EE 570: ocato ad Navgato: Thory & Practc Navgato Mathmatcs Thursay 7 F 2013 NMT EE 570: ocato ad Navgato: Thory & Practc Sld 1 of 15
Navgato Mathmatcs : Coordat Fram Trasformatos Dtrm th dtald kmatc rlatoshps tw th 4 major frams of trst Th Earth-Ctrd Irtal (ECI) Coordat Fram (fram) Th Earth-Ctrd Earth-Fxd (ECEF) Coordat Fram (-fram) Th ocal Navgato (Nav) Coordat Fram (-fram) Th Body Coordat Fram (-fram) Thursay 7 F 2013 NMT EE 570: ocato ad Navgato: Thory & Practc Sld 2 of 15
Navgato Mathmatcs : Coordat Fram Trasformatos - ECI/ECEF Rlatoshp Btw th ECI ad ECEF Frams ECI & ECEF hav co-locatd orgs o r = r = r = 0 Th x, y, ad z axs of th ECI & ECEF frams ar cocdt at tm t 0 Th ECEF fram rotats aout th commo z-axs at a fxd rat (ω ) o Igorg mor spd varatos (prcsso & utato) ω = 72.921151467 µrad/s (WGS84) whch s 15/hr Thursay 7 F 2013 NMT EE 570: ocato ad Navgato: Thory & Practc Sld 3 of 15
Navgato Mathmatcs : Coordat Fram Trasformatos - ECI/ECEF Th agular vlocty ad acclrato ar: 0 0 Th agl of rotato s tt 0 t GMST GMST: Grwch Ma Sdral tm Th ortato of fram {} wrt fram {} coms Cos( ) S( ) C R 0, S( ) Cos( ) 0 z 0 0 0 0 0 1 NOTE: ω = ω Thursay 7 F 2013 NMT EE 570: ocato ad Navgato: Thory & Practc Sld 4 of 15
Navgato Mathmatcs : Coordat Fram Trasformatos - ECEF/Nav Dscrptos wrt th Navgato fram Ortato of th -fram wrt th -fram C R R z y,, 90 cos s 0 s 0 cos s cos 0 0 1 0 0 0 1 cos 0 s Godtc at = ad Godtc o = s s c s c c cos s cos cos s s s s c c cos s cos s cos c 0 0 s s Compact Notato: cos θ = c θ & s θ = s θ Thursay 7 F 2013 NMT EE 570: ocato ad Navgato: Thory & Practc Sld 5 of 15
Navgato Mathmatcs : Coordat Fram Trasformatos - ECEF/Nav Agular vlocty of th -fram wrt th -fram rsolvd th -fram as a Skw-Symmtrc matrx T C C s s c c c c s c s c s s c c c s c s s s s c c s s c c s s 0 0 c c s 0 cos 0 s cos s 0 Th agular vlocty vctor s cos cos T C s C C C 0 0 3 2 3 1 2 1 0 T Thursay 7 F 2013 NMT EE 570: ocato ad Navgato: Thory & Practc Sld 6 of 15
Navgato Mathmatcs : Coordat Fram Trasformatos - ECI/ECEF/Nav Hc th ortato of th -fram wrt th -fram coms c s 0 s c s c c C CC s c 0 s s c c s 0 0 1 c 0 s cos 0 s s cos s cos cos s s cos cos s Thursay 7 F 2013 NMT EE 570: ocato ad Navgato: Thory & Practc Sld 7 of 15
Navgato Mathmatcs : Coordat Fram Trasformatos - ECI/ECEF/Nav Th agular vlocty of th -fram wrt th -fram rsolvd th -fram s: C??? 0 Cos( ) S( ) 0 s 0 S( ) Cos( ) 0 cos 0 0 1 s cos Thursay 7 F 2013 NMT EE 570: ocato ad Navgato: Thory & Practc Sld 8 of 15
Navgato Mathmatcs : Coordat Fram Trasformatos - ECEF/Nav Th vctor from org of th -fram to th -fram org rsolvd th -fram (from th last lctur) RE h Cos( ) Cos( ) r RE h Cos( ) S( ) 2 RE (1 ) h S( ) r Orgs of th {} ad {} frams ar th sam Th vlocty of th -fram wrt th -fram rsolvd th -fram (s hadout#1 for proof) v d r dt r r r h h Cos( ) S( ) S( ) Cos( ) Cos( ) RN h S( ) S( ) Cos( ) Cos( ) S( ) Cos( ) RE h Cos( ) 0 S( ) h Thursay 7 F 2013 NMT EE 570: ocato ad Navgato: Thory & Practc Sld 9 of 15
Navgato Mathmatcs : Coordat Fram Trasformatos - ECEF/Nav Rcallg th form of C suggsts that v ad hc, RN h Cos( ) C R h h E C v? v RN h Cos( )R h E h Thursay 7 F 2013 NMT EE 570: ocato ad Navgato: Thory & Practc Sld 10 of 15
Navgato Mathmatcs : Coordat Fram Trasformatos - ECEF/Nav Rstatg v as ad rcallg that Suggsts that v N, N Cos( ) RE h v, E h v D, R h v v ta cos s, E RE h v, R h v R h N N, E E Thursay 7 F 2013 NMT EE 570: ocato ad Navgato: Thory & Practc Sld 11 of 15
Navgato Mathmatcs : Coordat Fram Trasformatos Body Fram Dscrptos wrt th Body fram Ortato of th -fram wrt th -fram trms of rlatv yaw(), ptch(), th roll() agls C R R R z y x,,, c s 0 c 0 s 1 0 0 s c 0 0 1 0 0 c s 0 0 1 s 0 c 0 s c y c c c s s c s c c s s s c s c c s s s c s s c s s c s c c x z www.grc.asa.gov/www/k-12/arpla/rotatos Thursay 7 F 2013 NMT EE 570: ocato ad Navgato: Thory & Practc Sld 12 of 15
Navgato Mathmatcs : Coordat Fram Trasformatos Body Fram Th agular vlocty of th -fram wrt th -fram rsolvd/coordatzd th -fram C C C Thursay 7 F 2013 NMT EE 570: ocato ad Navgato: Thory & Practc Sld 13 of 15
Navgato Mathmatcs : Coordat Fram Trasformatos Body Fram Posto vctors to th org of th ody fram Th orgs of th ody ad Nav frams ar co-cdt r 0 Th orgs of th ECI ad ECEF frams ar co-cdt r r r r Vlocty of th -fram wrt th -fram rsolvd th - fram o Cas #2: A movg pot a rotatg fram v d r dt C r C v C r v d Cr dt Thursay 7 F 2013 NMT EE 570: ocato ad Navgato: Thory & Practc Sld 14 of 15
Navgato Mathmatcs : Coordat Fram Trasformatos Body Fram Acclrato of th -fram wrt th -fram rsolvd th -fram o Cas #2: A movg pot a rotatg fram a d v dt C r v C C r 2 v a d C r v dt r r v Thursay 7 F 2013 NMT EE 570: ocato ad Navgato: Thory & Practc Sld 15 of 15