Antti Lehtinen Doppler Positioning with GPS - Matematiikan laitos

Fortunately, findingthe matrix G by algebraic differentiation of the function
p(ˆx) is quite straightforward. Let us consider the differentiation in detail:
G = ∂p
∂ ˆx =
⎡
⎢
∂p ⎢
= ⎢
ˆru
∂
⎢
ˆd
⎢
⎣
∂p1
∂ ˆru
∂p1
∂ ˆ ∂p2
∂ ˆru
d
∂p2
∂ ˆ .
∂pn
d
.
∂pn
∂ ˆru ∂ ˆ ⎤
⎥ ∈ R
⎥
⎦
d
n×4
(4.23)
where pi are the components of the vector function p. Usingthe Definition 3,
one obtains an expression for a single component of the vector function p. This
can be written as
pi(ˆx) =vi • ri − ˆru
ri − ˆru + ˆ d − ˙ρi
(4.24)
Differentiatingthe ith component of the delta range residuals function p(ˆx) partially
with respect to the estimated receiver position ˆru yields
∂pi
∂ ˆru
= ∂
∂ ˆru
vi • ri − ˆru
ri − ˆru + ˆ d − ˙ρi
(4.25)
In equation 4.25 vi, ri, ˆ d and ρi are constants with respect to the estimated
receiver position ˆru. Applyingthe formula (fg) ′ = f ′ g + fg ′ for differentiating
products gives us
∂pi
∂ ˆru
= vi • (ri − ˆru) ∂ 1
∂ ˆru ri − ˆru +
vT i ∂
(ri − ˆru)
ri − ˆru ∂ ˆru
1
= vi • (ri − ˆru)
ri − ˆru 2
(ri − ˆru) T
ri − ˆru −
vT i
ri − ˆru I3×3
vi
=
ri − ˆru • ri
− ˆru ri − ˆru
ri − ˆru ri − ˆru
ri − ˆru
−
ri − ˆru • ri
T − ˆru vi
ri − ˆru ri − ˆru
where I3×3 is 3 × 3 identity matrix.
(4.26)
The equation 4.26 gives the partial derivative of the delta range residuals function
with respect to the receiver position estimate. The form of the equation is
satisfactory for numerical computation. However, the form is not very intuitive.
The right hand side of the equation 4.26 can be transformed into a geometrically
more illustrative form involvingcross products. Makinguse of the vector triple
28