The approach described in Section 4.9.4 and depicted in Figure 4.8 is an example of a feed-forward complementary filter implementation. For this example, the kinematics are given by eqn. (4.141),
represents the acceleration measurement in eqn. (4.142), the output prediction is ;
and the error estimator is defined in eqns. (4.145) anddesigned by the choice of L.
superscript is indicated on the
variable because the error state estimator can compute either variable at any time needed to form the rightmost summation in the figure. The error estimator can also propagate eqns. (4.125) and (4.127) to maintain an estimate of the system accuracy.
The feed-forward complementary filter approach has the navigation system integrating the variable u through the system kinematics to produce x_ and the error state estimator integrating
betweenaiding measurements to produce the corrected state estimate denoted as
in Figure 4.8.
As an alternative approach, over the sampling interval
starting from the initial condition
the navigation system could integrate
through the system kinematics to produce
At time 
the error state estimate
is added to the prior state estimate to produce the initial condition for the next interval of integration
Figure 4.9: Feedback complementary filter implementation diagram.
Because the navigation system state now accounts for the estimated error, the expected value of the error in
is zero and it is therefore proper to reset
This makes the time update portion of eqn. (4.145) trivial, so that it need not be implemented. This feedback approach is theoretically equivalent to the feed-forward approach. It is depicted in Figure 4.9 where the feedback of JX should be read as correcting the initial condition for each period of integration.
