The estimated error state vector and its covariance matrix are propagated from one

measurement time to the next.  The Kalman filter uses the state transition matrix

and process noise covariance matrix to perform the propagations via

x ( )   =   

  x (+)

k

k 1

k 1

T

T

P

k

( )   =   

k 1

  P

k 1

(+) 

k 1

  +   G

k 1

  Q  

k 1

G

k 1

Typically the impact of the propagation on the covariance matrix is to increase the

variances of the non bias estimated error states, although occasionally some of the

variances may decrease, for example when due to the Schuler effect.  The Schuler

effect is a sinusoidal oscillation of inertial navigation errors with an 84 minute

period.

9.2.2.2  Update

The Kalman filter incorporates measurements when they are available.  Since the

state carried in the Kalman filter is an error state, the measurement  z

k

 is a function

of the error state vector, and is usually referred to as the apriori measurement

residual.  The estimated error state vector is updated as

x (+)   =    x ( )  +  

 (z      

  x ( ))

k

k

K

k

k

H

k

k

The quantity z

k

   H

k 

x

k

( ) is the aposteriori residual, or equivalently the k

th

 element

of the innovations sequence, the sequence of new information from the

measurements.  The matrix K

k

 is the Kalman gain and is given by

T

 1

T

K

K

   =    P

k

( )  H

k

  (H

k

  P

k

( )  H

k

  +  

k

R )

The updated covariance matrix can be derived directly from the equation for the

updated state, yielding the symmetric Joseph form

T

T

P

k

(+)   =   (I      K

k

  H

k

)  P

k

( )  (I      K

k

  H

k

)   +   K

k

k

R   K

k

9 4