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A New Approach for Filtering Nonlinear Systems
Abstract
In this paper we describe a new recursive linear estimator for
filtering systems with nonlinear process and observation models. This
method uses a new parameterisation of the mean and covariance which can
be transformed directly by the system equations to give predictions of
the transformed mean and covariance. We show that this technique is more
accurate and far easier to implement than an extended Kalman filter.
Specifically, we present empirical results for the application of the
new filter to the highly nonlinear kinematics of maneuvering vehicles
... However, if the system is nonlinear, a parametrization method is not applicable and approximations are required. One widely used approach is the Extended Kalman Filter (EKF), which adopts a local linearization method around the predicted mean [48,49]. Among other drawbacks, severe nonlinearities in the system can significantly compromise the performance of the EKF. ...
... To avoid local linearization, the Unscented Kalman Filter and the Square Root Unscented Kalman filter approximate p x i |Y k by an ensemble of deterministically chosen weighted sample points denoted as χ in Algorithm 1 [27,49]. The weights of the sample points W (m) and W (c) are defined by the scaling parameters λ = n(α 2 − 1), η = (n + λ) where n is the is dimension of the system state space. ...
... For Gaussian distributions, β KF is optimally chosen as β KF = 2 [27]. When the methodologically weighted same points are propagated through the nonlinear transformation, denoted by φ(.), the true mean and covariance of the original distribution can be implemented for state estimation [49]. In comparison with UKF, the SRUKF provides better numerical stability by avoiding the computational step of the Cholesky factor at each time iteration. ...
Harsh operating conditions imposed by vehicular applications significantly limit the utilization of proton exchange membrane fuel cells (PEMFCs) in electric propulsion systems. Improper/poor management and supervision of rapidly varying current demands can lead to undesired electrochemical reactions and critical cell failures. Among other failures, flooding and catalytic degradation are failure mechanisms that directly impact the composition of the membrane electrode assembly and can cause irreversible cell performance deterioration. Due to the functional significance and high manufacturing costs of the catalyst layer, monitoring internal fuel cell states is crucial. For this purpose, a diagnostic-oriented multi-scale PEMFC catalytic degradation model is developed which incorporates the failure effects of catalytic degradation on cell dynamics and global stack performance. Embedded to the multi-scale model is a square root unscented Kalman filter (SRUKF)-based multiple-model fault diagnosis scheme. In this approach, multiple models are used to estimate specific internal PEMFC system parameters, such as the mass transfer coefficient of the gas diffusion layer or the exchange current density, which are treated as additional system states. Online state estimates are provided by the SRUKF, which additionally propagates model-conditioned statistical information to update a Bayesian framework for model selection. The Bayesian model selection method carries fault indication signals that are interpreted by a derived decision logic to obtain reliable information on the current-operating system regime. The proposed diagnosis scheme is evaluated through simulations using the LA 92 and NEDC driving cycles.
... Despite its simplicity this method has been a successful filtering means in the realm of nonlinear stochastic systems for decades (see Goodwin and Sin (1984); Grewal and Andrews (2001); Jazwinski (1970); Lewis (1986); Simon (2006)). Nevertheless, the first-order approximation provided by the EKF has been criticized in many studies resulted in the developments of the more accurate UKF designed in Julier et al. (1995Julier et al. ( , 2000; Wan and Van der Merwe (2001); Uhlmann (2002, 2004) and the CKF of Arasaratnam and Haykin (2009); Arasaratnam et al. (2010). We point out that a lot of evidence confirming the superiority of the latter filtering methods towards the EKF in estimating various continuous-and discrete-time nonlinear stochastic state-space systems have been presented in the above-cited papers. ...
... The Unscented Kalman Filtering (UKF) originates from the paper of Julier et al. (1995), which constructs the method for estimating discrete-time nonlinear stochastic systems. Later on, various issues related to the UKF have been explored by many authors, including Julier et al. (2000); Wan and Van der Merwe (2001); Uhlmann (2002, 2004); Gustafsson and Hendeby (2012); Morelande and García-Fernández (2013); Menegaz et al. (2015) and so on. ...
... Later on, various issues related to the UKF have been explored by many authors, including Julier et al. (2000); Wan and Van der Merwe (2001); Uhlmann (2002, 2004); Gustafsson and Hendeby (2012); Morelande and García-Fernández (2013); Menegaz et al. (2015) and so on. At the heart of the UKF is the Unscented Transform (UT) introduced by Julier et al. (1995Julier et al. ( , 2000; Uhlmann (2002, 2004). Here, the utilized UT is based on 2n + 1 deterministically selected sigma points (vectors) X i calculated by the rule ...
This brief technical note elaborates three well-known state estimators, which are used extensively in practice. These are the rather old-fashioned extended Kalman filter (EKF) and the recently-designed cubature Kalman filtering (CKF) and unscented Kalman filtering (UKF) algorithms. Nowadays, it is commonly accepted that the contemporary techniques outperform always the traditional EKF in the accuracy of state estimation because of the higher-order approximation of the mean of propagated Gaussian density in the time- and measurement-update steps of the listed filters. However, the present paper specifies this commonly accepted opinion and shows that despite the mentioned theoretical fact the EKF may outperform the CKF and UKF methods in the accuracy of state estimation when the stochastic system under consideration exposes a stiff behavior. That is why stiff stochastic models are difficult to deal with and require effective state estimation techniques to be designed yet.
... The state x comprises physical quantities (temperature, wind speed, air pressure, etc.) at many spatially distributed grid points, which often yields a state dimension n in the order of millions. Consequently, the Kalman filter (KF) [2,3] or its nonlinear extensions [4,5] that require the storage and processing of n × n covariance matrices cannot be applied directly. It is well-known that the application of particle filters [6,7] is not feasible either. ...
... A literature review highlights important EnKF papers with their respective contributions, and facilitates an easier access to the extensive and rapidly developing DA literature on the EnKF. Moreover, we put the EnKF in context with popular SP algorithms such as sigma point filters [4,5] and the particle filter [6,7]. Our presentation forms a solid basis for further developments and the transfer of beneficial ideas and techniques between the fields of SP and DA. ...
... With a leap in computing capacity, the 1990s saw major developments. The samplingbased sigma point Kalman filters [4,5] started to appear. Furthermore, particle filters [6,7] were developed to approximately implement the Bayesian filtering equations through sequential importance sampling. ...
The ensemble Kalman filter (EnKF) is a Monte Carlo based implementation of the Kalman filter (KF) for extremely high-dimensional, possibly nonlinear and non-Gaussian state estimation problems. Its ability to handle state dimensions in the order of millions has made the EnKF a popular algorithm in different geoscientific disciplines. Despite a similarly vital need for scalable algorithms in signal processing, e.g., to make sense of the ever increasing amount of sensor data, the EnKF is hardly discussed in our field. This self-contained review paper is aimed at signal processing researchers and provides all the knowledge to get started with the EnKF. The algorithm is derived in a KF framework, without the often encountered geoscientific terminology. Algorithmic challenges and required extensions of the EnKF are provided, as well as relations to sigma-point KF and particle filters. The relevant EnKF literature is summarized in an extensive survey and unique simulation examples, including popular benchmark problems, complement the theory with practical insights. The signal processing perspective highlights new directions of research and facilitates the exchange of potentially beneficial ideas, both for the EnKF and high-dimensional nonlinear and non-Gaussian filtering in general.
... As the performance of EKF degrades for systems with strong nonlinearities, researchers have been seeking better ways to conduct nonlinear state estimation. In the 1990s, UKF was invented [21,22]. Since then, it has been gaining significant popularity among researchers and practitioners. ...
... The width of spread is dependent on the covariance P and the scaling parameter λ, where λ = α 2 (n + κ) − n. Typically, α is a small positive value (e.g., 10 −3 ), and κ is usually set to 0 or 3 − n [21]. Then the sigma points are propagated through the nonlinear function g(·) to generate the sigma points for the transformed variable z, i.e., z i = g x i , i = 0, 1, · · · , 2n. ...
This article presents an up-to-date tutorial review of nonlinear Bayesian estimation. State estimation for nonlinear systems has been a challenge encountered in a wide range of engineering fields, attracting decades of research effort. To date, one of the most promising and popular approaches is to view and address the problem from a Bayesian probabilistic perspective, which enables estimation of the unknown state variables by tracking their probabilistic distribution or statistics (e.g., mean and covariance) conditioned on the system's measurement data. This article offers a systematic introduction of the Bayesian state estimation framework and reviews various Kalman filtering (KF) techniques, progressively from the standard KF for linear systems to extended KF, unscented KF and ensemble KF for nonlinear systems. It also overviews other prominent or emerging Bayesian estimation methods including the Gaussian filtering, Gaussian-sum filtering, particle filtering and moving horizon estimation and extends the discussion of state estimation forward to more complicated problems such as simultaneous state and parameter/input estimation.
... for g 1 , g 2 ∈ {f , h} need to be computed, a task that is rarely possible analytically. In engineering literature, the most prevalent numerical integration methods for approximating these integrals are the fully symmetric cubature integration formulas of McNamee and Stenger [17], used in the unscented transform of Julier et al. [18]. See [13] and [14,Ch. ...
Most Kalman filters for non-linear systems, such as the unscented Kalman filter, are based on Gaussian approximations. We use Poincar\'e inequalities to bound the Wasserstein distance between the true joint distribution of the prediction and measurement and its Gaussian approximation. The bounds can be used to assess the performance of non-linear Gaussian filters and determine those filtering approximations that are most likely to induce error.
... Although this assumption is valid for Brewer spectrometers, the GUF provides less accurate estimations than other uncertainty propagation techniques (González et al., 2024b), such as the Monte Carlo method (hereafter "MCM") 70 and the Unscented transformation (hereafter "UT"). The UT is an efficient technique that evaluates the uncertainty by applying the nonlinear model to a reduced set of points, referred to as sigma points (Julier et al., 1995;Julier and Uhlmann, 1997). These sigma points are carefully chosen using several parameters to ensure their statistics (first and second order) match those of the measurand. ...
Brewer spectroradiometers are robust, widely used instruments that have been monitoring global solar ultraviolet (UV) irradiance since the 1990s, playing a key role in solar UV research. Unfortunately, the uncertainties of these measurements are rarely evaluated due to the difficulties involved in the uncertainty propagation. This evaluation is essential to determine the quality of the measurements as well as their comparability to other measurements. In this study, eight double- and two single-monochromator Brewers are characterised and the uncertainty of their global UV measurements is estimated using the Monte Carlo method. This methodology is selected as it provides reliable uncertainty estimations and considers the nonlinearity of the UV processing algorithm. The combined standard uncertainty depends on the Brewer, varying between 2.5 % and 4 % for the 300–350 nm region. For wavelengths below 300 nm, the differences between single- and double-monochromator Brewers increase, due to stray light and dark counts. For example, at 295 nm, the relative uncertainties of single Brewers range between 11–14 % while double Brewers have uncertainties of 4–7 %. These uncertainties arise primarily from radiometric stability, the application of cosine correction, and the irradiance of the lamp used during the instrument calibration. As the intensity of the UV irradiance measured decreases, dark counts, stray light (for single Brewers), and noise become the dominant sources of uncertainty. These results indicate that the overall uncertainty of a Brewer spectroradiometer could be greatly reduced by increasing the frequency of radiometric calibration and improving the traditional entrance optics.
... However, in environments where model uncertainty increases due to motor noise and 6DoF motion, such as in this experiment, the performance of EKF cannot be guaranteed. Therefore, the UKF (Unscented Kalman Filter) was considered a more suitable filter for environments like the one in this study [11][12][13][14]. ...
This study introduces a pose estimation algorithm integrating an Inertial Navigation System (INS) with an Alternating Current (AC) magnetic field-based navigation system, referred to as the Magnetic Positioning System (MPS), evaluated using a 6 Degrees of Freedom (DoF) drone. The study addresses significant challenges such as the magnetic vector distortions and model uncertainties caused by motor noise, which degrade attitude estimation and limit the effectiveness of traditional Extended Kalman Filter (EKF)-based fusion methods. To mitigate these issues, a Tightly Coupled Unscented Kalman Filter (TC UKF) was developed to enhance robustness and navigation accuracy in dynamic environments. The proposed Unscented Kalman Filter (UKF) demonstrated a superior attitude estimation performance within a 6 m coil spacing area, outperforming both the MPS 3D LS (Least Squares) and EKF-based approaches. Furthermore, hyperparameters such as alpha, beta, and kappa were optimized using the Sequential Importance Resampling (SIR) process of the Particle Filter. This adaptive hyperparameter adjustment achieved improved navigation results compared to the default UKF settings, particularly in environments with high model uncertainty.
... Variations of the Kalman filter have been implemented to account for non-linear trajectories, including the common Extended Kalman Filter [37] and Unscented Kalman Filter [38]. These traditional methods for target tracking have been adapted in recent years for unmanned surface vehicles so they can avoid obstructions as they autonomously navigate on the water [39][40][41][42]. ...
Vessel speed reduction measures are a management tool used to reduce the risk of whale–ship strikes and mitigate their impacts. Large ships and other commercial vessels are required to publicly share tracking information, including their speed, via the Automatic Identification System (AIS), which is commonly used to evaluate compliance with these measures. However, smaller vessels are not required to carry AIS and therefore are not as easily monitored. Commercial off-the-shelf marine radar is a practical solution for independently tracking these vessels, although commercial target tracking is typically a black-box process, and the accuracy of reported speed is not available in manufacturer specifications. We conducted a large-scale measurement campaign to estimate radar-reported speed error by comparing concurrent radar- and AIS-reported values. Across 3097 unique vessel tracks from ten locations, there was strong correlation between radar and AIS speed, and radar values were within 1.8 knots of AIS values 95% of the time. Smaller vessels made up a large share of the analyzed tracks, and there was no significant difference in error compared to larger vessels. The results provide error bounds around radar-reported speeds that can be applied to vessels of all sizes, which can inform vessel-speed-monitoring efforts using radar.
... To this end, one may employ the Kalman filter [58], a well-established method for state estimation that recursively updates state estimates by combining information from previous and current measurements. In the literature, various extensions of the Kalman filter have been proposed to improve estimation accuracy [59,60]. In this work, we utilize the RTS smoother, further refining the state estimates by incorporating information from future measurements alongside past and current data. ...
In this paper, we address the identification problem for the systems characterized by linear time-invariant dynamics with bilinear observation models. More precisely, we consider a suitable parametric description of the system and formulate the identification problem as the estimation of the parameters defining the mathematical model of the system using the observed input-output data. To this end, we propose two probabilistic frameworks. The first framework employs the Maximum Likelihood (ML) approach, which accurately finds the optimal parameter estimates by maximizing a likelihood function. Subsequently, we develop a tractable first-order method to solve the optimization problem corresponding to the proposed ML approach. Additionally, to further improve tractability and computational efficiency of the estimation of the parameters, we introduce an alternative framework based on the Expectation--Maximization (EM) approach, which estimates the parameters using an appropriately designed cost function. We show that the EM cost function is invex, which ensures the existence and uniqueness of the optimal solution. Furthermore, we derive the closed-form solution for the optimal parameters and also prove the recursive feasibility of the EM procedure. Through extensive numerical experiments, the practical implementation of the proposed approaches is demonstrated, and their estimation efficacy is verified and compared, highlighting the effectiveness of the methods to accurately estimate the system parameters and their potential for real-world applications in scenarios involving bilinear observation structures.
... Statistical linearization is used by the unscented Kalman filter (UKF) instead of the mathematical linearization of the EKF. The unscented transform [78], which resembles a nonlinear function using a collection of points selected systematically to guarantee that higher-order components in the nonlinear function's Taylor series are approximated, is used in the statistical linearization process. To simulate the higher-order aspects of the underlying uncertainty in this case, more points can be employed [79]. ...
Citation: Sagar, M.M.; Konara, M.; Picard, N.; Park, K. State-of-the-Art Navigation Systems and Sensors for Unmanned Underwater Vehicles (UUVs). Appl. Mech. 2025, 6, 10. Abstract: Researchers are currently conducting several studies in the field of navigation systems and sensors. Even in the past, there was a lot of research regarding the field of velocity sensors for unmanned underwater vehicles (UUVs). UUVs have various services and significance in the military, scientific research, and many commercial applications due to their autonomy mechanism. So, it's very crucial for the proper maintenance of the navigation system. Reliable navigation of unmanned underwater vehicles depends on the quality of their state determination. There are so many navigation systems available, like position determination, depth information, etc. Among them, velocity determination is now one of the most important navigational criteria for UUVs. The key source of navigational aids for different deep-sea research projects is water currents. These days, many different sensors are available to monitor the UUV's velocity. In recent times, there have been five primary types of sensors utilized for UUV velocity forecasts. These include Doppler Velocity Logger sensors, paddlewheel sensors, optical sensors, electromagnetic sensors, and ultrasonic sensors. The most popular sensing sensor for estimating velocity at the moment is the Doppler Velocity Logger (DVL) sensor. DVL sensor is the most fully developed sensor for UUVs in recent years. In this work, we offer an overview of the field of navigation systems and sensors (especially velocity) developed for UUVs with respect to their use with tidal current sensing in the UUV setting, including their history, evolution, current research initiatives, and anticipated future.
... The Unscented Transform (UT), proposed by Julier and Uhlmann [36], aims to create nonlinear filters without the need for linearization. It approximates a probability distribution function (PDF) after it passes through a non-linear transformation using a set of sampled points, known as sigma points [7], [37]. ...
The classical Model Predictive Path Integral (MPPI) control framework, while effective in many applications, lacks reliable safety features since it relies due to its reliance on a
risk-neutral
trajectory evaluation technique, which can present challenges for safety-critical applications such as autonomous driving. Furthermore, if when the majority of MPPI sampled trajectories concentrate in high-cost regions, it may generate an
infeasible
control sequence. To address this challenge, we propose the U-MPPI control strategy, a novel methodology that can effectively manage system uncertainties while integrating a more efficient trajectory sampling strategy. The core concept is to leverage the Unscented Transform (UT) to propagate not only the mean but also the covariance of the system dynamics, going beyond the traditional MPPI method. As a result, it introduces a novel and more efficient trajectory sampling strategy, significantly enhancing state-space exploration and ultimately reducing the risk of being trapped in local minima. Furthermore, by leveraging the uncertainty information provided by UT, we incorporate a
risk-sensitive
cost function that explicitly accounts for risk or uncertainty throughout the trajectory evaluation process, resulting in a more resilient control system capable of handling uncertain conditions. By conducting extensive simulations of 2D aggressive autonomous navigation in both known and unknown cluttered environments, we verify the efficiency and robustness of our proposed U-MPPI control strategy compared to the baseline MPPI. We further validate the practicality of U-MPPI through real-world demonstrations in unknown cluttered environments, showcasing its superior ability to incorporate both the UT and local costmap into the optimization problem without introducing additional complexity.
... However, the local linearity assumption is often violated, and derivation of the Jacobian matrix is non-trivial and error prone. The Unscented Transform (UT) [16,17] was proposed to address these limitations. The key idea of UT is to approximate the distribution of the random variable using a set of Sigma points that can be transformed exactly, after which they can be used to re-estimate the statistics of the random variable in the target domain. ...
3D Gaussian Splatting (3DGS) has shown great potential for efficient reconstruction and high-fidelity real-time rendering of complex scenes on consumer hardware. However, due to its rasterization-based formulation, 3DGS is constrained to ideal pinhole cameras and lacks support for secondary lighting effects. Recent methods address these limitations by tracing volumetric particles instead, however, this comes at the cost of significantly slower rendering speeds. In this work, we propose 3D Gaussian Unscented Transform (3DGUT), replacing the EWA splatting formulation in 3DGS with the Unscented Transform that approximates the particles through sigma points, which can be projected exactly under any nonlinear projection function. This modification enables trivial support of distorted cameras with time dependent effects such as rolling shutter, while retaining the efficiency of rasterization. Additionally, we align our rendering formulation with that of tracing-based methods, enabling secondary ray tracing required to represent phenomena such as reflections and refraction within the same 3D representation.
... Then, we generate a set of sigma points by applying the Unscented transform (Julier et al., 1995) to the augmented state, X a 0 (k) =x a k X a i (k) =x a k + (L + λ) P a k i = 1, . . . , L X a i (k) =x a k − (L + λ) P a k i = L + 1, . . . ...
One of the most challenging tasks for a flying robot is to autonomously navigate between target locations quickly and reliably while avoiding obstacles in its path, and with little to no a-priori knowledge of the operating environment. This challenge is addressed in the present paper. We describe the system design and software architecture of our proposed solution, and showcase how all the distinct components can be integrated to enable smooth robot operation. We provide critical insight on hardware and software component selection and development, and present results from extensive experimental testing in real-world warehouse environments. Experimental testing reveals that our proposed solution can deliver fast and robust aerial robot autonomous navigation in cluttered, GPS-denied environments.
... The non-linearities that characterize both (22) and (23) imply that an analytical evaluation of the auxiliary likelihood via the Kalman filter (KF) is not feasible. Therefore, we use the augmented unscented KF (AUKF) (see Julier et al., 1995) as an computationally efficient means of evaluating the L a (y; β) and, hence, of producing the auxiliary score as the matching statistic within ABC. The precise form of the auxiliary likelihood function thus depends on both the first-order Euler discretization of the continuous-time state process and the particular specifications used to implement the AUKF. ...
A computationally simple approach to inference in state space models is proposed, using approximate Bayesian computation (ABC). ABC avoids evaluation of an intractable likelihood by matching summary statistics for the observed data with statistics computed from data simulated from the true process, based on parameter draws from the prior. Draws that produce a 'match' between observed and simulated summaries are retained, and used to estimate the inaccessible posterior. With no reduction to a low-dimensional set of sufficient statistics being possible in the state space setting, we define the summaries as the maximum of an auxiliary likelihood function, and thereby exploit the asymptotic sufficiency of this estimator for the auxiliary parameter vector. We derive conditions under which this approach - including a computationally efficient version based on the auxiliary score - achieves Bayesian consistency. To reduce the well-documented inaccuracy of ABC in multi-parameter settings, we propose the separate treatment of each parameter dimension using an integrated likelihood technique. Three stochastic volatility models for which exact Bayesian inference is either computationally challenging, or infeasible, are used for illustration. We demonstrate that our approach compares favorably against an extensive set of approximate and exact comparators. An empirical illustration completes the paper.
... We also propose a particle filter (PF) algorithm [11] based on the VMF measurement error mode, and EKF [12,Ch. 8.3] and UKF [13] algorithms that approximate the VMF update with the assumption that the unit vector measurement is the true direction's unit vector plus a trivariate normal noise. Our simulations show that the proposed positioning algorithms outperform the conventional algorithms in accuracy. ...
We propose modeling an angle-of-arrival (AOA) positioning measurement as a von Mises-Fisher (VMF) distributed unit vector instead of the conventional normally distributed azimuth and elevation measurements. Describing the 2-dimensional AOA measurement with three numbers removes discontinuities and reduces nonlinearity at the poles of the azimuth-elevation coordinate system. Our computer simulations show that the proposed VMF measurement noise model based filters outperform the normal distribution based algorithms in accuracy in a scenario where close-to-pole measurements occur frequently.
... This research has resulted in many approaches to the state-estimation problem including the much celebrated Kalman filter [6], extended Kalman filter [7], Unscented Kalman filter [8] and Sequential Monte-Carlo (SMC) approaches [9]. Each of these variants exploits different structural elements of the state-space model and each has known strengths and associated weaknesses. ...
A Bayesian filtering algorithm is developed for a class of state-space systems that can be modelled via Gaussian mixtures. In general, the exact solution to this filtering problem involves an exponential growth in the number of mixture terms and this is handled here by utilising a Gaussian mixture reduction step after both the time and measurement updates. In addition, a square-root implementation of the unified algorithm is presented and this algorithm is profiled on several simulated systems. This includes the state estimation for two non-linear systems that are strictly outside the class considered in this paper.
... Moreover, with increasing system nonlinearity, the EKF may face challenges in terms of the estimation accuracy and robustness [73]. To address this, the unscented Kalman filter (UKF) was developed [74,75]. Unlike the EKF's linearization method, the UKF uses the unscented transform (UT) to handle nonlinear systems, achieving an accuracy comparable to that of a second-order Taylor expansion linearization [76]. ...
Utilization of ocean resources and defense of national security heavily rely on underwater target tracking technology, which consequently holds significant strategic importance. The passive tracking technology for underwater target bearings, known for its extensive detection range, capability for long-term observation, and robust real-time capabilities, has emerged as a new focal point of research. This paper reviews the essential concepts, research developments, applications, and limitations of key technologies for passive underwater target bearing tracking, concentrating on three main areas: underwater target bearing estimation technology, target tracking technology, and comprehensive underwater target bearing tracking technology. Specifically, it discusses highly robust methods for tracking single or multiple underwater targets. Ultimately, this paper highlights the primary challenges currently facing research in this field and provides a perspective on future developments.
... Given a nominal catalog maintenance sensor-tasking plan P for 3600 seconds, with a time step of 60 seconds, the MDH-BMDP problemP seeks to determine the correct hypothesis with probability = 0.8 within 1600 seconds, with minimal change to the original plan. In this problem, when transitioned into an MDH-BMDP, the belief over the OOI state is approximated as a Gaussian distribution and propagated using an unscented Kalman filter (UKF) [15], one for each hypothesis. ...
When human operators of cyber-physical systems encounter surprising behavior, they often consider multiple hypotheses that might explain it. In some cases, taking information-gathering actions such as additional measurements or control inputs given to the system can help resolve uncertainty and determine the most accurate hypothesis. The task of optimizing these actions can be formulated as a belief-space Markov decision process that we call a hypothesis-driven belief MDP. Unfortunately, this problem suffers from the curse of history similar to a partially observable Markov decision process (POMDP). To plan in continuous domains, an agent needs to reason over countlessly many possible action-observation histories, each resulting in a different belief over the unknown state. The problem is exacerbated in the hypothesis-driven context because each action-observation pair spawns a different belief for each hypothesis, leading to additional branching. This paper considers the case in which each hypothesis corresponds to a different dynamic model in an underlying POMDP. We present a new belief MDP formulation that: (i) enables reasoning over multiple hypotheses, (ii) balances the goals of determining the (most likely) correct hypothesis and performing well in the underlying POMDP, and (iii) can be solved with sparse tree search.
... The UT is a method of estimating the statistical properties of variables by a finite set of parameters which was originally proposed in [10][11][12]. This method is more straightforward to implement than the traditional method of linear approximation methods for nonlinear mappings [11,13]. ...
Due to the complexity of the transmission tower structure and the correlation between wind and ice loads in the actual project, it is difficult to analyze the reliability of transmission towers with traditional methods. To solve this problem, the unscented transformation (UT) principle is presented concisely and used in the reliability analysis of transmission towers in this paper. Moreover, the finite element model of the target transmission tower is created. The reliability indices of the transmission tower under various loading cases are evaluated using UT and analyzed relative to the outcomes of the Monte Carlo method (MCS). Lastly, by analyzing and validating a wine-cup shape tangent tower, the simulation results show that the UT yields reliability indices with less than 6% relative error compared with MCS results for the transmission towers with lower reliability, which are more important in engineering. Variations in error caused by the change in correlation coefficients among variables are small. Consequently, the efficiency of calculations is improved by the UT-based reliability calculations for transmission towers in the case of correlated variables, which better meet engineering application requirements. It is proved that the method of reliability analysis for transmission towers based on the UT is applicable.
A method for integrated online identification of aerodynamic and thrust parameters has been proposed. Firstly, a motion model of an aircraft powered by a solid rocket ramjet engine was established in the longitudinal plane. Secondly, polynomial models of aerodynamic and thrust coefficients were established for the horizontal flight segment. Then, an integrated online identification simulation of aerodynamic and thrust parameters was carried out using the unscented Kalman filter method. The results indicate that the proposed method has high identification accuracy and efficiency.
The rise of CubeSats has highlighted the need for mission analysis that integrates
uncertainties inside the optimization phase. This work incorporates uncertainties
in initial conditions, maneuvers, and navigation into a stochastic trajectory optimization
process. It uses a novel implementation of the Unscented Transformation,
avoiding sigma points resampling by employing a continuous propagation
map. The problem is formulated as a nonlinear programming problem and applied
to Weak Stability Boundary trajectories, with the LUMIO mission as a case
study. The method yields more fuel-efficient and robust trajectories compared to
deterministic optimization. Monte Carlo analyses validate the effectiveness and
accuracy of the approach.
Results of using a state-space cohort model with smoothing unscented Kalman filter for assessment of Pacific cod stock abundance and the other population parameters in the Petropavlovsk-Komandorskaya Subzone are presented. Series of numerical experiments were carried out with various variants of the recruitment model. The results of the work performed can be used in preparing forecasts of the TAC of marine commercial fish in terms that the observational data allow the use of age-structured models in the state space.
Although the LiDAR SLAM technique has been already widely deployed on various robots, it may still suffers from degeneracy caused by inadequate constraints in scenes with sparse geometric features. If the degeneracy is not detected and properly processed, the accuracy of localization and mapping will significantly decrease. In this letter, we propose the P2d-DO method, which consists of a point-to-distribution degeneracy detection algorithm and a point cloud-weighted degeneracy optimization algorithm, to relieve the negative impact of degeneracy. The degeneracy detection algorithm outputs factors that characterize the degeneracy state by observing changes in the distribution probabilities within a local region. Factors reflecting the confidence of the point clouds are then fed to the degeneracy optimization algorithm, enabling the system to prioritize reliable point clouds by assigning larger weights during the matching process. Comprehensive experiments validate the effectiveness of our method, demonstrating significant improvements in both degeneracy detection and pose estimation in terms of accuracy and robustness.
Рассматривается система нелинейных стохастических функциональноразностных уравнений с ограниченным запаздыванием. Предполагается, что рассматриваемая систеМодель стохастической системы наблюдения, учитывающая случайные временные задержки между поступившим наблюдением и фактическим состоянием движущегося объекта, адаптирована для решения задачи идентификации параметров движения. Приведены уравнения для оптимальной байесовской идентификации. Для практического решения к задаче применен условно-минимаксный нелинейный фильтр (УМНФ). Подробно обсуждается синтез УМНФ, включая выбор структуры фильтра, на примере задачи позиционирования автономного подводного аппарата по наблюдениям стационарных акустических маяков. Выполнен вычислительный эксперимент на близкой к практическим потребностям модели с использованием трех вариантов фильтра - типовой аппроксимации обновляющего процесса, метода линейных псевдонаблюдений и геометрической интерпретации результатов угловых измерений.
This article introduces a Tensor Network Kalman filter, which can estimate state vectors that are exponentially large without ever having to explicitly construct them. The Tensor Network Kalman filter also easily accommodates the case where several different state vectors need to be estimated simultaneously. The key lies in rewriting the standard Kalman equations as tensor equations and then implementing them using Tensor Networks, which effectively transforms the exponential storage cost and computational complexity into a linear one. We showcase the power of the proposed framework through an application in recursive nonlinear system identification of high-order discrete-time multiple-input multiple-output (MIMO) Volterra systems. The identification problem is transformed into a linear state estimation problem wherein the state vector contains all Volterra kernel coefficients and is estimated using the Tensor Network Kalman filter. The accuracy and robustness of the scheme are demonstrated via numerical experiments, which show that updating the Kalman filter estimate of a state vector of length and its covariance matrix takes about 0.007s on a standard desktop computer in Matlab.
The global trajectory of the leading vehicle in GNSS-denied scenarios is important for vehicle formation. To achieve this, this paper proposes LiDAR-UWB Global Object Tracking (LUGOT), which fuses LiDAR-SLAM and UWB for object tracking in a global coordinate system within mobile-anchor scenarios. First, based on LiDAR-SLAM, the global high-frequency pose of the following vehicle is obtained. This pose is then synchronized with the UWB ultrahigh-frequency timestamp using PCHIP (Piecewise Cubic Hermite Interpolating Polynomial). Next, the synchronized data is combined with UWB measurements. Finally, the combined data is filtered using the introduced EKF/UKF with embedded outlier detection to determine the state of the object in the global coordinate system. To evaluate the filtering effect, this paper introduces cubic smoothing spline fitting to smooth the raw global path of the object as the ground truth. The experimental fleet platform was built to evaluate the effectiveness, outlier detection, and filtering effect of LUGOT. It avoids the long-tail effect and can be quickly commercialized with current technology. The source code for LUGOT is available at: https://github.com/ly3106/LUGOT.
We present a model of a stochastic observation system that allows for time delays between the received observation and the actual state of the observed object that formed these observations. Such delays can occur when observing the movement of an object in a water medium using acoustic sonars and have a significant impact on the accuracy of position tracking. We present equations to solve the optimal mean square filtering problem. Since the practical use of the optimal solution is barely feasible due to its computational complexity, we pay the main attention to an alternative, suboptimal but computationally efficient approach. Specifically, we adapted a conditional minimax nonlinear filter (CMNF) to the proposed model and formulated sufficient existence conditions for its estimate. We conducted a computational experiment on a model that is close to practical needs. The results of the experiment show the effectiveness of CMNF in the model considered. However, they also show a significant decrease in the quality of estimation compared to the model without random observation delays, which can be considered as a motivation for further research into the model and related problems.
The state estimation of the flexible multibody systems is a vital issue since it is the base of effective control and condition monitoring. The research on the state estimation method of flexible multibody system with large deformation and large rotation remains rare. In this investigation, a state estimator based on multiple nonlinear Kalman filtering algorithms was designed for the flexible multibody systems containing large flexibility components that were discretized by absolute nodal coordinate formulation (ANCF). The state variable vector was constructed based on the independent coordinates which are identified through the constraint Jacobian. Three types of Kalman filters were used to compare their performance in the state estimation for ANCF. Three cases including flexible planar rotating beam, flexible four-bar mechanism, and flexible rotating shaft were employed to verify the proposed state estimator. According to the different performances of the three types of Kalman filter, suggestions were given for the construction of the state estimator for the flexible multibody system.
The identification problem for a discrete stochastic system with anomalous errors in observations is considered. It is proposed to use algorithms for simultaneous assessment of the state and anomalous errors (unknown input in observations) to solve the identification problem. To increase the accuracy of identification, estimates of an unknown input are proposed to be calculated using additional smoothing algorithms. The simulation results are presented.
This article describes a method of implementation linearised
approximations to nonlinear systems that does not require the direct
derivation of Jacobian matrices. The approach is simpler to implement
than current techniques used with extended Kalman filters and other
linearised estimation and control algorithms
This paper describes a new approach for generalizing the Kalman
filter to nonlinear systems. A set of samples are used to parametrize
the mean and covariance of a (not necessarily Gaussian) probability
distribution. The method yields a filter that is more accurate than an
extended Kalman filter (EKF) and easier to implement than an EKF or a
Gauss second-order filter. Its effectiveness is demonstrated using an
example