Conference Paper

Reduced sigma point filters for propagation of means and covariances through nonlinear transformations

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Abstract

The Unscented Transform (UT) approximates the result of applying a specified nonlinear transformation to a given mean and covariance estimate. The UT works by constructing a set of points, referred to as sigma points, which has the same known statistics, e.g., first and second and possibly higher moments, as the given estimate. The given nonlinear transformation Is applied to the set, and the unscented estimate is obtained by computing the statistics of the transformed set of sigma points. For example, the mean and covariance of the transformed set approximates the nonlinear transformation of the original mean and covariance estimate. The computational efficiency of the UT therefore depends on the number of sigma points required to capture the known statistics of the original estimate. In this paper we examine methods for minimizing the number of sigma points for real-time control, estimation, and filtering applications. We demonstrate results in a 3D localization example.

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... Also, the variant with the smallest amount of sigma points, necessary to transfer the probability distribution with a sufficient accuracy, is called simplex sigma points. In [90], the authors proposed to construct these points based on minimal skew (third order moment), which results in recursive construction procedure. One option is to use a set of points equidistant from the mean value [146]. ...
... More information about unitary points generation is given in [90,146]. ...
... The idea to reduce the number of sigma points for better performance was already discussed in a certain number of papers. In [90], the authors propose to construct a set of sigma points the way to keep only the first statistical moments for the propagated data. The other ideas are dedicated to reduction of the state covariance matrix, so the modifications of the Kalman filter, namely reduced-rank square root Kalman filter [236] and, its variance, singular evolute interpolated Kalman filter [182] appeared. ...
Thesis
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The purpose of the work is to find a way to estimate the boundary conditions of the liver. They play an essential role in forming the predictive capacity of the biomechanical model, but are presented mainly by ligaments, vessels, and surrounding organs, the properties of which are "patient specific" and cannot be measured reliably. We propose to present the boundary conditions as nonlinear springs and estimate their parameters. Firstly, we create a generalized initial approximation using the constitutive law available in the literature and a statistical atlas, obtained from a set of models with segmented ligaments. Then, we correct the approximation based on the nonlinear Kalman filtering approach, which assimilates data obtained from a modality during surgical intervention. To assess the approach, we performed experiments for both synthetic and real data. The results show a certain improvement in simulation accuracy for the cases with estimated boundaries.
... Besides, other (non-classical) values of the scalars α, β and κ are possible and applied in practice. Furthermore, the number of the SP utilized can be also reduced to n + 1 [37,77,78]. ...
... Set the provisional size τ l+1 := min{τ * l , t k − t l+1 , τ max } of the next step; 2) Assemble the modified SP-matrix (29); 3) Compute the innovations covariance matrix by formula (30); 4) Compute the cross-covariance matrix in line with equation (31); 5) Compute the Kalman gain by formula (33); 6) Set the predicted state mean vector by formula (38); 7) Compute the predicted covariance matrix SR by formula (34); 8) Compute the predicted covariance matrix by formula (35); 9) Compute the requested filtering covariance matrix by formula (32); 10) Determine the measurement mean vector by the inner product (36); 11) Compute the requested filtering state mean vector by formula (37). ...
... , 2n, at time t end , where the subscript end always marks the last node in the generated mesh {t l } end l=0 (i.e. t end ≡ t k ), we compute the filtering state mean x k|k and covariance SR S k|k based on the measurement information z k sampled at the sampling instant t k as follows: 1) Reshape the SP-vector ¯ k|k−1 to the conventional SP-matrix form (28); 2) Assemble the modified SP-matrix (29);3) Cholesky-factorize the measurement noise covariance matrix R k = R the coupled innovations and filtering covariance pre-array A by formula (40); 5) Q R-factorize the transposed pre-array A for deriving the upper triangular post-array R; 6) Read-off the innovations covariance SR P 1/2 zz,k|k−1 of size m, the modified cross-covariance matrix P xz,k of size n × m and the filtering covariance SR S k|k of size n from the transposed post-array R in formula (41); 7) Compute the Kalman gain by formula (42); 8) Set the predicted state mean vector by formula (38); 9) Determine the measurement mean vector by the inner product (36); 10) Compute the requested filtering state mean vector by formula(37).Output. Save the filtering state mean vectors x k|k and covariance matrix SR S k|k calculated at all the sampling time instants t k .Appendix C. The orthogonal square-root accurate continuous-discrete unscented Kalman filter with decoupled measurement update Initialization. ...
Article
This paper further advances the idea of accurate Gaussian filtering towards efficient unscented-type Kalman methods for estimating continuous-time nonlinear stochastic systems with discrete measurements. It implies that the differential equations evolving sigma points utilized in computations of the predicted mean and covariance in time-propagations of the Gaussian distribution are solved accurately, i.e. with negligible error. The latter allows the total error of the unscented Kalman filtering technique to be reduced significantly and gives rise to the novel accurate continuous-discrete unscented Kalman filtering algorithm. At the same time, this algorithm is rather vulnerable to round-off and numerical integration errors committed in each state estimation run because of the need for the Cholesky decomposition of covariance matrices involved. Such a factorization will always fail when the covariance's positivity is lost. This positivity lost issue is commonly resolved with square-root filtering implementations, which propagate not the full covariance matrix but its square root (Cholesky factor) instead. Unfortunately, negative weights encountered in applications of the accurate continuous-discrete unscented Kalman filter to high-dimensional stochastic systems preclude from designing conventional square-root methods. In this paper, we address this problem with low-rank Cholesky factor update procedures or with hyperbolic QR transforms used for yielding J-orthogonal square roots. Our novel square-root algorithms are justified theoretically and examined numerically in an air traffic control scenario.
... In 1995, Julier et al. [35] gave rise a new class of state estimation algorithms termed the Unscented Kalman Filter (UKF), which were found to be a successful alternative to the EKF in various state and parameter estimation and machine learning tasks [21,22,30,31,33,34,36,63,82,85,86] . In particular, the cited authors stress a derivative-free fashion of the UKF and its more accurate approximation of filtering states. ...
... in which the parameters are set as follows: α = 1 , β = 0 and λ = α 2 (κ + n ) − n with κ = 3 − n . The constant λ regulates the spread of the SP around the mean ˆ x and the secondary scaling parameters β and κ can be optimized for matching higher moments of the random variable [33,34,36,63,82,85,86] . Other (non-classical) values of the scalars α, β and κ are also discussed below. ...
... Other (non-classical) values of the scalars α, β and κ are also discussed below. In addition, we stress that the number of the SP employed can be reduced to n + 1 [33,61,62] . However, reducing the SP set might affect the accuracy and robustness of the UKF and, hence, it is not addressed in this paper. ...
Article
This paper addresses the problem of square-rooting in the Unscented Kalman Filtering (UKF) methods rooted in the Ito^-Taylor approximation of the strong order 1.5. Since its discovery the UKF has become one of the most powerful state estimation means because of its outstanding performance in numerous stochastic systems of practical value, including continuous-discrete ones. Besides, the main shortcoming of this technique is the need for the Cholesky decomposition of covariance matrices derived in its time and measurement update steps. Such a factorization is time-consuming and highly sensitive to round-off and other errors committed in the course of computation, which can result in losing the covariance’s positivity and, hence, failing the Cholesky decomposition. The latter problem is usually overcome by means of square-root filter implementations, which propagate not the covariance itself but its square root (Cholesky factor), only. Unfortunately, negative weights arising in applications of the UKF to high-dimensional stochastic systems preclude from designing conventional square-root UKF methods. We resolve it with low-rank Cholesky factor update procedures or with hyperbolic QR transforms used for yielding J-orthogonal square roots. Our novel square-root filters are justified theoretically and examined and compared numerically to the existing UKF in a flight control scenario.
... Eine umfangreiche Übersicht über existierende UT findet sich in (Menegaz u. a. 2015). Erwähnenswert erscheint hier die Spherical Simplex Unscented Transformation (SSUT), die lediglich m UT = n + 2 Σ-Punkte benötigt und sich insbesondere für zeitkritische Anwendungen empfiehlt (Julier und Uhlmann 2002;Julier 2003;Lozano u. a. 2008). Durch Vorgabe des Gewichts w 0 , mit 0 ≤ w 0 < 1, ergeben sich die übrigen n + 1 Gewichte zu ...
... Für w 0 = 0 reduziert sich die Anzahl an notwendigen Σ-Punkten auf das absolute Minimum von n + 1 (Julier und Uhlmann 2002. ...
Thesis
Die Erde befindet sich in einem kontinuierlichen Wandel, der aus verschiedenen variierenden dynamischen Prozessen und einwirkenden Kräften resultiert. Die globale Erderwärmung, der Anstieg des Meeresspiegels oder tektonische Verschiebungen sind einige der globalen Phänomene, die diesen Veränderungsprozess sichtbar machen. Um diese Veränderungen besser zu verstehen, deren Ursachen zu analysieren und um geeignete Präventivmaßnahmen abzuleiten, ist ein eindeutiger Raumbezug zwingend notwendig. Der International Terrestrial Reference Frame (ITRF) als globales erdfestes kartesisches Koordinatensystem bildet hierbei die fundamentale Basis für einen eindeutigen Raumbezug, zur Bestimmung von präzisen Satellitenorbits oder zum Detektieren von Verformungen der Erdkruste. Die 2015 verabschiedete Resolution „A global geodetic reference frame for sustainable development“ (A/RES/69/266) der Vereinten Nationen (UN) verdeutlicht den hohen Stellenwert und die Notwendigkeit eines solchen globalen geodätischen Bezugssystems. Das Global Geodetic Observing System (GGOS) wurde 2003 durch die International Association of Geodesy (IAG) gegründet. „Advancing our understanding of the dynamic Earth system by quantifying our planet’s changes in space and time“ lautet die 2011 formulierte Zielsetzung, auf die alle Arbeiten von GGOS ausgerichtet sind, um die metrologische Plattform für sämtliche Erdbeobachtungen zu realisieren. Die Bestimmung eines globalen geodätischen Bezugsrahmens, der weltweit eine Positionsgenauigkeit von 1mm ermöglicht, ist eine der großen Herausforderungen von GGOS. Das Erreichen dieses Ziels setzt neben der technischen Weiterentwicklung und dem infrastrukturellen Ausbau geodätischer Raumverfahren das Identifizieren und Quantifizieren von systematischen Abweichungen sowohl im lokalen als auch im globalen Kontext voraus. Die Bestimmung eines globalen geodätischen Bezugsrahmens erfolgt durch eine kombinierte Auswertung aller geodätischen Raumverfahren. Da diese untereinander nur eine geringe physische Verknüpfung aufweisen, stellen lokal bestimmte Verbindungsvektoren, die auch als Local-Ties bezeichnet werden, eine der wesentlichen Schlüsselkomponenten bei der Kombination dar. Ungenaue, fehlerbehaftete und inaktuelle Local-Ties limitieren die Zuverlässigkeit des globalen geodätischen Bezugssystems. In der vorliegenden Arbeit werden ein Modell sowie verschiedene Lösungsverfahren entwickelt, die eine Verknüpfung der geometrischen Referenzpunkte von Radioteleskopen bzw. Laserteleskopen mit anderen geodätischen Raumverfahren durch prozessbegleitende lokale terrestrische Messungen erlauben. Während Radioteleskope zur Interferometrie auf langen Basislinien (VLBI) verwendet werden, ermöglichen Laserteleskope Entfernungsmessungen zu Erdsatelliten (SLR) oder zum Mond (LLR). Die Bestimmung des geometrischen Referenzpunktes von Laser- und Radioteleskopen ist messtechnisch herausfordernd und erfordert eine indirekte Bestimmungsmethode. Bestehende geometrische Methoden sind entweder auf eine bestimmte Teleskopkonstruktion beschränkt oder erfordern ein spezielles Messkonzept, welches ein gezieltes Verfahren des Teleskops voraussetzt. Die in dieser Arbeit hergeleitete Methode weist keine konstruktionsbedingten Restriktionen auf und erfüllt zusätzlich alle Kriterien der durch das GGOS angeregten prozessintegrierten in-situ Referenzpunktbestimmung. Hierdurch wird es möglich, den Referenzpunkt kontinuierlich und automatisiert zu bestimmen bzw. zu überwachen. Um die Zuverlässigkeit von VLBI-Daten zu erhöhen und um die Zielsetzung von 1mm Positionsgenauigkeit im globalen Kontext zu erreichen, wird das bestehende VLBI-Netz gegenwärtig durch zusätzliche Radioteleskope unter dem Namen VLBI2010 Global Observing System (VGOS) erweitert. Die hierbei entstehenden VGOS-Radioteleskope zeichnen sich u. a. durch eine sehr kompakte Bauweise und hohe Rotationsgeschwindigkeiten aus. Weitgehend ununtersucht ist das Eigenverformungsverhalten dieser Teleskope. Während für konventionelle Radioteleskope bspw. Signalwegänderungen von z. T. mehreren Zentimetern dokumentiert sind, existieren nur wenige vergleichbare Studien für VGOS-Radioteleskope. Hauptgründe sind zum einen die erhöhten Genauigkeitsanforderungen und zum anderen fehlende Modelle zur Beschreibung der Reflektorgeometrien, wodurch eine direkte Übertragung bisheriger Mess- und Analyseverfahren erschwert wird. In dieser Arbeit werden für VGOS-spezifizierte Radioteleskope Modelle erarbeitet, die eine geometrische Beschreibung der Form des Haupt- und Subreflektors ermöglichen. Basierend auf diesen Modellen lassen sich u. a. Änderungen der Brennweite oder Variationen der Strahllänge infolge von lastfallabhängigen Deformationen geometrisch modellieren. Hierdurch ist es möglich, wesentliche Einflussfaktoren zu quantifizieren, die eine Variation des Signalweges hervorrufen und unkompensiert vor allem zu einer systematischen Verfälschung der vertikalen Komponente der Stationskoordinate führen. Die Wahl eines geeigneten Schätzverfahrens, um unbekannte Modellparameter aus überschüssigen Beobachtungen abzuleiten, wird häufig als trivial und gelöst angesehen. Im Rahmen dieser Arbeit wird gezeigt, dass neben messprozessbedingten systematischen Abweichungen auch systematische Abweichungen durch das gewählte Schätzverfahren entstehen können. So resultieren aus der Anwendung eines Schätzverfahrens, welches ausschließlich in linearen Modellen Gültigkeit besitzt, i.A. keine erwartungstreuen Schätzwerte bei nichtlinearen Problemstellungen. Insbesondere in der Formanalyse des Hauptreflektors eines VLBI-Radioteleskops zeigt sich, dass die resultierenden Schätzwerte verzerrt sind, und diese Verzerrungen Größenordnungen erreichen, die als kritisch zu bewerten sind.
... Numerous approaches have been provided in many literatures. Among mainstream approaches, the most useful ones include the extended Kalman filter (EKF) [1]; deterministic sampling filters like unscented Kalman filter (UKF) [2] and cubature Kalman filter (CKF) [3]; random sampling filter like the particle filter (PF) [4]; converted measurement Kalman filter (CMKF) like the best linear unbiased estimation (BLUE) filter [5][6][7][8]. Compared with other approaches, EKF has the least computational burden, but may perform poorly under strong non-linearity situations. ...
... x k is the true state at time k, w k is white noise with covariance Q k , F k/k − 1 is the transfer matrix, and the state prediction x k and its covariance matrix P k are estimated with (2), which is the same as that of the Kalman filter. ...
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The non-linear filter is subject to divergence for close-range target tracking; an adaptive best linear unbiased estimation (BLUE) filter with fused range estimation is presented. After proposing the notion of predicted range, the weighted estimation of the predicted and measured ranges is used in the converted measurement model to track the target. The adaptive BLUE filtering parameters with Rayleigh range measurement are derived. Simulation results show the adaptive BLUE filter exhibits better robustness and accuracy with modest computational burden.
... It can approximate the probability distribution of the nonlinear system by the specified sampling strategy to obtain the mean and covariance matrices with the second-order accuracy of the nonlinear system without computing derivatives. Subsequently, a variety of sampling strategies and improved methods are derived, such as symmetric sampling [12,13], spherical simplex sampling [11] and minimum skew-simplex sampling (Julier, 2002) [10]. Among these methods, the scaled symmetric sampling strategy proposed by Merwe [16] avoids ''non-local effect", thus it is widely used (e.g., [21,24,22]). ...
... It can approximate the probability distribution of the nonlinear system by the specified sampling strategy to obtain the mean and covariance matrices with the second-order accuracy of the nonlinear system without computing derivatives. Subsequently, a variety of sampling strategies and improved methods are derived, such as symmetric sampling [12,13], spherical simplex sampling [11] and minimum skew-simplex sampling (Julier, 2002) [10]. Among these methods, the scaled symmetric sampling strategy proposed by Merwe [16] avoids ''non-local effect", thus it is widely used (e.g., [21,24,22]). ...
Article
The Scaled Unscented Transformation (SUT) method based on deterministic sampling strategy is introduced to nonlinear inversion and precision estimation of earthquake fault parameters. Firstly, the SUT method is used to estimate the precision of the curve fitting parameters obtained by Particle Swarm Optimization (PSO) algorithm and compared with least square method and Monte Carlo method to verify the effectiveness of the proposed method. Considering the inevitable error in the observations, we preprocess the simulated GPS data to obtain the approximate mean of observations, then contrast and analyze the fault parameter estimates and precision information obtained by SUT method and Monte Carlo method based on the hybrid PSO/Simplex algorithm (MPSO). The influence of the "adjusted" observations, positions and number of the observation points and the level of noise added to deformation observations are also considered. Finally, the methods in this paper are applied to the Lushan (China) earthquake. The experimental results show that both SUT method and Monte Carlo method can obtain better parameter estimates with "adjusted" observations; SUT method can obtain more accurate precision information of fault parameters; Monte Carlo method can only judge mean square errors of parameters from the order of magnitude and the correlations between parameters are not accurate; When using GPS measurement data to invert the fault parameters, the positions and number of GPS points have certain influence on some fault parameters; When the noise added to the GPS measurement data increases, the fitting of the fault parameter estimate become worse and the precision of fault parameter estimate decreases.
... Higher-order accuracy can be achieved by increasing the number of sigma points, but with an increased computational demand of the filtering process, e.g., skewed/third-moment approaches, conjugate unscented transform, or higher order filters (Julier 1998;Adurthi et al. 2018;Tenne and Singh 2003), because the computational cost of the filtering process is directly proportional to the number of sigma points. Recognizing this, Julier and Uhlmann (2002) and Julier (2003) suggested reduced asymmetric n þ 2 sigma point sets to quantify statistical properties of n-dimensional random variables. These methods could significantly reduce the computational burden, particularly for large-dimensional systems; however, the spread of the points and the skewness effects pose instability issues, particularly with increasing dimensions (Julier 2002;Julier and Uhlmann 2004). ...
... Papakonstantinou et al. (2022) proposed a scaled spherical simplex filter (S3F) that requires only n þ 2 sigma points for the unscented transformation and achieves similar accuracy and robustness as the UKF. The minimum set of sigma points that can be used to provide a nonsingular covariance is n þ 1 (Julier and Uhlmann 2002). However, the n þ 1 sigma points cannot achieve the same order of accuracy and robustness as the UKF. ...
... Although the use of UT and CKF eliminates the need for determining Jacobian matrices, it increases the computation time due to the increased number of sigma points. In [40], it is proved that for an n-dimensional space, the minimum number of sigma points required is n + 1 and proposed a set of minimal skew simplex points that can effectively determine the statistics. The proposal was further developed in [41] to result in spherical simplex unscented transformation (SSUT) which utilises n + 2 sigma points. ...
... These expressions are used to determine the sigma point outputs z^k (i) using (14) for each sigma point x k (i) . Where expressions for e d , e q , i d , and i q are given from (40), (41), (29), and (30), respectively, in which X represents the various reactance of machine and i r , i i are the real and imaginary components of the generator output current, which can be determined from PMU measurement ...
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Dynamic state estimation is essential in case of various monitoring, control, and protection strategies that are designed based on the state‐space model. Kalman filter‐based estimation algorithms are mainly used to estimate these states locally using the input and output measurements of the generator. However, in the case of wide‐area power system control and protection strategies, remote estimation of these states is required. This remote estimation relies upon phasor measurement units for measurement signals, which are limited to output measurements such as voltage, current, and frequency. For Kalman filter‐based techniques, apart from the output, input measurements such as field and torque input are also required to estimate the states. This study proposes an input invariant filter technique using unbiased minimum variance filter and spherical simplex unscented transform for remote estimation of generator states using the limited phasor measurement unit measurements. The estimation is performed in the absence of mechanical input torque and field voltage measurements using a minimum set of sigma points. The performance of the filter under various transient conditions and in the presence of measurement errors are analysed and compared with existing techniques.
... For real-time applications, Julier and Uhlmann [37] derive the minimal skew simplex unscented transformation (MSSUT), which only requires n UT = n obs + 1 Σ-points-the smallest possible number. However, this approach has a serious drawback, which limits the scope of applications. ...
... To estimate a second-order correction, the spherical simplex unscented transformation is applied as described in Section 2.3. Based on the observed image coordinates and w 0 = 0, n obs + 1 Σ-points Y are generated according to Equation (37). By concatenating both nonlinear functional models, i.e., ...
Article
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A global geodetic reference system (GGRS) is realized by physical points on the Earth’s surface and is referred to as a global geodetic reference frame (GGRF). The GGRF is derived by combining several space geodetic techniques, and the reference points of these techniques are the physical points of such a realization. Due to the weak physical connection between the space geodetic techniques, so-called local ties are introduced to the combination procedure. A local tie is the spatial vector defined between the reference points of two space geodetic techniques. It is derivable by local measurements at multitechnique stations, which operate more than one space geodetic technique. Local ties are a crucial component within the intertechnique combination; therefore, erroneous or outdated vectors affect the global results. In order to reach the ambitious accuracy goal of 1 mm for a global position, the global geodetic observing system (GGOS) aims for strategies to improve local ties, and, thus, the reference point determination procedures. In this contribution, close range photogrammetry is applied for the first time to determine the reference point of a laser telescope used for satellite laser ranging (SLR) at Geodetic Observatory Wettzell (GOW). A measurement campaign using various configurations was performed at the Satellite Observing System Wettzell (SOS-W) to evaluate the achievable accuracy and the measurement effort. The bias of the estimates were studied using an unscented transformation. Biases occur if nonlinear functions are replaced and are solved by linear substitute problems. Moreover, the influence of the chosen stochastic model onto the estimates is studied by means of various dispersion matrices of the observations. It is shown that the resulting standard deviations are two to three times overestimated if stochastic dependencies are neglected.
... Along these lines, a minimum-skew simplex sigma points filter is introduced by Julier and Uhlmann [16], utilizing n + 2 points to match the first two moments and to minimize third moment (skew) errors for a n-dimensional random variable, where n + 1 points form the vertices of a simplex. In the suggested approach in [16], however, the spread of the points increases exponentially with the state-space dimension, resulting in potential numerical stability problems even for relatively low dimensionality. Julier [17] introduced an alternate strategy for the n + 2 sigma points selection and transformation, named as spherical simplex Unscented Transform, where n + 1 points are now located on a hypersphere with radius proportional to p n. ...
... Julier [17] introduced an alternate strategy for the n + 2 sigma points selection and transformation, named as spherical simplex Unscented Transform, where n + 1 points are now located on a hypersphere with radius proportional to p n. In both [16,17], the sigma points are asymmetrically distributed about the origin, and therefore some symmetric distribution higher moment effects, especially for the skew and the other odd moments, are not captured now by the sigma points. To be able to limit the spread of the sigma points to some extent, in order to utilize the approach in [16] which can minimize third moment errors, the scaled Unscented Transformation is presented by Julier in [18], through the introduction of the scaling parameters a and b. ...
Article
The computational efficiency of a sampling based nonlinear Kalman filtering process is mainly conditional on the number of sigma/sample points required by the filter at each time step to effectively quantify statistical properties of related states and parameters. Efficaciously minimizing the needed number of points would therefore have important implications, especially for large n-dimensional nonlinear systems. A set of minimum number of n + 1 sigma points is necessary in each filtering application in order to provide mean and nonsingular covariance estimates. Incorporating additional sigma points than this minimum set improves the accuracy of the estimates and can take advantage of a richer information content that can possibly exist, but at the same time increases the computational demand. To this end, by adding one more sigma point to this minimum set, and assigning general, well defined weights and scaling factors, a new Scaled Spherical Simplex Filter (S3F) with n + 2 sigma points set size is presented in this work, and it is theoretically proven that it can practically achieve in all cases the same accuracy and numerical stability as the typical 2n + 1 sigma points Unscented Kalman Filter (UKF), with almost 50% less computational requirements. A comprehensive study of the suggested filter is presented, including detailed derivations, theoretical examples and numerical results, demonstrating the efficiency, robustness, and accuracy of the S3F.
... For completeness we here recall the complete algorithm in our case, before proceeding to the results section. Following [55,58], for θ discretized in R r , we introduce r + 1 so-called unitary simplex sampling points I [i] in the space R r and the associated weights α i with the following rules ...
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Tagged Magnetic Resonance images (tagged-MRI) are generally considered to be the gold standard of medical imaging in cardiology. By imaging spatially-modulated magnetizations of the deforming tissue, indeed, this modality enables an assessment of intra-myocardial deformations over the heart cycle. The objective of the present work is to incorporate the most valuable information contained in tagged-MRI in a data assimilation framework, in order to perform joint state-parameter estimation for a complete biomechanical model of the heart. This type of estimation is the second major step, after initial anatomical personalization, for obtaining a genuinely patient-specific model that integrates the individual characteristics of the patient, an essential prerequisite for benefitting from the model predictive capabilities. Here, we focus our attention on proposing adequate means of quantitatively comparing the cardiac model with various types of data that can be extracted from tagged-MRI after an initial image processing step, namely, 3D displacements fields, deforming tag planes or grids, or apparent 2D displacements. This quantitative comparison—called discrepancy measure—is then used to feed a sequential data assimilation procedure. In the state estimation stage of this procedure, we also propose a new algorithm based on the prediction–correction paradigm, which provides increased flexibility and effectiveness in the solution process. The complete estimation chain is eventually assessed with synthetic data, produced by running a realistic model simulation representing an infarcted heart characterized by increased stiffness and reduced contractility in a given region of the myocardium. From this simulation we extract the 3D displacements, tag planes and grids, and apparent 2D displacements, and we assess the estimation with each corresponding discrepancy measure. We demonstrate that—via regional estimation of the above parameters—the data assimilation procedure allows to quantitatively estimate the biophysical parameters with good accuracy, thus simultaneously providing the location of the infarct and characterizing its seriousness. This shows great potential for combining a biomechanical heart model with tagged-MRI in order to extract valuable new indices in clinical diagnosis.
... Compared with the EKF method, the estimation precision of UKF method is higher and it does not need to calculate the Jacobian matrix of nonlinear function. Julier and Uhlmann introduced the Spherical Simplex Unscented [19,20] into the UKF method, which reduced the number of sampling points and improved the computational efficiency of the algorithm. Merwe and Wan proposed the Square-Root UKF Filtering (SRUKF) algorithm [21], Lee and Alfriend [23] proposed the adaptive UKF algorithm which could dynamically estimate the noise characteristics. ...
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Periodic variable navigation can realize the integration of positioning, attitude determination, timing and other functions. As a novel autonomous navigation method, autonomous management and autonomous operation of spacecraft are the main direction of great significance to reduce the burden of ground measurement and control, to improve the viability of spacecraft, While, the precision of the navigation is very critical for the use of periodic variable navigation for the long range spacecraft. So, we propose a big-data inspired precision improvement algorithm in this paper. Period variable star phase time measurement is used as observation information, and accomplished by the orbital dynamics equation of spacecraft motion. According to the gathering of the sampling data and the sensor data, a self-learning system is trained with the parameters from the nonlinear filtering methods. Based on the Unscented Kalman Filtering, an autonomous navigation algorithm of period variable star is established to realize the navigation and positioning of spacecraft. Under the measurement conditions of a single sampling interval of 0.01s and the measurement precision of period variable star phase observation time of 10− 5s, It can be find that the navigation position determination precision can reach 400m, the speed determination precision can reach 0.3m/s, and the measurement precision can reach 10− 5s with our proposed algorithm.
... The computational cost of the unscented transformation is proportional to the number of sigma points employed, so there is a propensity to choose schemes with only few degrees of freedom. Among them, the simplex unscented approach requires a minimum number of N x + 1 samples to match the mean and covariance of a N x dimensional normally distributed random vector x [30,33]. On the other hand, additional sigma points can be introduced to reproduce higher-order moments of a Gaussian distribution, e.g. ...
Chapter
Accurately estimating the state of a dynamical system is of fundamental importance in a variety of applications, from engineering challenges to everyday life. This task is complex because uncertainties typically affect the dynamical behaviour as well as the available observations of the (hidden) state. The goal of this chapter is to provide a comprehensive overview, from the probabilistic problem statement to methods for its solution. In particular the focus will be on filtering problems for time-continuous state evolution equations and time-discrete observations. It will be shown that, except for very few cases, the filtering problem has no closed-form solution, which is generally infinite-dimensional. Hence, several practical algorithms to find an approximate solution are presented.
... These sigma points are computed via Cholesky-factorization [16] of the covariance matrix. An other approach decreases the number of sigma points to (n+1) [17], [18] via an optimization step but it is computationally more expensive than the Cholesky-factorization. For a systematic overview, see [19]. ...
Article
Advanced robotics and autonomous vehicles rely on filtering and sensor fusion techniques to a large extent. These mobile applications need to handle the computations onboard at high rates while the computing capacities are limited. Therefore, any improvement that lowers the CPU time of the filtering leads to more accurate control or longer battery operation. This article introduces a generic computational relaxation for the unscented transformation (UT) that is the key operation of the Unscented Kalman filter-based applications. The central idea behind the relaxation is to pull out the linear part of the filtering model and avoid the calculations for the kernel of the nonlinear part. The practical merit of the proposed relaxation is demonstrated through a simultaneous localization and mapping (SLAM) implementation that underpins the superior performance of the algorithm in the practically relevant cases, where the nonlinear dependencies influence only an affine subspace of the image space. The numerical examples show that the computational demand can be mitigated below 50% without decreasing the accuracy of the approximation. The method described in this article is implemented and published as an open-source C ++ library RelaxedUnscentedTransformation on GitHub.
... This is done by computing a set of 2 + 1 points, called sigma points, on the basis of the mean and variance of the original vector, transforming these points by the nonlinear map and then approximating the mean and variance of the transformed vector from the transformed sigma points. We refer to [2][3][4][5][14][15][16][17] for the theoretical aspects. ...
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In this paper we introduce a Cubic Root Unscented Kalman Filter (CRUKF) compared to the Unscented Kalman Filter (UKF) for calculating the covariance cubic matrix and covariance matrix within a sensor fusion algorithm to estimate the measurements of an omnidirectional mobile robot trajectory. We study the fusion of the data obtained by the position and orientation with a good precision to localize the robot in an external medium; we apply the techniques of Kalman Filter (KF) to the estimation of the trajectory. We suppose a movement of mobile robot on a plan in two dimensions. The sensor approach is based on the Cubic Root Unscented Kalman Filter (CRUKF) and too on the standard Unscented Kalman Filter (UKF) which are modified to handle measurements from the position and orientation. A real-time implementation is done on a three-wheeled omnidirectional mobile robot, using a dynamic model with trajectories. The algorithm is analyzed and validated with simulations.
... Note that the selection of the scaling parameter κ [10,40], which is used to calculate the deterministic sample state vectors in the UKF, affects the performance of the UKF. A general recommendation is to select κ = 3 − n [10] which eliminates the second-order Taylor series terms of the error as shown in (7). ...
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A mathematical framework to predict the Unscented Kalman Filter (UKF) performance improvement relative to the Extended Kalman Filter (EKF) using a quantitative measure of non-linearity is presented. It is also shown that the range of performance improvement the UKF can attain, for a given minimum probability depends on the Non-linearity Indices of the corresponding system and measurement models. Three distinct non-linear estimation problems are examined to verify these relations. A launch vehicle trajectory estimation problem, a satellite orbit estimation problem and a re-entry vehicle position estimation problem are examined to verify these relations. Using these relations, a procedure is suggested to predict the estimation performance improvement offered by the UKF relative to the EKF for a given non-linear system and measurement without designing, implementing and tuning the two Kalman Filters.
... It uses a collection of sample points to calculate covariance accuracy and mean accuracy. In [34], a variation of the UKF algorithm called Reduced Sigma Point Filter (RSPF) was presented, but it does not take linearization errors into account. ...
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The Extended Kalman Filter (EKF) has received abundant attention with the growing demands for robotic localization. The EKF algorithm is more realistic in non-linear systems which has an autonomous white noise in both the system and the estimation model. Also, in the field of engineering, most systems are non-linear. Therefore, the EKF attracts more attention than the Kalman Filter (KF). In this paper, we propose an EKF-based localization algorithm by edge computing, and a mobile robot is used to update its location concerning the landmark. This localization algorithm aims to achieve a high level of accuracy and wider coverage. The proposed algorithm is helpful for the research related to the use of EKF localization algorithms. Simulation results demonstrate that, under the situations presented in the paper, the proposed localization algorithm is more accurate compared with the current state-of-the-art localization algorithms.
... A simple FEM mesh of only a few hundred nodes would be too time-consuming for a clinical application, as it would require more than 300 simulations for each step of the assimilation process. To solve this issue, we use a reduced order unscented Kalman Filter (ROUKF) [26] instead of the UKF. This method significantly reduces the computation cost since only K + 1 simulations (in the best case) are required. ...
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Purpose: Augmented reality can improve the outcome of hepatic surgeries, assuming an accurate liver model is available to estimate the position of internal structures. While researchers have proposed patient-specific liver simulations, very few have addressed the question of boundary conditions. Resulting mainly from ligaments attached to the liver, they are not visible in preoperative images, yet play a key role in the computation of the deformation. Method: We propose to estimate both the location and stiffness of ligaments by using a combination of a statistical atlas, numerical simulation, and Bayesian inference. Ligaments are modeled as polynomial springs connected to a liver finite element model. They are initialized using an anatomical atlas and stiffness properties taken from the literature. These characteristics are then corrected using a reduced-order unscented Kalman filter based on observations taken from the laparoscopic image stream. Results: Our approach is evaluated using synthetic data and phantom data. By relying on a simplified representation of the ligaments to speed up computation times, it is not estimating the true characteristics of ligaments. However, results show that our estimation of the boundary conditions still improves the accuracy of the simulation by 75% when compared to typical methods involving Dirichlet boundary conditions. Conclusion: By estimating patient-specific boundary conditions, using tracked liver motion from RGB-D data, our approach significantly improves the accuracy of the liver model. The method inherently handles noisy observations, a substantial feature in the context of augmented reality.
... • sigma-points de type simplex : dans ce cas nous construit r = m + 1 particules qui est le nombre minimum de points nécessaires pour représenter la covariance d'une variable aléatoire. Ces sigma-points sont positionnés sur un polygone régulier de rayon √ m et sont construits de manière récursive en suivant la procédure décrite dans [102,72]. Cette procédure peut se résumer la manière suivante : r définie récursivement par : ...
Thesis
Les arythmies auriculaires constituent une pathologie majeure en cardiologie, et leur étude constitue un vaste sujet de recherche. Pour les étudier, de nombreux modèles mathématiques de la propagation du potentiel d'action dans les oreillettes ont été développés. La plupart de ces modèles génériques permettent de reproduire des séquences d'activations typiques des oreillettes. De tels modèles peuvent avoir un intérêt expérimental, voir clinique, par exemple dans l'aide à la localisation des foyers arythmiques ou encore dans l'analyse des échecs du traitement de ces arythmies. Néanmoins, pour atteindre ce but, il faut être capable de recaler au mieux le modèle, dans ses dimensions géométriques ou fonctionnelles, sur des données individuelles. L'assimilation de données, discipline mathématique dans laquelle nous cherchons à combiner de manière optimale théorie et observations, est alors un bon candidat à la personnalisation des modèles de la propagation du potentiel d'action. Dans cette thèse, nous proposons d'étudier différentes méthodes d'assimilation de données -- séquentielles et variationnelles -- dans le but de combiner les modèles de propagation avec des données électroanatomiques. Plus précisément, nous nous intéressons à deux applications possible de l'assimilation de données que sont l'estimation d'état et l'estimation de paramètres. Dans un premier temps, nous étudions un observateur d'état permettant de corriger la position du front de propagation simulé en se basant sur la position du front observé. Cet observateur est alors utilisé afin de compléter une carte d'activation obtenue lors d'une procédure clinique. Ensuite, ce même observateur est combiné à un filtre de Kalman d'ordre réduit afin d'estimer les paramètres de conductivités du modèle mathématique de propagation du potentiel d'action. Une étude de la stratégie d'estimation liée état-paramètre est ensuite réalisée pour voir comment la méthode se comporte face aux erreurs de modélisation. La méthode est ensuite testée sur un jeu de données acquis cliniquement. Puis, nous regardons du côté des méthodes d'assimilation de données variationnelles qui permettent l'estimation de paramètres spatialement distribués. Plusieurs problèmes de minimisation, permettant d'estimer un paramètre de conductivité distribué dans l'espace, sont alors introduits et analysés. Nous montrons alors que la discrétisation de ces problèmes de minimisation, dans le but d'obtenir des méthodes numériques de résolution, peut s'avérer complexe. Une méthode numérique est ensuite mise en place pour un des problèmes de minimisation étudié, et trois cas tests unidimensionnels sont analysés.Enfin, nous démontrons l'existence d'un minimum pour une des fonctions objectif étudiées en nous basant sur des résultats d'analyse fonctionnelle de la littérature.
... The sigma point Kalman filter (SPKF) method is a class of approximate nonlinear filtering methods based on the Gaussian distribution, including the unscented Kalman filter (UKF), central difference Kalman filter (CDKF), square-root unscented Kalman filter (SRUKF), etc. [17]. The ideas of these methods are roughly the same, of which the UKF is the most famous. ...
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State estimation is widely used in various automated systems, including IoT systems, unmanned systems, robots, etc. In traditional state estimation, measurement data are instantaneous and processed in real time. With modern systems’ development, sensors can obtain more and more signals and store them. Therefore, how to use these measurement big data to improve the performance of state estimation has become a hot research issue in this field. This paper reviews the development of state estimation and future development trends. First, we review the model-based state estimation methods, including the Kalman filter, such as the extended Kalman filter (EKF), unscented Kalman filter (UKF), cubature Kalman filter (CKF), etc. Particle filters and Gaussian mixture filters that can handle mixed Gaussian noise are discussed, too. These methods have high requirements for models, while it is not easy to obtain accurate system models in practice. The emergence of robust filters, the interacting multiple model (IMM), and adaptive filters are also mentioned here. Secondly, the current research status of data-driven state estimation methods is introduced based on network learning. Finally, the main research results for hybrid filters obtained in recent years are summarized and discussed, which combine model-based methods and data-driven methods. This paper is based on state estimation research results and provides a more detailed overview of model-driven, data-driven, and hybrid-driven approaches. The main algorithm of each method is provided so that beginners can have a clearer understanding. Additionally, it discusses the future development trends for researchers in state estimation.
... It uses a series of definite samples to approximate the posterior probability density of the state, instead of approximating the nonlinear function. The UKF does not need to calculate Jacobian matrix, and avoids linearization, which usually neglects higher order terms; thus, the calculation accuracy of UKF is higher than that of EKF [29,30]. UKF is widely used in many fields such as navigation, target tracking, signal processing and neural network learning. ...
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The unscented Kalman filter (UKF) can effectively reduce the linearized model error and the dependence on initial coordinate values for indoor pseudolite (PL) positioning unlike the extended Kalman filter (EKF). However, PL observations are prone to various abnormalities because the indoor environment is usually complex. Standard UKF (SUKF) lacks resistance to frequent abnormal observations. This inadequacy brings difficulty in guaranteeing the accuracy and reliability of indoor PL positioning, especially for phase-based high-precision positioning. In this type of positioning, the ambiguity resolution (AR) will be difficult to achieve in the presence of abnormal observations. In this study, a robust UKF (RUKF) and partial AR (PAR) algorithm are introduced and applied in indoor PL positioning. First, the UKF is used for parameter estimation. Then, the anomaly recognition statistics and optimal ambiguity subset of PAR are constructed on the basis of the posterior residuals. The IGGIII scheme is adopted to weaken the influence of abnormal observation, and the PAR strategy is conducted in case of failure of the conventional PL-AR. The superiority of our proposed algorithm is validated using the measured indoor PL data for code-based differential PL (DPL) and phase-based real-time kinematic (RTK) positioning modes. Numerical results indicate that the positioning accuracy of RUKF-based indoor DPL is higher with a decimeter-level improvement compared that of the SUKF, especially in the presence of large gross errors. In terms of high-precision RTK positioning, RUKF can correctly identify centimeter-level anomalous observations and obtain a corresponding positioning accuracy improvement compared with the SUKF. When relatively large gross errors exist, the conventional method cannot easily realize PL-AR. By contrast, the combination of RUKF and the PAR algorithm can achieve PL-AR for the selected ambiguity subset successfully and can improve the positioning accuracy and reliability significantly. In summary, our proposed algorithm has certain resistance ability for abnormal observations. The indoor PL positioning of this algorithm outperforms that of the conventional method. Thus, the algorithm has some practical application value, especially for kinematic positioning.
... Our choice of sample states (sigma points) follows equation 11 of Ref. [71]. Prescriptions that require fewer sigma points exist [134]. ...
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The ability to engineer high-fidelity gates on quantum processors in the presence of systematic errors remains the primary barrier to achieving quantum advantage. Quantum optimal control methods have proven effective in experimentally realizing high-fidelity gates, but they require exquisite calibration to be performant. We apply robust trajectory optimization techniques to suppress gate errors arising from system parameter uncertainty. We propose a derivative-based approach that maintains computational efficiency by using forward-mode differentiation. Additionally, the effect of depolarization on a gate is typically modeled by integrating the Lindblad master equation, which is computationally expensive. We employ a computationally efficient model and utilize time-optimal control to achieve high-fidelity gates in the presence of depolarization. We apply these techniques to a fluxonium qubit and suppress simulated gate errors due to parameter uncertainty below $10^{-7}$ for static parameter deviations on the order of $1\%$.
... In such non-linear models, the KF is used with an unscented transformation and hence the derivation of UKF. In order to carry out predictions, the UKF picks a finite set of points (called sigma points [174]) around the mean and generates a new mean by passing this set through the non-linear function that describes the system. Thus, the new estimate is obtained. ...
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The commercial availability of low-cost millimeter wave (mmWave) communication and radar devices is starting to improve the penetration of such technologies in consumer markets, paving the way for large-scale and dense deployments in fifth-generation (5G)-and-beyond as well as 6G networks. At the same time, pervasive mmWave access will enable device localization and device-free sensing with unprecedented accuracy, especially with respect to sub-6 GHz commercial-grade devices. This paper surveys the state of the art in device-based localization and device-free sensing using mmWave communication and radar devices, with a focus on indoor deployments. We first overview key concepts about mmWave signal propagation and system design. Then, we provide a detailed account of approaches and algorithms for localization and sensing enabled by mmWaves. We consider several dimensions in our analysis, including the main objectives, techniques, and performance of each work, whether each research reached some degree of implementation, and which hardware platforms were used for this purpose. We conclude by discussing that better algorithms for consumer-grade devices, data fusion methods for dense deployments, as well as an educated application of machine learning methods are promising, relevant and timely research directions.
... The weight ω (0) is associated with the mean sigma point X (0) and is parameterized for tuning. Standard choices of parameterization and parameter values are suggested in early publications on ut/ukf [42,44,45,47,48,115]. There exists no general guidelines for how to choose the parameter values and in ukf implementations the values are often either left unchanged from the original suggestions or chosen ad hoc [12,35,75,89,119,120]. ...
... In general for an nth order system the number of sigma points required for computation are 2n+1. In [36] it has been shown that the number of sigma point can be reduced to (n+1). The major advantage of this reduction is the reduction of computational load of the filter. ...
... • UKI-1 (J = N θ + 2) [82,12]. I N θ is defined recursively as ...
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We consider Bayesian inference for large scale inverse problems, where computational challenges arise from the need for repeated evaluations of an expensive forward model. This renders most Markov chain Monte Carlo approaches infeasible, since they typically require $O(10^4)$ model runs, or more. Moreover, the forward model is often given as a black box or is impractical to differentiate. Therefore derivative-free algorithms are highly desirable. We propose a framework, which is built on Kalman methodology, to efficiently perform Bayesian inference in such inverse problems. The basic method is based on an approximation of the filtering distribution of a novel mean-field dynamical system into which the inverse problem is embedded as an observation operator. Theoretical properties of the mean-field model are established for linear inverse problems, demonstrating that the desired Bayesian posterior is given by the steady state of the law of the filtering distribution of the mean-field dynamical system, and proving exponential convergence to it. This suggests that, for nonlinear problems which are close to Gaussian, sequentially computing this law provides the basis for efficient iterative methods to approximate the Bayesian posterior. Ensemble methods are applied to obtain interacting particle system approximations of the filtering distribution of the mean-field model; and practical strategies to further reduce the computational and memory cost of the methodology are presented, including low-rank approximation and a bi-fidelity approach. The effectiveness of the framework is demonstrated in several numerical experiments, including proof-of-concept linear/nonlinear examples and two large-scale applications: learning of permeability parameters in subsurface flow; and learning subgrid-scale parameters in a global climate model from time-averaged statistics.
... This is the so-called "linearized tangent model". As the linearization process leads to errors in the nonlinear system due to the calculation of the Jacobian matrix and therefore a decrease in the accuracy of the estimate, the unscented Kalman filter (UKF), central difference Kalman filter (CDKF), and square root unscented Kalman filter (SRUKF) [19] are proposed in the literature for solving this problem. These methods are based on similar ideas and belong to a class of approximate nonlinear filtering methods (sigma point Kalman filter (SPKF) method) based on Gaussian distribution. ...
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The monitoring of the main variables and parameters of biotechnological processes is of key importance for the research and control of the processes, especially in industrial installations, where there is a limited number of measurements. For this reason, many researchers are focusing their efforts on developing appropriate algorithms (software sensors (SS)) to provide reliable information on unmeasurable variables and parameters, based on the available on-line information. In the literature, a large number of developments related to this topic that concern data-based and model-based sensors are presented. Up-to-date reviews of data-driven SS for biotechnological processes have already been presented in the scientific literature. Hybrid software sensors as a combination between the abovementioned ones are under development. This gives a reason for the article to be focused on a review of model-based software sensors for biotechnological processes. The most applied model-based methods for monitoring the kinetics and state variables of these processes are analyzed and compared. The following software sensors are considered: Kalman filters, methods based on estimators and observers of a deterministic type, probability observers, high-gain observers, sliding mode observers, adaptive observers, etc. The comparison is made in terms of their stability and number of tuning parameters. Particular attention is paid to the approach of the general dynamic model. The main characteristics of the classic variant proposed by D. Dochain are summarized. Results related to the development of this approach are analyzed. A key point is the presentation of new formalizations of kinetics and the design of new algorithms for its estimation in cases of uncertainty. The efficiency and applicability of the considered software sensors are discussed.
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This paper compares the continuum evolution for density equation modelling and the Gaussian mixture model on the 2D phase space long-term density propagation problem in the context of high-altitude and high area-to-mass ratio satellite long-term propagation. The density evolution equation, a pure numerical and pointwise method for the density propagation, is formulated under the influence of solar radiation pressure and Earth's oblateness using semi-analytical methods. Different from the density evolution equation and Monte Carlo techniques, for the Gaussian mixture model, the analytical calculation of the density is accessible from the first two statistical moments (i.e., the mean and the covariance matrix) corresponding to each sub-Gaussian distribution for an initial Gaussian density distribution. An insight is given into the phase space long-term density propagation problem subject to nonlinear dynamics. The efficiency and validity of the density propagation are demonstrated and compared between the density evolution equation and the Gaussian mixture model with respect to standard Monte Carlo techniques.
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The unscented Kalman filter (UKF) is finding increased application in biological fields. While realizing a complex UKF system in a low-power embedded platform offers many potential benefits including wearability, it also poses significant design challenges. Here we present a method for optimizing a UKF system for realization in an embedded platform. The method seeks to minimize both computation time and error in UKF state reconstruction and forecasting. As a case study, we applied the method to a model for the rat sleep-wake regulatory system in which 432 variants of the UKF over six different variables are considered. The optimization method is divided into three stages that assess computation time, state forecast error, and state reconstruction error. We apply a cost function to variants that pass all three stages to identify a variant that computes 27 times faster than the reference variant and maintains required levels of state estimation and forecasting accuracy. We draw the following insights: 1) process noise provides leeway for simplifying the model and its integration in ways that speed computation time while maintaining state forecasting accuracy, 2) the assimilation of observed data during the UKF correction step provides leeway for simplifying the UKF structure in ways that speed computation time while maintaining state reconstruction accuracy, and 3) the optimization process can be accelerated by decoupling variables that directly impact the underlying model from variables that impact the UKF structure.
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This paper compares the continuum evolution for density equation modelling and the Gaussian mixture model on the 2D phase space long-term density propagation problem in the context of high-altitude and high area-to-mass ratio satellite long-term propagation. The density evolution equation, a pure numerical and pointwise method for the density propagation, is formulated under the influence of solar radiation pressure and Earth’s oblateness using semi-analytical methods. Different from the density evolution equation and Monte Carlo techniques, for the Gaussian mixture model, the analytical calculation of the density is accessible from the first two statistical moments (i.e. the mean and the covariance matrix) corresponding to each sub-Gaussian distribution for an initial Gaussian density distribution. An insight is given into the phase space long-term density propagation problem subject to nonlinear dynamics. The efficiency and validity of the density propagation are demonstrated and compared between the density evolution equation and the Gaussian mixture model with respect to standard Monte Carlo techniques.
Article
Nonlinear energy sink (NES) devices have recently been introduced as a means of passive structural control and have been shown to effectively dissipate energy from structural systems during extreme vibrations. Due to their essential geometric nonlinearities, time domain based methods are often applied for identifying their system parameters, which is a challenging task. The unscented Kalman filter (UKF) has been shown in numerical studies to be robust to highly nonlinear systems with noisy data and therefore presents a promising option for identification. In this study, the UKF is used to determine the model parameters of an experimental NES device whose behavior is governed by a geometric nonlinearity in its stiffness and a friction-based nonlinearity in its damping. The standard implementation of the UKF is compared with two implementation methods developed by the authors, which vary in their use of experimental responses to train the NES device model. The impact of choosing different prior distributions on the parameters is also analyzed through Latin hypercube sampling to enhance the quality of the identification for practical implementation, where the prior distribution on the parameters is often ill-defined. The identified models generated using one of the proposed UKF implementation methods is shown to provide a robust model of the NES, demonstrating that the UKF can be used for parameter identification with this class of devices.
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Reactivity is a key parameter in Nuclear Power Plants (NPPs). It reflects the balance between neutron generation and consumption inside the reactor core. Therefore, reactivity monitoring inside the reactor core is essential to ensure the safe operation of NPPs. However, no physical sensor is available for measurement of reactivity. It can be inferred indirectly either from the reactor period or estimated using the reactor flux variation. Inference of reactivity from reactor period has its own limitations. Therefore, reactor flux based reactivity computation is of more interest. Various techniques based on deterministic as well as stochastic approaches for on-line computation of reactivity using the reactor flux are reported in literature. In this paper, Rao-Blackwellised Unscented Kalman Filtering (RBUKF) based adaptive state estimator is proposed for on-line estimation of reactivity from the output signal of neutron detectors. The efficacy of proposed adaptive RBUKF over non-adaptive RBUKF is established through simulations under different transient scenarios.
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Ziel der vorliegenden Arbeit ist der Entwurf eines Zustandsbeobachters für einen pen- delfähigen Körper. Der Zustandsbeobachter soll dazu dienen, den Containerbrückenversuchsstand des Instituts für Mechanik und Meerestechnik der Technischen Universität Hamburg-Harburg in seiner Dynamik vollständig abzubilden und einen Reglerentwurf zur aktiven Dämpfung des räumlichen Lastpendelns zu ermöglichen. Es wird davon ausgegangen, dass hierfür Methoden der nichtlinearen Regelung Verwendung finden werden, da realisierte lineare Regler sich als wenig effektiv erweisen. Somit ist eine nichtlineare Systembeschreibung anzustreben. Aufgrund der Steuerbarkeit sowohl der Position der Aufhängung als auch der Pendellänge ändert sich das Systemverhalten während des Betriebs. Der Entwurf muss diese zeitvarianten Parameter berücksichtigen.
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This paper proposes an efficient data assimilation approach based on the sigma-point Kalman filter (SPKF). With a potential for nonlinear filtering applications, the proposed approach, designated as the local unscented transform Kalman filter (LUTKF), is similar to the SPKF in that the mean and covariance of the nonlinear system are estimated by propagating a set of sigma points—also referred to as ensemble members—generated using the scaled unscented transformation (SUT), while making no assumptions with regard to nonlinear models. However, unlike the SPKF, the LUTKF can reduce the influence of observations on distant state variables by employing a localization scheme to suppress spurious correlations between distant locations in the error covariance matrix. Moreover, while the SPKF uses the augmented state vector constructed by concatenating the model states, model noise, and measurement noise, the system state for the LUTKF is not augmented with the random noise variables, thereby providing an accurate state estimate with relatively few sigma points. In sensitivity experiments executed with a 40-variable Lorenz system, the LUTKF required only three sigma points to prevent filter divergence for linear/nonlinear measurement models. Comparisons of the LUTKF and the local ensemble transform Kalman filters (LETKFs) reveal the advantages of the proposed filter in situations that share common features with geophysical data assimilation applications. In particular, the LUTKF shows considerable benefits over LETKFs when assimilating densely spaced observations that are related nonlinearly to the model state and that have high noise levels—such as the assimilation of remotely sensed data from satellites and radars.
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Inferring the interactions between coupled oscillators is a significant open problem in complexity science, with multiple interdisciplinary applications. While the Kalman filter (KF) technique is a well-known tool, widely used for data assimilation and parameter estimation, to the best of our knowledge, it has not yet been used for inferring the connectivity of coupled chaotic oscillators. Here we demonstrate that KF allows reconstructing the interaction topology and the coupling strength of a network of mutually coupled Rössler-like chaotic oscillators. We show that the connectivity can be inferred by considering only the observed dynamics of a single variable of the three that define the phase space of each oscillator. We also show that both the coupling strength and the network architecture can be inferred even when the oscillators are close to synchronization. Simulation results are provided to show the effectiveness and applicability of the proposed method.
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The problem of obtaining long term relative orbit configurations for spacecraft clusters with realistic operational considerations such as safety, station keeping and inter-spacecraft distances is addressed. Two different approaches are developed for station keeping and safety objectives. In the first approach, relative orbit configurations, or relative TLEs, are found minimizing deviations from reference mean orbit which would maximize the station-keeping objective. In second one, relative configurations are found from a reference initial condition by minimizing probability of collision, hence maximizing the safety objective, between the spacecraft in the cluster which are propagated numerically through a high precision orbit propagator. For the design optimization, a derivative free algorithm is proposed. Effectiveness of the approaches is demonstrated through simulations. Using this design framework, several configurations can be found by exploring the limits of the clusters in terms of spacecraft number, distance bounds and probabilities of collision for long time intervals.
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An airship is lighter than an air vehicle with enormous potential in applications such as communication , aerial inspection, border surveillance, and precision agriculture. An airship model is made up of dynamic, aerodynamic, aerostatic, and propulsive forces. However, the computation of aerodynamic forces remained a challenge. In addition to aerodynamic model deficiencies, airship mass matrix suffers from parameter variations. Moreover, due to the lighter-than-air nature, it is also susceptible to wind disturbances. These modeling issues are the key challenges in developing an efficient autonomous flight controller for an airship. This article proposes a unified estimation method for airship states, model uncertainties , and wind disturbance estimation using Unscented Kalman Filter (UKF). The proposed method is based on a lumped model uncertainty vector that unifies model uncertainties and wind disturbances in a single vector. The airship model is extended by incorporating six auxiliary state variables into the lumped model uncertainty vector. The performance of the proposed methodology is evaluated using a nonlinear simulation model of a custom-developed UETT airship and is validated by conducting a kind of error analysis. For comparative studies, EKF estimator is also developed. The results show the performance superiority of the proposed estimator over EKF; however, the proposed estimator is a bit expensive on computational grounds. However, as per the requirements of the current application , the proposed estimator can be a preferred choice.
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In response to the ongoing COVID-19 pandemic caused by SARS-CoV-2, governments are taking a wide range of non-pharmaceutical interventions (NPI). These measures include interventions as stringent as strict lockdown but also school closure, bar and restaurant closure, curfews and barrier gestures i.e . social distancing. Disentangling the effectiveness of each NPI is crucial to inform response to future outbreaks. To this end, we first develop a multi-level estimation of the French COVID-19 epidemic over a period of one year. We rely on a global extended Susceptible-Infectious-Recovered (SIR) mechanistic model of the infection including a dynamical (over time) transmission rate containing a Wiener process accounting for modeling error. Random effects are integrated following an innovative population approach based on a Kalman-type filter where the log-likelihood functional couples data across French regions. We then fit the estimated time-varying transmission rate using a regression model depending on NPI, while accounting for vaccination coverage, apparition of variants of concern (VoC) and seasonal weather conditions. We show that all NPI considered have an independent significant effect on the transmission rate. We additionally demonstrate a strong effect from weather conditions which decrease transmission during the summer period, and also estimate increased transmissibility of VoCs.
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Constellation is required to be highly stable over several years for a space-based gravitational wave observatory. However, the stability of the constellation can be affected by orbit insertion errors. The effects of orbit insertion errors on the constellation are mainly studied in this paper. Firstly, Monte-Carlo, Unscented Transformation Covariance Analysis Method (UTCAM) and Spherical Simplex Unscented Transformation Covariance Analysis Method (SSUTCAM) are used for simulation. The results indicate that UTCAM and SSUTCAM are highly efficient in calculating, with a relative error of less than 6%. Therefore, it is concluded that because of their accuracy and high efficiency, UTCAM and SSUTCAM can be adequately used in orbit insertion error analysis for a space-based gravitational wave observatory. Secondly, SSUTCAM is used to study the effects of position and velocity errors on the constellation. For the case in this paper, when the position error does not exceed 300km, and the velocity error does not exceed 4cm/s, the constellation remains stable.
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The prior covariance calculated in the Reduced-rank Sigma-Point Kalman filter (RSPKF) data assimilation method can be suboptimal as a result of finite number of sigma points, effects of sampling error, process error, and other factors. In this study, we analyze the performance of RSPKF by applying a localization scheme that combines local analysis and global generation of sigma points. The analysis at each model grid for the entire domain are updated using only the observations local to the analysis grid. The localization enables a high rank approximation of the background error covariance in a local subspace of greatly lower dimension than the global domain using a small number of sigma points. The global-analysis vector is constructed combining the local analyses at all model grid points. The global analysis-covariance matrix is generated in the ensemble subspace and the global sigma points are constructed following the RSPKF algorithm. Numerical experiments of our method utilized the Lorenz-96 model. The performance of the localization scheme is assessed in the presence of varying parameters such as the number of sigma points (), inflation factor (), localization radius () and the number of model variables (). When the localization is implemented, the number of sigma points required to achieve the minimum RMSE is significantly reduced compared to a case where no localization is used, for three different cases of model variables (). We also show that the approximate number of sigma points used to obtain optimal estimate is independent of the state dimension of the model. This further highlights the importance of localization in RSPKF, making it a potential candidate for data assimilation in oceanic or atmospheric General Circulation Models (GCMs).
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Interception of the high velocity spiralling target requires accurate estimation of their relative states. In this work, the dynamics of spiralling target is modelled by representing the target acceleration through sinusoidal function in the inertial frame. A system model which consists of nine state variables is considered here with the three relative positions, three relative velocities, inverse of ballistic coefficient, manoeuvring coefficient and spiralling frequency of target. Non-linear state estimators, namely the unscented Kalman filter (UKF), new sigma point Kalman filter (NSKF) and cubature Kalman filter (CKF), are applied to track the spiralling target using the measurements obtained from an inbuilt seeker of the interceptor missile. Further, the estimated states are provided to a guidance block for generating interceptor missile accelerations. Performance comparison of state estimators in combination with PNG law is done by calculating the root mean square error(RMSE) of relative states and average miss-distance.
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Extensive clinical and experimental evidence links sleep–wake regulation and state of vigilance (SOV) to neurological disorders including schizophrenia and epilepsy. To understand the bidirectional coupling between disease severity and sleep disturbances, we need to investigate the underlying neurophysiological interactions of the sleep–wake regulatory system (SWRS) in normal and pathological brains. We utilized unscented Kalman filter based data assimilation (DA) and physiologically based mathematical models of a sleep–wake regulatory network synchronized with experimental measurements to reconstruct and predict the state of SWRS in chronically implanted animals. Critical to applying this technique to real biological systems is the need to estimate the underlying model parameters. We have developed an estimation method capable of simultaneously fitting and tracking multiple model parameters to optimize the reconstructed system state. We add to this fixed-lag smoothing to improve reconstruction of random input to the system and those that have a delayed effect on the observed dynamics. To demonstrate application of our DA framework, we have experimentally recorded brain activity from freely behaving rodents and classified discrete SOV continuously for many-day long recordings. These discretized observations were then used as the “noisy observables” in the implemented framework to estimate time-dependent model parameters and then to forecast future state and state transitions from out-of-sample recordings.
Article
A significant problem in tracking and estimation is the consistent transformation of uncertain state estimates between Cartesian and spherical coordinate systems. For example, a radar system generates measurements in its own local spherical coordinate system. In order to combine those measurements with those from other radars, however, a tracking system typically transforms all measurements to a common Cartesian coordinate system. The most common approach is to approximate the transformation through linearization. However, this approximation can lead to biases and inconsistencies, especially when the uncertainties on the measurements are large. A number of approaches have been proposed for using higher order transformation modes, but these approaches have found only limited use due to the often enormous implementation burdens incurred by the need to derive Jacobians and Hessians. This paper expands a method for nonlinear propagation which is described in a companion paper. A discrete set of samples are used to capture the first four moments of the untransformed measurement. The transformation is then applied to each of the samples, and the mean and covariance are calculated from the result. It is shown that the performance of the algorithm is comparable to that of fourth order filters, thus ensuring consistency even when the uncertainty is large. It is not necessary to calculate any derivatives, and the algorithm can be extended to incorporate higher order information. The benefits of this algorithm are illustrated in the contexts of autonomous vehicle navigation and missile tracking.© (1997) COPYRIGHT SPIE--The International Society for Optical Engineering. Downloading of the abstract is permitted for personal use only.
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The problem of tracking curves in dense visual clutter is challenging. Kalman filtering is inadequate because it is based on Gaussian densities which, being unimo dal, cannot represent simultaneous alternative hypotheses. The Condensation algorithm uses factored sampling, previously applied to the interpretation of static images, in which the probability distribution of possible interpretations is represented by a randomly generated set. Condensation uses learned dynamical models, together with visual observations, to propagate the random set over time. The result is highly robust tracking of agile motion. Notwithstanding the use of stochastic methods, the algorithm runs in near real-time.
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Mobile robot localization is the problem of determining a robot's pose from sensor data. This article presents a family of probabilistic localization algorithms known as Monte Carlo Localization (MCL). MCL algorithms represent a robot's belief by a set of weighted hypotheses (samples), which approximate the posterior under a common Bayesian formulation of the localization problem. Building on the basic MCL algorithm, this article develops a more robust algorithm called Mixture-MCL, which integrates two complimentary ways of generating samples in the estimation. To apply this algorithm to mobile robots equipped with range finders, a kernel density tree is learned that permits fast sampling. Systematic empirical results illustrate the robustness and computational efficiency of the approach.
Book
The product of a unique collaboration among four leading scientists in academic research and industry, Numerical Recipes is a complete text and reference book on scientific computing. In a self-contained manner it proceeds from mathematical and theoretical considerations to actual practical computer routines. With over 100 new routines bringing the total to well over 300, plus upgraded versions of the original routines, the new edition remains the most practical, comprehensive handbook of scientific computing available today.
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State estimators for nonlinear systems are derived based on polynomial approximations obtained with a multi-dimensional interpolation formula. It is shown that under certain assumptions the estimators perform better than estimators based on Taylor approximations. Nevertheless, the implementation is significantly simpler as no derivatives are required. Thus, it is believed that the new state estimators can replace well-known estimators, such as the extended Kalman filter (EKF) and its higher-order relatives, in most practical applications.
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"Pt.1. Consists of papers published in the 1960s -- Pt.2. Originally appeared as a special issue for Mar. 1983 of IEEE transactions on automatic control" Incluye índice
Conference Paper
This paper describes a generalisation of the unscented transformation (UT) which allows sigma points to be scaled to an arbitrary dimension. The UT is a method for predicting means and covariances in nonlinear systems. A set of samples are deterministically chosen which match the mean and covariance of a (not necessarily Gaussian-distributed) probability distribution. These samples can be scaled by an arbitrary constant. The method guarantees that the mean and covariance second order accuracy in mean and covariance, giving the same performance as a second order truncated filter but without the need to calculate any Jacobians or Hessians. The impacts of scaling issues are illustrated by considering conversions from polar to Cartesian coordinates with large angular uncertainties.
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An algorithm, the bootstrap filter, is proposed for implementing recursive Bayesian filters. The required density of the state vector is represented as a set of random samples, which are updated and propagated by the algorithm. The method is not restricted by assumptions of linearity or Gaussian noise: it may be applied to any state transition or measurement model. A simulation example of the bearings only tracking problem is presented. This simulation includes schemes for improving the efficiency of the basic algorithm. For this example, the performance of the bootstrap filter is greatly superior to the standard extended Kalman filter
Article
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
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In tracking applications target motion is usually best modeled in a simple fashion using Cartesian coordinates. Unfortunately, in most systems the target position measurements are provided in terms of range and azimuth (bearing) with respect to the sensor location. This situation requires either converting the measurements to a Cartesian frame of reference and working directly on converted measurements or using an extended Kalman filter (EKF) in mixed coordinates. An accurate means of tracking with debiased consistent converted measurements which accounts for the sensor inaccuracies over all practical geometries and accuracies is presented. This method is compared with the mixed coordinates EKF approach as well as a previous converted measurement approach which is an acceptable approximation only for moderate cross-range errors. The new approach is shown to be more accurate in terms of position and velocity errors and provides consistent estimates (i.e., compatible with the filter calculated covariances) for all practical situations. The combination of parameters (range, range accuracy, and azimuth accuracy) for which debiasing is needed is presented in explicit form
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The Kalman Filter has many applications in mobile robotics ranging from perception, to position estimation, to control. This report formulates a navigation Kalman Filter. That is, one which estimates the position of autonomous vehicles. The filter is developed according to the state space formulation of Kalman's original papers. The state space formulation is particularly appropriate for the problem of vehicle position estimation. This filter formulation is fairly general. This generality is possible because the problem has been addressed . in 3D . in state space, with an augmented state vector . asynchronously . with tensor calculus measurement models The formulation has wide ranging uses. Some of the applications include: . as the basis of a vehicle position estimation system, whether any or all of dead reckoning, triangulation, or terrain aids or other landmarks are used . as the dead reckoning element and overall integration element when INS or GPS is used . as th...
Article
A significant problem in tracking and estimation is the consistent transformation of uncertain state estimates between Cartesian and spherical coordinate systems. For example, a radar system generates measurements in its own local spherical coordinate system. In order to combine those measurements with those from other radars, however, a tracking system typically transforms all measurements to a common Cartesian coordinate system. The most common approach is to approximate the transformation through linearisation. However, this approximation can lead to biases and inconsistencies, especially when the uncertainties on the measurements are large. A number of approaches have been proposed for using higher order transformation models, but these approaches have found only limited use due to the often enormous implementation burdens incurred by the need to derive Jacobians and Hessians. This paper expands a method for nonlinear propagation which is described in a companion paper 3 . A disc...
A Skewed Approach to Filtering
  • S J Julier
S. J. Julier, "A Skewed Approach to Filtering," in The Proceedings of AeroSense: The 12th International Symposium on Aerospace/Defense Sensing, Simulation and Controls, Orlando FL, USA, April 1998, vol. 3373, pp. 54-65, SPIE, Signal and Data Processing of Small Targets.