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origin ORW of the runway-fixed reference frame is defined by the coordinates xt and
yt, measured relatively to the Earth-fixed reference frame, and the altitude of the
runway above sea level, Ht (the index t denotes the runway threshold).
The horizontal relation between FRW and FE has been shown in figure 5.7. The vertical
relation has been depicted in figure 5.9, using the intermediate reference frames
F0
E = O0
EX0
E Y0
E Z0
E and F00
E = O00
E X00
EY00
E Z00
E to help interpret the spatial orientation. F0
E
60 Chapter 5. Radio-navigation, sensors, actuators
has the same orientation as FE, but its origin has been moved to the projection point
of ORW on the horizontal plane at sea level. F00
E has the same orientation as F0
E , except
its origin has been shifted to runway level. Hence:
x00
e = x0
e = xe − xt (5.1)
x00
e = y0
e = ye − yt (5.2)
We can express the position of the aircraft relatively to the runway by introducing
the coordinates xf and yf , measured in the runway-fixed reference frame FRW (the
index f denotes ‘flight’), and the height of the airplane above the runway threshold,
Hf . Observing figures 5.7, 5.8, and 5.9, and substituting equations 5.1 and 5.2, we
find the following transformations from FE to FRW:
xf = (xe − xt) cos yRW + (ye − yt) sin yRW (5.3)
yf = −(xe − xt) sin yRW + (ye − yt) cos yRW (5.4)
Hf = H − Ht (5.5)
where yRW is the heading of the runway, measured relatively to the North, H is the
altitude of the airplane, and Ht is the elevation of the runway threshold. The latter
two variables are both referenced to sea level.
As can be seen from figure 5.7, Gloc can be computed from the coordinates xf and
yf as follows:
Rloc =
q
yf
2 + (xloc − xf )2
dloc = yf
)
Gloc = arcsin

dloc
Rloc

(5.6)
Gloc and dloc are positive if the aircraft flies at the right-hand side of the localizer reference
plane, heading toward the runway. The localizer current through the cockpit
instrument equals:
iloc = Sloc Gloc [μA] (5.7)
where Sloc is the sensitivity of the localizer system. According to ref.[2], Sloc has to
satisfy the following equation:
Sloc = 1.40 xloc [μA rad−1] (5.8)
where xloc is the distance from the localizer antenna to the runway threshold, shown
in figure 5.2. All distances are measured in [m], and all angles must be measured in
[rad]. The maximum value of the localizer current iloc is limited to ±150 μA, hence
±150 μA represents a full-scale deflection on the cockpit instrument [2].
The locations which provide a constant glideslope current lie on a cone, as shown in
figure 5.4. The nominal glide path has an elevation angle ggs which normally has a
value between −2 and −4. Obviously ggs is negative since the aircraft will descend
along the glide path. The magnitude of the glideslope current is proportional to the
glideslope deviation angle #gs [rad] (see figure 5.8):
igs = Sgs #gs [μA] (5.9)
Full-scale deflection on the cockpit instrument is again obtained at the limiting values
of the glideslope current igs = ±150 μA, see ref.[2]. Sgs is the sensitivity of the
glideslope system, which equals:
Sgs = 625
|ggs|
[μA rad−1] (5.10)
5.1. The Instrument Landing System 61
Runway
Position of
aircraft’s c.g.
at t = 0
Localizer
antenna
x
loc
xf (-)
d = y loc f (+)
X
RW
X"E
XE
Y" E
YRW
YE
G
loc(+)
R
loc
Y
RW
Figure 5.7: Localizer geometry and definition of X0
E , Y0
E , XRW, and YRW-axes
gs g (-)
gs(+) e
gsd (+)
y
gs(-)
X
RW
YRW
ZRW
Runway
yf (+)
Rgs
fx (-)
x (+) gs
Hf
Glideslope
antenna
Ggs (+)
Figure 5.8: Glideslope geometry and definition of XRW, YRW, and ZRW-axes
62 Chapter 5. Radio-navigation, sensors, actuators
Runway
Sea level
North
y
RW
y
RW
y
f
xf
Hf
H
Ht
y"
e x"
e
Y
RW
RW
X
Y’
E
X’
E
E
Z’  ,
 RW
Z
E
Z" ,

Y"
E

X"
E
Nominal glidepath
Figure 5.9: ILS geometry: expressing the aircraft position in the runway-fixed reference
　
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