Nonlinear Observation Equation Research Articles

Real-Time Attitude Estimation for Spinning Projectiles by Magnetometer Based on an Adaptive Extended Kalman Filter.

The attitude measurement system based on geomagnetic information offers advantages such as small space requirements, fast response times, excellent resistance to high-overload conditions, and cost-effectiveness. However, during the flight process of a high-mobility guided spinning projectile, calculating attitude based on geomagnetic information often leads to non-unique solutions. To address this challenge, this paper proposes the Adaptive Extended Kalman Filter (AEKF) attitude estimation algorithm, based on geomagnetic vector information. Based on the analysis of the short-term attitude motion characteristics of the projectile, the Kalman state system equation and the nonlinear observation equation are established, along with real-time correction of the yaw angle and adaptive updates of parameters. The effectiveness of the algorithm is verified by simulations and experiments, demonstrating its ability to eliminate the dual solution problem inherent in traditional Single-Epoch algorithms. It notably improves the accuracy of pitch and roll angle estimation while providing precise estimates of attitude angular rates. Furthermore, the algorithm effectively mitigates the impact of magnetic disturbances on attitude determination. The proposed method has many potential applications in attitude measurement and navigation using geomagnetic data.

An MPF Based Aerial Platform Tracking Method for Road Network Targets

This paper presents a new algorithm to improve the tracking accuracy of the aerial mobile observation platform by using road information constraints. Combined with the road information and Mixed Particle Filter(MPF) algorithm, the Road Constrained Mixed Particle Filter(RCMPF) algorithm is proposed. RCMPF algorithm can effectively solve the problem of non-linear observation equation of aerial observation platform and realize the tracking of ground targets. The simulation results show that the algorithm improves the tracking accuracy of the target.

Performance Evaluation of Cubature Kalman Filtering and Extended Kalman Filtering Based Phase Unwrapping For Insar

Phase unwrapping is one of the most critical steps in InSAR data processing. Filter-based phase unwrapping methods can perform phase unwrapping and noise removal simultaneously, and show a clear advantage in dealing with high noise and density fringe interferograms. Based on the precision estimation theory of nonlinear functions in Extended Kalman Filter (EKF) and Cubature Kalman Filter (CKF) algorithms, we derive and analyze in detail the estimated precision of the EKF and the CKF on nonlinear observation equations in the phase unwrapping model. Results show that both EKF and CKF algorithms have similar precision in areas with better phase quality. However, in poorer quality areas, the precision of the two shows a certain difference and there is no consistent superiority for each algorithm. Experiments using simulated and measurement data confirm the validity of our theoretical analysis.

3D target localization based on multi–unmanned aerial vehicle cooperation

Ground target three-dimensional positions measured from optical remote-sensing images taken by an unmanned aerial vehicle play an important role in related military and civil applications. The weakness of this system lies in its localization accuracy being unstable and its efficiency being low when using a single unmanned aerial vehicle. In this paper, a novel multi–unmanned aerial vehicle cooperative target localization measurement method is proposed to overcome these issues. In the target localization measurement stage, three or more unmanned aerial vehicles simultaneously observe the same ground target and acquire multiple remote-sensing images. According to the principle of perspective projection, the target point, its image point, and the camera’s optic center are collinear, and nonlinear observation equations are established. These equations are then converted to linear equations using a Taylor expansion. Robust weighted least-squares estimation is used to solve the equations with the objective function of minimizing the weighted square sum of re-projection errors from target points to multiple pairs of images, which can make the best use of the effective information and avoid interference from the observation data. An automatic calculation strategy using a weight matrix is designed, and the weight matrix and target-position coordinate value are updated in each iteration until the iteration stopping condition is satisfied. Compared with the stereo-image-pair cross-target localization method, the multi–unmanned aerial vehicle cooperative target localization method can use more observation information, which results in higher rendezvous accuracy and improved performance. Finally, the effectiveness and robustness of this method is verified by numerical simulation and flight testing. The results show that the proposed method can effectively improve the precision of the target’s localization and demonstrates great potential for providing more accurate target localization in engineering applications.

On the Theory of a Nonlinear Dynamic Circuit Filtering

Stochastic Differential Equations (SDEs) describe physical systems to account for random forcing terms in the evolution of the state trajectory. The noisy sampling mixer, a component of digital wireless communications, can be regarded as a potential case from the dynamical systems’ viewpoint. The universality of the noisy sampling mixer is attributed to the fact that it adopts the structure of a nonlinear SDE and its linearized version becomes a time-varying bilinear SDE. This paper develops a mathematical theory for the nonlinear noisy sampling mixer from the filtering viewpoint. Since the filtering of stochastic systems hinges on the structure of dynamical systems and observation equation set up, we consider three ‘filtering models’. The first model, accounts for a nonlinear SDE coupled with a nonlinear observation equation. In the second model, we consider a bilinear SDE with a linear observation equation to achieve the nonlinear sampling filtering. Note that the bilinear SDE coupled with the linear observation is a consequence of the Carleman linearization to the nonlinear SDE and the nonlinear observation equation. In the third model, we consider a Stratonovich SDE coupled with a nonlinear observation equation. The filtering equation of this paper can be further utilized to guide the design process of the noisy sampling mixer.

Distributed Error Correction of EKF Algorithm in Multi-Sensor Fusion Localization Model

In order to solve the problem that the standard extended Kalman filter (EKF) algorithm has large errors in Unmanned Aerial Vehicle (UAV) multi-sensor fusion localization, this paper proposes a multi-sensor fusion localization method based on adaptive error correction EKF algorithm. Firstly, a multi-sensor navigation localization system is constructed by using gyroscopes, acceleration sensors, magnetic sensors and mileage sensors. Then the information detected by the sensor is compared and adjusted, to reduce the influence of error on the estimated value. The nonlinear observation equation is linearized by Taylor, and the normal distribution hypothesis is carried out in two steps of prediction and correction respectively. Finally, the parameters of system noise and measurement noise covariance in EKF are optimized by using the evolutionary iteration mechanism of genetic algorithm. The adaptive degree is obtained according to the absolute value of the difference between the estimated value and the real value of EKF. The individual evaluation results of EKF algorithm parameters are used as the measurement standard for iteration to obtain the optimal value of EKF algorithm parameters. Experimental simulation results show that the improved algorithm proposed has higher real-time localization accuracy and higher robustness than those of the standard EKF algorithm.

Data Fusion of UWB and IMU Based on Unscented Kalman Filter for Indoor Localization of Quadrotor UAV

The indoor location technique plays a essential role during the application of quadrotor unmanned aerial vehicle (UAV). However, the control design problem for the quadrotor UAV is quite difficult in the indoor environment due to the weak GPS signal. Based on Ultra Wide Band (UWB), the related positioning issues can be solved of UAV through base station with known coordinate position and equipment with location tag, but it is difficult to meet the high-precision operation requirements. In this paper, an indoor positioning design method combined with the Inertial Measurement Unit (IMU) and UWB positioning technology is proposed, which can effectively suppress the error accumulation of the IMU and further improve the positioning accuracy. Moreover, the system architecture for a class of quadrotor UAV is designed. The multisensor fusion technology based on unscented Kalman filter (UKF) is used to avoid neglecting the high-order terms of the nonlinear observation equations of UWB and IMU, which can effectively improve the accuracy of solving the nonlinear equations. Finally, a hardware-in-the-loop simulation platform is designed to verify the effectiveness of the indoor positioning method and improve the positioning accuracy.

SAEnet: an application for adjustment of horizontal network using particle swarm optimization algorithm

The horizontal network of control stations plays an important role in geometric measurement activities, such as determining positions of point, distance, area, or even volume. One of the methods of determining horizontal network values is the least squares method. The method of least squares is one of the most frequently used methods to obtain unique estimates for a set of redundant measurements. The equations that are used in the horizontal network surveys are the nonlinear equations so it must be linearized in order to be computed using least squares method. The problem that arises when linearizing the nonlinear observation equation is the approximate coordinates of the network stations are needed for its computations. In order to handle this problem, the Particle Swarm Optimization (PSO) algorithm is applied as an algorithm for determining the approximate coordinate values of the network stations. The algorithm of PSO is a heuristic function that optimizes a problem by iteratively trying to improve a candidate solution toward the best solutions. The advantages of PSO are that PSO is designed for solving a problem more quickly when classic methods are too slow and finding an approximate solution when classic methods fail to find any exact solution. This paper presents the design of an application for horizontal network adjustment using PSO algorithm, namely SAEnet. The Algorithm was written in Matlab which is very effective in matrix manipulation and has been widely used in various fields of numerical computations. The computation of horizontal network adjustment using SAEnet application consists of determination of initial approximation values using PSO and the adjustment computation using the least squares method. SAEnet is also equipped with save menu to Microsoft Excel format and load menu from Microsoft Excel format. The test shows that the input and output of the program can run as expected. The results are then compared to other program, like STAR*NET, using a variety of data sets. The results of both computations show the maximum difference between both programs is 0.000040 while the average difference is 0.000020. The test shows that there is no significant difference between the results of the SAEnet and STAR*NET computation. Hence, SAEnet can be used for horizontal network adjustment computation.

Square‐root‐extended complex Kalman filter for estimation of symmetrical components in power system

The paper presents a square-root-extended complex Kalman filter (SRECKF) by decomposing covariance matrix with its square-root forms to improve stability of the filter for estimating complex number. αβ transformation is used to map three-phase instantaneous voltages in the abc phases into instantaneous voltages on the αβ axes, and a non-linear state equation and observation equation of the three-phase voltages are built by introducing a complex vector and defining state variables. Positive symmetrical component, negative symmetrical components, and frequency of the three-phase voltages are estimated using traditional extended complex Kalman filter (ECKF), the estimation results show that the method proposed here are superior to traditional extended complex Kalman filter on estimation accuracy and convergence rate.

An Advanced Cooperative Positioning Algorithm Based on Improved Factor Graph and Sum-Product Theory for Multiple AUVs

In this research, a novel autonomous underwater vehicle (AUV) cooperative positioning algorithm is proposed to solve the implementation problem of multi-sensor-fusion applications. Different from the traditional methods [i.e., the extended Kalman filter (EKF), unscented Kalman filter (UKF), and iteration extended Kalman filter (IEKF)], which have large linearity error under the condition of nonlinear observation equation when multiple AUV are cooperative positioning, the proposed algorithm utilized the Baysis filter to solve the AUV cooperative problem. Factor graph and sum-product (FGS)-based cooperative positioning algorithm is established to mathematically implement the Bayse filter by converting the global function estimation problem into a local function sum-product estimation problem. Furthermore, to improve the performance of the proposed algorithm, a robust data processing method is presented by introducing a transform matrix to the estimated position information. To demonstrate and verify the proposed methods, the simulation and real tests in different scenarios are performed in this research. Compared with the traditional EKF, UKF, and IEKF cooperative positioning algorithm, the positioning error of the proposed improved FGS (IFGS) cooperative positioning algorithm is obviously smaller than that of the other three algorithms. Moreover, the IFGS algorithm can reduce the complexity of the algorithm, available improving the computational speed of the whole system. This proposed algorithm has important theoretical and practical value for the both industry and academic areas.

M-estimator based robust unscented Kalman filter through statistical linearization

The robust unscented Kalman filter (UKF) is revisited in this paper from a new point of view, namely the statistical linear regression (SLR) perspective of the unscented transformation (UT). When the actual distribution of the observation noise deviates from the assumed Gaussian distribution, the performance of the filter may degrade significantly owing to lacking robustness. After linearizing the nonlinear observation equation using the SLR perspective, the M-estimator is employed to replace the weighted least-squares one, resulting in the robust UKF. Besides providing new theoretical aspects, the advantageous performance of the proposed filter is re-emphasized in terms of the following three: (1) robust, the proposed filter is robust against deviations from the assumed Gaussian distribution; (2) efficient, by elaborately selecting the tuning parameter of the M-estimator, the robust filter will have a slight efficiency loss compared with the UKF when the Gaussian assumption does hold; (3) accurate, the proposed filter achieves robustness without compromising the accuracy of the UT.

Approximate Closed-Form TDOA-Based Estimator for Acoustic Direction Finding via Constrained Optimization

This paper focuses on estimating the azimuth and elevation of a sound emitter in 3-D space based on time-difference-of-arrival (TDOA) measurements with an array of acoustic sensors. The TDOA-based direction finding problem is significant because in a range of scenarios the source only emits a transient signal and only TDOA measurements can be used to find the direction. The linear least squares estimator provides a suboptimal solution, since there is nontrivial information loss in the linearization of the nonlinear observation equation. To avoid the information loss, the Lagrange multiplier method is usually used to realize the constrained optimization, but the computational complexity is rather high. This paper proposes a constrained least squares estimator to deal with the direction finding problem. The proposed method makes use of both Lagrange multiplier and quadratic constraints to form the cost function. The resultant estimator is shown to be approximate closed-form so that the computational complexity is reduced greatly, but contributes little under a small noise level. Mathematical formula is derived to evaluate the theoretical accuracy of the proposed estimator in terms of mean square error. Both simulation and field experimental results demonstrate that the proposed estimator can outperform the traditional linear and nonlinear estimators.

Probabilistic damage detection of structures with uncertainties under unknown excitations based on Parametric Kalman filter with unknown Input

System identification and damage detection for structural health monitoring have received considerable attention. Various time domain analysis methodologies based on measured vibration data of structures have been proposed. Among them, recursive least-squares estimation of structural parameters which is also known as parametric Kalman filter (PKF) approach has been studied. However, the conventional PKF requires that all the external excitations (inputs) be available. On the other hand, structural uncertainties are inevitable for civil infrastructures, it is necessary to develop approaches for probabilistic damage detection of structures. In this paper, a parametric Kalman filter with unknown inputs (PKF-UI) is proposed for the simultaneous identification of structural parameters and the unmeasured external inputs. Analytical recursive formulations of the proposed PKF-UI are derived based on the conventional PKF. Two scenarios of linear observation equations and nonlinear observation equations are discussed, respectively. Such a straightforward derivation of PKF-UI is not available in the literature. Then, the proposed PKF-UI is utilized for probabilistic damage detection of structures by considering the uncertainties of structural parameters. Structural damage index and the damage probability are derived from the statistical values of the identified structural parameters of intact and damaged structure. Some numerical examples are used to validate the proposed method.

Kalman particle filtering algorithm for symmetric alpha‐stable distribution signals with application to high frequency time difference of arrival geolocation

In this study, a non-linear filtering algorithm for state estimation with symmetric alpha-stable (SαS) noise is presented. The dynamic system model investigated here can be described by a linear state-space equation and a non-linear observation equation. The contribution of this study can be summarised as follows. First, particle filtering approach is employed for coarse estimation of the unknown parameters and then Kalman filter is performed to achieve better estimation. Second, SαS noise is considered as the additive disturbance in the observed signal and Gaussian approximation is used to compute the characteristics. Third, the calculation complexity is analysed according to the proposed algorithm. The proposed method is compared with the standard particle filter, extended Kalman filter and unscented Kalman filter for static parameter estimation of a periodic signal. As a practical application, the proposed method is used in high frequency source localisation based on time difference of arrival measurements.

On a Non-linear Electronic Circuit Filtering

The stochastic versions of non-linear dynamic circuits are formalized using non-linear stochastic differential equations. Stochastic differential equations (SDEs) are exploited to analyse dynamical systems in noisy environments. A potential application of the SDEs can be regarded as `stochastic processes in electronic circuits'. The noisy sampling mixer, a component of digital wireless communications, is an appealing and standard case from the dynamical systems' viewpoint. It assumes the structure of a non-linear SDE, and its linearized version becomes time-varying bilinear SDE. This paper derives the filtering equations for the noisy non-linear sampling mixer circuit utilizing the filtering density evolution equation. The filtering model for the stochastic problem of concern here comprises the following: (1) a non-linear SDE describing the noisy sampling mixer and (2) a non-linear noisy observation equation. It is interesting to note that the filtered estimate accounts for observations. On the other hand, the predicted estimate does not account for the observation terms in evolution equations. As a result of this, the filtered estimate confirms the greater accuracy of estimated state trajectory in contrast to the predicted trajectory. The filtering equation of this paper can be further utilized for control of the noisy sampling mixer, where the observations are available.

Improving water quality forecasting via data assimilation – Application of maximum likelihood ensemble filter to HSPF

Summary An ensemble data assimilation (DA) procedure is developed and evaluated for the Hydrologic Simulation Program – Fortran (HSPF), a widely used watershed water quality model. The procedure aims at improving the accuracy of short-range water quality prediction by updating the model initial conditions (IC) based on real-time observations of hydrologic and water quality variables. The observations assimilated include streamflow, biochemical oxygen demand (BOD), dissolved oxygen (DO), chlorophyll a (CHL-a), nitrate (NO 3 ), phosphate (PO 4 ) and water temperature (TW). The DA procedure uses the maximum likelihood ensemble filter (MLEF), which is capable of handling both nonlinear model dynamics and nonlinear observation equations, in a fixed-lag smoother formulation. For evaluation, the DA procedure was applied to the Kumho Catchment of the Nakdong River Basin in the Republic of Korea. A set of performance measures was used to evaluate analysis and prediction of streamflow and water quality variables. To remove systematic errors in the model simulation originating from structural and parametric errors, a parsimonious bias correction procedure is incorporated into the observation equation. The results show that the DA procedure substantially improves predictive skill for most variables; reduction in root mean square error ranges from 11% to 60% for Day-1 through 3 predictions for all observed variables except DO. It is seen that MLEF handles highly nonlinear hydrologic and biochemical observation equations very well, and that it is an effective DA technique for water quality forecasting.

Third-Order Continuous-Discrete Filtering for a Nonlinear Dynamical System

Approximate higher-order filters are more attractive and popular in control and signal processing literature in contrast to the exact filter, since the analytical and numerical solutions of the nonlinear exact filter are not possible. The filtering model of this paper involves stochastic differential equation (SDE) formalism in combination with a nonlinear discrete observation equation. The theory of this paper is developed by adopting a unified systematic approach involving celebrated results of stochastic calculus. The Kolmogorov–Fokker–Planck equation in combination with the Kolmogorov backward equation plays the pivotal role to construct the theory of this paper “between the observations.” The conditional characteristic function is exploited to develop “filtering” at the observation instant. Subsequently, the efficacy of the filtering method of this paper is examined on the basis of its comparison with extended Kalman filtering and true state trajectories. This paper will be of interest to applied mathematicians and research communities in systems and control looking for stochastic filtering methods in theoretical studies as well as their application to real physical systems.

Loosely Coupled INS/GPS Integration with Constant Lever Arm using Marginal Unscented Kalman Filter

A loosely coupled Inertial Navigation System (INS) and Global Positioning System (GPS) are studied, particularly considering the constant lever arm effect. A five-element vector, comprising a craft's horizontal velocities in the navigation frame and its position in the earth-centred and earth-fixed frame, is observed by GPS, and in the presence of lever arm effect, the nonlinear observation equation from the state vector to the observation vector is established and addressed by the correction stage of an unscented Kalman filter (UKF). The conditionally linear substructure in the nonlinear observation equation is exploited, and a computationally efficient refinement of the UKF called marginalized UKF (MUKF) is investigated to incorporate this substructure where fewer sigma points are needed, and the computational expense is cut down while the high accuracy and good applicability of the UKF are retained. A performance comparison between UKF and MUKF demonstrates that the MUKF can achieve, if not better, at least a comparable performance to the UKF, but at a lower computational expense.

Estimation of Traffic Densities for Multilane Roadways Using a Markov Model Approach

Inductive loop detectors are widely deployed in strategic roadway networks. This paper investigates recursive estimation of traffic densities using the information provided by loop detectors. The existing studies for multilane roadways mainly focus on the scenario where vehicles' lane-change movements are not common and can be ignored. This paper, however, takes into consideration lane-change effect in traffic modeling and incorporates a Markov chain into the state space model to describe the lane-change behavior. We update the traffic density estimate using the Kalman filter. To avoid the approximation due to the linearization of the nonlinear observation equation in the extended Kalman filter, we have considered a suitable transformation. Numerical studies were carried out to investigate the performance of the developed approach. It is shown that it outperforms the existing methods.

Real time dynamic precise point positioning with mixture filtering

The linearisation of non-linear observation equations in precise point positioning (PPP) can always bring out additional errors. To weaken such linearisation errors and improve the performance of unscented Kalman filtering (UKF), a kind of mixture filtering based on combined UKF and extended Kalman filtering (EKF) is presented in this contribution. First, the PPP solutions estimated on the base of EKF are taken as predicted information for UKF, which can avoid the impossibility of matrix decomposition during UKF implementation; second, the accuracy of PPP solutions can be optimised by rerunning the UKF. An airborne test is used to demonstrate the performance of the new mixture filtering concept. The experimental results show that the accuracy of dynamic PPP with mixture filtering is better than their counterparts as estimated with EKF only.