Within the field of Guidance, Navigation, and Control, the navigation process refers to accurately determining one's position in space. The quest of accurate navigation has shaped human history. Early navigation techniques involved dead reckoning with infrequent measurement updates from line-of-sights to stars or landmarks on the shore. The first practical inertial navigation system (INS) attributed to the German V-2 missile in 1942. In the early 1960's the Kalman filter was developed to aid in the merging of mathematical models and measurement updating. Throughout the space age and continuing into today's remote systems the hardware has made vast improvements; however, the navigation filtering theory has remained mostly stagnant with the multiplicative Extended Kalman Filter (MEKF) still being the workhorse of most modern INS applications.
In most INS applications, the state vector usually consists of the attitude, position, velocity, and Inertial Measurement Unit (IMU) calibration parameters such as biases and scale factors. Because position-type measurements are usually only given, such as pseudoranges to GPS satellites, the observability of the attitude and gyroscope calibration parameters is weak. Since the early days of employing the MEKF for INS applications, and even modern-day applications, the state errors are defined as a simple algebraic difference between the truth and the estimate.
It has been argued that a new state-error definition is required in which state-error quantities are defined using elements expressed in a common frame. This provides a realistic framework to describe the actual errors. In previous work, the errors were put into a common frame using the estimated attitude error, which led to the ``geometric EKF'' (GEKF). The GEKF provides extra transport terms, due to error-attitude coupling with the states, in the filter. This previous work focused strictly on attitude estimation, which incorporated only ``body-frame" errors. In this work the GEKF is extended to the INS formulation. Here, errors are considered for both the body frame and the filter's navigation frame.
In this work, it is shown how these body-frame and navigation-frame errors are related through a similarity transformation. The body-frame error, and the navigation-frame error through this similarity transformation, are first examined in a Planar Inertial Navigation (PIN) problem. For the PIN problem, the MEKF and GEKF algorithms are derived using the same kinematic and measurement models. These algorithms are then studied for a single simulation test case; these results were then verified via Monte Carlo analysis. For this example, it was shown that the body-frame errors had a faster convergence for the GEKF; however, the navigation-frame states showed slower convergence. It is argued here that the direct comparison of these results is inconclusive since both filters employ different error definitions; therefore, the errors being examined utilize a different error metric. It can be stated that the errors realized by the GEKF are more representative of the real errors experienced by the system.
The body-frame and navigation-frame errors are also used to derive the GEKF for three navigation filters. Specifically, this work examines the absolute Earth-Centered-Earth-Fixed (ECEF) navigation, relative ECEF navigation, and North-East-Down (NED) navigation filters. The counterpart MEKF filters are also derived in this work to illustrate the differences seen in the state matrices due to the additional coupling terms. These filters are also studied via simulation studies. However, now the body-framed vectors do not show faster convergence for the GEKF. This is because the measurement update for this specific example is unaffected by the new error definition. The measurements are not affected by the new error definition because these filters only utilize pseudo-GPS position measurements, and it is shown that the position error still utilizes the old error definition due to its kinematic relation to the velocity error.
Finally, this work conducts a linearize analysis of a simplified INS where the GEKF in the NED frame is considered. It is shown via a stationary analysis that the fundamental frequencies of the GEKF are the same as those of the MEKF. This is because during the stationary analysis all of the additional coupling terms seen in the state error matrix are neglected due to assumptions made about the vehicle's motion.
|Advisor:||Crassidis, John L.|
|Commitee:||Mook, Joseph, Sudit, Moises|
|School:||State University of New York at Buffalo|
|Department:||Mechanical and Aerospace Engineering|
|School Location:||United States -- New York|
|Source:||DAI-B 80/07(E), Dissertation Abstracts International|
|Subjects:||Engineering, Aerospace engineering|
|Keywords:||Common frame error definition, GEKF, Geometric extended Kalman filter, Inertial navigation system|
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