measurement.  This is an inherent danger in Kalman filters; once K becomes

sufficiently close to zero, the filter will respond very slowly or not at all to new

information.  Most Kalman filters incorporate an artificial lower limit on K to prevent

this from happening.

9.2.2.3  Initial Conditions

The Kalman filter state estimate and covariance matrix need to be established at

the initial time.  Then the propagations and updates proceed as noted.  The initial

state estimate 

x

0

 is set to the expected value of the state at t

0

.  Without any prior

knowledge, the initial error state estimate is often set to the zero vector.  The initial

covariance matrix P

0

 reflects the uncertainty associated with the way in which the

total navigation state is initialized.

9.3  KALMAN FILTERING FOR UNAIDED GPS

9.3.1  The GPS Navigation Process

GPS signals are timed at their arrival at the receiver by the code loop correlation

process.  The total slew of the bit edge that achieves maximum correlation with the

incoming code is the time offset from the local reference time.  The time of broadcast is

contained in the navigation message, which is decoded in the receiver after correlation.

The difference between the time of broadcast and the time of arrival is the transit time

from the satellite to the receiver.  This includes the receiver and satellite clock offsets

from the GPS time.

Multiplication of the calculated time of transit by the speed of transmission (light)

results in a measurement of pseudorange. Corrections may be made to this

pseudorange for the assumed, modeled, or measured tropospheric and/or ionospheric

delays.  In stand alone operation of single frequency receivers, the modeled

corrections are generally applied.  Dual frequency receivers will measure the

ionospheric delay directly and apply a smoothed value.

The measurement process is corrupted by noise which introduces errors into the

calculations.  Examples of errors are receiver clock drift, errors in the ionospheric

corrections and system dynamics not considered during the measurement process.

With four pseudorange measurements to four different satellites, the absolute position

and user clock offset could be found.  Algorithms exist to analytically find the user

position and clock offset from a set of four pseudorange measurements.  A solution is

usually implemented with an assumed rough location and iterative updates to that

location, essentially an application of Newton s root finding method as implemented in a

Kalman filter.

The velocity of the GPS receiver is computed by processing the relative velocity along

the line of sight between the satellite and the receiver.  This relative velocity is usually

obtained by measuring the Doppler offset of the incoming carrier signal.  The

measurement is called deltarange which includes the receiver clock frequency drift.

Similar to the position computation, the receiver clock error (drift in the case of

deltarange measurements) is an unknown parameter and should be resolved along

with the absolute velocity.

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