NMT EE570 Location and Navigation

Transcription

1 NMT EE57 Location and Navigation Som Dfinitions Handout# Th ECI to ECEF Fram Orintation and Angular Vlocity: Th angular vlocity of th ECEF fram with rspct to th ECI fram (Ωi Μrad/sc): In[4]: Ωi, ; Ωi Th angular displacmnt of th ECEF fram with rspct to th ECI fram is th Grnwich Man Sidral Tim (GMS) angl (ΘGMST ) Ωi t. Θi ΘGMST Ωi t; In[5]: In[6]: Ci D; PrintA"Ci ", MatrixFormACi E E Cos@Θi D - Sin@Θi D Ci Sin@Θi D Cos@Θi D In[7]: In[8]: Ωi, TransposACi E.Ωi, ; PrintA"Ωi, ", MatrixForm@Ωi, D E Ωi, Ωi Position, Vlocity, and Acclration: Th position of th ECEF fram with rspct to th ECI fram: In[9]: ri, ; Th Navigation Fram Orintation and Angular Vlocity: Th orintation of th Navigation fram with rspct to th ECEF fram as a DCM is Lb is th goic latitud of th body is th goic longitud of th body In[4]: @- 9 -

2 coord_frams.nb In[4]: PrintA"Cn D -Lb D 9 - " ", MatrixForm@Cn D. PrittyMatE Cos@ D - Sin@ D Sin@ D Cos@ D Cn D -Lb D - Sin@Lb D - Cos@Lb D Cos@Lb D - Sin@Lb D - c slb - s - clb c - slb s c - clb s clb - slb Th angular vlocity of th Nav fram with rspct to th ECEF fram coordinatizd in th ECEF fram: Cn W,nCn Cn Wn,n In[4]: In[4]: Cn Simplify@D@Cn, tdd; PrintA"Cn ", MatrixForm@Cn D. PrittyMat E - clb c Lb slb s - c c slb Lb clb s Cn - clb s Lb - c slb - s slb s Lb - clb c - slb Lb - clb Lb Comput th angular vlocity of th n-fram with rspct to th -fram coordinatizd in th -fram: W,n In[44]: In[45]: W,n SimplifyACn.Transpos@Cn DE; PrintA"W,n DT ", MatrixFormAW,n E E W,n In[46]: In[47]: Cn DT - Cos@ D Lb Sin@ D Lb DD - Cos@ D Lb - Sin@ D Lb DD ; DD PrintA"Ω,n ", MatrixForm@Ω,n D E Sin@ D Lb Ω,n - Cos@ D Lb Comput th angular vlocity of th n-fram with rspct to th -fram coordinatizd in th n-fram: Cn In[48]: In[49]: Wn,n SimplifyATranspos@Cn D.Cn E; PrintA"Wn,n ", MatrixFormAWn,n EE Wn,n - Sin@Lb D Lb In[5]: Ω,n Sin@Lb D Cos@Lb D DD DD ; DD - Lb - Cos@Lb D

3 coord_frams.nb In[5]: PrintA"Ω,n ", D E n Ω,n A quick chck: In[5]: Cos@Lb D - Lb - Sin@Lb D PrintA"Ω,n Cn Ω,n ", MatrixFormASimplifyACn. Ω,n E EE Sin@ D Lb n Ω,n Cn Ω,n - Cos@ D Lb Th orintation of th Navigation fram with rspct to th ECI fram as a DCM is: Cni Ci Cn In[5]: In[54]: Cin SimplifyACi. Cn E; PrintA"Cin Ci Cn D ", MatrixFormACi E. PrittyMat, MatrixForm@Cn D. PrittyMat, " ", MatrixFormACin E E Cin Ci Cn D cθi - sθi sθi cθi - c slb - s - clb c - slb s c - clb s clb - slb - Cos@Θi D Sin@Lb D - Sin@Θi D - Cos@Lb D Cos@Θi D - Sin@Lb D Sin@Θi D Cos@Θi D - Cos@Lb D Sin@Θi D Cos@Lb D - Sin@Lb D Comput th angular vlocity of th n-fram with rspct to th i-fram coordinatizd in th i-fram: Ωi,n Ωi, Ci.Ω,n In[55]: In[56]: Ωi,n SimplifyA Ωi, Ci.Ω,n E; PrintA"Ωi,n ", MatrixForm@Ωi,n D E Sin@Θi D Lb i Ωi,n - Cos@Θi D Lb Ωi Position, Vlocity, and Acclration: Dfin a fw Variabls R is th arth s smi-major axis ~ mtrs is th goic hight of th body RE is th transvrs radius of curvatur In[57]: R RE ; - Sin@Lb D RN is th mridian radius of curvatur In[58]: RN R H - L I - Sin@Lb D M ; Th position of th Nav fram origin with rspct to th ECEF fram origin rsolvd in th -fram (from nots): In[59]: r,n HRE L Cos@Lb D Cos@ D HRE L Cos@Lb D Sin@ D ; HH - L RE L Sin@Lb D

4 4 coord_frams.nb Comput th Vlocity : First, comput v,n r,n d r,n r,n dlb r,n d r,n d In[6]: PrintB" r,n ", Trm MatrixFormASimplifyADAr,n, Lb EEE. 9Cos@ Lb D - Sin@Lb D, " Trm "F Trm r,n - Cos@ D Sin@Lb D J Sin@Lb D Sin@ D J - I- I- Sin@Lb D MM 4 I- Sin@Lb D M I- I- Sin@Lb D MM 4 Cos@Lb D - I- Sin@Lb D M J- J- Sin@Lb D NN I- Sin@Lb D M 4 R -4 R N 4 R -4 R N I- M R Trm Trm Trm Looking at th top (Trm ) and midl (Trm ) trms I - I - Sin@Lb D MM In[6]: Out[6] SimplifyB 4 I - Sin@Lb D M \ Trm - Cos@ D Sin@Lb D \ Trm - Sin@Lb D Sin@ D Looking at th bottom (Trm ) trm In[6]: Out[6] SimplifyB - Cos@ D Sin@Lb D H RN L - Sin@Lb D Sin@ D H RN L F H - L R F I - Sin@Lb D M R H - L I - Sin@Lb D M HRN L I - Sin@Lb D M I- I- Sin@Lb D MM \ Trm - Cos@Lb D - - r,n I - R H - L Sin@Lb D M Cos@Lb D H RN L -Cos@ D Sin@Lb D -Sin@Lb D Sin@ D Cos@Lb D

5 coord_frams.nb Nxt, comput In[6]: PrintB" r,n r,n r,n r,n ", MatrixFormASimplifyADAr,n, EEE. : - Cos@Lb D Sin@ D HRE L Cos@Lb D Cos@ D HRE L In[64]: PrintB" r,n r,n r,n r,n ", MatrixFormASimplifyADAr,n, EEE F Cos@Lb D Cos@ D Cos@Lb D Sin@ D Sin@Lb D Cos@Lb D Cos@ D Cos@Lb D Sin@ D Sin@Lb D Composing th final rsult : v,n d r,n v,n r,n dlb r,n d Λ b r,n d B r,n r,n r,n F Lb -Cos@ D Sin@Lb D - Sin@ D -Cos@Lb D Cos@ D -Sin@Lb D Sin@ D Cos@ D -Cos@Lb D Sin@ D Cos@Lb D -Sin@Lb D HRN L Lb Cos@Lb D HRE L - v,n Cn HRN L Lb Cos@Lb D HRE L - Which suggsts that v,n - Sin@Lb D -Sin@ D Cos@ D HRE L Cos@Lb D Finally, comput R HRN L Lb Cos@Lb D HRE L - Cn v,n "RE ">F 5

6 6 coord_frams.nb Th Body Fram Orintation and Angular Vlocity: Th orintation of th b-fram wrt th n-fram: Dscribd in trms of rlativ yaw(ψ), pitch(θ), thn roll(φ) angls In[65]: In[66]: @ΦD; PrintA"Cnb PrittyMat, PrittyMat, " ", D. PrittyMatE Cnb cφ - sφ sφ cφ cψ - sψ sψ cψ cθ sθ - sθ cθ cθ cψ cψ sθ sφ - cφ sψ cφ cψ sθ sφ sψ cθ sψ cφ cψ sθ sφ sψ - cψ sφ cφ sθ sψ - sθ cθ sφ cθ cφ Position, Vlocity, and Acclration: hnc Th origin of th b-fram wrt is th sam as that of th n-fram r,b r v,b v,n,n