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In this paper, two observability measures are introduced for a discrete linear system. The degrees of observability of both the system and its subspaces can be examined with these measures. The measures are well conditioned to perturbation and applicable to multi-input/multi-output time-varying systems. The relations among observability, observability measures, error covariance, and the information matrix are presented. It is shown that the measures have direct connections with the singular value decomposition of the information matrix. In contrast to the error covariance, the measures are determined by the system model and independent of the initial error covariance. An example of the observability analysis of the Global Positioning System/Inertial Navigation System is given. The measures are confirmed to be less sensitive to the system model perturbation. It is also shown that the vertical component of the gyro bias can be considered unobservable with a tactical-grade inertial measurement unit for a horizontal constant-speed motion. Index Terms--Global Positioning System (GPS), inertial navigation, observability measure, singular value decomposition (SVD).

The psi-angle model and the equivalent tilt(ET) model have been widely used for in-flight alignment (IFA) to align and to calibrate a strapdown inertial navigation system (SDINS) on a moving base. However, these models are not effective for a system with large attitude errors because the neglected error terms in the models degrade the performance of a designed filter. In this paper, with an odometer as an external aid, a velocity-aided SDINS is designed for IFA. And equivalent error models applicable to IFA with large attitude errors are derived in terms of rotation vector error and additive and multiplicative quaternion errors. It is found that error models in terms of additive quaternion error (AQE) become linear. Thus the proposed error models reduce unmodeled error terms for a linear filter. From a number of van tests, it is shown that the proposed error models effectively improve the performance of IFA.