Where am I? Sensors and Methods for Mobile Robot Positioning

124 Part I Sensors for Mobile Robot Positioning
f(t) = f + at (4.7)
0
where
a = constant
t = elapsed time.
f
2d/c
This signal is reflected from a target
and arrives at the receiver at
time t + T.
fo
t
2d
T = — (4.8)
c
where
T = round-trip propagation time
d = distance to target
c = speed of light.
Figure 4.34: The received frequency curve is shifted along the time
axis relative to the reference frequency [Everett, 1995].
The received signal is compared with a reference signal taken directly from the transmitter. The
received frequency curve will be displaced along the time axis relative to the reference frequency
curve by an amount equal to the time required for wave propagation to the target and back. (There
might also be a vertical displacement of the received waveform along the frequency axis, due to the
Doppler effect.) These two frequencies when combined in the mixer produce a beat frequency F : b
F = f(t) - f(T + t) = aT (4.9)
b
where
a = constant.
This beat frequency is measured and used to calculate the distance to the object:
d
F c b
4 F r
F d
(4.10)
where
d = range to target
c = speed of light
F = beat frequency
b
F = repetition (modulation) frequency
r
F = total FM frequency deviation.
d
Distance measurement is therefore directly proportional to the difference or beat frequency, and
as accurate as the linearity of the frequency variation over the counting interval.