where⑀measis the vector containing the pseudorange measurement errors and xis the vector representing errors in the user position and receiver clock offset.
The error contribution⑀xcan be minimized by making measurements to more than four satellites, which will result in an overdetermined solution set of equations similar to (2.33). Each of these redundant measurements will generally contain inde- pendent error contributions. Redundant measurements can be processed by least squares estimation techniques that obtain improved estimates of the unknowns.
Various versions of this technique exist and are usually employed in today’s receiv- ers, which generally employ more than four user-to-satellite measurements to com- pute user PVT. Appendix A provides an introduction to least squares techniques.
radial component of the relative velocity vector along the line of sight to the satel- lite. Vectorvris given as the velocity difference
vr = −v u& (2.37)
wherevis the velocity of the satellite, andu& is the velocity of the user, both refer- enced to a common ECEF frame. The Doppler offset due to the relative motion is obtained from these relations as
( )
∆f f f f
R T T c
= − = − v−u& ⋅a
At the GPS L1 frequency, the maximum Doppler frequency for a stationary user on the Earth is approximately 4 kHz, corresponding to a maximum line-of-sight velocity of approximately 800 m/s.
There are several approaches for obtaining user velocity from the received Doppler frequency. One technique is described herein. This technique assumes that the user positionuhas been determined and its displacement (∆xu,∆yu,∆zu) from the linearization point is within the user’s requirements. In addition to computing the three-dimensional user velocity u& =(x&u,y&u,z&u), this particular technique deter- mines the receiver clock driftt&u.
For thejth satellite, substituting (2.37) into (2.36) yields
( )
[ ]
f f
Rj = Tj1−c1 j − ⋅ j
v u& a (2.38)
The satellite transmitted frequency fTj is the actual transmitted satellite fre- quency.
As stated in Section 2.4.1, satellite frequency generation and timing is based on a highly accurate free running atomic standard, which is typically offset from sys-
f+ ∆f
f− ∆f
t
Figure 2.18 Received Doppler frequency by user at rest on Earth’s surface.
tem time. Corrections are generated by the ground-control/monitoring network periodically to correct for this offset. These corrections are available in the naviga- tion message and are applied within the receiver to obtain the actual satellite transmitted frequency. Hence,
fTj = f0 +∆fTJ (2.39)
wheref0is the nominal transmitted satellite frequency (i.e., L1), and∆fTjis the cor- rection determined from the navigation message update.
The measured estimate of the received signal frequency is denotedfjfor the sig- nal from thejth satellite. These measured values are in error and differ from thefRj values by a frequency bias offset. This offset can be related to the drift ratet&u of the user clock relative to GPS system time. The valuet&uhas the units seconds/second and essentially gives the rate at which the user’s clock is running fast or slow relative to GPS system time. The clock drift error,fj, andfRj,are related by the formula
( )
fRj =fj 1+t&u (2.40)
wheret&uis considered positive if the user clock is running fast. Substitution of (2.40) into (2.38), after algebraic manipulation, yields
( )
c f f f
cf t f
j Tj
Tj
j j j
j u Tj
− +v ⋅a = ⋅u a& − &
Expanding the dot products in terms of the vector components yields
( )
c f f
f v a v a v a x a y a z a
j Tj
Tj
xj xj yj yj zj zj u xj u yj u
− + + + = & + & +& zj j u
Tj
cf t
− f&
(2.41)
wherevj=(vxj,vyj,vzj),aj=(axj,ayj,azj), andu& =(x&u,y&u,z&u). All of the variables on the left side of (2.41) are either calculated or derived from measured values. The compo- nents ofajare obtained during the solution for the user location (which is assumed to precede the velocity computation). The components ofvjare determined from the ephemeris data and the satellite orbital model. ThefTjcan be estimated using (2.39) and the frequency corrections derived from the navigation updates. (This correction, however, is usually negligible, andfTjcan normally be replaced byf0.) Thefjcan be expressed in terms of receiver measurements of delta range (see Chapter 5 for a more detailed description of receiver processing). To simplify (2.41), we introduce the new variabledj, defined by
( )
d c f f
f v a v a v a
j
j Tj
Tj
xj xj yj yj zj zj
=
− + + + (2.42)
The termfj/fTjon the right side in (2.41) is numerically very close to 1, typically within several parts per million. Little error results by setting this ratio to 1. With these simplifications, (2.41) can be rewritten as
dj = x a&u xj +y a&u yj +z a&u zj −ct&u
We now have four unknowns:u& = x&u,y&u,z t&u,&u which can be solved by using measurements from four satellites. As before, we calculate the unknown quantities by solving the set of linear equations using matrix algebra. The matrix/vector scheme is
d= H
= d
d d d
a a a
a a a
a a
x y z
x y z
x y
1 2 3 4
1 1 1
2 2 2
3 3
1 1 a
a a a
x y z
ct
z
x y z
u u u
u 3
4 4 4
1 1
=
−
g
&
&
&
&
Note thatHis identical to the matrix used in Section 2.4.2 in the formulation for the user position determination. In matrix notation,
d= Hg
and the solution for the velocity and time drift are obtained as g= H d−1
The phase measurements that lead to the frequency estimates used in the veloc- ity formulation are corrupted by errors such as measurement noise and multipath.
Furthermore, the computation of user velocity is dependent on user position accu- racy and correct knowledge of satellite ephemeris and satellite velocity. The rela- tionship between the errors contributed by these parameters in the computation of user velocity is similar to (2.35). If measurements are made to more than four satel- lites, least squares estimation techniques can be employed to obtain improved esti- mates of the unknowns.