Antti Lehtinen Doppler Positioning with GPS - Matematiikan laitos

More bad news is that the user velocity is extremely hard to measure. A speedometer
will not do, since it produces no information about the direction of the
velocity. And even if the user velocity vector could be determined, say with a
combination of gyroscopes and three-dimensional compasses, transforming the
vector into the ECEF coordinate system would complicate the computation.
5.1.4 Biased Time Estimate
In Chapter 4 we assumed that the ephemeris and an accurate time estimate were
available. In practise, however, the receiver only has an estimate for the current
time. In conditions where pseudorange measurements are unavailable the GPS
satellites cannot provide time information. Thus, the receiver time estimate is
probably biased. The consequence of the biased time estimate is that the satellite
positions and velocities are calculated wrong.
The biased time estimate effect can again be thought of as a measurement error.
The positioning algorithms utilise the delta range difference function p(ˆx).
Accordingto Definition 3, we have
p(ˆx) =ˆ˙ρ − ˙ρ (5.5)
Evidently, the biased time estimate causes error to the expected delta range vector
ˆ˙ρ. Accordingto the equation 5.5, however, the effect is the same as if there were
additional error in the delta range measurement vector ˙ρ. The resultingerror can
be accurately approximated by examiningthe differences of the expected delta
range vectors at time t and at the biased time t + △t. When△t is small we can
use the first order Taylor series:
ˆ˙ρ(t + △t) − ˆ˙ρ(t) = ∂ ˆ˙ρ △t (5.6)
∂t
The symbolic differentiation of the i th component gives
∂ ˆ˙ ρi
∂t = ri − ru
ri − ru3 • vi × (ri − ru × vi)+ri − ru 2
ai
(5.7)
where ai is the relative acceleration vector of the i th satellite. Theoretical bounds
∂
ˆ˙ρi
for are hard to derive from the equation 5.7. This is because the two vectors
∂t
in the square brackets are pointingto nearly opposite directions. Numerical
∂
ˆ˙ρi
simulations show that the maximum value of
∂t is about 15 cm/s2 . Applying
the equation 5.6, wee see that for example a time bias error of 0.1 seconds may in
the worst case cause a positioningerror similar to 1.5 cm/s measurement error.
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