Antti Lehtinen Doppler Positioning with GPS - Matematiikan laitos

equations in the system of equations 3.5. Physically this means pseudorange
measurements from at least four different satellites at the same time. If there
are more than four equations, the methods developed for solvingoverdetermined
systems of equations may be applied.
3.3.2 Solution Procedure
There are numerous methods for solvingthe system of equations 3.5. Both closedform
and iterative solutions have been extensively studied. A good overview of
different methods is provided in [Korvenoja, pp. 31-50]. Here we will introduce
the most commonly used method, namely the iterative least squares algorithm.
For notational simplicity, let us denote the user position and time vector as
n =[xuyu zu ctu] T where xu, yu, zu are the user x, y, z coordinates, respectively,
c is the speed of light and tu is the user clock bias. Moreover, let us
denote the estimated user position and time vector as ˆn = T ˆxu ˆyu ˆzu c ˆtu .The
expected pseudorange is denoted by ˆρi and is the theoretical pseudorange calculated
assuming n = ˆn. With the pseudorange model 3.2 one has the expected
pseudorange as a function of ˆn
ˆρi =ˆρi(ˆn) =ri − ˆru + c ˆtu
(3.6)
Usingthe vector notation, we have the pseudorange measurement vector ρ =
[ρ1 ρ2 ... ρn] T and the expected pseudorange vector ˆρ =[ˆρ1 ˆρ2 ... ˆρn] T
Usingthe first order Taylor series, one can approximate the theoretical pseudorange
vector ˆρ at any point ˆn + △n:
ˆρ (ˆn + △n) =ˆρ (ˆn)+ ∂ˆρ
△n (3.7)
∂ ˆn
where n is the linearization point and △n is an arbitrary vector. Naturally, the
approximation 3.7 is most accurate when △n is small.
Obviously, the ultimate purpose of the algorithm is to find the receiver’s position.
Thus, the user needs to find a correction term △n such that the theoretical
pseudoranges match the measured ones. This can be achieved by setting the
right hand side of the equation 3.7 to equal ρ:
ˆρ (ˆn)+ ∂ˆρ
△n = ρ (3.8)
∂ ˆn
Now the equation 3.8 can be solved. Accordingto [Kaplan, p. 521], the least
squares solution for the correction term is
△n = −
∂ˆρ
∂ ˆn
T ∂ˆρ
∂ ˆn
12
−1 T ∂ˆρ
(ˆρ − ρ) (3.9)
∂ ˆn