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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).

Lever-arm uncertainty can be an important error source in the integration of the Global Positioning System (GPS) and inertial navigation system (INS). This paper presents both numerical and experimental studies on the estimation of the lever arm in the integration of a very-low-grade inertial measurement unit (IMU) with an accurate single-antenna GPS measurement system. Covariance simulation results showed that maneuvers play an important role on the estimation of the lever arm and attitude. The length of the lever arm has a rather insignificant effect on the estimation of these. Experimental tests conducted with a low-cost microelectromechanical system (MEMS) IMU and a carrier-phase differential GPS (CDGPS) measurement system showed that the lever arm can be estimated with centimeter-level accuracy. The test results confirmed that angular motions and horizontal accelerations improve the estimates of the lever arm and yaw angle, respectively. Index Terms--Global positioning system (GPS), inertial measurement unit (IMU), inertial navigation, inertial navigation system (INS), lever arm, observability.

Observability properties of errors in an integrated navigation system are studied with a control-theoretic approach in this paper. A navigation system with a low-grade inertial measurement unit and an accurate single-antenna Global Positioning System (GPS) measurement system is considered for observability analysis. Uncertainties in attitude, gyro bias, and GPS antenna lever arm were shown to determine unobservable errors in the position, velocity, and accelerometer bias. It was proved that all the errors can be made observable by maneuvering. Acceleration changes improve the estimates of attitude and gyro bias. Changes in angular velocity enhance the lever arm estimate. However, both the motions of translation and constant angular velocity have no influence on the estimation of the lever arm. A covariance simulation with an extended Kalman filter was performed to confirm the observability analysis. Index Terms--Global Positioning System, GPS/INS, inertial measurement unit (IMU), inertial navigation, lever arm, observability.