colorado Center for Aslrodynamics Research The University of Colorado p0ltyltJltzltl y n Zfxydxdy f g WX Hnweverx and y are mdependent mm poltyltgWlpmyz hydy f x13 y 5 cawaaacemer Var Aarmynams Pesaarch The mwsuyamwaaa 1mm men Sun in treasury sq Apnvn mfunmum s g m m 3 Infant 7 mm Myriam Emma afzhe slaw mm Km 5m aux and ms at X 1 m x m cm m39egnm LUJ w1ltwe rgtyu 5 Ydemtneedmwhcmyequumhulshuwallequaa mm A esmm aftthand 1 thinqu mud cawaaacemer Var Aarmynams Pesaarch The mwsuyamwaaa ASENSW Xam 2110806 E x 1 A spancim F mm m 1 dmgqmls acimdmg m m diffuuml zqnman shmm m 1h gure Rang magnum ohsn39uuaas m bang tikza m a slannn mm m m was 11 lb 1 ue g m mu m Asmm dmlherusna nb39enatmn u m 17 Exam 2 Key II We wish to use a Kalman Filter to estimate x Xy u To obtain the proper state transition matrix for this state vector we must augment the state vector to include position velocity and any dynamical parameters which are to be estimated Hence we define X to be Where i7 3ux2 yiAy2A39 J39cu yv 6 Li3uxzy 39y2 Colorado Center for Astrodynamics Research The University of Colorado Exam 2 Key 6 Then u 6v 6 6x 6y 6F x A 6 May 6V 6V 6 6x ay an Then 0 0 1 0 0 0 0 0 1 0 Az 6M 3M 0 0 3xzyz 0 2y 0 0 0 0 0 0 0 0 Colorado Center for Astrodynamics Research The University of Colorado Exam 2 Key Note that the measurement parameters do not affect Af also PZxxs2yys2 1967 6X 616161 6x 6y6u y y 0 we may now generatecI t to by integrating numerically FX andq39 at Atcp at subject to the initial conditions given by XS Colorado Center for Astrodynamics Research The University of Colorado Exam 2 Key There are two options we may use to solve for the 3 element state vector Both options should yield the same results 1 Select the proper elements of q and use a 3 element state deviation vector From the 5x5 matrix I 4 choose CD11 cD12 CD15 13 tut 121 122 1325 0 0 1 2 Use a 5 element state vector but set the diagonal elements of I70 that correspond to u vto very small numbersD1010 Note that if option 2 is used H must be defined as 0 0 0 0 0 H p l l l l where alal0 0x 0y Bu 0v 0M Bu 0v Colorado Center for Astrodynamics Research The University of Colorado 4 Exam 2 Key The Kalman Filter algorithm is as follows 1 Do the time update to t7 integrate X FX and lt15 11 AZCIgt 11 from to to 1 f1 I I110 E cIgt t1t0ic1gtt1t0 This will be 3x3 or 5x5 matrix depending on yl pl pl which option is chosen where P1 is the actual observation and P is the computed observation p1 J9xzyfy2 K H1EH1P71Tai4 Colorado Center for Astrodynamics Research The University of Colorado Exam 2 Key 2 Do the measurements update at 17 5amp1 fl K1y1H1f1 P1 1 K1H1 131 X1 X 5amp1 WhereX1 isa3x l or 5x1 depending on which option is use Colorado Center for Astrodynamics Research The University of Colorado SEN5070 Exam r4 2110306 5 mm Amm the o mvmg 9 4 Iaus exmlufrhefw mungmmcexquot n a cm 27 cm 1m3 3 9 r 54 Fm m n when m n 4 m ohsm mans and n is VJ number vista e mm In lube mumhemfmdzpcadml nb39ermnmn wuun 1er 3 my LheKahmn llualuaprqmreusz of mu 4943mmer nu umvnnmyuaa 1513 2 eqns uv mmmn and ubsrstam ave hneav lt mmmmnm mm Mum m F w hmmm mmmmmu K m m 4 commie cam m harem ames Ramon m mwwsuymcmormo The J Examp e An Emma w summed m mum m m 01b 5mm m m um szu man u 3m gun km null mm m pimpaim a m m epl39ltmens gemnmn used gmumm when m minded 1 Ind 5 we 5 msz an Obemcnn arm and large mm mm 4 mm mmn m Geumcamxym un We gemn ad for a rm 1W data m Um 3914 xmuhtnd abymcnn commng Gm m 4mm mm 3914 page sveth m Emma funk mum and compnd m 3914 Vina commie cam m harem ames Ramon m mwwsuymcmormo as comm Cmmr w Aswadyvsmms Haisrm m m mm Comrade The J3 Problem Gvavnatmna Putarma Fundmn VuHhe J3 Mudg U zap l135m1 gt 7 1 7Eljlgsml gt7 35m Wham 42 sm z z z z xyz J J3 Equations of Motion J l Taking the partials ofthe Gravitational Potential to get Xyz accelerations 1 Copyright 2005 Conventional Kalman Filter CKF J I mm as am K 5n mm D u 2 m j g in mi 15 m musing 25 m u 50 m M 39 Ennis Aim mu mm i Dagmar y hf g EWNAvw 5 2m m in mm x 39 y 18 CopynghtZOOS Conventional Kalman Filter CKF Position RSS 50662 m RSS errors are based on XT X x at each stage t Copyright 2005 Extended Kalman Filter EKF No State Noise Added i m E A I WAVM I I w 000 v D m yquot l we 2 m D in ma tin V we 25 mu m y HY Note that prefit residuals forthe CKF are much largerthat the EKF but postfit residuals are identical indicating that the reference trajectory forthe CKF is in the linear range 20 L Copyright 2005 Extended Kalman Filter EKF J l xmv ms Ems mm 120 N Na n ma 5 quotl 2 25 w Velocity RSS 4332 x 10392 ms v Emir ms Ems Mm 100 m n 5 out tg M39v vy mm Position RSS 50279 m mm RSS position and velocity errors for CKF and EKF are nearly identical 1 21 CopynghtZOOS Band Diagonal State Nonse Matrl A a 0 0 L ff 0 0 0 ATfa 0 0 A7120 0 0 0 A713022 0 0 A7120 FQFT 3 2 2 AT 1 0 0 ma 0 0 0 A7120 0 0 Ataz 0 2 0 0 A7120 0 0 Am 2 2 See web page handouts or previous slides for derivation 22 39 Copyright 2005 Conventional Kalman Filter with SNC Band Diagonal State Noise Matrix results for optimum value 02 1013 Inu I mu 7 5 n r m 7 Sn mo sn 207 an Jan 39 a V a J V 39 5n tun n 25D mu Pier mvm in 20 mm in my y h Copyright 2005 y ch paw l Ivy um mm Rummy x 39n 2m m 330 3 in iumesl Conventional Kalman Filter with SNC Band Diagonal State Noise Matrix results for optimum value 02 1013 x a gr lws E39vu39 All u m as swam new i Cum 7 Walgg wmaw n 5 mi ms 200 zsu v s m HMS Ews m 39m m z 2 1232 Velocity RSS 162 X 10392 ms mm lawsivo39s mm 1cm in 15 amalgam so mu m milmum Position RSS 843 m Copyright 2005 so we a ma 2m mum Hrs sun mm m m 393 uun amne 0 Kalman gain is so large that data is fit almost perfectly well below noise levels of 1cm and 1mmls Conventional Kalman Filter with SNC Band Diagonal State Noise Matrix results for optimum value 02 1013 a mmng 15v 200 mm quotmum This is what happens if an input error A inject a huge amount of process noise 3 7 ie 2 1013 instead of 1013 K t a W Copyright 2005 Conventional Kalman Filter with input error Band Diagonal State Noise Matrix results for optimum value 02 1013 m mwum 39 20 standard deviations of estimation error covariance are red lines Velocity RSS 122 x 10392 ms x ml luvs mm m we w cumvssail am am Position RSS 2084 m Small residuals do not a good orbit Make RSS errors are larger than Previous results m Copyright 2005 13 Extended Kalman Filter with State Noise SNCJ Band Diagonal State Noise Matrix A13 Atz Ta 0 0 Ta 0 0 A13 Atz 0 To 0 0 To 0 3 Z 0 0 AT U 0 0 AT U rQr 2 Ta 0 0 Ara 0 0 Z 0 AL 0 0 A102 0 2 Atz Z 0 0 702 0 0 A102 quot1 i 5 Copyright 2005 Extended Kalman Filter with State Noise SNCJ I Band Diagonal State Noise Matrix Looked for optimum values of 02 by analyzing residuals and errors Optimum value to minimize position errors 02 1013 ml mm m nss lbmlwn mm rm 3 Copyright 2005 Extended Kalman Filter with State Noise SNCJ Band Diagonal State Noise Matrix results for optimum value 02 1013 mm Rama y recall that Y O 1 5n me ion 25 m u Yum lmvmmil Copyright 2005 Extended Kalman Filter with State Noise SNCJ I Band Diagonal State Noise Matrix results for optimum value 02 103913 Velocity RSS 1484 x 10392 ms x Errol ms EmvsAlmr mu nuns umnm 5 J WW a in am zen mi 5 ma 25o v E39vm was EvmsAluv mu m nucmzqui mu 15c m 25 1 2w was am Alta 1w m us 0 male Position RSS 8520 m The CKF and EKF results are 3 fl Nearly identical indicating that D Initial conditions are in the linear 1 m range CopydngOOS 30 Extended Kalman Filter with Fading Adds a fading term to the time update this downweights earlier data by Keeping the Kalman gain elevated T Sqti ti 1Pi 1qTti ti l where At time between measurements 20 seconds 1 ageweighting time constant mi 31 Copyrigl1L2005 Copyright 2005 M mm m V 9 Extended Kalman Filter with Fading Looked for optimum value of s to minimize the residuals and errors Optimum value of s 10339 This corresponds to a 1 of 10 minutes n v at am nllml Valnclly Ermr may nu m 1025 Im ms ms 155 mu 1055 sum m2 H125 mm um um ms ms was ms was SHAle I Extended Kalman Filter with Fading Optimum values of s to minimize position and velocity errors s 10339 Pinyin mum Rummy n D m my lamsammim mu maznaumag g nm 4 E a gram 5W 5 ilmnmmil39a 2 3 E E M 33 CopynghtZOOS Extended Kalman Filter With Fading Optimum values of s to minimize position and velocity errors s 10339 in x Enuv mus Zr Allm mm quotH 1549670l m7 Velocity RSS 1545 x 10392 ms D a 25 mmquot am 2 v m m Winmnzl I t 34 CopynghtZOOS Kalman Filter Results Comparison J Conclusions The CKF and EKF produce comparable results indicating that the reference orbit for the CKF is in the linear range for this example Fading produces comparable results to SNC rm Cupynght znns Dynamic Model Compensation DMC J I DMC accounts for unmodeled or inaccurately modeled accelerations acting on the spacecra JS in this problem The state vector was augmented to he following Xxyzkyzmmnz where nx ny and 711 are the accelerations A GaussMarkov process is used to account for these accelerations iit 43170 W where ut is white Gaussian noise with Em 0 Eutu1 026t 1 and 3 Where I is a time constant 39i a Cupynght znns Dynamic Model Compensation B and 0 were optimized to give the lowest position and velocity errors 13 02 1016 orbital period m l amt NM mummy an mm 15 am 25a ian u nmc immmsi 199 am mm mm am an mum z lmszy E g 39E in m 1m ion 25 m a nine lullMail f x 37 Copyright 2005 Dynamic Model Compensation Optimized Results mm 21 mum xmmms zummm 1m m quot5 mm Velocity RSS 1429 x 10392 ms 25 w 5 x Eum W5 Elms Allm mu mus u unsszazl u my w my m7 is m m m l am 5 w 12 quotmg 25 m 35 mm my me mm mummy Position RSS 8195 m l x 38 Copyright 2005 Dynamic Model Compensation 1 I Optimized Results x W XEvvui cams Em39sAllnv um nu m ammrnsl Actual vs estimated accelerations u an i i a m m 2 a 39 2 D x mquot z E H gays Eums Allul mu quotnus 2 uasau E 2 u in wc u m m Nu m 4 Errors in acceleration estimates in xy 39 m m D And 2 directions Note the DMC did a poorjob of recovering accelerations More work is needed on optimizing c and 3 We should do a better job of recovering accelerations and the state while reducing tracking residuals 39 Copynght zoos Filter Comparisons J I All filters achieved comparable results however we should do better with the DMC m3 5 40 Copynght 2005 2O t a 0 Added deviations to the initial conditions so that they are outside linear range to show Convergence for EKF with SNC Divergence for CKF with SNC Original Initial Conditions 757700301 5222606566 485149977 2213250611 4678372741 5371314404 m and ms Deviation 8 5 5 8 5 5 m and ms Perturbed Initial Conditions 757708301 5222611566 485150477 2221250611 4683372741 5366314404 m and ms ll Copyright 2005 0 Conventional Kalman Filter Band Diagonal State Noise Matrix results for optimum value 02 103913 Rmum mmme 39 LAW w van mm Deviated Initial Conditions a ms 155 am 2 m mail Emviwsarmsmmnan nmmum 15a 20 mm mm y hf Copyright 2005 25m m a WWW 21 Conventional Kalman Filter Band Diagonal State Noise Matrix results for optimum value 02 1013 x Ere MN m 5Atlc391D339nli52537il4Bl Deviated Initial Conditions 39 Velocity Rss 288 ms 4 x Evvn39 was Eucu N c39 tDC nuns a 15595 m m M a m an my t as zsu mm W Liczmlt39 Dawns imam y m3 0 0 ma lt r m zsu 09 an 5 r m in may Position RSS 235349 m Copyright 2005 Extended Kalman Filter Band Diagonal State Noise Matrix results for optimum value 02 103913 39 Deviated Initial Conditions jE M A gt y Hx am m 0 mm mums mmntm 5 mi 151 2m 7m innmast As ErrovsAlter mu v m n mnaana Zv JLJMMW Rawalml Rnnqn mquot W5 a Z WMWM 39 an mu 15a 2m 25a an n we in meal in ma quot 15 a zoo 25m m m lrm mle i mg an my y Copyright 2005 22 Extended Kalman Filter Band Diagonal State Noise Matrix results for optimum value 02 103913 xsiiai lRMS Emmy im iii iis r 527m E Deviated Initial Conditions im t5 2m 25 mu m vsiarimssiimmm ma iii iii f W L su iuc in m 25H m 3m 2m imam mm inn Velocitv RSS 2 53 x 10 2 ms mi my arms Altai inn imzii HDEWZZl 25a m 35H fin 200 nm in mil5i Position RSS 1779 m 45 Copyright 2005 Filter Comparisons CKF EKF Initial Conditions Linear Range Nonlinear Range Linear Range n iiiiii dl Kallg Ran e Residual o 834 Po iti m X Velocity Error rns v Velocitv Error mS Z Veloclt Error rn S 1 21 r 2 3D Veloci RSS m S 1622E 02 2528E 02 While the CKF does not diverge its solution is significantly In error 46 Copyright 2005 0 Added deviations to the initial conditions to show Convergence for EKF with SNC Divergence forthe Batch Processor Original nitia Conditions 757700301 5222606566 485149977 2213250611 4678372741 5371314404 m ms Deviation 1000 1000 1000 500 500 500 mms Perturbed nitia Conditions 758700301 5223606566 485249977 2713250611 5178372741 4871314404 m ms t a 39 a Copyright 2005 Batch Processor 1 Pass 1 Pass 3 Range RMS 445733 km Range rate RMS 4710 kms Range RMS 367236 km Range rate RMS 4304 kms f Runyu mm Hg aw mam quntmv nv39qntmi i Ranqunmntws 37 m M5 3 mi Note that the batch processor is not converging and successive Iterations show divergence 43 0 Copyright 2005 24 Extended Kalman Filter Band Diagonal State Noise Matrix results for various values of 2 xl 39 Deviated Initial Conditions Trajectory updated after 30 minutes 2m 1 m39 mquot m39 mquot Note Values ofo2 from 10395 to 102m were tested wever the residuals and errors were orders of magnitude higher for value of 02 between 105 to 109 Therefore those values are not shown on the plots a 4 ma 4 mrv 5 Note that the optimal value for 0392 is the same as for small initial condition errors n 49 Copyngm 2005 Extended Kalman Filter Band Diagonal State Noise Matrix results for 02 1013 Deviated lnitial Conditions Trajectory updated after 30 minutes 4 y Hx m we A 4 we Copyright 2005 Extended Kalman Filter Band Diagonal State Noise Matrix results for 02 1013 Mime mm incl 20 Km imvnnmw Deviated Initial Conditions m zen Position Errors after 250 minutes XEmr ms E mum 253 m is dummy l r yumquot m 7 an Kain 5U m an em gt a 5n mu tsn 2m Yime mi must Position Errors Position RSS 8643 m V x 51 Copyright 2005 Extended Kalman Filter Band Diagonal State Noise Matrix results for 02 1013 g a ia 39 wwl w e DeVIated lnItIal Conditions imam an m 1 m m m mymEwmmmzmm Velocity Errors after 250 minutes w mm mm a or z w my ma 15a 2m Velocity Errors Velocity RSS 1258 x 10 2 ms Note that position and velocity R88 are comparable to those on slide 40 If IC errors are large it may be Advantageous to use the EKF to Copyrigl1L2005 Obtain ICs for the batch 52 L IE STATISTICAL ORBIT DETERMINATION The Minimum Variance Estimate ASEN 5070 LECTURE 16 100406 Colorado Center for Astrodynamics Research The University of Colorado THE MINIMUM VARIANCE ESTIMATEQ The least squares and weighted least squares methods do not include any information on the statistical characteristics of the measurement errors or the a priori errors in the values of the parameters to be estimated The minimum variance approach is one method for removing this limitation Colorado Center for Astrodynamics Research The University of Colorado wIMUM VARIANCE ESTIMATE Gum The swam af nIepmpagmmn cquanam and 0mm Ion mm cqum Hans an mx 44 m ufhxn k 442 Fmd Th Imch unbmm mmunum nnance cmmalc m 0mm smle x The salmon m Ihls mbIcm proceed b oIIov5 Us slaw rmmno mam Jud II de nmarb argq 43 m reduce Eq mu m me foIIowmg folm an 1 4431 he I EM 0 Rum Rn EM D RE R24 51 I 0 RE v R1 Cnlnli n Cum In Axln ynimms FESHIEII quotI2 nlvusllvnl Canaan WIMUM VARIANCE ESTIMATE I nu ma nmhh m cmcu Incmmmo me h I hum mhm mmnuum NHan mum Tm cnnmwmmcs I mm m Iluw wqmmnmm m AIdm cLI m In IoIImvmg Ix px m be I lem The chmrcmcm n1 1 Immrcwmmc ImpIm mm 11w csmumo IS m m mch up 0 u InkAr cmnbumlmn I In ubwmvuoum I39IK n x m mm M I nhxpeCI L II and I m m mm m Imam me best mum Z LIVHum II III annual Ix Imbmacd men I Adimnmv m uhmmlmg qus H 4 v and I4 411mm Eq w m mu m w mng umcm Hem EM EIMyI mum Hn cmmmn Cum In Axln ynlmms Hessmu me nlvusllvnl Crlnunn wIMUM VARIANCE ESTIMATE Rm we Er D 1qu mm m U onm39mch IIcI nIIuwmg cumlminl an M IS uhlumud M IH 7 mm m mm mm Is m m mbmcd me Imam mnm mg marm M mm mm Eq 4471 Tm sandman xcqmrce the row of M In In anhngnnm Ia IIw qumns NH cmmmn Cam In Asunnwamms nus m quotIn Llllwlslvnlcrluinn wIMUM VARIANCE ESTIMATE m Mmmmm HumIce Hum bum 5 unbmicd mm m mamnan sum ommmce quotmm can be expressed 15 I569 Appende Al PkE IXA 73x7EIxI7xu11Ix E L Hm nusfymg eq H 4m and 4 47 B39 mmumzmg P we mm um P 7 PA 5 um mm 01 any P ma results mm m 11 um qlu c E H 4 7 4Deutsch 10b57 SubsumungEquu 9mm Imm Eq I448Hca me mIInw g Ir esu i pA EM wk 7 M 7 Emmm 7 mean 7 r 7 x V EMIzLTJIT hem e have um MH 7 I I mm Irom Eq H 447 um me manmxce mamx can bc wnllen a HIOV cmmmn Cam In Asunnwamms nus m quotIn Llllwlslvnlcrluinn HE MINIMUM VARIANCE ESTIMATE E WIK IL III mnhc wIcquI llmIIIsII Cq 44 7139 Tu mvuIvelIImuMmml nupnmI IWIA 4 7I u Id In krcp Ihc canxtnuncd rcmmn fnr PI xymmclw Eq 447 Ix mIInumI m III 144 1 m IIIC InIImvmg Imm PA mm AW 7 MW I r IIIm 44 In Mm A IN 4 n A H mm uI39umpml ch Lbe ngc munmlm The mm term I MM Im uxum Um PI I rmuumymmm M u umumum of I n I w xamy that H lirs I wmhnn WIIII IMVWI In M IIIIIN II amI I131 I MH I cor m39 HTJMIT mrnMT HA 44 m 611 0 MR Cnlnladn 3mm in Asundynmus Resendquot The mumsIv nl Cnlnladn ampNIMUM VARIANCE ESTIMATE No mm IS I mush rm an amnmySM one mm ouan conmuom musI I m I RAF 7 HA a 2 5M and arRJIT 7 HA mm not Leo uII mnk k m Impos condmon I and Show Hm Ins mm a nummum m m P Hence n IS mquurd ma MR 7 WI 7 MH n IIIIZ len the rim of mm commons M ATHTRquot MAM mac R1 assumed Io lu posmu de nuc s mmung Eq 44 I3 I mm the 5W and a Eqs 4412 Icnds to me fuIIowmg n A LHTIT HI I Cnlnladn 3mm h Asundynmus Resendquot The mumsIv nl Cnlnladn vb su HA H IWIMUM VARIANCE ESTIMATE g Now mm mamx H RHH ls mll rank wlmlr mqulrcx mm m 3 n hell xlm lllwr c quotmm wlll cm and AT HTR H IJ Mm Then m we urElr r4413 lllHTR H HTHquot H415 2 H 3 a mu l l 7 mm Sulumunmr of lzq r4 41mm Eq lllur lead lo we following Hprcs mu rm m coumncc mulllx Pki HTRquotHF H4l39ll 39llh Eq 44 Im arm 44 5 lhc luau mamall muruuum llrlllllucsllmllln oka r gm as M HTIT39HV H R HAIR Colorado Caner lorAsIrouynamlcs Research The umversuyol Colorado 44 PROPAG low OF THE ESTIMATE AND COVARMNCF nun ll rlrc cmmm all a mm r r mural Ixy usm Eq rl 4 m ch cltllmlnc may lu mapped 0 any lam um hy lng Lq HA1 m all ng ruler E Elm a sum 7 xA l lull In Vic ol q 44 um q H lm lwcnlnca E l39nl x rXHx l W39mrr 4441 ltmce lllc srme Inlllslllml nuler r dclcrmlanr n mum mm m H 4 xr mquot FA 39HAJJFquotrrr M415 qumllonx H 4 l9 and H 4 Lil can he used In map me eltllmmc or rm slnlc and llbilhocuucd L AIV dndlh c mum rmm l m u Colorado Caner lorAsIrouynamlcs Research The umversuyol Colorado MUM LIKICLIIIOOD AND BAYESIAN IM39TIN The method of rlllliilliil Ltkl liliunrl Exilllrilmil Tar nelelnlllihig the heal es timate til a Variable l due to Fuller 19121 The Maxrnlllm Likelihood Estimate MLEl of ll llzlrnlileter 9glvcn nbaervnllons yr yg myk and the Joint proba blth dcllell function fll1ll2 quotin St Hill lh de ned to be llial value of 9 lhal nlaxrnllzes llle prohali39lllw densny function l39nlpnle and Men 19W Iowever ll ls a random nrnilile and 1 have knowledge ol llw39 probability delirin fllnelioli tlle MLE of 9 li defined to be the l which mtlxllllizex llie prublibllily dellzlty function at a conditioned on knowledge ol39tlleobienxlltollsy yz y WEIluau in rum Colorado Cenrer ior Aeirodynamlce Research The univerelry oi Colorado 15 MAXIMUMLIKICLIIIUUDJ llfl39l ION Tlle Brim t Jlllltllt lol39 6 I de ned lo he lhe lilenll of llie rotilllllonnl detisrly l39nnctloil glleii by Eq 451illl39nlpole and Myers 108W TllL JOllll density flinc tloii El H S i lllltl the conditional density function Eq 1452i are referred to as lhe Ilrelillanil rln llnli L The loglt liehilld minimizing L is that of nll the pomhle value or 9 we alionld choose the one llnil lllaxmilzes the probability nt ohlalning llle UliSCI39lleloll lhat llr lllally were observed If 9 la a random variable i i r i i K or lien 39 or r39nneuuu n the 39 39 llllll lmll39ll 39 39 T 39 39 Ilil l39lll correspond lathe liiean oflhe conditional denstt function Hence the MLE and the Bayes estimate for a lansslan rlerlsny lllliellon tire ldenlicnl Colorado Cenrer ior Aeirodynamlce Research The univerelry oi Colorado l Review problem X 3XZ 4X0 ll Given the system X with the state vector defined by X and the deviation vector defined by 51 6X Where 6 indicates a small deviation from a reference value a Write the linearized equations in state space form 6 A6 b How would you determine the state transition matrix for this system What Additional information is needed to generate the state transition matrix Colorado Center for Astrodynamics Research The University of Colorado 7 Review problem lll Circle the correct answers or answers a Given the observation state equation yt t zaxUthlcxl 11210 Where ax X1 and c are constants and t is given Which of the following state vectors are observable Colorado Center for Astrodynamics Research The University of Colorado Review problem b The differential equation x56 afcz t4 0 1 1st order and 1st degree 2 2nd order and 1st degree 3 linear 4 nonlinear c Given two uncorrelated observations and the second is twice as accurate as the first we would use the following weighting matrix 10 2 o 10 11 1W 2W 3 W 4W o I 01 o 2 12 Colorado Center for Astrodynamics Research The University of Colorado 7 Review problem d If the differential equation for the state is given by 56bx20 It would not be necessary to use a state deviation vector T or F 639 t e The state transition matrix will contain terms such as X this partial derivative are 6Jz n The units of lLT 2 L2T 3 lT 4 It is dimensionless f If the state transition matrix is symplectic it can be inverted by inspection TorF Colorado Center for Astrodynamics Research The University of Colorado Inclusion of Apriori Information in the Batch 39 teessor Computational Algorithm If apriori information X and0 with attendant covariancef 0 is given this Information should be maintained when iterating the batch algorithm ie for the first iteration X 0 0 XI E X X 20 Hence 9a A X0 x0 X0 x0 x1 Solving for E yields X f and f3 should be held constant to begin each iteration Hence Colorado Center for Astrodynamics Research The University of Colorado Inclusion of Apriori Information in the Batch g aggressor Computational Algorithm Thus for the nm iteration the apriori value of in is given by fquot EA xn1 464 and X X 1 x 1 n n n Finally m 11 5cm EHlTwH 130 12HTwy 1304 1 I Colorado Center for Astrodynamics Research The University of Colorado ORBIT DETERMINATION ASEN 5070 LECTURE 6amp7 91106 Symbolic Toolbox Example EDUgtgt syms x y 2 M0 EDUgtgt r sqrtx 2 y 2 2 2 r x 2y 2z 2 12 EDUgtgt u MUr u Mux 2yquot2z 2 12 EDUgtgt dudx diffu39x39 dUdX Mux 2y 2z 2 32x EDUgtgt dudx subsdudx39x 2y 2z 2 323939r 3 dUdX MUr 3x Symbolic Toolbox Example EDUgtgt statevec x y z state vec x Y z EDUgtgt dU jacobianUstatevec dU Mux2y2z232x MUx2y 2z232iy Mux2y2z2u 322 mu dU subsdv x 2yquot2z 2 3239 r 339 it MUrquot3x MUrquot3y MUrquot3z EDUgtgt Integration Error State Error u my i i Ivhriiy pm Artninr mn my 2 ma imm Default integration tolerance ReITol 1e3 IN Integration Error Energy Change r w 2 r me imst Default integration tolerance RelTol 1e3 Change on the order 105 m2s2 Error in n ma EuengCmvge quotswag Integrator n 398600 km3s2 Energy Calculation n 3986005 km3s2 I00 Observation State Relation Chapter 4 Fundamentals of Orbit Determination ta 42 LINEARIZATION OF THE ORBIT DETERMINATION PROCESS For the general case the governing relations involve the nonlinear expression X FX t Xm E Xk 421 i 422 YGXif6i 11 where X is the unknown ndiniensional state vector at the time 7 and Y for i 1 l is a pdimensional set of observations that are to be used to obtain a best estimate of the unknown value of X 1e XL In general 1 lt n and m p x gtgt 1 The formulation represented by Eqs 421 and 422 is char acterized by l the inability to observe the state directly 2 nonlinear relations between the observations and the state 3 fewer observations at airy time epoch than there are state vector components 1 lt n and 4 errors in the observations represented by 6 Then the nonlinear orbit determination problem in which the complete state vector is to be estimated can be replaced by a linear orbit determination problem in which the deviation from some reference solution is to be determined To conduct this linearization procedure let the n x 1 state deviation vector x and the p x 1 observation deviation vector y be de ned as follows xr Xr X t yt Yt Yt 423 It follows that U Xf Xf e Xquot t 424 Expanding Eqs 421 and 422 in a Taylor s series about the reference trajec tory leads to Xr FXfFX tUFm 0Xr in Km 7 KW 402 Km 7 KW XUil XiaUH 1 0G quot Y 7 GXtve4GX tl0X i 0X1 7 X fag Jr 6 lo If the terms of order higher than the rst in Eq 425 are neglected under the assumption that the higher order products are small compared to the rst order terms and if the condition Xquot FXt and CXf ii are used Eq 425 can be written as X 447 X7 426 yiHzxi i V am OG fl ml HI laxl Hence the original nonlinear estimation problem is replaced by the linear estima tion problem described by Eq 426 where xt Xm X t X X fi 139 s where Generally in this text uppercase X and Y will represent the state and the observation vectors and lowercase X and y will represent the state and observa tion deviation vectors as de ned by Eq 423 However this notation will not always be adhered to and sometimes x and y will be referred to as the state and observation vectors respectively IO Example 421 Compute the A matrix and the H matrix for a satellite in a plane under the in uence of only a cenn alfbrce Assume that the satellite is being tracked with range observations p from a single ground station Assume that the station coor dinates X5 Y5 and the gravitational parameter are unknown Then the state vector X is given by where U and V are velocity components and X 5 and Y3 are coordinates of the tracking station From Newton s Second Law and the law of gravitation or in component form Xamp 73 f Y ILL 73 If these equations are exp gassed i1 rst rcler fq m th folIowinuexpressi0n is bt 39 d 7 7 0 alne A F1 L Y F2 V v 1X 0 F3 r i lquot X x F1 3 1 F5 0 A75 E 0 its FT 0 131 aFl OFI 0131 0131 aFl 8131 aFa 0X 0Y3 OFXquot1 617 W 6ng 15 0 0 1 0 0 0 0 0 0 0 1 0 0 0 3X2 3XY X i r l 0 0 0 o In 7 lm In 3 XY 3 Y2 Y I 1 H H 0 u 5 0 0 0 0 0 0 0 0 0 o 0 0 0 0 u o 0 0 0 0 u u 0 The I matrix is given by H Op 0p 0 01 0 0 0 Op 139 0X 7 0X OY 0U 0V 0p 0X3 0Y3 where r 7 r j r 7 2 1 p a xsr 3 45 It follows then that HXXg VAT XXs YYg p p p p State Transition Matrix 5dr Afxt 426 yiz iX5 i OFU AW lml HT laxl 421 THE STATE TRANSITION MATRIX The rst of Eq 426 represents a system of linear differential equations with timedependent coef cients The symbol l indicates that the values ofX are derived from a particular solution to the equations X F Xi which is generated with the initial conditions XUU X6 The general solution for this system Xf Az xf can be expressed as X I U fklxr 427 where X is the value of X at m that is xi x1 The matrix Igtf ft is called the state transition matrix and was introduced in Chapter 1 Section 125 19 1 DOLLft I 2 DUITtl I1Lifil1tjstk 428 Mn11 ltIgt 1fT out Arlt1gttm 4210 with initial conditions ITLfL 20 mansition Matrix for a Linear System Given the system c ax by y39 ky Find the state transition matrix for Xlil Equations are linear in the dependent variables Xy and their derivatives in space state form XAXiiia hm 3838523233310 y 0 k y ltIgttt0AltIgttt0 E 339 9 D Colorado Center for Astrodynamics Research The University of Colorado msition Matrix for a Linear System Since the differential equations for 1055 are linear with constant coefficients we can solve them using Laplace Transforms CIgtIt0 L1 SI A 1 b SA S a b SA1 s a s as b 0 s b 1 s k b at at k1 ltIgttt0 e ake e 0 e Colorado Center for Astrodynamics Research The University of Colorado mansition Matrix for a Linear System If A is a constant matrix there are anumber of ways including Laplace Transforms to solve the equation lt1 m0 AC1 mo For example we may integrate the equations directly l 11 12 A11 A12 11 12 0 21 22 A21 A22 21 22 A11 11 A12 21 A11 12 A12 22 A21 11 A22 21 A21 12 A2Z ZZ Note that the columns of 1 are independent Hence ifA At and we Must use numerical integration we could integrate the rows of 139 independently as n systems of nxl equations as opposed to an n x 11 system of simultaneous equations Colorado Center for Astrodynamics Research The University of Colorado msition Matrix for a Linear System Evaluating the equations for d yields 11 a 11 b 21 1 21 k 21 2 12 a 12 b 22 3 22 k 22 4 with re CIgttt0 1 Note that the columns of 1 are independent ie Eqns 1 and 2 are independent of3 and 4 Colorado Center for Astrodynamics Research The University of Colorado Winsition Matrix for a Linear System Solutions From Eq 2 451 a 1n 21kt1nC 21 ek2 4 ZICekt C tt0ov 21O C0and CDZIO Colorado Center for Astrodynamics Research The University of Colorado Wsition Matrix for a Linear System Likewise from Eq 4 k 22 Ce zz00 221 k1 IC1 and 22e Thus Eq 1 becomes 11 11 and 11 eat Colorado Center for Astrodynamics Research The University of Colorado mansition Matrix for a Linear System Finally from Eq 3 1512 a 12 bequott 5 The homogeneous equation 12 a 12 has the solution 12 ceal To get a particular solution note that 6 has the derivative kek Colorado Center for Astrodynamics Research The University of Colorado Wsition Matrix for a Linear System So try kt 5121 C16 substitute the particular equation into Eq 5 Clkek aClek be Then Clk aCl b0 a Cl b k a Colorado Center for Astrodynamics Research The University of Colorado mansition Matrix for a Linear System The general solution is the sum of homogeneous and particular solutions 17 u Ce e a tt00 4112 0 C L k 1 Hence b b b 21 eateki eatekt k a k a a k and b eal eat ekz Note that 10 lt1 I to a k 0 eh lt1 I to I Colorado Center for Astrodynamics Research The University of Colorado 7 423 RELATING THE OBSERVATIONS TO AN EPOCH STATE X7 IgtftLx 427 YifjXei j1h YT H1ltIgtlrlrx 51 Y2 32ml 2 kX l 62 3937 yl Hr l frfrX ru If the following de nitions are used 11 Hi I UiJ Ll 1 y E g H g i 3 4238 yr H bm n a and if the subscript on X is dropped for convenience then Eq 4237 can be expressed as follows Ill m ll y Hx 6 4239 where y is an m gtlt 1 vector x is an n gtlt 1 vector 6 is an m X 1 vector H is an m gtlt n mapping matrix where m p X 6 is the total number ofobsewations If p or f is suf ciently large the essential condition m gt n is satis ed However we are still faced with m unknown observation errors resulting in m 71 total unknowns and only m equations The least squares criterion provides us with conditions on the m observation errors that allow a solution for the 72 state variables x at the epoch time 21 Least Squares 43 THE LEAST SQUARES SOLUTION y HX 6 4239 x 126T6 431 M 2 1236 X 1246 12y Hx y Hx 432 Note that Eq 432 is a quadratic function of X and as a consequence the ex pression will have a unique minima When see Appendix B Eq 34 H 373 0 T on U and OX 90x gt I dx x for all 6x 75 0 The second condition implies that the symmetric matrix 02 0x2 is positive de nite Carrying out the rst operation on Eq 432 see Appendix B Eq B73 yields 6 7 a 0 7y 7 HxTH iHTy 7 Hx 433 x The value of X that satis es Eq 433 will be the best estimate of x which we will call 2 Hence M r H1 Hp ny 434 34 Also from Eqs 433 and 873 it follows that 3 0x3 which will be positive de nite if H is full rank Equation 434 is referred to as the normal equation and H TH is referred to as the normal nmlrz x Note that the matrix H TH is an n X n symmetric matrix and if this matrix is positive de nite H is rank n then the solution for the best estimate of X is given by 2 HTH39HTy 436 HTH 435 Equation 43 6 is the well known least squares solution for the best estimate of x given the linear observation state relationship expressed by Eq 4239 With the observations y and a speci ed value of 2 the value for the best estimate of the observation errors E can be computed om Eq 4239 as 2 y 7 Hi 437 35 Least Squares Example of least squares assume w I and we have no apriori Let K a tl 8 Note that this is a linear system Assume we wish to estimate Colorado Center for Astrodynamics Research The University of Colorado Ln Least Squares A a 1 X A HTH HTY B where 1 r1 Y1 H 1 t2 Y 12 1 a In 1 1 I IX X Note that HTH will always 7 1 x asymmetric matrix H H 1 z 2 HTY 1 a a ax Colorado Center for Astrodynamics Research The University of Colorado Least Squares 1i 9 Assume 4 3 t123amp Y 5 6 A l3 iii ie we have chosen perfect observations Colorado Center for Astrodynamics Research The University of Colo ad Minimum Norm 431 THE MINIMUM NORM SOLUTION For the solution given by Eq 436 to exist it is required that m 2 n and H have rank 77 Consider the case where m lt n that is H is of rank lt n In other words there are more unknowns than linearly independent observations One could choose to specify any n 7 m of the 72 components of X and solve for the remaining m components of X using the observation equations with E 0 However this leads to an in nite number of solutions for 2 As an alternative to obtain a unique solution one can use the minimum norm criterion to determine x Generally a nominal or initial guess for x exists Recall that the differential equations have been linearized and x X 7 X The nn nimzzm norm criterion chooses X to minimize the sum of the squares of the difference between X and X subjectto the constraint that e 0 that is y H x Hence the performance index becomes x A 12x739x ATy Hx 438gt XHTHHT1y 431 where HHT is an m X 77 matrix of rank 77 The quantities HTHHT 1 of Eq 4313 and HTHF39HI of Eq 436 are called the pseudo memes ofH in the equation H 2 y They apply when there are more unknowns than equations or more equations than unknowns respectively Eq 4313t is the solution for x of minimum length In summary XHTH 1HTy if m gt17 5 H ly if m n X HTHHT1y ifm lt11 4314 432 SHORTCOMINGS OF THE LEAST SQUARES SOLUTION Three major shortcomings of the simple least squares solution are 1 Each observation error is weighted equally even though the accuracy of observations may differ E The observation errors may be con elated not independent and the simple least squares solution makes no allowance for this a The method does not consider that the errors are samples from a random process and makes no attempt to utilize any statistical information The rst of these objections is overcome through the use of the weighted least squares approach 433 WEIGHTED LEAST SQUARES SOLUTION Y1 H1XLv611 1391 yg ngl 632 U g Y HzXL Eli quotIf H RamaJ Y1 H1 yz H2 y H y E 4316 6 w 0 U 52 0 U3 0 e U39 y HXLI 6 nquot 4317 x 12eTll396 Z 12eg39uviei 4318 139l Jx 12y Hx1Tll39y Hx 4319 7 0 ll Hkan vH HTll39y er 4320 XL HTil39Hx HTU39y 4321 5C1 HTll39vHFlHTll39y 4322 Hgosing the Weighting Matrix The Weighting Matrix may be chosen by using the RMS of the observation residuals 5y H5c Compute the RMS of the observation errors for each type of observation Colorado Center for Astrodynamics Research The University of Colorado 4 Moosing the Weighting Matrix Let i represent the observation typesay j 1 gt range 139 2 gt range rate so for two observation types let 1 W M 1 RMS We use the mean square MS so that J x y HxT W y Hx will be dimensionless This can enhance numerical stability of the normal equations Colorado Center for Astrodynamics Research W i The University of Colo a L IE STATISTICAL ORBIT DETERMINATION EKF with large IC errors Combining estimates Information Filter Concth Exam 3 ASEN 5070 LECTURE 38 121106 Colorado Center for Astrodynamics Research The University of Colorado Added deviations to the initial conditions to show Convergence for EKF with SNC Divergence for the Batch Processor Original Initial Conditions 757700301 5222606566 485149977 2213250611 4678372741 5371314404 m ms Deviation 1000 1000 1000 500 500 500 mms Perturbed Initial Conditions 758700301 5223606566 485249977 2713250611 5178372741 4871314404 m ms M K Cupyngm inns rm Copyright 2605 Batch Processor Pass 1 Range RMS 367236 km Range rate RMS 4304 kms Pass 3 Range RMS 445733 km Range rate RMS 4710 kms l l l quot su tau min 24 2w em a n 5 s39u mu m m m l E M l l l 5 ma 197cm muff 25 Jan 1 n 7 Sn WE Sfm mm uzja 25m 33 3 Note that the batch processor is not converging and successive Iterations show divergence 3 Extended Kalman Filter Band Diagonal State Noise Matrix results for various values of 02 m39 menu was WM rim W was l 74 WA may 1 mlucs y mquot Note Values ofa2 from 10395 to 102m were tested owever the residuals and errors were orders of magnitude higher for value of 02 between 1Cr5 to 109 Therefore those values are not shown on the plots Note that the optimal value for 02 is the same as for small initial condition errors Va Copyright 2005 as am mam lm Trajectory updated after 30 minutes Deviated Initial Conditions mu Extended Kalman Filter Band Diagonal State Noise Matrix results for 02 103913 Deviated Initial Conditions Trajectory updated after 30 minutes so am m m mm mm mm mm 1 mammalm lsn am we mms 5n ms 151 am 250 mu n 5 m WWW mmmmm 5 isBmsAucrzsaumsnann27713l a m E g 0 WM 5 r 42 a 5 we x51 2m 25 m n V ruuuluummsl Copyright 2005 Extended Kalman Filter Band Diagonal State Noise Matrix results for 02 103913 u Mumzsamkscm Deviated Initial Conditions Position Errors after 250 minutes xsvorRMS 2 mm 251 m m scm32tl 0 25a Jan sn mu 5 z z z Enov ms mums mvvszschm GE 2 me inquot Mes Position Errors Position RSS8643m m m 2m 2m zun m We lvvmulssl Copy g39ht 2005 Extended Kalman Filter Band Diagonal State Noise Matrix results for 02 1013 x Ever lms Errors Altar zsn nuns z a Human Velocin Deviated Initial Conditions Velocity Errors after 250 minutes x Enm lams Errars Aim zsu m z u un mzn l my am 2 mm chanzm um 159 gm m mlnmesl Velocity Errors Velocity RSS1 258 X 10392 ms Note that position and velocity R33 are compara e to those on slide 40 Of lecture 32 If IC errors are large it may be Advantageous to use the EKF to Obtain le for the batch 7 VI lmsl Copyrtgit zoos 417 COMBINING ESTIMATES Assume we are given two unbiased and uncorrelated estimates 31 and 2 for the nvector x Assume that the associated estimation errors 711 and 172 are Gaussian with covariance matrices P1 and P2 Our objective is to establish a performance index and determine the combination of 21 and 32 that is an optimal unbiased E x estimate of x 771517x 4171 E 771 E l772l 0 E W1 77 Pi E 772 77ng Pz E in 713 0 4177 Statistical Orbit Determination Copyright 2006 University ofColorado at Boulder Treat estimates as observations Write observation equations Then use Linear Unbiased Minimum Variance Estimate Equation as follows Statistical Orbit Determination Copyright 2006 University ofColorado at Boulder yHxE quot I y 2 H a 77 M I 772 R 0 R 0 PZ x HTR le HTR l y Pf PzquotquotRquot xxPzquot xz P Pf Pzquotquot Stamstma ormtDaermmatmn CupynghtZElElE U mversm ur cmurauu at Bummer If there are n independent solutions to combine 1 71 H x ZR 23 44711 1 1 Stamstma ormtDaermmatmn CupynghtZElElE U mversm ur cmurauu at Bummer 410 INFORMATION FILTER A sequential estimation algorithm can be developed by propagating the infor mation matrix A E P 1 Maybeck 1979 This form of the lter offers some numerical properties With better characteristics than the covariance lter Writing Eq 473 A T 1 1 1 NT 71 1 m Ht Rt Ht Pk gt Ht Rt yk Pk x1e 473 in terms of the information matrix gives Ac HERngk k Am Hnglyk 4101 0139 Akik Kka l gRilyk Statistical Orbit Deterrriination Copyright 2006 University of Colorado at Boulder Recall that the equations for the standard sequential algorithm also called the covariance filter are Time UpdureiCawriunce Filter n1 ltIgtr 11 5 41017 1 P1 1 7 1 1P I TLHit 1 1quot1 12 1 Q1V1 ILL1tL zlIeasmz mcm UpdaIECOWIriancv F flier 39 1 1 1 Ak1 P11H1RI 1 Hlv1P11Hk11 41019 41018 X1 A1 IX39LHLJ ykdrl HL1YL1 41020 v V 7 Pk1I IU1H11PL1 MU Statistical Orbit Determination Copyright 2006 University of Colorado at Boulder The equations for the information filter are as follows The procedure starts with a priori values n and Y0 From these compute e1 A PU 41015 D11 AUX Statistical Orbit Determination Copyright 2006 University of Colorado at Boulder Time Updareillszrnlurion Filfer Maw PTOt1ALltDtt1 41022 L1c1 1quot17 11Ft111 1 gtlt 11 111111k1r1k11 Q1 1 41023 D141 I Lk1FTTL1Jk gtlt Tt t 1j 41024 71 A141 Pk I Lk1FTtc1 1 gtlt 1111 41025 Staustma Orb1tDEtEnmnat10n C0pynght 2006 Unwersm 4 00107300 at E0u1der Measznvmem UPI1197111fbrmmion Filrer 131 511 Hf1Rg11yk1 41026 Pg l AH Km HEHRQI 151111 41027 kw AH 41028 Staustma Orb1tDEtEnmnat10n C0pynght 2006 Unwersm 4 00107300 at E0u1der We can initialize the information lter with PD 00 or with P0 singular and obtain valid results for 1 once P7 becomes nonsingular The Cholesky Algo rithm of Chapter 5 may be used to solve for 2 However the solution for X is not required by the algorithm and needs to be computed only when desired The con ventional sequential estimation algorithm fails in these cases Also as indicated in Section 471 the conventional sequential estimator can fail in the case where very accurate measurements are processed which rapidly reduce the estimation error covariance matrix This can be mitigated with the information lter Statistical Orbit Determination Copyright 2006 University of Colorado at Boulder Consider the example discussed in Section 471 Pa21 7216 H1 1 6 R where 6 ltlt 1 and we assume that our computer word length is such that 1 6 y l 1 e r The objective is to nd P1 that is the estimation error covariance matrix after processing one observation using the information lter Statistical Orbit Determination Copyright 2006 University of Colorado at Boulder The information lter yields 2 0 P1 2 U 039 This is the same symmetric positive de nite result obtained from the batch pro cessor and agrees with the exact solution to 06 Because we are accumulating the information matrix at each stage the accuracy of the information lter should be comparable to that of the batch processor The conventional Kalmau lter yields 0 0 P1 r 4725 039 0 Hence the conventional covariance lter fails for this example but the information lter does not Statistical Orbit Determination Copyright 2006 University of Colorado at Boulder Concept Exam 3 ASEN 5070 121106 Under what condition is the sum of the squares of the error vector elements of the orthogonal transformation technique equal to the sum of squares of the observation residuals Y b 0 a m J quotUl at n Io In order to use the orthogonal transformation approach it is necessary to form the information matrix T or F 9 The estimation error covariance matrix generally yields a realistic estimate of the solution accuracy T or do In order to implement the xed interval smoother it is necessary to compute the smoothed estimation error covariance T or F 39139 The matrix relating deviations in the Cartesian coordinates to deviations in the orbit elenrents is orthogonal T or F 9 The equation for the family of probability ellipsoids 9 7x01134873 11 in the ECI frame and Altquot39713 71iquotinquotll in the principal axis system yield the same results T or F 7 The Eigenvalues of a positive defmite matrix may be imaginary T or F Statistical Orbit Determination Copyright 2006 University of Colorado at Boulder STATISTICAL ORBIT DETERMINATION Givens Examgle Apriori Information and Initialization Addition of new data ASEN 5070 LECTURE 28 110606 Colorado Center lor Astrodynamics Research The University oi Colorado 54 GIVENS TRANSFORA ATIONS I L c r 1 th r mini otrr r a 39 39 hu 39 39 diagonal a rank a matrix can he reduced to an upper triangular n y 7 matrix with a lower in e n x n null matrix Ifthe element to he nulled already has a Value of t e transformation matrix will be the identity matrix and the corresponding transformation may be skip ed s an example the transformation to null the fourth element in the third col umu is shown as follows 1 hm h12 7113 11 31 0 7122 7123 hzu 32 0 0 l133l13 ya I o 0 lug 7141314 1 0 0 15315 yr 1 0 0 has thl yo 1 0 0 Irmolrym Colorado Center lor Astrodynamics Research The University oi Colorado Accuracy Comparison for Batch and Givens Einite Precision Computer conSlder 1 1 Notice that a vector of H 1 1 Observations is not needed Why 1 1 5 Machine precision is such that 1 i 82 1 The normal matrix is given by T 3 3 s H H 3 3 2 2 exact solution s s 5 our computer will drop the 52 and To order s HTH 3 3 8 HTH0 hence it 3 5 3 25 is singular Colorado Center for Astrodynamics Research 39 quot i The University of Co Accuracy Comparison for Batch and Givens 39Einite Precision Computer Consequently the Batch Processor will fail to yield a solution Note that this Illustrates the problem with forming HTH ie numerical problems are amplified The Cholesky decomposition yields 3 3 6 R J3 0 0 R is singular and will not yield a solution for 2 Use the Givens transformation to determine R lSt zero element 21 of H C9S9011 S9C9011 00111 2 odynamics Research Colorado Center for Astr The University of Co Accuracy Comparison for Batch and Givens Einite Precision Computer x11 x21 S6 x2 i C x1 1 1N5 IN 0 1 1 2H 245 1J 1J 0 1 1 0 0 0 0 1 1 1 5 1 1 5 3x2 3x3 Note that the magnitude of the columns of H y are unchanged 2 Next zero element 31 x1 J51 x2 1 Colorado Center for Astrodynamics Research The University of Colorado Accuracy Comparison for Batch and Givens 39Einite Precision Computer 1 0 m J 9 6 quotJ8 2N8 0 1J 2N5 2N5 JE 3 sJ 0 1 0 0 0 0 0 1J 0 2N8 1 15 0 J sJ J5 Nextzero element 32 XI 01xZ SE 1C9 0 1 0 0 J5 3 e J5 3045 0 0 1 0 0 0 JigJ3 o 1 o 0 gs 15 o o Colorado Center for Astrodynamics Research The University of Colorado Accuracy Comparison for Batch and Givens Einite Precision Computer The Givens transformations yield R 0 In fact J mm 3 0 3 5 3 2 s2 so Which will yield a valid solution for fc l 75 6 MW F is 3 0 Colorado Center for Astrodynamics Research The University of Colorado Accuracy Comparison for Batch and Givens Einite Precision Computer Which is the exact solution result forHTH Hence for this example the orthogonal transformations would yield the correct solution However the estimation error covariance matrix would be incorrect Colorado Center for Astrodynamics Research The University of Colorado 541 A Plume INFORMATIOV AND IMTMLILATION A The formulation given enl39liel39 does hut speci cally address the qrrestreu um pl39l uri irrl srrruururr Assurrre tr priori irrl orrrrrliurr E urul T are rruriluhle The u39ocedure is initialized by writing the rr prior information in the form ofa dam equation that is ill the form ofy Hx r This is accomplished by writing xrr 5415 where x is the true mine and ii is the error his We assume that Elrr 0 EimT P 5416 Compute the upper triangular square mm of 5417 Colorado Center lor Astrodynamics Research The University ol Colorado Q 541 4 PRIOR ile0Rl39I39ION AND iMTIALIZATIOi r If is not diagonali the Cholesk decomposition may be used ta accomplish this Next compute R the square root ofthe u priori infm nmtion matrix s 5418 hence 5419 5420 De ne TEEXUE l 542 Colorado Center lor Astrodynamics Research The University ol Colorado 541 A Pm INFORMATION an INITIALIZATION then I xTI 5422 whereT w 0Il Note Illzll El 15422 expresses the llpl39iol39l39 information ill the form of the lam equation A H x F Hence the equations we wish to solve for x using orthogonal lrallaforlumions nl39e E XTI yHxr where x is all 7 vector and y is all m vector Colorado Cenlerlor Amrodynamics Research The University ol Colorado 1 541 4 PRIORIINFORMATION NINTALIZTION The leasl squares solmioii for x in Eq 5413 IS found by minimizing rlie J lhau lias39 quot J mt lO if 3 not replace 6 will W1 5 in J Till2 l ll2 llanEll2 lleayllZ 5424 llelquot Colorado Cenlerlor Amrodynamics Research The University ol Colorado 41 4PIl0illNl0RMlT10N an IMTIALIZATION 5 After multiplying by an orthogonal lnlusfol maliull Q Eq 54l H may be written as E T T F T i J x7 r Tr xi 15425 Choose Q so lJlat R T l q and Winn 13 1 2 15426 H n y whch R is upper triangular Eq 542 can now he linen as i l J R x71 15427 0 P or n n 1 le 7 bH HoH 5418 Colorado Center for Amrodynamics Research o The Universiiy ol Color ad 54I A PRIORINIl39OIWA39I39ION ANDINITIALIZA39I ION W The whe be d inve as nu x could be determined at any poim in he process The minimum value of is given by substituting x into Eq 15424 7 2 J HeHZZ HinbH ZHxiylz 5430 1 1 Nola mm the rst term on the righthand side of E corresponds o lhe nbmi of le error in he a pl iari Value for x multip ted before The minimum mlne of J is found by choosing x so rlmr in b i 5429 ecmr x is obtained b the backward subsrirunbn described by Eq 518 re and are replaced by b and R respecrivsly Obsewe Ilatx usually would elermilled a er processing all obssnuribus However inrermerlime Values of 1 30 lied by lhe square root ofthe rse oflhe a prior covariance mamx Mix 7 l ll lx 7 all2 Colorado The Universiiy ol Colorado Cenler lor Amrodynamics Research 54 l I ll0R INFORMATION ND NITlALlZATION 39 whieh also can be expressed as 2 r lnrx iill x TP lrx ir 34m From Eqs 5430 and 543 I it is seen that m D new 2 Hat 7 in 1 28ll hllHZ E 5433 1 1 TTr 2 n R Rnze 1 Where ii7 r g y Hjc 5434 Colorado Center lor Astrodynamics Research The University or Colorado L5 l PRIORllNFORMA39I ION lNIl lNl39l lrllZ lTl0N Consequently a z 5 that is the elements of the en or vector 9 contain a contribution from errors in the u priorl information us well as the observation residuals The RMS of the observation residuals q is given by 5435 and from Eq 5433 l2 1 Colorado Center lor Astrodynamics Research The University or Colorado

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