Square-root adaptive Wave Filtering for marine vessels

SummaryThis paper addresses the numerical aspects of adaptive filtering (AF) techniques for simultaneous state and parameters estimation arising in the design of dynamic positioning systems in many areas of research. The AF schemes consist of a recursive optimization procedure to identify the uncertain system parameters by minimizing an appropriate defined performance index and the application of the Kalman filter (KF) for dynamic positioning purpose. The use of gradient‐based optimization methods in the AF computational schemes yields to a set of the filter sensitivity equations and a set of matrix Riccati‐type sensitivity equations. The filter sensitivities evaluation is usually carried out by the conventional KF, which is known to be numerically unstable, and its derivatives with respect to unknown system parameters. Recently, a novel square‐root approach for the gradient‐based AF by the method of the maximum likelihood has been proposed. In this paper, we show that various square‐root AF schemes can be derived from only two main theoretical results. This elegant and simple computational technique replaces the standard methodology based on direct differentiation of the conventional KF equations (with their inherent numerical instability) by advanced square‐root filters (and its derivatives as well). As a result, it improves the robustness of the computations against round off errors and leads to accurate variants of the gradient‐based AFs. Additionally, such methods are ideal for simultaneous state estimation and parameter identification because all values are computed in parallel. The numerical experiments are given. Copyright © 2015 John Wiley & Sons, Ltd.

This paper describes an adaptive Kalman filter based method for accurate estimation of forward path geometry of an automobile. The forward geometry is modeled as two contiguous clothoid segments with different geometries and continuous curvature across the transition between them. This results in a closed-form parametric expression of the same polynomial order as previous models. Instead of using a conventional Kalman filter with fixed process model parameters based on a compromise between noise and filter lag, we adaptively tune the process model parameters. This results in the better filter performance with stable estimates during constant geometry scenarios and faster response during abrupt geometry transitions. Performance evaluation of the proposed method on various simulated road geometries and comparing with previous approaches demonstrate the feasibility and higher accuracy of the proposed method. The high accuracy estimation of forward path or road geometry is directly useful in applications that rely on detecting targets in the forward path of the host vehicle, e.g., adaptive cruise control and automotive collision warning.

The satellite clock plays a key role in the global navigation satellite system (GNSS). The accuracy of GNSS and its applications depend on the quality of the satellite clock. Therefore, precisely estimating and predicting the satellite clock is an important issue in the fields of GNSS and its application. As an optimal estimation algorithm, Kalman filter has been used to estimate and predict the satellite clock. However, in a conventional Kalman filter algorithm, the noise covariance matrices of satellite clock need to be predetermined, which restricts its further applications since the noise covariance matrices, especially the process noise covariance matrix, are usually unknown in the real cases. With inappropriate noise covariance matrices, the state estimation of conventional Kalman filter is suboptimal. To cope with this problem, a new noise covariance matrix estimation method of Kalman filter is proposed, and then we apply it to the problem of satellite clock estimation and prediction. Considering the fact that the process noise covariance matrix depends on the unknown noise parameters, the problem of estimating process noise covariance matrix can be solved by estimating the unknown noise parameters. First, the correlation between the Kalman innovations is used to establish a linear relationship with the unknown noise parameters. Then the unknown parameters can be estimated by least-squares estimation. Finally, the satellite clock can be estimated and predicted with the estimated noise parameters. In the new method, no prior information about the noise parameters is needed. Even with some extreme prior noise parameters, the new method can also work very well and has good convergence properties. For comparison, we conduct two experiments using the new method and the adaptively robust Kalman filter with classified adaptive factors based on opening windows separately, both results are consistent with each other very well, which verifies the correctness and effectiveness of this new method.

For multi-target tracking, the IMM (interactive multiple model) algorithm has been proposed. The IMM is expected to reduce tracking errors for both nonmaneuvering and maneuvering targets. However, the IMM requires a heavy computational burden, because it utilizes multiple Kalman filters in parallel. On the other hand, the Kalman filter with a turning acceleration estimator, which can adapt a maneuvering target, has been proposed. The Kalman filter with a turning acceleration estimator is a single two-stage type filter and has a problem of setting threshold for the maneuver detector. In this paper, we propose the hybrid filter with a constant-velocity (CV) filter and a turning acceleration estimation filter. The proposed filter does not require a maneuver detector, because it integrates the outputs of two filters using the likelihood of each filter. And its computational requirement is smaller than the IMM, since it consists of only two Kalman based filters. The proposed method can prevent deteriorating tracking accuracy by reducing the risk of maneuver misdetection when a target maneuvers. We evaluate the performance of the proposed filter by computer simulation, and show the effectiveness of the proposed filter, comparing with the conventional Kalman filter and the two-stage Kalman filter.

The Global Navigation Satellite System (GNSS) is a widely used positioning technique. Computational efficiency is crucial to applications such as real-time GNSS positioning and GNSS network data processing. Many researchers have made great efforts to address this problem by means such as parameter elimination or satellite selection. However, parameter estimation is rarely discussed when analyzing GNSS algorithm efficiency. In addition, most studies on Kalman filter (KF) efficiency commonly have defects, such as neglecting application-specified optimization and limiting specific hardware platforms in the conclusion. The former reduces the practicality of the solution, because applications that need such analyses on filters are often optimized, and the latter reduces its generality because of differences between platforms. In this paper, the computational cost enhancement of replacing the conventional KF with the information filter (IF) is tested considering GNSS application-oriented optimization conditions and hardware platform differences. First, optimization conditions are abstracted from GNSS data-processing scenarios. Then, a thorough analysis is carried out on the computational cost of the filters, considering hardware–platform differences. Finally, a case of GNSS dynamic differencing positioning is studied. The simulation shows that the IF is slightly faster for precise point positioning and much faster for the code-based single-difference GNSS (SDGNSS) with the constant velocity (CV) model than the conventional KF, but is not a good substitute for the conventional KF in the other algorithms mentioned. The real test shows that the IF is about 50% faster than the conventional KF handling code-based SDGNSS with the CV model. Also, the information filter is theoretically equivalent to and can produce results that are consistent with the Kalman filter. Our conclusions can be used as a reference for GNSS applications that need high process speed or real-time capability.

This article proposes a novel initial bias estimation method using a trajectory generator (TG). The accuracy of attitude and position estimation in navigation after using the inertial navigation system/Doppler velocity log (INS/DVL) and INS/DVL/gyrocompass (IDG) for 1 h were evaluated, and the results were compared to those obtained using the conventional Kalman filter (KF) estimation method. The probability of a horizontal position error < 1852 m (1 nautical mile) with a bias interval > 400 s was 100% and 9% for the TG and KF, respectively. In addition, the IDG average horizontal position errors over 1 h were 493 m and 507 m for the TG and KF, respectively. Moreover, the amount of variation was 2 m and 27 m for the TG and the KF, respectively. Thus, the proposed method is effective for initial bias estimation of INS/DVL and IDG using micro-electro-mechanical system sensors on a constantly moving vessel.

The article concerns remote measurement. Aiming to obtain current state variables of a mechanical system with time delay, we estimated current state variables from the delayed measurement signal by using a Kalman filter. In order to increase the stability of the estimation system, we have proposed a novel Kalman filter with dual mathematical models, and also have shown that it was very effective because it was more robust and precise than a conventional Kalman filter with a single model. We have examined an influence for the estimation error that was caused by modeling errors of the plant, and have examined one for the sampling time error. From these results it is safer to set the variance of the measurement noise and input noise higher if uncertain, because the increment of the estimation error is smaller. Moreover the increment of the estimation error caused by the sampling time error keeps almost constant from the experiment and the simulation, when the sampling time is set longer.

In this paper, after reviewing the traditional Kalman filter formulation, a development of a fuzzy logic-based adaptive Kalman filter is outlined. The adaptation is in the sense of adaptively tuning, on-line, the measurement noise covariance matrix R or the process noise covariance matrix Q. This improves the Kalman filter performance and prevents filter divergence when R or Q are uncertain. Based on the whiteness of the filter innovation sequence and employing the covariance-matching technique the tuning process is carried out by a fuzzy inference system. If a statistical analysis of the innovation sequence shows discrepancies with its expected statistics then a fuzzy inference system adjusts a factor through which the matrices R or Q are tuned on line. This fuzzy logic-based adaptive Kalman filter is tested on a numerical example. The results are compared with these obtained using a conventional Kalman filter and a traditionally adapted Kalman filter. The fuzzy logic-based adaptive Kalman filter showed better results than its traditional counterparts.

A continuous glucose monitoring system is composed of a glucose sensor and an estimation (calibration) algorithm. The glucose sensor is composed of an electrode that is inserted under the skin and generates a noisy electrical current in response to interstitial glucose levels. The relationship between the electrical current and the interstitial glucose levels varies between individuals and within the same individual (with sensor wear time). The estimation algorithm infers the glucose levels in the blood from the noisy electrical current signal and uses intermittent capillary glucose measurements to account for intra-and inter-individual variability. In this article, we propose a novel real-time Kalman filter-based estimation algorithm that is composed of three steps: 1) noise filtering step; 2) compartment matching step using sequential Kalman filter; and 3) parameters estimation step using the conventional Kalman filter. The initial estimate of the parameters and covariance matrix are extracted offline using a nonlinear cubature Kalman filter. Our algorithm is compared with four alternative algorithms using 20 sensor data sets. Each data set was generated over a seven-day sensor wear period, during which patients were tested on three in-clinic days (days 1, 4, and 7). The comparison is based on the mean absolute relative difference (MARD) between frequent reference glucose measurements and the sensor glucose levels. MARD for days 1, 4, and 7 were 10.3%, 10.7%, and 11.9% for our algorithm and 12.46%, 15.75%, and 14.17% for the alternative algorithm, respectively. Our algorithm is currently being integrated into a commercial continuous glucose monitoring system (AgaMatrix, Inc., Salem, NH, USA).

According to the poor adaptive ability of traditional filter algorithm in the estimatio n for traffic state and travel time with Kalman filter, an improved fuzzy adaptive Kalman filtering method was pro posed. The new interest of observation noise was defined, and the fuzzy logic was used to adjust the importance weights of system noise and observation noise through on-line monitoring the interest changes, which changed the trust and utilization degree of the model for the observation information, and this made the filter eventually tend to be stable. To guarantee the real-time performance of system, a direct input - output fuzzy membership function matching method was put forward to take the place of fuzzy reasoning. The method was tested on the urban expressway in Guangzhou by using real-time detection data, and the result show that the traffic state es timation model had better tracking ability than conventional Kalman filter, and results of travel time prediction show that there was a sligh t difference between the prediction value and that of actual observation in free traffic flow state, and the relative error was under 1 5% in traffic congested state. The precision and applicability of this method were acceptable, and it can be used to provide a basis for travel time of urban expressway in traffic control and guidance system.

A high accurate tracking technique with the use of intelligent approach on matrix covariance resetting is proposed in this paper. In practice, the conventional Kalman filters have a fast convergence rate at the beginning. However, after some iteration the Kalman filter steps become very small. To overcome this defect and to make use of Kalman filter abilities, the matrix covariance resetting idea is used. The matrix covariance presetting usually is used to improve the tracking algorithm result especially for high maneuvering targets. To determine the optimal value of the unknown resetting parameter in each step, the intelligent fuzzy block is used. In this paper, an innovative technique is presented, which resets covariance matrix by using fuzzy logic. It is demonstrated by means of numerical acceleration examples that the tracking capability of the proposed method is essentially as good as that of the traditional methods, especially for high maneuver targets.

In Global Navigation Satellite System (GNSS) positioning, gross errors seriously limit the validity of Kalman filtering and make the final positioning solutions untrustworthy. Thus, the detection and correction of gross errors have become indispensable parts of Kalman filtering. Starting by defining an incremental Chi-square method of recursive least squares, this paper extends this definition to Kalman filtering to detect gross errors, explains its nature and its relation with the currently adopted Chi-square variables of Kalman filtering in model and data spaces, and compares them with the predictive residual statistics. Two robust Kalman filtering models based on an incremental Chi-square method (CI-RKF) were established, and the one with a better incremental Chi-square component was selected based on a static accuracy evaluation experiment. We applied the selected robust model to the GNSS positioning and the GNSS/inertial measurement unit (IMU) / visual odometry (VO) integrated navigation experiment in an occluded urban area at the East China Normal University. We compared the results for conventional Kalman filtering (CKF) with a robust Kalman filtering constructed using predictive residual statistics and an Institute of Geodesy and Geophysics (IGGШ) weight factor, abbreviated as “PRS-IGG-RKF”. The results show that the overall accuracy of CI-RKF in GNSS positioning was improved by 22.68%, 54.33%, and 72.45% in the static experiment, and 12.30%, 7.50%, and 16.15% in the kinematic experiment. The integrated navigation results indicate that the CI-RKF fusion method increased the system positioning accuracy by 66.73%, 59.59%, and 59.62% in one of the severe occlusion areas, and 42.04%, 59.04%, and 52.41% in the other.

A photoplethysmography (PPG) signal provides very useful information about a subject's hemodynamic status in a hospital or ubiquitous environment. However, PPG is very vulnerable to motion artifacts, which can significantly distort the information belonging to the PPG signal itself. Thus, the reduction of the effects of motion artifacts is an important issue when monitoring the cardiovascular system by PPG. There have been many adaptive techniques to reduce motion artifacts from PPG signals. In the present study, we compared a method based on the fixed-interval Kalman smoother with the usual adaptive filtering algorithms, e.g. the normalized least mean squares, recursive least squares and the conventional Kalman filter. We found that the fixed-interval Kalman smoother reduced motion artifacts from the PPG signal most effectively. Therefore, the use of the fixed-interval Kalman smoother can reduce motion artifacts in PPG, thus providing the most reliable information that can be deduced from the reconstructed PPG signals.

In federated design of ultra-tight GPS/INS integrated system, the baseband signal pre-processing is completed in a single pre-filter assigned for each channel. As the state space model of this single pre-filter includes the code tracking errors coupled with carrier tracking errors, ionospheric errors and normalized signal amplitude, the carrier tracking process may be destroyed. Also, the measurement noises are not independent any longer after passing through the code and carrier discriminators. Therefore, we propose a double-filter-based pre-filter model that distributes the carrier and code tracking into two independent filters: a conventional pre-filter, where the normalized signal amplitude is excluded from the state space and tracks only the code signal, and a 3-dimension state filter, tracking the carrier signal. The measurement information from both filters is a scalar quantity, which removes most of the noise correlation. To further improve the performance of the double-filter-based pre-filter model, we propose a modified Kalman filter algorithm. Simulation and field tests have been conducted, and the performance analysis has been done for the following configurations in a vector-tracking mode: double-filter model with modified Kalman filter, double-filter model with conventional Kalman filter and traditional single-filter model. The preliminary analysis indicates that the double-filter model with modified Kalman filter shows the best performance in tracking and navigation domains, while the traditional single-filter model shows a sub-optimal performance.

The aim of this paper is to improve the tracking performance of an aeronautical object with manoeuvring flight. The conventional Kalman filter may induce unsatisfactory tracking errors when applied to the manoeuvring object, since the parameters of the filter cannot adapt to the manoeuvring object. For this reason, two adaptive techniques are presented in this paper. The first is based on the covariance matching technique and the other on the Kumar adaptive Q theory. These tracking algorithms both give satisfactory performance from the simulation results. The comparison between the performance of these two tracking algorithms is also presented.