Nonlinear Estimation for Autonomous Optical Navigation

Nonlinear state estimation is a pre-requisite for advanced process control and fault diagnosis tasks. In literature, various recursive nonlinear filtering techniques have been proposed and used for state estimation of nonlinear systems. Over the last few years, Moving Horizon Estimation (MHE) is increasingly being used for state estimation of nonlinear systems. Moving horizon estimation works with a window of data and hence requires additional online computation compared to recursive nonlinear filters. However, MHE performs both smoothing and filtering and thus has the potential to obtain more accurate state estimates as compared to the recursive filters. Most of the available comparisons of MHE with recursive filters are based on simulation case studies where the true states and parameters, as well as noise processes are exactly known. In this work, we apply MHE to a three-tank experimental setup and compare its performance with various nonlinear filters available in literature such as Extended Kalman Filter (EKF), Unscented Kalman Filter (UKF) and Gaussian Sum EKF (GSEKF). We present some of the challenges in experimental implementation of state estimation approaches. These include presence of unknown disturbances, non-whiteness of noise signals as well as lack of accurate measurements. We then discuss the approach followed for obtaining model parameters and noise characterization to make the models amenable for filter implementation. It is found that EKF, GSEKF and UKF perform as well as MHE as far as accuracy is concerned, but require significantly lower computational efforts.

Industrial process monitoring tools require robust and efficient estimation techniques that maintain a high service factor by remaining online during abnormal operating conditions, such as during loss of measurements, changes in control status, or maintenance. Constraints incorporate additional process knowledge into estimation by bounding estimated disturbances within feasibility limits thereby providing robustness to faulty measurements or conditions that violate process models. Moving horizon estimation (MHE) and unscented Kalman filtering (UKF) are two estimation techniques that permit incorporation of constraints prior to evaluating the a priori estimate. This paper evaluates both constrained nonlinear estimators versus the extended Kalman filter (EKF) using industrial process data provided by ExxonMobil Chemical Company. Results provide short-term insight into the fouling process, and parameter estimates produced by UKF and MHE are shown to be more accurate than EKF.

Accurate crop start date estimation is crucial for crop yield forecasting which is important not only for a government but also for agriculture-based or trading companies. The estimation can be done using the Normalized Difference Vegetation Index (NDVI) computed from radiant energy from the crops of interest. The NDVI collected from the Moderate Resolution Imaging Spectroradiometer (MODIS) aboard the Terra satellite is chosen in this study thanks to its free availability which is suitable for a developing country. In a tropical country as Thailand, the NDVI data is very noisy due to high density of clouds. An appropriate estimation technique must therefore be implemented. In this paper, the NDVI is modelled by a triply modulated cosine function with the mean, the amplitude and the initial phase as state variables. The state and the NDVI of single rice crop in the northeast Thailand are estimated using the Extended Kalman Filter (EKF), the Unscented Kalman Filter (UKF), the Moving Horizon Estimator (MHE) and the Moving Horizon Estimator with Pre-Estimation (MHE-PE). The MHE-PE, recently proposed in the literature, is an optimization-based estimator using an auxiliary estimator to describe the dynamics of the state over the horizon which has been shown to overcome the classical MHE strategy in terms of accuracy and computation time. The EKF and the MHE-PE provide the smallest start date estimation error compared to the others, which is 0 day in mean and 18 days in standard deviation. However, the EKF fail to detect the NDVI of preplant crops and parasite weeds while the MHE-PE does not.

This chapter deals with the problem of non-linear and non-Gaussian states and parameters estimation using Bayesian methods. The performances of various conventional and state-of-the-art state estimation techniques are compared when they are utilized to achieve this objective. These techniques include the Extended Kalman Filter (EKF), Unscented Kalman Filter (UKF), and Particle Filter (PF). In the current work, the authors consider two systems (biological model and power system) to perform evaluation of estimation algorithms. The results of the comparative studies show that the UKF provides a higher accuracy than the EKF due to the limited ability of EKF to accurately estimate the mean and covariance matrix of the estimated states through lineralization of the nonlinear process model. The results also show that the PF provides a significant improvement over the UKF because, unlike UKF, PF is not restricted by linear-Gaussian assumptions which greatly extends the range of problems that can be tackled.

We demonstrate experimentally that inference in a complex hybrid dynamic Bayesian network (DBN) is possible using the 2-time slice DBN (2T-DBN) from (D. Koller et al., Sequential Monte Carlo Methods in Practice: p.445-464, Springer-Verlag, NY, 2000) to model fault detection in a watertank system. In this, a generic particle filter (PF) is used for inference. We extend the experiment and perform approximate inference using The extended Kalman filter (EKF) and the unscented Kalman filter (UKF). Furthermore, we combine these techniques in a 'non-strict' Rao-Blackwellisation framework and apply it to the watertank system. We show that UKF and UKF in a PF framework outperform the generic PF, EKF and EKF in a PF framework with respect to accuracy and robustness in terms of estimation RMSE (root-mean-square error). Especially, we demonstrate the superiority of UKF in a PF framework when our beliefs of how data was generated are wrong. We also show that the choice of network structure is very important for the performance of the generic PF and the EKF algorithms, but not for the UKF algorithms. Furthermore, we investigate the influence of data noise in the watertank simulation. Theory and implementation is based on the theory presented in (R. v.d. Merwe et al., Tech. Rep. CUED/F-INFENG/TR-380, Dept. of Eng., Cambridge Uni., 2000).

State of Charge (SOC) estimation plays a crucial role in managing the performance and longevity of electric vehicle (EV) batteries. Various estimation techniques, including various Kalman Filters (KF), have been employed for this purpose. The mathematical process and limits of different KF family algorithms are analysed. In this study, an impact of nonlinear KF parameters on SOC estimation accuracy for EV batteries is analyzed. Specifically, the performance of Extended Kalman Filter (EKF), Unscented Kalman Filter (UKF), and Particle Filter (PF) is examined. EKF is most preferred for SOC estimation whereas introducing sigma points in UKF is the basic difference between EKF and UKF. In comparison to EKF and UKF, PF is more flexible since it applies to nonlinear systems with non-Gaussian distributions. Utilising a sequence of noisy measurements, KF is a recursive method that approximates the state of a dynamic system. It requires accurate parameters such as process noise covariance (Q) and measurement noise covariance (R) to work effectively. In this work, the effect of these parameters on state of charge estimation using nonlinear KFs has been analysed under different settings of noise covariance matrices. Tuning these parameters is essential for optimizing the performance of the KF and ensuring accurate and reliable state estimation in real-world applications. To find parameter values that provide the best balance between accuracy, robustness, and computational efficiency, the tuning process involves experimentation, simulation, and validation using real-world data. According to the results observed, the filter parameters such as three diagonal elements of Q are selected as 2e-8, 0.005, 0.003 and R as 2e-6. By evaluating the state estimation accuracy under different driving scenarios, robustness and sensitivity of filtering technique in estimating SOC for EV batteries has been studied. This analysis contributes to the optimization of SOC estimation algorithms, enhancing the reliability and efficiency of EV battery management systems.

The focus of this work is the extension of nonlinear state estimation methods to gas-lifted systems. The extended Kalman filter (EKF), unscented Kalman filter (UKF) and particle filter (PF) were used to estimate the nonlinear states. Brief descriptions of the filters were first presented starting from the linear Kalman filter. Hypothesis tests on the expectation of the residuals were performed to show how close to optimal the estimation methods are and it showed the UKF estimates to be slightly better than EKF while PF performs the worst. The PF has poor accuracy using residual visualisation, hypothesis test and the root mean squared error (RMSE) values of the residuals. The gas-lifted system exhibits casing heading instability where the states show oscillatory behaviour depending on the value of the input but the results here do not change in a known way for each filter as the input is changed from the non-oscillatory region to the oscillatory region. Therefore, for this noise distribution and model assumption, either the EKF or UKF can be used for nonlinear state estimation with UKF better preferred if computational cost is not considered when control solutions are used in gas-lifted system.

Computation of arrival cost for moving horizon estimation via unscented Kalman filtering

Several types of nonlinear filters (EKF – extended Kalman filter, UKF – unscented Kalman filter, PF – particle filter) are widely used for location estimation and their algorithms are described in this paper. In the article filtering accuracy for non-linear form of measurement equation is presented. The results of complex simulations that compare the quality of estimation of analyzed non-linear filters for complex non-linearities of state vector are presented. The moves of maneuvering object are described in two-dimensional Cartesian coordinates and the measurements are described in the polar coordinate system. The object dynamics is characterized by acceleration described by the univariate non-stationary growth model (UNGM) function. The filtering accuracy was evaluated not only by the root-mean-square errors (RMSE) but also by statistical testing of innovations through the expected value test, the whiteness test and the WSSR (weighted sum squared residual) test as well. The comparison of filtering quality was done in the MATLAB environment. The presented results provide a basis for designing more accurate algorithms for object location estimation.

Combined Kalman and sliding innovation filtering: An adaptive estimation strategy

This paper presents the efficient Unscented Kalman filter (UKF) based estimation and control of a 3-DOF Omni bundle robot. The angular trajectory tracking becomes more difficult under highly non-linear measurement noise and process noise. A rapidly converging Unscented Kalman Filter (UKF) is presented in the paper. The extended Kalman filter (EKF) provides less accurate results in the highly non-linear system and sometime may become unstable because of convergence related issues, whereas in UKF the results in the non-linear system are better than EKF. UKF is more accurate in state and noise estimation. The computed torque technique is taken as a model to apply UKF. After, estimation the noise is removed. The experiment is conducted on Omni bundle with process and measurement noise. The results prove the superiority of presented UKF over conventional UKF and EKF.

Induction machine is highly nonlinear model with states that change with operating point and temperature. In these cases, estimating these variables from other easily obtained measurements can be extremely useful. This paper deals with the problem of state estimation of induction machine on the basis of a third-order electrical model using Bayesian methods. The performances of Bayesian estimation techniques are compared when they are utilized to achieve this objective. These techniques include the extended Kalman filter (EKF), the unscented Kalman filter (UKF), the particle filter (PF), and the developed improved particle filter (IPF). The estimation results, which are validated using simulations, show that IPF provides improved estimation performance over PF, even with abrupt changes in estimated states, and both of them can provide improved accuracy over UKF and EKF. These advantages of the IPF are due to the fact that it uses a better proposal distribution that takes the latest observation into account.

PurposeAutonomous orbit determination using geomagnetic measurements is an important backup technique for safe spacecraft navigation with a mere magnetometer. The geomagnetic model is used for the state estimation of orbit elements, but this model is highly nonlinear. Therefore, many efforts have been paid to developing nonlinear filters based on extended Kalman filter (EKF) and unscented Kalman filter (UKF). This paper aims to analyze whether to use UKF or EKF in solving the geomagnetic orbit determination problem and try to give a general conclusion.Design/methodology/approachThis paper revisits the problem and from both the theoretical and engineering results, the authors show that the EKF and UKF show identical estimation performances in the presence of nonlinearity in the geomagnetic model.FindingsWhile EKF consumes less computational time, the UKF is computationally inefficient but owns better accuracy for most nonlinear models. It is also noted that some other navigation techniques are also very similar with the geomagnetic orbit determination.Practical implicationsThe intrinsic reason of such equivalence is because of the orthogonality of the spherical harmonics which has not been discovered in previous studies. Thus, the applicability of the presented findings are not limited only to the major problem in this paper but can be extended to all those schemes with spherical harmonic models.Originality/valueThe results of this paper provide a fact that there is no need to choose UKF as a preferred candidate in orbit determination. As UKF achieves almost the same accuracy as that of EKF, its loss in computational efficiency will be a significant obstacle in real-time implementation.

Surveillance of underwater for maritime warfare is traditionally being carried out by bearings-only tracking from many decades. The measurements used for state estimation here are nonlinear. Also the noise in the measurements and the process cannot be always Gaussian. The traditional nonlinear filtering algorithms like extended Kalman filter and modified gain extended Kalman filter use the linearisation of the system. The unscented Kalman filter (UKF) uses the sigma point approach based on Gaussian distribution to deal with nonlinearity. The particle filter (PF) uses the randomly generated particles based on the pdf of the state. PF is highly complex to implement and it also suffers from sample impoverishment. Hence, ensemble Kalman filter (EnKF) which is a simplified form of PF will be tried out for bearings-only tracking in this research work. The performance of EnKF is compared with PF and UKF and the results obtained using these filters in Matlab are presented.

This paper studies three-dimensional multi-robot Cooperative Localization (CL) problem. Most of existing CL strategies adopt Extended Kalman Filter (EKF) or Maximum a Posteriori (MAP). In this paper, a novel approach based on Moving Horizon Estimation (MHE) is proposed. The main contribution of this paper is twofold: 1) MHE is integrated with EKF for three-dimensional CL using single mobile beacon, which can bound localization error, impose various constraints on states and noises, and make use of previous range mea- surements for current estimation. 2) A sufficient condition on observability of multi-robot CL is derived by using Fisher Information Matrix. Simulation is conducted to verify that the proposed MHE based CL algorithm outperforms EKF based method in terms of localization accuracy, and two scenarios where our algorithm is superior to EKF are discussed.