A Noise Estimation Algorithm based on Modified System Model and its Application on Backtracking
Xuan Xiao, Xiang Guo, Meiling Wang, Tong Liu, Songtian Shang, Beijing Institute of Technology, China
As post-processing method could enhance navigation accuracy, backtracking algorithm is widely used in SINS/GPS integrated system. Through fusing the information from two directions repeatedly by Kalman filter (KF), namely the Forward Kalman Filter (FKF) and Backward Kalman Filter (BKF), backtracking algorithm can make the best use of data provided by SINS and GPS in Two Filter Smoother (TFS).
However, the performance of backtracking algorithm is briefly determined by the priori information of process and observation noise statistics. Since the priori statistics of noise is involved in both the FKF and BKF, the inaccurate noise parameter selection will deteriorate the performance more significantly than the conventional KF. To mitigate the side effect of inaccurate noise parameter, scholars have done many investigations. Multiple Model Adaptive (MMA) filtering approach is proposed in adaptive Kalman filtering, which is composed of several Kalman filters with different models. However, MMA filtering requires massive computation and the performance of filter is affected by the capacity of models and the rules of data fusion. Additionally, TFS will increase the computation complexity of MMA because the data is implemented from two directions. Sage-Husa adaptive filtering algorithm is proposed to distinguish the Kalman filters parameters online when the statistical information of noise changes remains unknown in some cases. However, this algorithm does not separate the system variables estimation from the noise estimation process, which may cause filter divergence. And this coupling variables and noise estimation would undermine TFS. These adaptive filtering approaches have apparent shortcomings in backtracking process, which clearly limit their applicability.
A method for estimating the statistical information on noise real-timely is proposed. By constructing a transform matrix and separating the system states from the noise, the remodeled observation sequence only contains process noise and observation noise. Meanwhile, according to the law of large numbers, it is possible to estimate the covariance matrix of noise statistics through the remodeled observation sequence. This algorithm has been proved to acquire the covariance matrix of noise statistics effectively. Using the estimation results offered by this method, it is without doubt that TFS could perform better forward and backward independently, which is beneficial to navigation accuracy.
Moreover, it is worth noting that the most critical step of noise estimator is to calculate accurate remodeled coefficient, which relies on the accuracy of system model. However, there are extra computational error item existing in system model. When the noise estimator applied in TFS, these extra error items would be accumulated in system model, causing the inaccuracy of remodeled coefficient, as well as the estimation result finally. This paper presents a modified noise estimator in TFS to get a better estimation result of process noise and observation noise in isolation of extra error items.
The common system model with respect to navigation frame (n frame) and its inverse are both strongly coupled with ideal values of navigation states. In common system model, all state vectors with respect to n frame are ideal values, and specific force, as well as angular rate, are ideal outputs of IMU. However, in engineering practice, the ideal n frame and ideal outputs of IMU are unavailable. While in practical navigation, there is a perturbation in n frame because of the existence of errors caused by inertial instruments. Actually n frame is an uncertain frame on account of the absence of ideal tracking angular rate. The actual frame that the navigator modeling is computational frame (c frame) rather than n frame. And differing from ideal outputs of IMU, the actual output of IMU contains drift and bias errors. These practical navigation results with respect to c frame and practical outputs of IMU are generally used as substitutes to calculate in system model with respect to n frame. This is the cause of generating extra computational error item. In order to reduce the influence of extra error items, this paper proposes a modified system model by selecting c frame as the reference frame and removing all ideal values from system model. The attitude, velocity and position equations in SINS error propagation equations, which is the basis of system model, will be derived in c frame without ideal values. This method would reduce the coupling relation between system model and practical state vectors. The derivation of this modified system model will be introduced in final paper in detail. And in noise estimator, system model would be stored in each sampling time to calculate remodeled coefficient. If extra computational error item cannot be eliminated, the calculation result of remodeled coefficient would be far from the true value. Applying this modified system model in noise estimator can mitigate the adverse effects of extra computational error.
To investigate the performance of modified noise estimator in TFS, some simulations are carried out. From simulation results, it is clear that the noise estimator can achieve a better statistical estimation result in TFS. Particularly, this estimator can get a better performance in TFS with modified system model. It has an improved estimation of statistics compared with traditional adaptive Kalman filter. With more accurate estimation of process and observation noise statistics, the improvement of most navigation state estimation accuracy is raised up to 30%, some even 70%. The simulation test results can verify that, by using modified system model without extra computational error items, the modified noise estimator can get an satisfactory estimation of process noise and observation noise statistics forward and backward, which help improving the navigation estimation results in TFS.
The present work is continued by carrying out a practical experiment. The experimental platform is Unmanned Ground Vehicle (UGV) refitted by Inertial Navigation and Intelligent Navigation Laboratory (ININ Lab) in Beijing Institute of Technology (BIT). The UGV was equipped with IMU and GPS receiver. IMU data used in this test was collected from the PNS100-BGI integrated navigation system, produced by OlinkStar Company, containing a MEMS-IMU called ADIS16405. The NovAtel OEM628 receiver configured in RTK mode is used to collect GNSS data, and the acquisition rate is set to 1 Hz. These experimental devices are capable of data collection for algorithm validation.
In conclusion, the main contributions of this paper are twofold: (I) a modified system model, in which considers the extra computational error items, is established theoretically; (II) a modified noise estimator based on modified system model above is proposed in TFS, and simulation test results indicate that the navigation results are improved effectively by using the proposed approach.