Antti Lehtinen Doppler Positioning with GPS - Matematiikan laitos

For notational simplicity let us denote c˙tu, the effect of user clock drift to the
delta range measurement, by d. The new variable d has unit of velocity, that is
m/s. The true delta range can now be written in the form
˙ρTi = vi • ri − ru
+ d (4.14)
ri − ru
Now approximate the true delta range ˙ρTi with the measured delta range ˙ρi. Using
the equation 4.14 we obtain the followingequation with 4 unknowns, namely
ru ∈ R3 and d ∈ R:
vi • ri − ru
ri − ru + d − ˙ρi = 0 (4.15)
Assume we have simultaneous delta range measurements from n different satellites.
Each measurement contributes one equation similar to 4.15. This yields us
the system of equations for multiple satellite Doppler positioning:
where
⎧
⎪⎨
⎪⎩
v1 • r1 − ru
r1 − ru + d − ˙ρ1 =0
v2 • r2 − ru
r2 − ru + d − ˙ρ2 =0
.
vn • rn − ru
rn − ru + d − ˙ρn =0
• vi,i=1..n are the satellite relative velocity vectors, assumed known
• ri,i=1..n are the satellite position vectors, assumed known
• ˙ρi,i=1..n are the delta range measurements, assumed known
• ru is the receiver position vector, unknown
(4.16)
• d is the receiver clock drift effect to the delta range measurement, unknown
Thus, there are four unknowns, namely xu,yu,zu and d. To simplify notations,
let us make the followingdefinition:
Definition 1. The receiver position and drift vector
x =
ru
d
⎡
⎢
= ⎢
⎣
22
xu
yu
zu
d
⎤
⎥
⎦ ∈ R4