Exact approximation of Rao-Blackwellised particle filters

Particle methods are a category of Monte Carlo algorithms that have become popular for performing inference in non-linear non-Gaussian state-space models. The class of 'Rao-Blackwellised' particle filters exploits the analytic marginalisation that is possible for some state- space models to reduce the variance of the Monte Carlo estimates. Despite being applicable to only a restricted class of state-space models, such as conditionally linear Gaussian models, these algorithms have found numerous applications. In scenarios where no such analytical integration is possible, it has recently been proposed in Chen et al. [2011] to use 'local' particle filters to carry out this integration numerically. We propose here an alternative approach also relying on \local" particle filters which is more broadly applicable and has attractive theoretical properties. Proof-of-concept simulation results are presented

Rao-Blackwellization for Adaptive Gaussian Sum Nonlinear Model Propagation

When dealing with imperfect data and general models of dynamic systems, the best estimate is always sought in the presence of uncertainty or unknown parameters. In many cases, as the first attempt, the Extended Kalman filter (EKF) provides sufficient solutions to handling issues arising from nonlinear and non-Gaussian estimation problems. But these issues may lead unacceptable performance and even divergence. In order to accurately capture the nonlinearities of most real-world dynamic systems, advanced filtering methods have been created to reduce filter divergence while enhancing performance. Approaches, such as Gaussian sum filtering, grid based Bayesian methods and particle filters are well-known examples of advanced methods used to represent and recursively reproduce an approximation to the state probability density function (pdf). Some of these filtering methods were conceptually developed years before their widespread uses were realized. Advanced nonlinear filtering methods currently benefit from the computing advancements in computational speeds, memory, and parallel processing. Grid based methods, multiple-model approaches and Gaussian sum filtering are numerical solutions that take advantage of different state coordinates or multiple-model methods that reduced the amount of approximations used. Choosing an efficient grid is very difficult for multi-dimensional state spaces, and oftentimes expensive computations must be done at each point. For the original Gaussian sum filter, a weighted sum of Gaussian density functions approximates the pdf but suffers at the update step for the individual component weight selections. In order to improve upon the original Gaussian sum filter, Ref. [2] introduces a weight update approach at the filter propagation stage instead of the measurement update stage. This weight update is performed by minimizing the integral square difference between the true forecast pdf and its Gaussian sum approximation. By adaptively updating each component weight during the nonlinear propagation stage an approximation of the true pdf can be successfully reconstructed. Particle filtering (PF) methods have gained popularity recently for solving nonlinear estimation problems due to their straightforward approach and the processing capabilities mentioned above. The basic concept behind PF is to represent any pdf as a set of random samples. As the number of samples increases, they will theoretically converge to the exact, equivalent representation of the desired pdf. When the estimated qth moment is needed, the samples are used for its construction allowing further analysis of the pdf characteristics. However, filter performance deteriorates as the dimension of the state vector increases. To overcome this problem Ref. [5] applies a marginalization technique for PF methods, decreasing complexity of the system to one linear and another nonlinear state estimation problem. The marginalization theory was originally developed by Rao and Blackwell independently. According to Ref. [6] it improves any given estimator under every convex loss function. The improvement comes from calculating a conditional expected value, often involving integrating out a supportive statistic. In other words, Rao-Blackwellization allows for smaller but separate computations to be carried out while reaching the main objective of the estimator. In the case of improving an estimator's variance, any supporting statistic can be removed and its variance determined. Next, any other information that dependents on the supporting statistic is found along with its respective variance. A new approach is developed here by utilizing the strengths of the adaptive Gaussian sum propagation in Ref. [2] and a marginalization approach used for PF methods found in Ref. [7]. In the following sections a modified filtering approach is presented based on a special state-space model within nonlinear systems to reduce the dimensionality of the optimization problem in Ref. [2]. First, the adaptive Gaussian sum propagation is explained and then the new marginalized adaptive Gaussian sum propagation is derived. Finally, an example simulation is presented

AUV SLAM and experiments using a mechanical scanning forward-looking sonar

Navigation technology is one of the most important challenges in the applications of autonomous underwater vehicles (AUVs) which navigate in the complex undersea environment. The ability of localizing a robot and accurately mapping its surroundings simultaneously, namely the simultaneous localization and mapping (SLAM) problem, is a key prerequisite of truly autonomous robots. In this paper, a modified-FastSLAM algorithm is proposed and used in the navigation for our C-Ranger research platform, an open-frame AUV. A mechanical scanning imaging sonar is chosen as the active sensor for the AUV. The modified-FastSLAM implements the update relying on the on-board sensors of C-Ranger. On the other hand, the algorithm employs the data association which combines the single particle maximum likelihood method with modified negative evidence method, and uses the rank-based resampling to overcome the particle depletion problem. In order to verify the feasibility of the proposed methods, both simulation experiments and sea trials for C-Ranger are conducted. The experimental results show the modified-FastSLAM employed for the navigation of the C-Ranger AUV is much more effective and accurate compared with the traditional methods

Adaptive Approximate Bayesian Computational Particle Filters for Underwater Terrain Aided Navigation

International audienceTo perform long-term and long-range missions, underwater vehicles need reliable navigation algorithms. This paper considers multi-beam Terrain Aided Navigation which can provide a drift-free navigation tool. This leads to an estimation problem with implicit observation equation and unknown likelihood. Indeed, the measurement sensor is considered to be a numerical black box model that introduces some unknown stochastic noise. We introduce a measurement updating procedure based on an adaptive kernel derived from Approximate Bayesian Computational filters. The proposed method is based on two well-known particle filters: Regularized Particle Filter and Rao-Blackwellized Particle Filter. Numerical results are presented and the robustness is demonstrated with respect to the original filters, yielding to twice as less non-convergence cases. The proposed method increases the robustness of particle-like filters while remaining computationally efficient

A class of fast exact Bayesian filters in dynamical models with jumps

In this paper, we focus on the statistical filtering problem in dynamical models with jumps. When a particular application relies on physical properties which are modeled by linear and Gaussian probability density functions with jumps, an usualmethod consists in approximating the optimal Bayesian estimate (in the sense of the Minimum Mean Square Error (MMSE)) in a linear and Gaussian Jump Markov State Space System (JMSS). Practical solutions include algorithms based on numerical approximations or based on Sequential Monte Carlo (SMC) methods. In this paper, we propose a class of alternative methods which consists in building statistical models which share the same physical properties of interest but in which the computation of the optimal MMSE estimate can be done at a computational cost which is linear in the number of observations.Comment: 21 pages, 7 figure

Simultaneous State and Parameter Estimation of Distributed-Parameter Physical Systems based on Sliced Gaussian Mixture Filter

This paper presents a method for the simultaneous state and parameter estimation of finite-dimensional models of distributed systems monitored by a sensor network. In the first step, the distributed system is spatially and temporally decomposed leading to a linear finite-dimensional model in state space form. The main challenge is that the simultaneous state and parameter estimation of such systems leads to a high-dimensional nonlinear problem. Thanks to the linear substructure contained in the resulting finite-dimensional model, the development of an overall more efficient estimation process is possible. Therefore, in the second step, we propose the application of a novel density representation - sliced Gaussian mixture density - in order to decompose the estimation problem into a (conditionally) linear and a nonlinear problem. The systematic approximation procedure minimizing a certain distance measure allows the derivation of (close to) optimal and deterministic results. The proposed estimation process provides novel prospects in sensor network applications. The performance is demonstrated by means of simulation results