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MAE142_Lecture5

MAE142_Lecture5 - Geospatial Position Local Frame...

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Geospatial Position Local Frame Transformations Great Circle Navigation Rhumb Course MAE 142 HL-10 and B-52 (NASA Image)

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2 MAE 142 ECEF and Local Transformations Transformations between ECEF and local frames require vector addition and a rotation matrix. X Y Z Latitude Longitude p Known: Position vector for point p in local NED coordinates (brown) Position of local frame origin in ECEF coordinates (blue) Unknown: Position vector of point p in ECEF coordinates (pink) N E
3 MAE 142 ECEF to Local Rotation Matrix ECEF to Local transformation matrix is obtained as the product of three rotations. First rotate the angle of longitude about the Z-axis of ECEF This first rotation converts from (X, Y, Z) in ECEF to an intermediate frame (X', Y', Z') Y ' =− X sin  Y cos X Y X' Y' Z, Z' Y' Y X' X X ' = X cos  Y sin Z ' = Z [ X ' Y ' Z ' ] = [ cos sin 0 sin cos 0 0 0 1 ] [ X Y Z ]

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4 MAE 142 Latitude Angle Rotation The second rotation is defined by the angle of latitude. X' Y', Y'' X'' Z'' Z' Z' Z'' X' X'' Z ' ' =− X ' sin  Z ' cos X ' ' = X ' cos  Z ' sin Y ' ' = Y ' [ X ' ' Y ' ' Z ' ' ] = [ cos 0 sin 0 1 0 sin 0 cos ] [ X ' Y ' Z ' ]
5 MAE 142

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MAE142_Lecture5 - Geospatial Position Local Frame...

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