Positioning and Navigation Using the Russian Satellite System

− c · δt
S,Iono
R
=
x
0
− x
S
S
0
· (x
R
− x
0
) +
y
0
− y
S
S
0
· (y
R
− y
0
) +
z
0
− z
S
S
0
· (z
R
− z
0
) +
(8.1.8)
c · [(δt
R
+ L
R,GLO
) − (δt
0
+ L
R,GLO,0
)] + ε
S
R
where known and modelled values have been shifted to the left-hand side of the equation.
Having measurements to a number of satellites 1, 2, . . . , n, one can summarize the resulting set of
observation equations in matrix notation:
l = A · x + ε
(8.1.9)
with
l =






P R
1
R
−
1
0
− c · (δt
R,0
+ L
R,GLO,0
) + c · δt
1
− c · δt
1,T rop
R
− c · δt
1,Iono
R
P R
2
R
−
2
0
− c · (δt
R,0
+ L
R,GLO,0
) + c · δt
2
− c · δt
2,T rop
R
− c · δt
2,Iono
R
..
.
P R
n
R
−
n
0
− c · (δt
R,0
+ L
R,GLO,0
) + c · δt
n
− c · δt
n,T rop
R
− c · δt
n,Iono
R






(8.1.10)
the vector of the known values,
A =











x
0
− x
1
1
0
y
0
− y
1
1
0
z
0
− z
1
1
0
1
x
0
− x
2
2
0
y
0
− y
2
2
0
z
0
− z
2
2
0
1
..
.
..
.
..
.
..
.
x
0
− x
n
n
0
y
0
− y
n
n
0
z
0
− z
n
n
0
1











(8.1.11)
the design matrix,
x =





(x
R
− x
0
)
(y
R
− y
0
)
(z
R
− z
0
)
c · [(δt
R
+ L
R,GLO
) − (δt
R,0
+ L
R,GLO,0
)]





(8.1.12)
the vector of the unknowns, and
ε =






ε
1
R
ε
2
R
..
.
ε
n
R






(8.1.13)
the noise vector.
This system of equations can then be solved using the conventional methods, e.g. a least squares
adjustment or Kalman filtering.
Since the satellite positions as computed from GLONASS ephemeris (or almanac) data are expressed
in the PZ-90 frame, the resulting receiver position is also given in this coordinate frame. To get the receiver
position in a different coordinate frame, the resulting coordinates must be transformed as desired.
In a combined GPS/GLONASS receiver, signals from GPS satellites are also delayed in the HF part
of the receiver. But since GPS employs identical frequencies for all satellites, these delays are equal for
all satellites and thus form part of the clock term. However, this GPS delay is different from the common
GLONASS delay in Eq. (8.1.4), due to the different frequencies. This leads to different realizations of
GPS and GLONASS system times in the receiver. In addition, some combined receivers even use different
clocks for GPS and GLONASS reception. Thus, in a combined GPS/GLONASS scenario different receiver
clock errors with respect to GPS and GLONASS system times must be accounted for.

88
8 OBSERVATIONS AND POSITION DETERMINATION
Thus, rewriting Eq. (8.1.4) for a GPS satellite i and a GLONASS satellite j yields:
P R
i
R
=
i
R
+ c · (t
R,GP S
− t
GP S
+ L
R,GP S
) − c · δt
i
+ c · δt
i,T rop
R
+ c · δt
i,Iono
R
+ ε
i
R
(8.1.14)
P R
j
R
=
j
R
+ c · (t
R,GLO
− t
GLO
+ L
R,GLO
) − c · δt
j
+ c · δt
j,T rop
R
+ c · δt
j,Iono
R
+ ε
j
R
(8.1.15)
where the receiver clock errors δt
R
with respect to GPS and GLONASS system times have been inserted
as t
R,GP S
− t
GP S
and t
R,GLO
− t
GLO
, respectively.
Introducing the respective GPS times into Eq. (8.1.15) yields:
P R
j
R
=
j
R
+ c · (t
R,GP S
−t
GP S
+L
R,GP S
) + c · (t
GP S
−t
GLO
) + c · (L
R,GLO
−L
R,GP S
) +
c · (t
R,GLO
−t
R,GP S
) − c · δt
j
+ c · δt
j,T rop
R
+ c · δt
j,Iono
R
+ ε
j
R
(8.1.16)
Denoting t
R,GP S
− t
GP S
as the receiver clock error δt
R
(with respect to GPS system time), t
GP S
− t
GLO
as the difference in system times δt
Sys
and (L
R,GLO
− L
R,GP S
) + (t
R,GLO
− t
R,GP S
) as the receiver
inter-system hardware delay δt
R,HW
, Eqs. (8.1.14) and (8.1.16) transform to
P R
i
R
=
i
R
+ c · (δt
R
+ L
R,GP S
) − c · δt
i
+ c · δt
i,T rop
R
+ c · δt
i,Iono
R
+ ε
i
R
(8.1.17)
P R
j
R
=
j
R
+ c · (δt
R
+ L
R,GP S
) + c · δt
Sys
+ c · δt
R,HW
− c · δt
j
+ c · δt
j,T rop
R
+
(8.1.18)
c · δt
j,Iono
R
+ ε
j
R
Regarding the pair of Eqs. (8.1.17) and (8.1.18), we notice six unknowns in the combined GPS/GLO-
NASS single point solution: the three coordinates of the receiver position (implicitly contained in
S
R
),
the receiver clock offset (including GPS hardware delay) δt
R
+ L
R,GP S
, the time difference between GPS
and GLONASS system times δt
Sys
and the receiver inter-system hardware delay δt
R,HW
. However, in
a single point solution the latter unknown cannot be separated from the difference in system times,
effectively leaving five unknowns to solve for, as already discussed in Section 4. These different receiver
hardware delays therefore contribute to the difference in system times as determined by the observer.
The estimation of the difference in system times thus yields only an approximation that in addition will
be dependent on the receiver.
Thus, for a set of m GPS and n GLONASS satellites, after linearization of the geometric range the
observation equations in matrix form read:
l = A · x + ε
(8.1.19)
with
l =


























P R
1
R
−
1
0
− c · (δt
R,0
+ L
R,GP S,0
) + c · δt
1
− c · δt
1,T rop
R
− c · δt
1,Iono
R
..
.
P R
m
R
−
m
0
− c · (δt
R,0
+ L
R,GP S,0
) + c · δt
m
− c · δt
m,T rop
R
− c · δt
m,Iono
R
P R
m+1
R
−
m+1
0
− c · (δt
R,0
+ L
R,GP S,0
) − c · (δt
Sys,0
+ δt
R,HW,0
) + c · δt
m+1
−
c · δt
m+1,T rop
R
− c · δt
m+1,Iono
R
..
.
P R
m+n
R
−
m+n
0
− c · (δt
R,0
+ L
R,GP S,0
) − c · (δt
Sys,0
+ δt
R,HW,0
) + c · δt
m+n
−
c · δt
m+n,T rop
R
− c · δt
m+n,Iono
R


























(8.1.20)

8.1 Pseudorange Measurements
89
the vector of the known values,
A =


















x
0
− x
1
1
0
y
0
− y
1
1
0
z
0
− z
1
1
0
1 0
..
.
..
.
..
.
..
.
..
.
x
0
− x
m
m
0
y
0
− y
m
m
0
z
0
− z
m
m
0
1 0
x
0
− x
m+1
m+1
0
y
0
− y
m+1
m+1
0
z
0
− z
m+1
m+1
0
1 1
..
.
..
.
..
.
..
.
..
.
x
0
− x
m+n
m+n
0
y
0
− y
m+n
m+n
0
z
0
− z
m+n
m+n
0
1 1


















(8.1.21)
the design matrix,
x =








(x
R
− x
0
)
(y
R
− y
0
)
(z
R
− z
0
)
c · [(δt
R
+ L
R,GP S
) − (δt
R,0
+ L
R,GP S,0
)]
c · [(δt
Sys
+ δt
R,HW
) − (δt
Sys,0
+ δt
R,HW,0
)]








(8.1.22)
the vector of the unknowns, and
ε =












ε
1
R
..
.
ε
m
R
ε
m+1
R
..
.
ε
m+n
R












(8.1.23)
the noise vector.
Please note that t
R
and δt
R
now denote the receiver clock reading and offset with respect to GPS
system time, as described above.
Satellite coordinates x
1
, . . . , x
m
, x
m+1
, . . . , x
m+n
must be given in the same coordinate frame to
obtain a valid receiver position (cf. Section 5). Coordinates of the receiver position are then expressed
in the frame used for the satellite positions.
An example of positioning results using GPS and GLONASS absolute positioning with pseudoranges
is shown in Figure 8.1. Positions were computed from data logged by a 3S Navigation R-100/R-101
receiver, which was set up at a known location at the Institute of Geodesy and Navigation. Pseudorange
and carrier phase measurements were logged every second for approximately one hour. The plot shows the
deviation from the known location of the antenna in the horizontal plane. GPS positions were computed
from carrier smoothed L
1
C/A-code pseudorange measurements. GLONASS positions were computed
from carrier smoothed dual-frequency P-code measurements. Wherever possible, the ionospheric free
linear combination was formed. These observables used are not really identical for GPS and GLONASS,
but with P-code and dual-frequency measurements readily available on GLONASS, the best possible
results for each system are determined. GLONASS satellite positions were converted from PZ-90 to
WGS84 using the transformation according to (Roßbach et al., 1996).
The large deviations from the true position due to GPS S/A can be clearly seen. Standard deviations
of the computed positions are 25.4 m in North/South direction and 10.0 m in East/West direction.
Due to the lack of S/A on GLONASS, positions computed only from GLONASS range measurements
scatter much less. Here the standard deviations are 4.6 m in North/South direction and 7.5 m in
East/West direction. For the combined GPS/GLONASS positioning, all satellite measurements were

90
8 OBSERVATIONS AND POSITION DETERMINATION
Position Deviation [m] from Center E 11 37’ 43.783” N 48 04’ 39.911”
◦
GPS
×
GLONASS
GPS+GLONASS
East/West Deviation [m]
-100
-80
-60
-40
-20
0
20
40
60
80
100
North/South
Deviation
[m]
-100
-80
-60
-40
-20
0
20
40
60
80
100
◦
◦
◦◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦ ◦
◦
◦
◦
◦◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦◦◦◦◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦◦
◦
◦◦
◦
◦◦
◦
◦
◦◦
◦◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦◦
◦
◦◦
◦
◦◦
◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦◦
◦
◦
◦◦
◦
◦
◦◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦ ◦◦◦◦◦◦◦◦◦
◦ ◦◦◦◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦◦
◦
◦
◦◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦◦
◦◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦◦◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦