Tightly Coupled GNSS/INS Using Fisher Information Matrix Based Observability Analyzing for Urban Scenarios
Zihuan Hao, Jian Li, Yiran Luo, School of Information and Electronics, Beijing Institute of Technology, Key Laboratory of Electronic and Information Technology in Satellite Navigation, Ministry of Education, China
Location: Pavilion Ballroom East
Date/Time: Wednesday, Apr. 22, 5:14 p.m.
The integration of the inertial navigation system (INS) and the global navigation satellite system (GNSS) is now widely used in position and attitude determination applications. The GNSS/INS integrated algorithms provide good performance in open area with adequate availability of satellites. However, challenges arise in urban scenarios. Shadowing effect and multipath signals degrade the performance of Kalman type filter used in integrated GNSS/INS system. This paper investigates performance improvement of tightly coupled GNSS/INS using observability analysis result. In this work, observability analysis results based on Fisher information is used as weighting matrix in the derived least-square mannered Kalman filter. The performance of proposed Observability Weighted Kalman Filter is evaluated by comparing with classical Extended Kalman Filter (EKF) and Adaptive Kalman Filter (AKF). Simulate results showed that the observability based tightly-coupled method is more efficient in minimizing the effect of variation of line of sight (LOS) satellites and measurement errors. Field experiments is designed and will be conducted next to further evaluate the algorithm.
Tightly coupled GNSS/INS system is an attractive option in many navigation applications. This kind of system uses the GNSS pseudo-range and pseudo-range rate measurements to evaluate and correct errors of INS which enables the system to work with less than 4 LOS satellites in GNSS restricted environment. In order to reduce the effect of varying measurement noise, the AKF have been investigated to estimate the noise covariance matrix online using innovation or residual information. But the accuracy of the estimated noise covariance matrix is coupled with the state estimation error. In biased state estimation situations, divergence can even be caused. Especially, in urban scenarios with canyons and bridges, the GNSS measurements suffer from varying noise level, multipath, and poor geometry. Considerable studies focused on different issues have been made to improve the performance of tightly coupled GNSS/INS in challenging urban scenarios, like the robust estimation technique for abnormal measurements rejecting, the redundant measurement noise covariance estimation method for multipath mitigating. and Multi-GNSS based tightly coupled system. However, inaccurate state estimation will still degrade the performance of these method. Besides, the effect of carrier to noise ratio (CNR) and constellation is neglected, which can be used to evaluate the quality of measurements.
In this work, the observability analyzing result is used to improve the EKF update equation due to the fact that the observability analyzing result reflects the accuracy of the filtered results. Firstly, the Fisher information matrix based observability analyzing method of tightly coupled GNSS/INS is introduced, which provides theoretically quantitative results to evaluated the accuracy of state estimation. Thus it is used as weighting approach for measurements of different satellite. Secondly, the equivalent least-squares formulation of classical EKF is derived, which indicates that a new weighting function is allowed to reshape the EKF update process. Then the complete algorithm of observability weighted Kalman filtering approach is presented.
For a Kalman type filter, the precision of filtered result depends on the observability of system state. For liner system, singular value decomposition (SVD) is commonly used. Generally, the nonlinear tightly coupled GNSS/INS system satisfies the piece-wise constant system (PWCS) theorem and the SVD method can also be used. While in urban areas, the number of LOS satellites varies quickly and the PWCS condition is not satisfied. To solve this problem, the Fisher information matrix based method is introduced. For tightly coupled system, the Fisher information matrix consist of two parts, the variance of pseudo-range (and pseudo-range rate) and the cross multiplication of unit vector in LOS between the receiver and the satellite. The previous part can be derived from the CNR and the latter is associated with geometry of LOS satellites. Based on the derived 17 states transform and observation equation of tightly coupled system, the normalized weighted matrix is then established from the Fisher information matrix.
After establishing transform and observation equation, the prediction and update step of EKF for classical tightly coupled GNSS/INS are derived. A new measurement equation is built by stacking together the perdition and measurement equation of EKF. Reshaping the equation and estimating the states in least square sense, a weighted solution is obtained. It is proved that the weighted solution is equivalent to the update step of classical EKF when choosing a unit matrix as weighting matrix. Finally, the general design of the observability weighted EKF for tightly coupled GNSS/INS is presented and the filter steps of the proposed algorithm is listed.
In simulation experiment, three different tightly coupled algorithms are compared, including classical Extend Kalman Filter (EKF), Innovation-based Adaptive Extend Kalman Filter (IAEKF), and the proposed Observability Weighted Extend Kalman Filter (OWEKF). The GNSS signal is generated by the simulator and a Novatel receiver is used to obtain GNSS measurements. The IMU data is simulated at 100 Hz with Gauss-Markov error respect to a tactical grade. Three changeling periods are set in the whole 1000 seconds simulation: a) number of LOS satellites variation from 6 to 2; b) CNR deceasing from 45 to 39 dB/Hz of 2 chosen satellite; c) number of LOS satellites variation from 6 to 4, in the same time 2 of the left 4 satellites has an CNR decease from 45 to 39 dB/Hz. In other epochs 6 satellites are available with 45 dB/Hz CNR. The simulation result indicates that the OAEKF is more alert to the change of constellation and CNR and correspondingly holds better performance in the designed changeling periods and reaches steady state faster when the scene returns to normal state. Filed experiment has been designed and is expected to further prove the efficiency of the proposed algorithm.
In this paper an observability weighted EKF is proposed for tightly coupled GNSS/INS. The Fisher information matrix based method is used to analyze the states’ observability. When signal power and constellation vary, the proposed algorithm shows a more stable performance. In challenging urban scenarios, the algorithm can be an alternative to classical EKF and AKF.