Unscented Transformation Unscented Kalman Filter
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1 Usceed rsformo Usceed Klm Fler Usceed rcle Fler Flerg roblem Geerl roblem Seme where s he se d s he observo Flerg s he problem of sequell esmg he ses (prmeers or hdde vrbles) of ssem s se of observos become vlble o-le
2 Flerg roblem Soluo of sequel esmo problem gve b oseror des p X X Mrgl of he oseror p Dmc Se Spce Model Geerl dscree me o-ler o-guss dmc ssem Assumpo: Ses re mrov.e.
3 Flerg Algorhms Flerg Algorhms Klm Fler (KF) eded Klm Fler (KF) KF L l f G KF: Ler evoluo fucos; Guss ose KF: No-ler evoluo fucos; No-guss ose Moe Crlo Mehods rcle fler: o-ler evoluo fucos; No-guss ose Soluo b eded Klm Fler Soluo b eded Klm Fler d d d me upde d mesureme upde frmewor (corol pu u s egleced) U d f me Upde: G G Q F F Mesureme Upde: h K H H U RU H K f F f h H h K H K s Klm G Q s he vrce process ose R s he v v v v f G h U K s Klm G Q s he vrce process ose R s he vrce of he mesureme ose
4 Crude Appromo b KF KF uses frs order erms of he lor seres epso of he oler fucos. Lrge errors re roduced whe he models re hghl o-ler Locl ler ssumpo bres dow whe he hgher order erms become sgfc. Beer Appromo? Oule Usceed rsformo Scled Usceed rsformo Usceed Klm Fler Usceed rcle Fler
5 Usceed rsformo he usceed rsformo (U) s mehod for clculg he sscs of rdom vrble whch udergoes oler rsformo d bulds o he prcple h s eser o pprome probbl dsrbuo h rbrr oler fuco. he roblem ropgg dmesol rdom vrble hrough oler fuco g : R R o geere g Usceed rsformo (co d) Formulo Assume hs me d covrce A se of weghed smples or sgm pos S re chose s follows: where s sclg prmeer d s he h row or colum of he mr squre roo of. s he wegh ssoced wh he h po such h
6 ropgo of Sgm os ch sgm po s propged hrough h he oler fuco g d he esmed me d covrce of re compued s follows hese esmes of he me d covrce re ccure o he secod order of he lor seres epso of g for oler fuco ropgo of Sgm os (co d) Comprso he KF ol clcules he poseror me d covrce ccurel o he frs order wh ll hgher order momes ruced; however U clcules he me d covrce o he secod order. Deled proof: Ref: Smo Juler d Jeffre K. Uhlm A Geerl Mehod for Appromg Noler rsformos of robbl Dsrbuos
7 mple A cloud of 5 smples drw from Guss pror s propged hrough rbrr hghl oler fuco d he rue poseror smple me d covrce re clculed whch c be regrded s groud ruh of he wo pproches KF d U. he sclg fcor As he dmeso of he se-spcespce creses he rdus of he sphere h bouds ll he sgm pos creses. ve hough he me d covrce of he pror dsrbuo b re sll cpured correcl does so he cos of smplg o-locl effecs. hese pos re smmercll dsrbued bou he me. Hgher order effecs such s sew become more sgfc s he dmeso creses. he sgm pos c be scled owrds d w from he me of he pror dsrbuo b proper choce of
8 Scled Usceed rsformo (SU) Scled Usceed rsformo (SU) hs mproveme overcomes dmesol sclg effecs b clculg he rsformo of scled se of sgm pos of he form where s posve sclg prmeer whch c be mde rbrr smll o mmze hgher order be mde rbrr smll o mmze hgher order effecs. SU Formulo SU Formulo he scled usceed rsformo c be wre s: he scled usceed rsformo c be wre s: S S : f s prmeer whch mmzes he effecs from hgh order erms Ref: Smo J. Juler he Scled Usceed rsformo order erms
9 SU Formulo SU Formulo h l d l b b d l he sgm po seleco d sclg c be combed o sgle sep b seg d selecg he sgm pos re se b: m c m c Usceed Klm Fler (UKF) Usceed Klm Fler (UKF) h d l fl h f d l f h he usceed lm fler s srghforwrd pplco of he scled usceed rsformo where he se vrble s ugmeed vecor d he covrce s lso ugmeed mr v R Q
10 UKF Algorhm UKF Algorhm he lgorhm. Ilzo R Q R. Updg () Compug Sgm os UKF Algorhm (co d) UKF Algorhm (co d) (b) U d (b) me Upde: v f m f c h m
11 UKF Algorhm (co d) UKF Algorhm (co d) (c) Mesureme upde c ~ ~ c K K ~ ~ K K ~ ~ where v v compose sclg prmeer = + v + Q s process ose cov. R s mesureme ose cov. K s lm g s weghs Usceed rcle Fler Oule Usceed rcle Fler--Oule he seme of he problem Sequel mporce smplg Sequel mporce smplg Geerc prcle fler (GF) Choces of proposl dsrbuo GF Usceed rcle Fler (UF) Usceed rcle Fler (UF)
12 he problem For he pcl Bes Iferece problem: gve he observos up o me de ( : ) we w o cosruc he poseror of he ssem se p : : Usg Moe Crlo mehods he poseror could be ppromed b cloud of weghed dscree N suppors s p : : w : : N s s he umber of he suppors. Imporce Smplg Uforuel s ofe o possble o drecl smple from he poseror des fuco; herefore proposl q : : s roduced. For obecve fuco of : g( : ) we hve g : g : p : : d : p : : g : q : : d : q : : p : : p : g : q : : d : p : q : : w : g : q : : d : p : where w : s he uormlzed mporce weghs. w : p : : p : q : : *
13 Sequel Imporce Smplg I order o compue sequel esme of he poseror dsrbuo b me whou modfg he prevousl smuled ses :- proposl dsrbuos of he followg form c be used q q q : : : : : : Uder he ssumpos h he ses correspod o Mrov process d h he observos re codoll depede gve he ses p : p p p : : p p p (*) w w q q : : : q : : p p w w q : # Geerc rcle Fler (GF) Algorhm For =:N s Drw prcles from he proposl q Assg he prcle wegh ccordg o (#) d For Normlzg he weghs <Re-smplg> w
14 Choces of roposl q Opml mporce des p p q op p w w p w p p # d Se rso pror p q w w p # p No curre observo corpored Ref: og Ru d uqg Che Beer roposl Dsrbuos: Obec rcg Usg Usceed rcle Fler CVR Beer roposl UKF he usceed Klm fler s ble o ccurel propge he me d covrce of he Guss ppromo o he se dsrbuo. Bg overlp bewee dsrbuo b UKF d he rue poseror dsrbuo. UKF+GF=UF
15 Usceed rcle Fler (UF) Usceed rcle Fler (UF) Al h Algorhm. Ilzo For =:N s drw he ses (prcles) from he pror d se p s (p ) p p R Q Usceed rcle Fler (co d) Usceed rcle Fler (co d) F. For = () Imporce smplg sep For = N: ---upde he prcles wh he UKF: * Clcule sgm pos: * ropge prcle o fuure (me upde) v f m c h m
16 Usceed rcle Fler (co d) Usceed rcle Fler (co d) *I b ( d ) *Icorpore ew observo (mesureme upde) c ~ ~ c K K ~ ~ --Smple K K ~ ~ ˆ N q ˆ ˆ Smple --Se Updg weghs ccordg o (#) d ormlzg hem N q ~ : : AND ˆ ˆ ˆ ˆ : : : : Updg weghs ccordg o (#) d ormlzg hem
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