Basic GPS Concept - Fundamentals of Global Positioning System Receivers: A Software Approach - page 14

where

x u
ρ i − b u

x i − y i − y u

ρ i − b u

z u
ρ i − b u

z i −

α i 1 = α i 2 = a i 3 = ( 2 . 9 )

Equation (2.13) can be written in a simplified form as

δρ = αδx ( 2 . 14 )

where δρ and δx are vectors, α is a matrix. They can be written as

= δρ 1 δρ n T

δρ δρ 2 ···

= δx u δb u T

δx δy u δz u

α 11 α 12 α 13 1

α 21 α 22 α 23 1

α 31 α 32 α 33 1

α = (2.15)

α 41 α 42 α 43 1

.
α n 1

α n 2 α n 3 1

where [ ] T represents the transpose of a matrix. Since α is not a square matrix,
it cannot be inverted directly. Equation (2.13) is still a linear equation. If there
are more equations than unknowns in a set of linear equations, the least-squares
approach can be used to find the solutions. The pseudoinverse of the α can be
used to obtain the solution. The solution is ( 3 )

= [ α T α ] − 1 α T δρ δx ( 2 . 16 )

From this equation, the values of δx u , δy u , δz u ,and δb u can be found. In general,
the least-squares approach produces a better solution than the position obtained
from only four satellites, because more data are used.
The following steps summarize the above approach:

A. Choose a nominal position and user clock bias x u 0 , y u 0 , z u 0 , b u 0 to represent
the initial condition. For example, the position can be the center of the earth
and the clock bias zero. In other words, all initial values are set to zero.
B. Use Equation (2.5) or (2.6) to calculate the pseudorange ρ i .These ρ i values
will be different from the measured values. The difference between the
measured values and the calculated values is δρ i .
C. Use the calculated ρ i in Equation (2.9) to calculate α i 1 , α i 2 , α i 3 .
D. Use Equation (2.16) to find δx u , δy u , δz u , δb u .
E. From the absolute values of δx u , δy u , δz u , δb u and from Equation (2.11)
calculate δv .