Global Positioning System Reference
In-Depth Information
8.4.4 Attitude Determination
An additional application of the interferometric techniques described earlier in this
section is
attitude determination
. If antennas are placed on a rigid body, such as an
aircraft, then the baseline vectors between each pair of antennas are known quanti-
ties within the body-frame coordinate frame defined in Figure 8.19. The
x
-axis
extends through the nose of the vehicle, the
z
-axis points downward, and the
y
-axis
is mutually perpendicular to the
x
- and
z
-axes to form a right-handed coordinate
system (e.g., through the right wing as viewed by the pilot for an aircraft). Typically,
the nominal center of mass of the platform is chosen as the origin.
If carrier-phase measurements are taken from each of the antennas, the integer
ambiguities may be resolved, as discussed earlier, to determine the baseline vectors
within the local north, east, down (NED) coordinate frame. This may be mecha-
nized by first solving for these quantities in an ECEF coordinate frame (e.g.,
WGS-84) and then applying the appropriate transformation (see Chapter 2). At any
given time, the relationship between the coordinates of three antennas expressed in
the body frame (known from the installation) and expressed in the NED frame
(computed from GPS carrier-phase measurements) may be written as [25]:
RR
ned
=
(8.45)
body
where
R
ned
is the matrix of antenna coordinates in the NED frame,
R
body
is the matrix
of antenna coordinates in the body frame, and
T
is the 3
×
3 transformation matrix:
cos
ψθ
cos
−
sin
ψφ
cos
+
cos
ψθφ
sin
sin
sin
ψφ
sin
+
cos
ψθ
sin
c
os
φ
T
=
sin
ψθ
cos
cos
ψφ ψθφ
cos
+
sin
sin
sin
−
cos
ψφ ψθ
sin
+
sin
sin
cos
φ
(8.46)
−
sin
θ
cos
θ
sin
φ
cos
θ
cos
φ
) that represent
heading
,
pitch
, and
roll
(more formally heading,
elevation
, and
bank angle
[37]), respec-
The desired end quantities are the
Euler angles
(
ψ
,
θ
,
φ
x
z
y
Figure 8.19
Body-frame coordinate system.
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