GPS navigation processing using the quaternion-based divided difference filter

In this paper, we propose a second-order divided difference (DD2) filter for ballistic target tracking in the re-entry phase. The DD2 filter linearizes the nonlinear dynamic and measurement functions by using a second-order interpolation formula to approximate the derivatives of nonlinear functions. We compare the new method, extended Kalman filter (EKF) and unscented Kalman filter (UKF) using a Monte Carlo simulation approach. The simulation results show that DD2 filter outperform EKF and UKF in terms of estimation accuracy and filtering credibility. We conclude that the DD2 filter is a strong candidate for the ballistic target tracking problem being studied.

Neural networks have been proposed as nonlinear filters in a variety of applications that involve nonlinear processing of input signals; examples include blind signal separation, image registration, and blind deconvolution. The Global Positioning System (GPS) navigation equations are nonlinear (quadratic) in nature, and a direct closed form solution of the GPS navigation equations does not exist. This article presents a new approach to solving the GPS pseudorange equations using three-layer neural networks. A three-layer radial basis function (RBF) neural network is designed, which solves the non-linear GPS pseudorange equations directly as opposed to the linear least squares or extended Kalman filter approaches in traditional GPS receivers. For training the neural network, a carefully selected cost function is minimized using a variation of the classical conjugate gradient algorithm such that training time for the neural network is reasonable. Simulations have been performed at SiRF Technology Inc. that show stable behavior even under bad geometry conditions where the traditional recursive least squares and extended Kalman filter approaches show high sensitivity to measurement errors. Under good geometry conditions the neural network solution shows slightly improved noise performance compared to the expected performance of traditional leas squares solution. Simulations have been performed with additive white Gaussian noise and correlated noise models to evaluate the performance of the trained neural network. © 2000 John Wiley & Sons, Inc.

The extended Kalman filter (EKF) has been used widely in Global Position System (GPS) and Strapdown Inertial Navigation Systems (SINS) integrated navigation systems,which simply linearizes all nonlinear models.However, careful treatment of the nonlinearity of the system models is particularly critical when the integrated systems use low cost micro-electro-mechanical (MEMS) inertial sensors. Hence application of the EKF in MEMS inertial sensor-based integrated navigation systems will likely lead to inaccurate results. To overcome the shortcomings of the EKF, the unscented Kalman filter (UKF) has been proposed.The UKF does not require the linearization of the system models. Alternatively it uses a set of deterministically selected "sigma-points",which completely capture the true mean and covariance of the original random vector. Then these sigma-points are propagated through the nonlinear models.This captures the mean and covariance to second order accuracy for arbitrary nonlinear functions. In this paper the EKF and UKF are applied in GPS/SINS integrated systems respectively. Simulation result demonstrates that the performance of the UKF is better than the EKF in MEMS inertial sensor-based SINS/GPS integrated systems.

High dynamic aircraft is a very attractive new generation vehicles, in which provides near space aviation with large flight envelope both speed and altitude, for example the hypersonic vehicles. The complex flight environments for high dynamic vehicles require high accuracy and stability navigation scheme. Since the conventional Strapdown Inertial Navigation System (SINS) and Global Position System (GPS) federal integrated scheme based on EKF (Extended Kalman Filter) is invalidation in GPS single blackout situation because of high speed flight, a new high precision and stability integrated navigation approach is presented in this paper, in which the SINS, GPS and Celestial Navigation System (CNS) is combined as a federal information fusion configuration based on nonlinear Unscented Kalman Filter (UKF) algorithm. Firstly, the new integrated system state error is modeled. According to this error model, the SINS system is used as the navigation solution mathematic platform. The SINS combine with GPS constitute one error estimation filter subsystem based on UKF to obtain local optimal estimation, and the SINS combine with CNS constitute another error estimation subsystem. A non-reset federated configuration filter based on partial information is proposed to fuse two local optimal estimations to get global optimal error estimation, and the global optimal estimation is used to correct the SINS navigation solution. The χ 2 fault detection method is used to detect the subsystem fault, and the fault subsystem is isolation through fault interval to protect system away from the divergence. The integrated system takes advantages of SINS, GPS and CNS to an immense improvement for high accuracy and reliably high dynamic navigation application. Simulation result shows that federated fusion of using GPS and CNS to revise SINS solution is reasonable and availably with good estimation performance, which are satisfied with the demands of high dynamic flight navigation. The UKF is superior than EKF based integrated scheme, in which has smaller estimation error and quickly convergence rate.

T HIS Note describes a robust modification of the divided difference filtering technique. The robust technique relies on Huber’s generalized maximum likelihood approach to estimation [1]. Specifically,Huber’smethod is a combinedminimum ‘1and ‘2norm estimation technique, which exhibits robustnesswith respect to deviations from the commonly assumed Gaussian error probability density functions, for which the least–squares or minimum ‘2-norm technique exhibits a severe degradation in estimation accuracy [2]. TheHuber-based estimates are robust in the sense that theyminimize the maximum asymptotic estimation variance when applied to contaminated Gaussian densities. The Huber technique was originally developed as a generalization of maximum likelihood estimation, applied first to estimating the center of a probability distribution in [1] and generalized tomultiple linear regression in [2– 4]. The Kalman filter is a recursive minimum ‘2-norm technique and therefore exhibits sensitivity to deviations in the true underlying error probability distributions [5]. For this reason, theHuber technique has been further extended to dynamic estimation problems. Boncelet and Dickinson [6] first proposed to solve the robust filtering problem by means of the Huber technique at each measurement, by expressing the discrete-time Kalman filter as a sequence of linear regression problems. The authors do not provide any simulation results to validate the proposed technique. Kovacevic et al. [7] follow thework of [6] and develop a robust Kalman filter using the Huber technique applied to a linear regression problem at each measurement update. References [8–10] express the dynamic filtering problem as a sequential linear regression to be solved by the Huber technique and apply the filter to underwater vehicle tracking, power system state estimation, and spacecraft rendezvous navigation, respectively. The increase in computation due to the use of the Huber technique was found in [10] to be small. It should be noted that [6–10] apply the Huber methodology to linearized filters. The divided difference filter is one of several new estimation techniques that are collectively known as sigma-point Kalman filters (SPKF). The first-order (DD1) and second-order (DD2) divided difference filters [11,12] are generalizations of the filter introduced by Schei [13] and are two examples of SPKF-class estimators; other examples can be found in [14–16]. Like the basic Kalman filter, the SPKFs seek to determine a state estimate that minimizes the ‘2 norm of the residuals. The SPKF technique differs from the standard Kalmanfilter in the sense that the SPKFs do not linearize the dynamic system for the propagation, but instead propagate a cluster of points centered around the current estimate to form improved approximations of the conditional mean and covariance. Specifically, the divided difference filters make use of multidimensional interpolation formulas to approximate the nonlinear transformations. As a result of this approach, the filter does not require knowledge or existence of the partial derivatives of the system dynamics and measurement equations. SPKFs have the additional advantage over the basic Kalman filter in that they can easily be extended to determine second-order solutions to the minimum ‘2-norm filtering problem, which increases the estimation accuracywhen the system andmeasurement equations are nonlinear. It is important to note that the SPKFs use a minimum ‘2-norm measurement update and are therefore subject to the same sensitivity to non-Gaussian measurement errors as the Kalman filter. Therefore, the purpose of this Note is tomodify the DD1 andDD2measurement update equations by making use of the Huber technique to provide robustness against deviations from Gaussianity without a large increase in computation. This Note first provides a short review of the DD1 and DD2 filters and then shows how the measurement update can be expressed in terms of a standard regression problem, which can be solved using the robust Huber technique. The filtering techniques are then applied to a benchmark problem that involves estimating the trajectory of an entry body from discrete-time range data measured by a radar tracking station. The simulation is conducted using Monte Carlo techniques for both Gaussian and non-Gaussian cases. The computational cost associated with each filter is provided.

In this paper we use four nonlinear blending filters in order to integrate global positioning system and strapdown inertial navigation system (GPS/SDINS). As we will see in this paper, the Unscented Kalman filter (UKF) in comparison with extended Kalman filter (EKF), central difference Kalman filter (CDKF) and particle filters (PFs) has the best performance both in estimation accuracy and computation time. It will also give us more reliable solution when encountering satellite signal blockage as a probable problem in land navigation. The results have clearly demonstrated that the UKF would improve the guidance from the point of accuracy and computation time to the mentioned problems.

A smart city’s main purpose is to provide intelligent responses to different problems of the rapid urban population growth. For instance, integrating fleet management solutions into intelligent transportation systems (ITS) can efficiently resolve transportation problems relying on each vehicle information. Usually, the position estimate is ensured by the integration of the Global Positioning System (GPS) and Inertial Navigation Systems (INS). For multisensor data fusion, the Extended Kalman Filter (EKF) is generally applied using the sensor’s measures and the GPS position as a helper. However, the INS are expensive and require more complex computing which induces restrictions on their implementation. Furthermore, the EKF performance depends on the vehicle dynamic variations and may quickly diverge because of environmental changes. In this paper, we present a robust low cost approach using EKF and Support Vector Machines (SVM) to reliably estimate the vehicle position by limiting the EKF drawbacks. The sensors used are a GPS augmented by a low cost wireless sensor network. When GPS signals are available, we train SVM on different dynamics and outage times to learn the position errors so we can correct the future EKF predictions during GPS signal outages. We obtain empirically an improvement of up to 94% over the simple EKF predictions in case of GPS failures.

This paper investigates the kernel entropy based extended Kalman filter (EKF) as the navigation processor for the Global Navigation Satellite Systems (GNSS), such as the Global Positioning System (GPS). The algorithm is effective for dealing with non-Gaussian errors or heavy-tailed (or impulsive) interference errors, such as the multipath. The kernel minimum error entropy (MEE) and maximum correntropy criterion (MCC) based filtering for satellite navigation system is involved for dealing with non-Gaussian errors or heavy-tailed interference errors or outliers of the GPS. The standard EKF method is derived based on minimization of mean square error (MSE) and is optimal only under Gaussian assumption in case the system models are precisely established. The GPS navigation algorithm based on kernel entropy related principles, including the MEE criterion and the MCC will be performed, which is utilized not only for the time-varying adaptation but the outlier type of interference errors. The kernel entropy based design is a new approach using information from higher-order signal statistics. In information theoretic learning (ITL), the entropy principle based measure uses information from higher-order signal statistics and captures more statistical information as compared to MSE. To improve the performance under non-Gaussian environments, the proposed filter which adopts the MEE/MCC as the optimization criterion instead of using the minimum mean square error (MMSE) is utilized for mitigation of the heavy-tailed type of multipath errors. Performance assessment will be carried out to show the effectiveness of the proposed approach for positioning improvement in GPS navigation processing.

In underwater navigation, the global positioning system is unavailable; hence, other solutions must be pursued in the development of navigation systems. In this paper, filters for the position, linear velocity, and acceleration of underwater vehicles are derived by combining a long baseline acoustic positioning system with an inertial navigation system. A dynamic model is devised via state augmentation, including the bias of the pseudo-ranges, which accounts for the effect of the unknown offset between the emitters’ and receivers’ clocks. With this technique, the resulting dynamics are linear, in spite of the original nonlinear nature of the problem. The proposed solution, which includes the explicit dynamic estimation of the bias of the pseudo-ranges, is a linear Kalman filter. The observability of the system is assessed, which allows to establish globally exponentially stable error dynamics. The performance of the solution is evaluated with realistic simulation results, considering sensor noise and discrete-time measurements. Finally, the comparison with the extended Kalman filter (EKF), the unscented Kalman filter (UKF), and the Bayesian Cramer–Rao bound is presented, including Monte Carlo simulations. This comparison shows the goodness of the proposed solution, which converges for all initial conditions and exhibits performance comparable to the EKF and the UKF, whereas the EKF and UKF are shown to lack global convergence.

The Kalman filter is an optimal state estimator used in many of the engineering applications. In general, Kalman filter has two typical forms for non-linear estimation they are, the extended Kalman filter and unscented Kalman filter. This paper discusses the implementation of GPS-based Inertial navigation system (a combination of accelerometers and gyroscopes) using an extended Kalman filter (EKF). Both GPS and INS sensors are used to update the position and velocity in a land vehicle so that real time guidance and navigation tasks are performed. Due to the low sampling rate of GPS, continuous and reliable position cannot be obtained regularly. Similarly, the INS output tends to accumulate error over time which makes it unreliable. These limitations are overcome by integrating these systems due to their complementary characteristics in order to achieve long term accuracy. The method discussed involves collecting the data from GPS and INS sensors and processing them using EKF to obtain better estimates on the position and velocity. The vehicle orientation is linearized using the quaternion in the prediction and correction stages of EKF.

For the guidance, control and navigation of autonomous vehicles, both positioning and orientation are important for mission success and surveillance. Global positioning system (GPS) and strapdown inertial navigation system (SINS) have complementary characteristics to overcome the demerits of each other and can well perform as integrated navigation system. The observability analysis of low-cost SINS/GPS from many researches shown that the heading and yaw rate gyro bias is unobservable in straight motion and with constant acceleration. This paper presents integration of low-cost SINS/GPS aided with odometer and heading from two antennae GPS receiver to overcome this problem and to control the accuracy degradation during GPS outage. The carrier phase measurements are processed to obtain the heading from two antennae position measurement. The obtained heading and position from GPS and velocity measurement from odometer are used as observation input to extended Kalman filter (EKF) for the heading and positioning accuracy improvement of SINS/odometer/GPS integrated system for land vehicle navigation. When GPS signal is unavailable, the filter switches to SINS/odometer integration to control the position and velocity errors growth during this period. The observation model for heading measurement from two antennae is derived for the designed system. A field test was conducted to check the effectiveness using proposed method for heading and positioning accuracy improvement. The test results showed that Low-cost IMU/Odometer/ Two Antennae GPS integrated systems' accuracy both in position and heading is improved significantly.

With the development of the vehicle industry, controlling stability has become more and more important. Techniques of evaluating vehicle stability are in high demand. Integration of Global Positioning System (GPS) and Inertial Navigation System (INS) is a very practical method to get high-precision measurement data. Usually, the Kalman filter is used to fuse the data from GPS and INS. In this paper, a robust method is used to measure vehicle sideslip angle and yaw rate, which are two important parameters for vehicle stability. First, a four-wheel vehicle dynamic model is introduced, based on sideslip angle and yaw rate. Second, a double level Kalman filter is established to fuse the data from Global Positioning System and Inertial Navigation System. Then, this method is simulated on a sample vehicle, using Carsim software to test the sideslip angle and yaw rate. Finally, a real experiment is made to verify the advantage of this approach. The experimental results showed the merits of this method of measurement and estimation, and the approach can meet the design requirements of the vehicle stability controller.

In this paper, we propose a systematic framework for the autonomous navigation system based on distributed filtering for an Unmanned Aerial Vehicle (UAV). The proposed framework consists of the design and algorithm of the autonomous navigation. Therein, the camera mounted on the UAV functions as a navigation sensor targeted for navigation and positioning. In order to reduce the computational complexity and exclude the risk caused by Global Positioning System (GPS) outage, an autonomous navigation system based on distributed filtering is designed and realized. When GPS is available by monitoring the GPS integrity, sensor information from Strapdown Inertial Navigation System (SINS) and GPS is fused using a 7-state Conventional Kalman Filter (CKF) to estimate the full UAV state (position, velocity and attitude); when GPS is unavailable, sensor information from gyroscopes, accelerometers and magnetometer is fused using a 4-state Extended Kalman Filter (EKF) to estimate the attitude and heading of the UAV, and sensor information from SINS and vision positioning system is fused using a 7-state Incoordinate Interval Kalman Filter (IIKF) to estimate the position and velocity of the UAV. In addition, the second-order vertical channel damping loop is adopted to fuse measurements from a barometer with those of SINS, which suppresses the divergence of the vertical channel error and makes the altitude information calculated by SINS trustable. Both ground and flight experiments of the autonomous navigation system have been carried out. The test results show that the system can provide stabilized attitude information in long durations, and can realize the automatic flight control of UAV.

Combined Kalman and sliding innovation filtering: An adaptive estimation strategy

In present days, real time navigation depends on Kalman filter to fuse data from Global Positioning System (GPS) and Inertial Navigation System (INS). But there exist drawbacks like long design time, requirement of prior knowledge in fusing data from GPS and INS using Kalman Filter. In order to overcome the drawbacks of Kalman filter, GPS and INS integration are done using Artificial Neural Networks (ANN) like FFNNs. But still there exists certain drawbacks like less accuracy in FFNNs for GPS and INS data integration. Therefore, this paper introduces Higher Order Neural Networks (HONNs) approach for INS and GPS data integration to overcome the drawback. A higher order feed forward neural network architecture with optimum number of nodes is used. In the given architectures, the replacement of summation at each node by multiplication results in more powerful mapping, because of its capability of processing higher-order information from training data. Performance comparison of HONNs with FFNNs shows that the proposed architecture provides satisfactory results in terms of error rate and number of epochs. The HONNs like Multiplicative Neural Network (MNN) module and Sigma-Pi-ANN module and Traditional Feed Forward Neural Networks(FFNNs) like Radial Basis Function Neural Network (RBF) module and Back propagation Neural Network (BPN) module are trained to predict the INS position error and provide accurate position of the moving aircrafts. Sigma-Pi ANN is found to be the best in terms of accuracy, number of epochs and execution time among the two HONNs.