Positioning and Navigation Using the Russian Satellite System

of observations to m GPS and n GLONASS satellites (not including the respective reference satellites),
we obtain a system of observation equations in matrix notation:
l = A · x + ε
(8.1.76)

100
8 OBSERVATIONS AND POSITION DETERMINATION
with
l =












∆P R
1r
GP S
U R
−
∆
1r
GP S
U R
− c ·
∆δt
1r
GP S
,T rop
U R
..
.
∆P R
mr
GP S
U R
−
∆
mr
GP S
U R
− c ·
∆δt
mr
GP S
,T rop
U R
∆P R
m+1,r
GLO
U R
−
∆
m+1,r
GLO
U R
− c ·
∆δt
m+1,r
GLO
,T rop
U R
..
.
∆P R
m+n,r
GLO
U R
−
∆
m+n,r
GLO
U R
− c ·
∆δt
m+n,r
GLO
,T rop
U R












(8.1.77)
the vector of the known values,
A =




















x
0
− x
1
1
0
−
x
0
− x
r
GP S
r
GP S
0
y
0
− y
1
1
0
−
y
0
− y
r
GP S
r
GP S
0
z
0
− z
1
1
0
−
z
0
− z
r
GP S
r
GP S
0
..
.
..
.
..
.
x
0
− x
m
m
0
−
x
0
− x
r
GP S
r
GP S
0
y
0
− y
m
m
0
−
y
0
− y
r
GP S
r
GP S
0
z
0
− z
m
m
0
−
z
0
− z
r
GP S
r
GP S
0
x
0
− x
m+1
m+1
0
−
x
0
− x
r
GLO
r
GLO
0
y
0
− y
m+1
m+1
0
−
y
0
− y
r
GLO
r
GLO
0
z
0
− z
m+1
m+1
0
−
z
0
− z
r
GLO
r
GLO
0
..
.
..
.
..
.
x
0
− x
m+n
m+n
0
−
x
0
− x
r
GLO
r
GLO
0
y
0
− y
m+n
m+n
0
−
y
0
− y
r
GLO
r
GLO
0
z
0
− z
m+n
m+n
0
−
z
0
− z
r
GLO
r
GLO
0




















(8.1.78)
the design matrix,
x =



(x
R
− x
0
)
(y
R
− y
0
)
(z
R
− z
0
)



(8.1.79)
the vector of the unknowns, and
ε =












∆ε
1r
GP S
U R
..
.
∆ε
mr
GP S
U R
∆ε
m+1,r
GLO
U R
..
.
∆ε
m+n,r
GLO
U R












(8.1.80)
the noise vector.
Compared to the system of Eqs. (8.1.65) to (8.1.69), in system Eqs. (8.1.76) to (8.1.80) one unknown
(the single difference receiver inter-system hardware delay ∆δt
U R,HW
) has cancelled. This advantage
was obtained by sacrificing one more satellite measurement and using this satellite as reference satellite.
In case when there is only one GPS or one GLONASS satellite among the set of observed satellites (the
latter seems more likely, if the GLONASS constellation dwindles further), a combined GPS/GLONASS
solution therefore cannot be calculated, since there will be no reference satellite for this system. Anyhow,
with only one satellite of one system, in the system of Eqs. (8.1.65) to (8.1.69) this one measurement
is employed to calculate the single difference receiver inter-system hardware delay. This will cause the
calculated combined GPS/GLONASS positioning solution to be identical to the one possible single system
positioning solution. Thus, both systems – Eqs. (8.1.65) to (8.1.69) and (8.1.76) to (8.1.80) – are
equivalent.
An example of positioning results using GPS and GLONASS double difference positioning with pseu-
doranges is shown in Figure 8.5. Positions were computed from data logged by two 3S Navigation
R-100/R-101 receivers, which were set up on observation pillars at known locations near the Institute

8.1 Pseudorange Measurements
101
Position Deviation [m] from Center E 11 37’ 43.783” N 48 04’ 39.911”
◦
GPS
×
GLONASS
GPS+GLONASS
East/West Deviation [m]
-25
-20
-15
-10
-5
0
5
10
15
20
25
North/South
Deviation
[m]
-25
-20
-15
-10
-5
0
5
10
15
20
25
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