Basic GPS Concept - Fundamentals of Global Positioning System Receivers: A Software Approach - page 13

The solution of Equation (2.8) is

− 1

δx u
δy u
δz u
δb u

α 11 α 12 α 13 1 δρ 1

δρ 2
δρ 3
δρ 4

=

α 21 α 22 α 23 1

( 2 . 10 )

α 31 α 32 α 33 1

α 41 α 42 α 43 1

where [ ] − 1 represents the inverse of the α matrix. This equation obviously does
not provide the needed solutions directly; however, the desired solutions can be
obtained from it. In order to find the desired position solution, this equation must
be used repetitively in an iterative way. A quantity is often used to determine
whether the desired result is reached and this quantity can be defined as

δx u +

δv = δy u + δz u + δb u ( 2 . 11 )

When this value is less than a certain predetermined threshold, the iteration will
stop. Sometimes, the clock bias b u is not included in Equation (2.11).
The detailed steps to solve the user position will be presented in the next
section. In general, a GPS receiver can receive signals from more than four
satellites. The solution will include such cases as when signals from more than
four satellites are obtained.

2.7 POSITION SOLUTION WITH MORE THAN FOUR SATELLITES ( 3 )

When more than four satellites are available, a more popular approach to solve
the user position is to use all the satellites. The position solution can be obtained
in a similar way. If there are n satellites available where n> 4, Equation (2.6)
can be written as
ρ i = (x i − x u ) 2 + (y i − y u ) 2 + (z i − z u ) 2 + b u ( 2 . 12 )

= 1 , 2 , 3 ,...n . The only difference between this equation and Equation
(2.6) is that n> 4.
Linearize this equation, and the result is

where i

δρ 1
δρ 2
δρ 3
δρ 4
.
δρ n

α 11 α 12 α 13 1

α 21 α 22 α 23 1

δx u
δy u
δz u
δb u

α 31 α 32 α 33 1

= ( 2 . 13 )

α 41 α 42 α 43 1

.
α n 1 α n 2 α n 3 1