Detailed Examples (GPS) Part 3

One Dimensional Position Aided INS

Next, consider the same system as in the previous example, but with a position measurement

available at T =1 second intervals. The measurement noiseis white and Gaussian with variance

During each intervalthe INS integrates the INS mechanization equations as described in eqn. (4.143). At timethe INS position estimateis subtracted from the position measurementto form a position measurement residualwhich is equivalent towhere H = [1,0, 0]. With this H and the F from the previous example, the observability matrix is

which has rank 3; therefore, the error state should be observable from the position measurement.

This residual error will be used as the measurement for an error state estimator

where

Figure 4.7: Error standard deviation of 1-d inertial frame INS with position measurement aiding.

Either of the methods for determining Qd results in

Figure 4.7 displays plots of the error standard deviation for each state as a function of time. In this example,which produces eigenvalues for the discrete-time system nearand 0.85. Due to the integration of the noise processesthe variance grows between the time instants at which position measurements are available. The position measurement occurs at one second intervals. Typically, theposition measurement decreases the error variance of each state estimate, but this is not always the case. Notice for example that atthe standard deviation of all three states increase. Therefore,the gain vector L selected for this example is clearly not optimal. At least at thistime, the gain vector being zero would have been better. Eventually, byan equilibrium condition has been reached where the growth in the error variance between measurements is equal to the decrease in error variance due to the measurement update.

Figure 4.8: Feed-forward complementary filter implementation diagram.

Although each measurement has an error variance ofthe filtering of the estimates results in a

steady state position error variance less thatIn comparison with the results in the previous section, the position measurement results in bounded variance for the velocity and accelerometer bias estimation errors.

Notice that there is a distinct difference in the state estimate error variance immediately prior and posterior to the measurement correction. Performance analysis should clearly indicate which value is being presented.

The simulation is for an ideal situation where the initial error variance is identically zero. In a typical situation, the initial error variance may be large, with the measurement updates decreasing the error variance until equilibrium is attained.