and where (S

xi

, S

yi

, S

zi

) and (V

xi

, V

yi

, V

zi

) are the components of the computed

satellite position and velocity respectively.  The matrix H has two different types of

rows, one type for pseudorange

[ (S

xi

     U

x

) / R

i

,    (S

yi

     U

y

) / R

i

,     (S

zi

     U

z

) / R

i

,  1, 0,  0, 0,  0]

and a similar row for pseudorange rate

[0,  0, 0,  0,    (S

xi

     U

x

) / R

i

,     (S

yi

     U

y

) / R

i

,     (S

zi

     U

z

) / R

i

,  1]

9.3.4  GPS Augmented Kalman Filter

A modification of this formulation is to include three acceleration states in addition

to the position and velocity states.  Although there is no direct measurement of

acceleration in the unaided GPS receiver, these augmented states aid the filter in

sorting out non zero mean errors. Specifically, if these states are included, and the

vehicle undergoes constant acceleration, the apparent discrepancy in the velocity

data will build up as a bias in the acceleration states, and the resultant filter

accuracy will improve.  In essence, these states represent an unknown bias error in

the states related to the velocity terms by their first difference, so the filter assumes

that any such errors belong in these states.  Of course, if the acceleration is not

constant, the acceleration states will not perfectly track the error, and in fact the

filter will respond more sluggishly to the velocity changes.  But for the case of an

aircraft with constant acceleration turns, the augmented state filter will outperform

the eight state filter.

9.3.5  GPS Kalman Filter Tuning

It is important to note that the covariance matrix is actually an estimate of the

statistics of the estimation error vector.  Mismodeling of the system dynamics or of

the process or measurement noises can cause the true estimation error

uncertainties to be quite different from the covariance matrix computed by the

Kalman filter.  Modeling only a subset of the total set of errors (suboptimal Kalman

filter) will also cause an inaccurate covariance matrix.  When this occurs, the

accuracy of the navigation system may be substantially degraded.  The process

whereby the covariance matrix of the mechanized filter is made to closely

approximate the true covariance matrix is referred to as Kalman filter tuning.

One way in which GPS Kalman filters are often tuned is through the use of adaptive

tuning.  Specifically, this refers to dynamically setting the process noise Q as a function

of vehicle motion. This approach is used to account for mismodeling in the state

dynamics model.  In this case, the errors are not Gaussian noise, but may be biases in

turns as already shown.  Therefore, the correct Q depends on the vehicle profile.  For

straight and level flight, a small  Q is appropriate.  For turns or higher dynamics, Q must

be larger.  For filter stability reasons,  Q must be set to the highest level of uncertainty

expected. This means that in

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