Re: velocity calculations from a GPS
- From: Sam Wormley <swormley1@xxxxxxxxx>
- Date: Fri, 30 Nov 2007 17:51:52 GMT
DWilson wrote:
Hi all,
Does anyone know internally how the GPS unit derives the velocity
measurements?
Is it straight differencing of subsequent lat/long positions or is it via
some sort of Doppler
on the received frequency of the L1 signal?
I have heard the latter, but that would not explain why the velocity
measurments tend to get noisier as soon as you stop, but the signal to noise
ratio improves rapidly once you are moving, even if only at a walking pace.
That would indicate differencing with a possible Kalman filter type
correction.
Also the differential velocity for Doppler is, of course, extremely small
for many applications (e.g. walking/car/marine).
Ref: Misra & Enge "GPS: Signals, Measurements, and Performance" (2001)
Sec. 5.2.1 (pgs 196-197) Velocity Estimation
"The relative motion of a satellite and the user results in changes in
the observed frequency of the satellite signal. This Doppler shift is
measured routinely in the carrier tracking loop of a GPS receiver
[Section 9.6]. Given the satellite velocity, the Doppler shift can be
used to estimate the user velocity. The Doppler shift, or equivalently,
the range rate [Section 1.3.3], can be written as a projection of the
relative velocity vector on the satellite line-of-sight vector. The
measurement, however, is biased by the receiver clock bias rate (i.e.,
frequency offset), and what's actually measured is the pseudorange
rate.
"The delta pseudoranges obtained from carrier phase measurements are
proportional to the average pseudorange rates or the line-of-sight
velocity of the user relative to the satellite over the time interval.
The model for pseudorange rates can be obtained by differentiating
(5.1). It is left as an exercise to show that
[equation 5.28 is true]
where v_sup(k) [a vector quantity] is the satellite velocity vector,
known from the navigational message broadcast by the satellite; v is
the user velocity vector, to be estimated. Both v_sup(k) and v are
expressed in the ECEF coordinate frame. The user-to-satellite unit
vector 1_sup(k) is determined from an estimate of the user position;
b_dot is the bias of the receiver clock (m/s), and the
epsilon_sub_phi_sup(k) denotes the combined error doe to changes during
the measurement interval in the satellite clock, ionosphere and
troposphere. Note that the velocity of an object attached to the earth
is zero in the ECEF coordinate frame.
"The principal source of error in (5.28) throughout the 1990s was the
satellite clock frequency dithering due to SA. Now with SA gone, the
remaining errors arise from changes in the ionospheric and tropospheric
delays and in multipath, and are generally small. Problems, however,
can arise if the user dynamics are excessive. The delta ranges give
only average velocity over a time interval. High accelerations and
jerks would clearly be problematic. The PPS performance specifications
for velocity estimation (0.1 m/s rms in any direction; 0.2 m/s 2drms)
are based on a constant-velocity scenario [JPO(1991)].
"Equation (5.28) is linear in user velocity components, and can be
rewritten...
the combined set of measurements from K satellites can be written as a
set of equations compactly in matrix notation as
[equation 5.29]
where matrix G characterizes the user-satellite geometry, as defined
previously (5.10). It is interesting that the problem of estimation of
user velocity based on pseudorange rates is identical in structure to
that of estimation of user position from pseudoranges (5.9). A
least-squares solution and the DOP parameters can be defined, as
before, and related to the rms error in these estimates"
.
- References:
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- From: DWilson
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