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although the origin OV moves relatively to the Earth-fixed reference frame.
Measurement reference frame FM: This is a left-handed orthogonal reference frame
that is used for correcting stability and control derivatives of the aircraft if position
of the center of gravity differs from the one used during the flight tests. For
the Beaver aircraft, the origin OM lies in a point, resulting from the perpendicular
projection of the foremost point of the wing chord, parallel to the OBXBZBplane
[34]. The XMOMZM-plane coincides with the OBXBZB-plane. The positive
XM-axis points to the tail of the aircraft, the positive YM-axis points to the left
wing (‘port’), and the positive ZM-axis points upwards.
Special body-fixed reference frame for the Beaver, FR: This is a special reference
frame for the Beaver aircraft. It is identical to FB with one exception: its origin
OR is placed in a body-fixed reference point, which has been selected to coincide
with a c.g. position that was actually used during one flight. It has the following
coordinates in FM: x = 0.5996m, y = 0m, z = −0.8815m, see ref.[34]. FR is used
only to define the moments and products of inertia for the aircraft condition on
which the aerodynamic model was based; see table B.5 in appendix B.
Runway-fixed reference frame, FRW: This reference frame is used to define the geometry
of the final approach to a runway. Its origin lies on the runway threshold,
and it is aligned in the direction of the runway centerline. The XRW-axis coincides
with the runway centerline, pointing in the direction of landings and take-offs.
ZRW is directed downwards and YRW is directed points to the right, as seen from
an aircraft on final approach. See also figures 5.9 and 5.7 from chapter 5.
A.7.2 Summary of the application of the reference systems
The equations for translational and angular velocities, which were derived in chapter
2, have been referenced to the body axes FB. The wind reference frame FW is often
used to express aerodynamic forces and moments (the stability reference frame FS is
closely related).
The position of the airplane has been defined with respect to the Earth-fixed reference
frame FE, while the definition of the aircraft attitude in terms of the Euler angles
y, q, and j required the introduction of the vehicle-carried vertical reference frame
FV. The runway-fixed reference frame is used for the definition of the final approach
geometry, i.e. the position of the airplane in relation to the landing runway.
Finally, two special body-fixed reference frames FM and FR are used to define
the reference flight condition for the Beaver model in table B.5 of appendix B; these
reference frames are not relevant for other purposes.
A.7. Reference frames and sign conventions 293
X (north)
Y (east)
Y (east)
X (north)
Z (down)
Z (down)
E
V
V
E
V
E
g
c.g.
r
Figure A.1: Relationship between the vehicle-carried vertical reference frame FV and the
Earth-fixed reference frame FE
A.7.3 Relationships between the reference frames
In figure A.1 the relationship between the Earth-fixed and vehicle-carried vertical
reference systems is shown. As can be seen from this figure, FE and FV differ only in
the position of their respective origins.
The relationship between the vehicle-carried vertical and the body-axes has been
shown in figure A.2. The Euler angles y, q, and j define the orientation of FB with respect
to FV, hence they define the attitude of the aircraft relatively to the surface of the
Earth. Each of these Euler rotations can be expressed by a separate transformation
matrix:
TY =
2
4
cos y sin y 0
−sin y cos y 0
0 0 1
3
5 (A.1)
TQ =
2
4
cos q 0 −sin q
0 1 0
sin q 0 cos q
3
5 (A.2)
TF =
2
4
1 0 0
0 cos j sin j
0 −sin j cos j
3
5 (A.3)
The combined transformation from FV to FB then becomes:
TV!B = TF · TQ · TY =
=
2
4
cos y cos q sin y cos q −sin q
cos y sin q sin j − sin y cos j sin y sin q sin j + cos y cos j cos q sin j
cos y sin q cos j + sin y sin j sin y sin q cos j − cos y sin j cos q cos j
3
5
(A.4)
294 Appendix A. Symbols and definitions
t -









?
Horizontal plane
ZV
XV
YV
y
y
X0
Y0
• A rotation by y about the ZV-axis
to the intermediate position X0Y0ZV
t
ZV
XV
YV
　
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