Sliding Mode Estimation Scheme For Missile Homing Guidance

Copyright ~ IFAC Automatic Control in Aerospace. Palo Alto. California. USA. 1994

SLIDING MODE ESTIMATION SCHEME FOR MISSILE HOMING GUIDANCE K.R.BABU·, I.G.SARMA·, K.N.SWAMY" 'Indi4n In.ttitute 01 Sdenc:e, DepArtment 01 Computer Science And AutomAtion, BangtUore, Indi4 .. IndiAn ITLltituU 01 Science, Adllcmeed TecJmolomI Progrumme, B4flgGlore, IndiA

Abetract: A Dew approach to man«m.rinc \arpJ\ trackinc uainc the SlicliDc Mode ObeerYer theory, iD the apecific coMett of bomiD« miaaile pidanc.e. is deoveloped. Tbia 8Cheme requiNa a priori information abou\ the taqet aoceIeration bouDd.e aIooe, and theft is DO Deed for a dyDamk: mocld to npreeem the taqet JDaDeuver dynamics. The propoeed estima&or Is as almpJe to lmplemeat .. the COIM!IItioD&l ExteDded Kalman Fiher(EKF) ibclf. The llimulatioD reaulta, iD the praeDCe of rea1iaUc DOiee eourcea, demoDstra&e the luperiority of the propoeed estimation acheme CMJr the EKF.

Key Words :

1

ata~ estima.tlon; ~Ie atrudure ayatema; taqet

INTRODUCTION

tracJd.Dc. mlaa1Iee; aeroepaoe encmeeriDg

Another approach to t.aqet t.raddng ill throuch the uae of Adaptiw KaimaD FUerlDg. The adaptivlty ill bued on either the covariaDce mat.chiDg priDdple or on filter reiDitiaHut101l. EwDthouch they perform reuouably better t.hao the noaadapdve methods, the IIipiflc:aot time lac iDhereD.t with theee eIJtimatee ~ result iD leas than the d5lred perfonnance. AD excel)em auney of the target treckiD& met.hoda P8l1able iD open Bterature tUl1988, iD the broader conte:xt of miaIIile IUidaDce and control, caD be fOWld iD C10uUet et al. (1989).

wen

It is known that many of the modem guid.aDce Iawa for short raqe homing miaailes are capable of pnwidin& significant performaDCe Improvement OYer the cJaaejca1 ProportiODal Navigation (PN). However, they require additional information iD the form of tar&'et acceleration and time to 10 estimates, for their Implemem.atioD. Moreover, their performance is critically 8eIlBitJw to the accuracy of Lhi. iDformatJoD. This is pert.icularly true iD the mdpme phase, I. . iDI for about ODe aecoDd or 10 before iDterceptioD. Any miacoDception about the tarpt maoeUWtB iD Lb.ia phaae may l-t to lup miIIB diIItauce wJUeB, becaWIe of leek of adequate time to correct the ~ fectB of tbeee erron. Thus, the IICC1ll1ICY of t.he t&rpt acceJeratioD iDformation bec:omeB crucial for ally ~ formaDce ImpaOtemeDt OYer the PN. S'mce t.he t&rpt acceIerat.iou caD DOt be measured directly, it haa to be eatim&t.ed wdn, the availaNe IIeDIIOC meaeuremeDta.

Recently, St.epbaa aDd Geeriq (1991) haw propo.d the UIe cl the Iinpl&l' perturebeUOD apporadl to d. alp aD adaptiw Ueckin& filt.er. By ~ the target related informat.iOD at a rate r.t.er thaD the relatJw beariDc dynamica, they obtalD improved ~ fol'1D&llCe. The maiD diaadvaDtace of thIa approec:h, howewr, is that., wbeD the target ceeaea to maDeUwr, It .. hiJhly likely that. the iDcreMed buc:hridth of the filter may result iD Iarp eatimatiOll errora. CIoutier et al. (1003) U8I thelDMlllW'elDellt CODcat&Dation t.ec:hDlque, to efl'ecUve1y u1Wze the DIIe4IIIUremeotll .lI8llable iD between the alprithimie 8&lDpJiq periods, to device • batch amocKhiD& &Dd filt.eriDg tec:hDique to _imate the \arp\ eeceleraUOD information aceura&ely. LiD et al. (1992) U8II a difl'eren.. tial pme t.hecnUe approec:h, to ct.ip • combined pidance aDd elJtimatiOD ec:heme, iD which the w.rpt maDeUwr is treUed . . . proc:ea DoiIe &Dd the filter ill made lObaR with reapect to the wont upected Wpt maaeuver.

Many of the pre88Dt 88tlmatJoo echemea are bued either the KalmaD PUw (KF) or the Extended KaImau FUter (EKF) IJtructUC'ell, bued on IIOJDe . . wmed model for the tarpt maaeuWlI. The mOIIt widely I18ed teclmJque modell the t.aqet dynamica . . . first order GauJ.Mukov proce8I ( Mooee et al. 1979) , driven by white DOi8e. This ..umption which ill maiDly reeorted to for the II&b of IdmpUficatioIl iD the calc:u1adona &Dd raultll iD computation time _\0mp, howner, mode. poorly the target lICCIeIeratioa aDd thua n!IIUIta iD poor trecldng perf'oramance. K1m et at (HI89) WIe. drmlac target model &Dd obtain aD improved perf'ormauce, at the e:xpenae of extra eomput.atioDal eft'ort. OD

ID abort, the current approach_ to the target trec:k-

iDI caD be IIU!DID8rized .. foUowr. 405

"here C, la the ,CIa row cl the m )( n matrix C.

1. Maay of them require IIOIDt! Idnd of a model to repreaent the tarpt mueuftl dyumica. aDd it la poeBlble UW the model may Ilot accurately repreaent the actual tarpt behaviour durin& the ftJPt. The neultlq modeJJng errot8 may eauee lea of performaDce of the estimation lCheme and may eftD lead to Ioae of stability.

Vi =

iT

"', =

~

= ~ .. = C,(:z: -

~

.9'1(:Z:) aDd

tainity in

RecoDBt.ructJon of the the state of the 8yIItem from the mee811I"fJd outputa and Inputa CODBtltutes the b&Ik: obeervv problem. The desip of nonlinear ~ aerftl'I h. been epproeched throuch EKF and Uneerized Kalman Filten, etc, the robuml_ properties of"hich are not well eRabIlahed. The reeemly ~ alidin& obeenoen, bued OIl Variable Strue> ture S,.t.emII(VSS), are Inherently Iloallnear and offer robu8tD_ with respect to unmodelled dynamica. In thiII aediOll, a brief review ol.Udiq ot.erwr theory (S1otine et al.. 1986). praeuted.

={

I -1

(6)

o

(7)

If:z:>O If:z: < 0 If:z:=O

III = /(:z:, t) - i(~, t)

(9)

/(:z:, t), Ill.• a.umed to aatiIfy :

IAII < et{:z:, t) 0(., .) bein& • mown quamlty

(10)

Uain& ( 1) and ( 7) we can "rite :

£

=

1lI- Km - K.I.

iC,I.6I- K~i -

K.I.) < 0

(11)

(12)

The pin KI muat be c:boeen auch that it aatJd. the above inequality. I>uriq IIlkliD& the 1lWitchin& term in ( 1) eeta to bep. == 0 aud couequmtly•• == o .

Coa8ider the following pneraJ noaJlnear Iystem

yE R'"

i}C,(i: - ~} < 0

"here, Jrepetllelltl the mmrn pen of / . The uncet'-

SLIDING OBSERVERS

= C:z:( t}

(5)

= j(~, t) + Km + K.I.

"here

llol8e inteDsitiea.

y

.\. e,e,• = ""

For the inequaUty in (6) to hold. the 8tate traJectories are requ.lred to eHde &Iona: the lUlface and thla la called the eHding condlUon. If the inequality la .... iafled, thea the eHd1n& alOll& ~ la ~ aDd is will ID to RIO. Keepiq iD view the above requIr. meDt, the obeerver at.ructure can be deftDed:

In thJa p&per". a n_ tarpt maIleuftl e8timation deme bued OD Slidln& Mode Obeerver theory 11 pt-. 1IeIlted. LUre the differential game theoretic estJma.. Uon IICbeme ( LiD et oL lW2), the propoeed method aJao requires information only about the bound. OIl the maDeUwrablIity of the tar&et, rather than it.I exact behaviour. Mote0get. . . obaervered by Slot.ine et al. (1986). the IIlidin& obeerver ha inw.t.iD& IObwrtn_ properties in the presence cl modeling ~ rora and meuurement nol8e. It IB very almple to u. p1ement when competed with any of the above cited lChemea. The almulatlon nwlta preeent.ed iD SectloG IV demonstrate the superiority of the propeed &Cheme in the pn8eDC8 of rapid tarlet ID8II8UftI'II and bi&h

:z: E R"

(4)

The IIUCface e, will be attract! w: if:

3. The eDOnDOUl computation ~ with IIOIDt! of the adat.pdve estimatioD ..bellV!l IleVedy coanraiD their oabo.ni aa.

i: = f(:z:,t)

1 -e,e, 2

'nt.Idn& the tot.a1 time derivative of Vj

2. The Bipificant time Ia& ..-ociated with the praeut eetimatiOll 1Cbemee, may DOt BerW 8I1y uaeful purpoee, whea u.ed in the cuidance Jaw.

2

followin& Lyapunov function:

Coaaider the

(1)

•

(2)

"here r .. the output of the aynem whkh .. taken for coaftDience, to be a linear combination of the 1ItataI.

=

yC£=O c(1lI- KJ.) =

(13)

0

(14)

the mc' that j = 0 can be t.ed to "dte the apN8Iion:

I>urlnc eHd1nc,

The obeerwr problem 18 to obtain aD eItlmUoc of :z:(t), namely ~(t). audl that the error betweell the eetual output y(t) aDd the en1mated output j .. driftD to r.ero .. quickly .. poaribie. A andlng audace ~, i 1 to m eaD be de8ned .. foUon:

=

i

Ci 406

= =

[1- Ks(CK.)-lq~ 0

(16) (17)

From (16) it fDI1OW11 that duriq 1lkiiDc, the aytem dyoamial are redw:ed from the n lla otdec to the (nm)tA order Md the liDear plDa KJ haft DO iDfIueuce on the dynamica of the error on the .Udiq 111lface. They aIfec:t only the capture pbaae by Incteel8in& the recion of direct attnlet.lon.

Md

Seneor Nolee EffectB

I If the me8lW'8Dlf1Di1 are corrupted by DOiae, then "ft haw

.,.

y= Ci+tI

a:

.,

+ v.) < 0

6)

+ a!./v...coe('Y", - 8) - a. sin('Y, - 6) + /tlc ooe{-yc - 6) (24) = (-3N - 3;.6 + r(b)S - a...(ooe{-y", - 8) + a!./tIm 8in{-y", - 6) + Gc coe{-yc - 6)

For .Uding to occur, "ft require

(C,i + tIt)(C,i

:w(re + ri) + a... sin{-y... -

=

(18)

(19)

- at/tit sin{-yc -

(C, i+tI,)( Cd6.f-KJi-K.sgn(Ci+tI»)+Vi) < 0

6»/r

(25)

(20) Md for lIUIt.a1ned .HdiDg,

where

mUlt I8t.Wy

"ft

v.... tit : miIIlle Md te..r&a wlodt.lel 'Y"" 'Y' : the miBsile and target fI1cht path angles

(21) ThiI condition iI hard to aatiafy in paeral, ace the bound on v may Dot be !mown or it may not eW!D be bounded. 'I"bem'ore, the .yatem CaD DOt. remain in the pure 1HdiD& mode in the plMeDCe of arbit.raly meuuremezs1 noile. However, if we 8IIUIDe thai tI iI bounded, then the II}'Item will remain in the boundwy layer of width tlo, where iJ{t) ~ tlo. There are two approechea to IOlvin& thia problem. The ftra 18 to use & low ~ filter to att.enuate the high frequency Doi8e Md the IeCODd one la to rep)ece the di8cODtiDUOUI IIipum function by & CODtinUOUI appnndmation of it, lib the l8t.uration function.

3

ESTIMATOR

by Ooatier d 01. (1_), the polar imp'.......tadon producea a aimpler &lid more aecurUoe e8tfmU0r, UlaD ita ~~. MorecMIl', the ~ of IlkliD& obaener theory ill relatiwly . . complicated in the caM of a linear DMIM11ftmem model. ID t.hia coordIn.ate frame, the eltlmator Itate and ~

x y where X

:0::

(r

J=

be

~med

VI

mJaile and ~ acc::eleradoal COIDID8Dded aec::elerat.ion of mIIIile mIaUe autopilot darnpln, Md natural frequendea additive ~ noiIe

UIiq theatate apeee re~ deftlopeci iD the pre\liOUl IaCtiOll, a DODlInear maDluftr eRlmator 11 Den preMDted. WIth the f!8tImatIoa erIOI'II . . the 1"Iritc:hiD& 8UI'faces, appUc:atlon of the SM theocy r. .wtll In the follcnrin& tIItimatoc 1ItrUetw'e:

The po&ar coordinate frame bM been .leeted for the impelement&*ion of the e8tim&tor bec:a.e, _ me.ed

caD

relatJve raup LOS angle to the tarpt.

The abaft meMUJ'elDeDt model 8IBUDHII thM the DU. M iI equipped with M active eeeRr, which provides meMUn!!DeDtl of reap, ita rat.e Md LOS aqIes. Further, it iI IIIIUIDed thM the miIale iI equipped with M acc::eleromet.er to meeBwe the millile ~rMion. The meuwement Do-. are eaumed to be ",uaiM di8tn"buted, with r.ero meUl end ftriUlCtll, - &i?eD in the nen eeetion on IimuIation Itudiel.

TARGET MANEUVER

auremeat mode18

r 6

x

=

J(X,O.O) + KJ{i) +K.Sgn(y) (26)

"here

_:

0 0 ~ A:c. 0 0 ks. 0 ke. 0 0 0 0 0

ku 0 =

=

f(X. Gc. 0,) HX+VI

(22) (23)

0 ~

K.=

r r 6 ; i a... a...)' Md r r

.,.;

.

0 0 0 0 0

»r.

0 0 0 0 0 0

(27)

ke. A-..

repc. . . . the nritchin& pin mairix. From the above atruclun, it may be DOt.ed tbM the era. Ilpwn termI are ~ly omitted in the expn..ioa. (26) . ThIa la doae for the __ ollimptidty in the ob.enw deaip. The Dai 8t.ep ill to c:hooee the piDI which

;

,

a...

-(w"a... +w!(ae - a...) 407

The above Slidiq Mode Rate.umatoc la In the Randard Luenberpr form, except foe the eddlt.ioo:lal DOlllinear switcb.lnc terms. It .. theae tenDII which impart robustDaa to the ...imator. The main cWrereDCe be-tween the Sliding Mode artimator and the EKF is that the former explJcit.ly Labs lmo MXOUIlt the effect of dlsturbanca and compellllMe8 for them, while m the latter, t.he disturbalM:ea are accommoda&ed m

result In satisfactory performaace. A. obaen>ed previously, the linear pin terma KJ do not affect the estimation error dynamiCII, when the aywtem .. In the sliding mode. They can be dUJ8eIl sw:h that t.he 11)"1tem is quicldy forced Into the slJdin, mode. In t.he present. context, they are moeen 88 the corresponding ,alns of an EKF. The nonlineu- switching pins, kw, i = 1 to 9 ahouJd be moeen such that they satisfy the sliding condition. When the sliding is echieved, the reduced order dynamial on the slJding patch is defined by the set of equations:

R = 0 R = 0 A = - kJ. R - Jc.. R+ ll.!I ku

j

=

kou

0

the focm of ~ ncMee. This is the main te88Ol1 for their superior performance owr the conventional EKF based schemes. Wlt.h thia, the Sliding Mode Elrtimator (SME) can be written 88 :

x

= p = KJ =

(28) (29) (30)

(32)

j

(33)

= a... = a". =

=

3M -

+ (a". -

(34) (35)

=

wheR Gtn and a", denote the tar&et acceleration camponenta, aloq the LOS and normal to the LOS respectively.

2

3RlJ + 3RiJ8 - 3RM ~)sinh",

a: Iv, cosh, - 9)

IlI2

- 9) - lie sinC'Y, - 9)

The performaDee of the aboge fUt.er mthe ple8eDCe of no_ 80W'C8S typical at a reaJatic mJaIdle engapmem aituatlon hu been mW8t.lgated m ilie in t.he foI1owIng eecUon through aimulatlon.

(36)

il. . R- R'I R~ -3-9-3-8-3-9-3-9 R

R

R

R

a: sinh·, - 9) + a, cosh, - 9)

Ills =

(~)

Once the Rate est.imuea are obtained from the above equatJOI1I, the target. acceleration can be conat.rw:t.ed from them .. fo1lowa:

where

Ill!

p~R-l

where P, Q and R are the at.ate eetimatioD error, prooe. noise and meuurement. aoiIte covviance matrices rapecti\l'8ly. F is the Jacobian f1 I with reapect to the estimated st&te X.

(31)

9 =

+ KJ{;) + K.Sgn(i)

(39) F(£)P+ pF(£f - PHTJrl HP+ (40) I(X,O.O)

v,R R (a". - ~)coe('Y", - 9) -(w,,(a... - ~)

4

SIMULATION RESULTS

(37) Th lnvest1pte die performance of the propoeed ~ matJon echeme, the follcnriq mladJe..target. enppment lIItwU;lon la COIIaidered.

(38)

By examining the equatjoaa (30),(32),(33) and(35), it. can be inferred that on the elidiDg pet.ch, the . . Umation error dynamiaa behave .. a law peI8 fi.1t.er, whoee natural frquency la dependent. on the DODliDeu- switching piu. Moreo\IU, it. can be ot.erwd t.hat the amal1er the denominat.or terma on the ricbt hand Bidee of (30), (32), (33) and (35), the amaUer is the eatimat.iOD error. UIiq Iimllar equmenta, it. is in Slotine et al. (11.) that, thouch the eff~ of addIn& a ailDDDl term to the I8t.lma&or equa. tiona la to mcreue the eff~ve bandwidth of the fllter, the BM beaed eatlmator pcamu the rema.rJrable property of inaeeBi.D& the eff~ve bandwiddi without pot.eDtially iDcleaaiDl the DOiM IIeUlthrity. 'IbiII ia peRicularly true at. h.ich data rMeI aDd law noiee intenaity, 88 br'Oqht out by aimulMioo renha ~ IIeDUId in the lieU eect.ion.

Bngagement Scenario : Thia la a pl8D8l' eqepment., in whim both the tar&et 8Ild the miIIiJe are UI1IIDfld to be palm ID8I8 modeIe moriDg with COIlltant apeed. The tar&et initiat.ea an 8g evuive ID&Deuftr at the .wt of the ~ switchee to a diw: maneuvec of 8g aAer two IleCCDda 8Ild continues with it till the end of the . . . .ement. The miaUe ill uaumed to be guided by the Proport.ioaal Navipt;ion law. In ord.. to provide c:oaaiBtency in e'V8lualin& the performance of the eatllMt.of, the eetual aat.e lnforDWion ill ued m the pldaDce law, ra&h.. t.haD the estimated RateB. A umpIin& time of 0.02 eeconda ill uecl. A. mentlODed earlier, it. la ..umed that the meaBUreIIIeDY of LOS ancIe, renp Uld Iu. ra&e are av.ilahle. Tbe rap dependeace of the meMUrelDem noi8e cbanM:teratlca (1abIe 1), apeclficeUy in the LOS

mown

408

brute OIlt the IUperiorIICIu!me OYer the EKF, in term. of ita robustDeae under hiP aoiae imeaBky coadJtiou. planar ~ atuatioD,

aqle meMUftIIIe~, .. coaaidend to ~ lato IICCOWIt tbe effect of gliDt. ~.

ity of the

For the purpoae of compuWoD, the r.ulta with the EKF are alao preaeDted. nu. EKF ........tleJly the OIle ~med by equaUoaa (3G) - (41), with the t.erm8 abaeDl. HoweV'U, the proc:e8IllOi8e ~ wviaDce matrix, Q, .. choeeD cWfereaily &om that of the SM .tlmatoc. It la tuned to provide good traddDg periDrmaace eYen in tbe preaence of target maaeuW:lII. nu. matrix, as clftlll in n.ble 1, la choIIeIl to be a dlagooal matrix with entriaa, 9fi i = 1 to 8. All the IIt8te8 of the e8tImatora are IDlIeJIMd to their true valu.. TIle initial -imatlnD enoc ~ variance matrix, P, la &et to a diagoaal matrix. The entri. of thia matrix, correapoading to tbe raage aDd t.he acceleratJoD rela&ecl quantJei, are eet to 100 aDd tbe 8Ilt.riee correapoding to the LOS are _ to 10-41. As t.he idgnum tenDs upUdtly tab into 8CCOUIlt the effect of target manlUwra, the ~ ncUe COftri... anoe matrix Q Is eet to zero for the SM eet.lmator. Table 1 cl'" the Iimulatlon data.

~

8

manum

C1outier, J. R., J.H. Even and J.J. FMley (1_). AI. nomt of Air to Air MiIKile and Guidaace TedmoIOC)'. IEEE Control S~temll Magazine, 8,27-34. Ooutier, l.R., C.F. Lin &Dd C. Yang (1W3). MenlUvving 'farp& 'I'r1Idd.q VIa SII1OOthln& aDd Flltering ThrouP Me8II11teID8IIt conea&elWioD.

Jotl1Tl4l oJ Guidance, Control and Dtr namia, 16, 377-385. Hepnet, S.A.R. and H.P. GeeriD& (1991). Adaptive Two TIme Sc:ale FIlter for EatimatioD. Journal

n.r.- AcceleratiOD

oJ Guidance, Control

and DJlnamia,14, 581-588. Kim, K.D., J.L.S~r aDd M. 1bak (1989). ~ pt ManeuV'U Mode1i for 'IhIddD& EItimat.orI. Proc. oJ InternatiDn4l Conference on

1"he resulte preaellied here are tbe aw:raga obtained OYer ten fUIUI. Figure 1 giWII tbe LOS rate _Un.Uon proftls for the two estlmatioIl dems. 'The noisy nature of tbe _imatee t.owarda the ead of the enppment Is due to the I'8IIp dependence of the LOS anele noise COIUIidered. Fi&ura 2 and 3 preaeDl tbe estimates of the tarpt ~ion component. normal to tbe LOS and aIon& the LOS, nspec:tlwly. The larpr time 181 In the estlm8tell in 2 C8Il be Mttributed to the absence of the LOS rate Dle8llW'emente. Thou&h the resulte for the other atMee are not included here, they follow a almilar tnnd. 'The performance of the SM eBtima&or C8Il be obeer Peel to be quite amiIar to that of the EKF. ThiI ia mainly due to a carefu) fine tu.nin& of the latter.

Control Th.eorJJ and AppZicatio7U. LiD., C.F., Q. Wang, l.L. Speyer, J.H. Even and l.R. Cloutier ( 1992). In&egrated Eetlmation, Guidance and Control S,-m d..tp Uaq Game Theoretic Approach. Proc. DJ American Control ConJerm«,

3220-3224.

Mooee, R.L, H.F. Vanlandingbam aDd D.H. McCabe

(1979). Modelin& and EltimatIOD for 'I'teddnI Maneuverin& Tarpta. IEEE 7ranac:atioTu

Ft,.

on Aeroqace and Elecmmic SlI.tenu, AES-1&, 448-456. Slot.ine, l. J. E, J.K. Hedrick and E.A. MIaawa (1986). NonUnear State Ot-nation UtdDc SIld-

iD& Obeerven.Proc. oJ!5th. CDC,323-329. ]able I. Simulation Data

ID order to .tudy the robWltn_ propertJes of the tBtimaton, the me.urement noi8e inWMItJe. are iDc:nIII8MI three foIda. The lIlmuIatiOll data remabls the 88Ol8 • in the previOUl C888, ucept that the measuremeDl no_ covari8Ilce matrix R la IlUitably modWed. Flguree 4-6 preaent the corl'88pOlldiq IIimuJa.. Uon tsUlta. NotJce that, whJie there Is no appreciabie depadatioD iD the performance of the SM estimator, the perf'ormance of the EKF degrades sharply aDd , . wita in larp ....lmatioD erron in the LOS rate &Dd &alpt ecceIeratJon compoDeDt8 normal to the LOS. ThIs prcwidee a cleac indicat.ion u the robuRa_ u the SM estimator.

&

REFERENCES

NW. Co~DDCeI

R.an&e ~8361 + 8.014 )( 10-ZJli>ji:lt

m'

Raup Rat.e(0.8361 + 3.207 )( 10-s R') /i:lt m' / t? LOS angle (O.2090/]{Jo 2.~. 10-10)!i:lt rJ

+

Vc

300m/.

R

4500m 105" 45° 00 2Onu(umpliq lime)

l'

l ... , At

Kt. K.,. Ks. K..

CONCLUSIONS

A new maDlUftrin& tarpl veddn& dame, b.ed OD the SIidln& Mode Obaenw theory ie preeented. It .. Itrucured around the EKF with IIdditiOMl mtc:hIn& terma, which uplicitly &ab into 8CCOU.Dl the efl'td of the tarpIllD8lllln1wn. It"" aimple to lmplemem .. the EKF i_If. The Ilm"IeJ;jOD reaulta preaemed for a

Ks. K .. KT.

KA. K.. 409

7.0 41.0 5.0 110 0.005 0.1

qu ll22 qu

'"

9l5s 4IIe

8O/it. Qn 1.0 4.0

Q.

10.0 1000.0 5000.0 10-41 10'" 10'" 0 0

0 . 10

0 .10

rOO/s

rod/a

---- Actual ...... St.4E - - - EKF

,I~Iv',

- - - Actual ....... SME

J

- - - EKf

.....Cl>

0 .05

.....Cl>

0 Cl:::

0 Cl:::

(/)

(/)

0 .05

0

0

-1

-1

-0.00

-0.00

J

'-~ -0.05

O.

2.

-0.06 0

l.

Time( seconds) Fig . 1

200.00

LOS Rate Estimates

Fig . 4

200.00

m/s2

LOS Rate Estimates

2

m/s

I ~

100.00

I

,,

\

100.00

c:

c:

0 0 0

0 0 0

0 .00

.....Cl>

0.00

.....Cl>

0> ....

.3

--,

'-,I \

0>

....

0 ..... -100.00

-100.00

+-o............................................................................~.........................-t o 2. l. Time(seconds)

-200.00 +-O.................,...,..................,...,..................,...,..................,...,.................j ~ 2. ~

-200.00

Fig . 2

Time( seconds) Fig. 5

Normal Component

~.oo~--~~----------------------,

m/al

- - - Actual ........ St.4E - - - EKF

c:

- - - Actual

....... SUE

80.00

20.00

Normal Component

- - - EKf

c:

o

~ 20.00

o o

o

-80.00

+-o'""........,....,...............,....,...............,....,...............,....,................j

o

Fig . 3

LOS Component

Fig. 6

410

LOS Component