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Performance is analyzed for an airborne tightly coupled inertial navigation system (INS)/GPS/electro‐optical imaging (EO) system that simultaneously estimates platform states, sensor biases, and unknown ground object coordinates using a single Kalman filter. Analysis is done by (1) comparing the performance of the INS/GPS/EO system with that of conventional tightly coupled INS/GPS navigation systems, (2) changing INS and GPS performance to determine the dependency on individual sensor performance, and (3) investigating the benefits of tracking a known ground object (control point). Simulation results show that (1) poor INS/GPS yaw angle accuracy is significantly improved by tracking an unknown ground object with the INS/GPS/EO system; (2) GPS performance has effects on platform position, velocity, and orientation accuracy, while INS performance has effects mainly on platform orientation accuracy; and (3) tracking a control point results in better navigation accuracy than tracking an unknown ground object, suggesting the possibility of using control points as an alternative to GPS.

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Phase (deg) Fig. 5. Amplitude-phase plot of the describing function for the vernier jet phase-plane code with r = 0.1 degls and d = 0.02 deg. A is in degrees.It is possible to numerically compute the describing function by taking the actual computer code for the phase-plane and driving it with the inputs w, = Awcos$ and Be = Asin$. The output is integrated using Simpson's rule and the results plotted in the gain-phase form.This permits the linearization, as defined by the describing function, of the exact computer code. The describing function for the vernier jet phase-plane for w = 0.04 Hz is illustrated in Fig. 5 . ffd has been set to zero for this analysis. The amplitude phase plot is qualitatively similar to the plot for the simplified phase-plane but there are several differences. In this case the phase angle never reaches -90 deg. This is due to Region 6 and Region 2 which always set the jets to off. The trajectory of the input completely encircles Regions 4 and 8 and never enters them for large values of A , but must always pass through Regions 6 and 2. This ensures that n, will always be less than zero thus causing the phase shift to be slightly more negative.In the upper left-hand comer the plot of the describing function forms a loop. This is due to the shelves formed by Region 9. This asymmetry about = x12 causes n4 to be nonzero even if the rate limits are not encountered. Because it is a function of A , it varies along with the gain, causing the loop. The numerically computed describing function produces a useful frequency-domain interpretation of the phase-plane.In the on-orbit autopilot the angular rate and attitude errors are calculated by a state estimator that derives those values from the gimbal angles of the inertial measurement unit on the orbiter. The state estimator also produces a disturbance acceleration estimate, a d . This estimate is based on a model that assumes that the external disturbance acceleration is a constant. The state estimator rate estimate has a second-order filter characteristic with a corner frequency of ~0 . 0 4 Hz. Gimbal angle measurements are input into the estimator at 6.25 Hz. The autopilot cycle time is 0.08 s. Every pass it estimates rate and disturbance acceleration. Every other pass it extrapolates attitude from the previous cycle's measurements. As a result, the rate and attitude errors are not simple functions of the measured attitude. The effects of the state estimator may be incorporated into the describing function analysis by feeding the input, Asin$, into the input of the state estimator, which drives the phase plane, and measuring the output at the output of the phase-plane logic, as was done above. The describing function for the combination state estimator and phase-plane is illustrated in Fig. 6. The primary effect of the state estimator is to increase the phase shift by 90 deg. IV. CONCLUSIONSThe use of frequency-dependent describing functions for the analysis of digital controller stability is described. This method permits the determination of li...

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