Anticipative grid design in point-mass approach to nonlinear state estimation

Šimandl, Miroslav; Královec, Jakub; Söderström, Torsten

doi:10.1109/9.995053

Cited by 70 publications

(23 citation statements)

References 5 publications

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“…The direct numerical solution of the Fokker-Planck equation has been proposed as one implementation of nonlinear filtering; published results use variants of a fixed grid in the target's state-space [4,18]. However, the "moving mesh" methods developed for the solution of partial differential equations [2,5,9,10] have significant computational advantages over fixed grids: they place sample points where they are required by the underlying target density, and adapt these sample points as the density evolves.…”

Section: Solution Of the Density Equationmentioning

confidence: 98%

“…Bayesian filters are implemented using either Monte Carlo methods (MCMs), i.e., "particle filters" [3,6,8,15], or by direct numerical computation [4,18,7]. A new approach to nonlinear Bayesian filtering is proposed in this paper that uses the direct numerical approach.…”

Section: Introductionmentioning

confidence: 99%

“…Target density computed using the jump tracker forward equation(12) and the moving mesh partial differential equations(18).…”

mentioning

confidence: 99%

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The Jump Tracker: Nonlinear Bayesian Tracking with Adaptive Meshes and a Markov Jump Process Model

Smith

2006

2006 Fortieth Asilomar Conference on Signals, Systems and Computers

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Many tracking problems lie outside of the traditional situation of linear (or linearizable) measurements and dynamics addressed by the Kalman filter and its variants. This is especially true in applications where target measurements are highly ambiguous and visibility is affected by unpredictable phenomena such as intermittent interference and low signal-to-noise ratios. Bayesian tracking provides the general solution to this more general class of problems. Conceptually, Bayesian tracking is straightforward: given the target measurements, apply Bayes' rule to compute the probability density of the target location at any given time, all the while assuming a target motion model. Bayesian trackers are computationally expensive; there are two basic approaches to their implementation: sequential Monte Carlo methods such as particle filtering, and deterministic methods that compute the target density directly. This paper presents a new approach to the direct method that is theoretically and computationally novel in several ways. First, recent results from the theory of adaptive moving meshes are modified and applied, an approach that is distinctly different from previously published direct methods that use fixed meshes to solve the Fokker-Planck equation. Second, a straight-line motion model based on a Markov jump process for the velocity is assumed. Straight-line motion punctuated by jumps in target velocity may be a more suitable assumption for some target dynamics than the traditional random walk assumed in the Kalman filter and many Bayesian trackers. The resulting linear partial differential equation that describes the target position density is relatively easy to solve numerically, especially compared to the FokkerPlanck equation that results from the random walk motion assumption. The proposed Bayesian tracking algorithm is a promising alternative to competing methods. It is also shown that, like particle filters, the adaptive mesh approach necessarily suffers from the curse of dimensionality. Simulation results are shown using the example of bistatic radar.

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Section: Solution Of the Density Equationmentioning

confidence: 98%

Section: Introductionmentioning

confidence: 99%

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The Jump Tracker: Nonlinear Bayesian Tracking with Adaptive Meshes and a Markov Jump Process Model

Smith

2006

2006 Fortieth Asilomar Conference on Signals, Systems and Computers

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“…Other possible approaches are statistical linearization [17], grid based methods [35], [37] and multiple model methods [7], [35], Gaussian sum approximations [38], [39], and numerical solving of the Kolmogorov forward equation [40], [41].…”

Section: B Optimal Continuous-discrete Filteringmentioning

confidence: 99%

On Unscented Kalman Filtering for State Estimation of Continuous-Time Nonlinear Systems

Särkkä¹

2007

IEEE Trans. Automat. Contr.

474

157

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Abstract-This article considers the application of the unscented Kalman filter (UKF) to continuous-time filtering problems, where both the state and measurement processes are modeled as stochastic differential equations. The mean and covariance differential equations which result in the continuous-time limit of the UKF are derived. The continuous-discrete unscented Kalman filter is derived as a special case of the continuous-time filter, when the continuous-time prediction equations are combined with the update step of the discrete-time unscented Kalman filter. The filter equations are also transformed into sigma-point differential equations, which can be interpreted as matrix square root versions of the filter equations.

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“…Unfortunately, the resulting Gaussian densities are not capable of representing arbitrary density functions that may appear in estimation problems of arbitrary nonlinear systems. Other density representations include Gaussian mixture densities [7], gridbased approaches [8], simple moments of probability density functions [9], exponential densities [3], fourier series [10], [11], the representation by means of sample sets [12], or Dirac mixture densities [13], which are capable of representing more general densities.…”

Section: Introductionmentioning

confidence: 99%

Direct fusion of Dirac mixture densities using an efficient approximation in joint state space

Klumpp¹,

Hanebeck²

2008

2008 IEEE International Conference on Multisensor Fusion and Integration for Intelligent Systems

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Abstract-In this paper, we present a direct fusion algorithm for processing the combination of two Dirac mixture densities. The proposed approach allows the multiplication of two Dirac mixture densities without requiring identical support and thus enables the fusion of two independently generated sample sets. The resulting posterior Dirac mixture density is an approximation of the true continuous density that would result from the processing of the underlying true continuous density functions. This procedure is based on a suboptimal greedy approximation of the joint state space by means of a Dirac mixture that iteratively increases the resolution of the fusion result while considering only the relevant regions in the joint state space, where the fusion constraint holds.

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Anticipative grid design in point-mass approach to nonlinear state estimation

Cited by 70 publications

References 5 publications

The Jump Tracker: Nonlinear Bayesian Tracking with Adaptive Meshes and a Markov Jump Process Model

The Jump Tracker: Nonlinear Bayesian Tracking with Adaptive Meshes and a Markov Jump Process Model

On Unscented Kalman Filtering for State Estimation of Continuous-Time Nonlinear Systems

Direct fusion of Dirac mixture densities using an efficient approximation in joint state space

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