f

u 

=

contribution to deltarange caused by the frequency bias, to

be found

Figure 9 2.  Geometry for GPS Measurement

Note that since the deltarange measurements are found by a scaled sum of VCO

commands from the carrier tracking loops, the average receiver and satellite

velocities over the deltarange dwell time should be used in the above equations

instead of instantaneous velocities.  The average velocity can be approximated by

the velocity at the midpoint of the deltarange interval.

With four pseudorange measurements, there are four simultaneous quadratic

equations with four unknowns, the three coordinates of user position and the user s

receiver clock offset.  Except in unusual geometric conditions, there exists a

solution.  In practice, there are many computations to be made before arriving at

this equation.  For example, the satellite positions are broadcast as orbital

parameters (ephemerides) and are a function of current time.  In all, 24 variables

must be computed or solved from the available information.

9.3.3  The GPS Kalman Filter Model

To cast the Kalman filter equations in GPS form for the unaided receiver, the state

vector must be defined and the system dynamics and measurement m atrices must

be formulated.  As a minimum, typically an eight state error vector is chosen:

position error (dx, dy, dz), receiver clock phase error (b

u

), velocity error (dv

x

, dv

y

,

dv

z

) and receiver clock frequency error (f

u

):

9 8