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y
y
1
-
?

Z0
X0
Y0
XB
q
q
• A rotation by q about the Y0-axis
to the intermediate position XBY0Z0
t
ZV
XV
YV
y
y
1
-
?

Z0
X0
Y0
XB
q
q
A
A
A
A
A
A
A
A
A
A
A
A
AU
























j
j
YB ZB
• A rotation by j about the XB-axis
to the final position XBYBZB
Figure A.2: Relationship between the vehicle-carried vertical reference frame FV and the
body-fixed reference frame FB
A.7. Reference frames and sign conventions 295
t
ZV
XV
YV
c
c
X0
Y0
• A rotation by c about the ZV-axis
to the intermediate position X0Y0ZV
?
-









Horizontal plane
t
ZV
XV
YV
c
c
X0
Y0
• A rotation by g about the Y0-axis
to the intermediate position XWY0Z0
:
-
?

g
g
Z0
XW
t
ZV
XV
YV
c
c
X0
Y0
• A rotation by μ about the XW-axis
to the final position XWYWZW
:
-
?

g
g
Z0
XW








/
B
B
B
B
B
B
B
B
B
B
B
B
B
B
BN
μ
μ
ZW
YW
Figure A.3: Relationship between the vehicle-carried vertical reference frame FV and the
flight-path reference frame FW
296 Appendix A. Symbols and definitions
c.g.
ao
bo
ao
V
XB
XS
XW
ZB
ZS = ZW
YB = YS
bo YW
Figure A.4: Relationship between the body-fixed reference frame FB, flight-path reference
frame FW and stability reference frame FS
d (+)
r
d (+) e
d
aleft
( )
d
aright
(+)
d
aleft
d
aright
d
a =
Figure A.5: Sign conventions for control surface deflections
A.7. Reference frames and sign conventions 297
Thus, we find the following relation between a vector yB in the body reference frame
and yV in the vehicle-carried vertical reference frame:
yB = TV!B · yV (A.5)
The orientation of the flight-path axes with respect to the vehicle-carried vertical
axes can also be expressed in terms of Euler angles, denoted by c, g, and μ. This is
shown in figure A.3. The Euler angles y, q, and f define the orientation body axes
FB in relation to the vehicle-carried vertical reference frame FV; the angles c, g, and
μ define the orientation of the flight-path axes FW in relation to FV.
Figure A.4 shows the relationships between the body, flight-path, and stability
reference frames. These three reference systems all have their origin in the aircraft’s
center of gravity. The XW-axis is aligned with the velocity vector of the aircraft. The
orientation of the flight-path axes with respect to the body-fixed reference frame is
defined by the angle of attack a and the sideslip angle b. The stability reference
system is displaced from the flight-path axes by a rotation b and from the body axes
by a rotation −a.
For the orientation of the runway-fixed reference frame relatively to the Earthfixed
reference frame, refer to section 5.1.1. This section also shows how the aircraft
coordinates can be expressed in terms of the runway-axes.
A.7.4 Sign conventions for deflections of control surfaces
Figure A.5 shows the positive directions of control surface deflections. To summarize:
• The positive elevator deflection is measured downwards; a positive value of de
results in a pitch-down moment for the aircraft.
• The deflections of the rudder and ailerons are positive if they force the aircraft
to turn to the left. If one aileron deflection is positive, the deflection of the
opposite aileron consequently will be negative. The ‘total’ aileron deflection is
defined as: da = da right − da left .
• The flap angle is positive if the flaps deflect downwards (which they always
do), similar to the elevator deflection. A positive value of df results in an increase
in lift and drag of the aircraft.
Please be aware that other literature may use different definitions, especially for
　
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