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Appendix C / Axis Transforms
_ _
−1
YE XE
−1
ZE hE
( µ , λ ) : = ( tan ( P P ) , tan ( P P ) )
r,
o
r,
o
16-11
o
o
o
o
Equation 16.9-2
The transform from the Earth to Alignment frame (TRETOA) is obtained by
substituting the Euler angles into Equation 16.1-4. Introducing the
functional form from LGA to direction cosines for this transform,
T
A
E
: =
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎣
− sin λ
cos λ
A
E
DC
LGA
( µ , )
T : = ϕ
λ
− sin µ
r,
o
r,
o
r,
o
⋅ cosµ
⋅ cosµ
r,
o
r,
o
,
,
,
r,
o
− sin λ
cos λ
cosµ
16.10 LGA to Earth Position Vector Transformation
r,
o
r,
o
r,
o
r,
o
⋅ sin µ
⋅ sin µ
r,
o
r,
o
Equation 16.9-3
,
,
,
0
cos λ
sin λ
r,
o
r,
o
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎦
Equation 16.9-4
The position of a point (p) with respect to Earth axes is determined from its
geodetic orientation with respect to the Earth frame, and its altitude along
the geodetic vertical through point (d),
P
E
p
: =
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎝
ZG ( P + P )
r,
d
ZG ( P + P )
r,
d
d,
p
d,
p
⋅ cosµ
⋅ sin µ
⋅ cosλ
⋅ cosλ
( ( )
) ⎟ ⎟⎟⎟⎟⎟⎟
2
ZG
1 − e ⋅ P + P ⋅ sin λ
r,
d
p
p
d,
p
p
p
p
⎞
⎠
Equation 16.10-1
The Earth’s eccentricity (e) and the WGS84 values for the Earth’s semimajor
and semi-minor radii (RA and RB) are defined in §18.1.
16.11 Earth to LGA Transformation
A review of a number of “exact” and iterative techniques developed by
Paul, Heikkinen, Barbee, Borkowski, and Zhu for the inverse transformation
from ECEF to LGA axes is provided by Zhu [Z.4] . Although the most
accurate of the non-iterative techniques for near-Earth studies is
Heikkinen’s method, Olsen’s iterative algorithm taken from [0.7] has been
selected and defined in §22.5.4.