Category Archives: Bayesian Filtering

Using EKF estimation in a PI controller for improving its performance under noise

Y. Zhou, Q. Zhang, H. Wang, P. Zhou and T. Chai, EKF-Based Enhanced Performance Controller Design for Nonlinear Stochastic Systems, IEEE Transactions on Automatic Control, vol. 63, no. 4, pp. 1155-1162, DOI: 10.1109/TAC.2017.2742661.

In this paper, a novel control algorithm is presented to enhance the performance of the tracking property for a class of nonlinear and dynamic stochastic systems subjected to non-Gaussian noises. Although the existing standard PI controller can be used to obtain the basic tracking of the systems, the desired tracking performance of the stochastic systems is difficult to achieve due to the random noises. To improve the tracking performance, an enhanced performance loop is constructed using the EKF-based state estimates without changing the existing closed loop with a PI controller. Meanwhile, the gain of the enhanced performance loop can be obtained based upon the entropy optimization of the tracking error. In addition, the stability of the closed loop system is analyzed in the mean-square sense. The simulation results are given to illustrate the effectiveness of the proposed control algorithm.

Probabilistic SLAM is still the way to go for dynamic environments (according to this paper)

C. Evers and P. A. Naylor, Optimized Self-Localization for SLAM in Dynamic Scenes Using Probability Hypothesis Density Filters, IEEE Transactions on Signal Processing, vol. 66, no. 4, pp. 863-878, DOI: 10.1109/TSP.2017.2775590.

In many applications, sensors that map the positions of objects in unknown environments are installed on dynamic platforms. As measurements are relative to the observer’s sensors, scene mapping requires accurate knowledge of the observer state. However, in practice, observer reports are subject to positioning errors. Simultaneous localization and mapping addresses the joint estimation problem of observer localization and scene mapping. State-of-the-art approaches typically use visual or optical sensors and therefore rely on static beacons in the environment to anchor the observer estimate. However, many applications involving sensors that are not conventionally used for Simultaneous Localization and Mapping (SLAM) are affected by highly dynamic scenes, such that the static world assumption is invalid. This paper proposes a novel approach for dynamic scenes, called GEneralized Motion (GEM) SLAM. Based on probability hypothesis density filters, the proposed approach probabilistically anchors the observer state by fusing observer information inferred from the scene with reports of the observer motion. This paper derives the general, theoretical framework for GEM-SLAM, and shows that it generalizes existing Probability Hypothesis Density (PHD)-based SLAM algorithms. Simulations for a model-specific realization using range-bearing sensors and multiple moving objects highlight that GEM-SLAM achieves significant improvements over three benchmark algorithms.

A new approach to SLAM based on KF but without linearization

Feng Tan, Winfried Lohmiller, and Jean-Jacques Slotine, Analytical SLAM without linearization, The International Journal of Robotics Research
Vol 36, Issue 13-14, pp. 1554 – 1578, DOI: 10.1177/0278364917710541.

This paper solves the classical problem of simultaneous localization and mapping (SLAM) in a fashion that avoids linearized approximations altogether. Based on the creation of virtual synthetic measurements, the algorithm uses a linear time-varying Kalman observer, bypassing errors and approximations brought by the linearization process in traditional extended Kalman filtering SLAM. Convergence rates of the algorithm are established using contraction analysis. Different combinations of sensor information can be exploited, such as bearing measurements, range measurements, optical flow, or time-to-contact. SLAM-DUNK, a more advanced version of the algorithm in global coordinates, exploits the conditional independence property of the SLAM problem, decoupling the covariance matrices between different landmarks and reducing computational complexity to O(n). As illustrated in simulations, the proposed algorithm can solve SLAM problems in both 2D and 3D scenarios with guaranteed convergence rates in a full nonlinear context.

A new method for estimating inertial sensor signals

M. Ghobadi, P. Singla and E. T. Esfahani, Robust Attitude Estimation from Uncertain Observations of Inertial Sensors Using Covariance Inflated Multiplicative Extended Kalman Filter, IEEE Transactions on Instrumentation and Measurement, vol. 67, no. 1, pp. 209-217, DOI: 10.1109/TIM.2017.2761230.

This paper presents an attitude estimation method from uncertain observations of inertial sensors, which is highly robust against different uncertainties. The proposed method of covariance inflated multiplicative extended Kalman filter (CI-MEKF) takes the advantage of non-singularity of covariance in MEKF as well as a novel covariance inflation (CI) approach to fuse inconsistent information. The proposed CI approach compensates the undesired effect of magnetic distortion and body acceleration (as inherent biases of magnetometer and accelerometer sensors data, respectively) on the estimated attitude. Moreover, the CI-MEKF can accurately estimate the gyro bias. A number of simulation scenarios are designed to compare the performance of the proposed method with the state of the art in attitude estimation. The results show the proposed method outperforms the state of the art in terms of estimation accuracy and robustness. Moreover, the proposed CI-MEKF method is shown to be significantly robust against different uncertainties, such as large body acceleration, magnetic distortion, and errors, in the initial condition of the attitude.

Extending bayesian fusion from Euclidean spaces to Lie groups

Kevin C. Wolfe, Michael Mashner, Gregory S. Chirikjian, Bayesian Fusion on Lie Groups, JOURNAL OF ALGEBRAIC STATISTICS Vol. 2, No. 1, 2011, 75-97, DOI: 10.18409/jas.v2i1.11.

An increasing number of real-world problems involve the measurement of data, and the computation of estimates, on Lie groups. Moreover, establishing confidence in the resulting estimates is important. This paper therefore seeks to contribute to a larger theoretical framework that generalizes classical multivariate statistical analysis from Euclidean space to the setting of Lie groups. The particular focus here is on extending Bayesian fusion, based on exponential families of probability densities, from the Euclidean setting to Lie groups. The definition and properties of a new kind of Gaussian distribution for connected unimodular Lie groups are articulated, and explicit formulas and algorithms are given for finding the mean and covariance of the fusion model based on the means and covariances of the constituent probability densities. The Lie groups that find the most applications in engineering are rotation groups and groups of rigid-body motions. Orientational (rotation-group) data and associated algorithms for estimation arise in problems including satellite attitude, molecular spectroscopy, and global geological studies. In robotics and manufacturing, quantifying errors in the position and orientation of tools and parts are important for task performance and quality control. Developing a general way to handle problems on Lie groups can be applied to all of these problems. In particular, we study the issue of how to ‘fuse’ two such Gaussians and how to obtain a new Gaussian of the same form that is ‘close to’ the fused density.This is done at two levels of approximation that result from truncating the Baker-Campbell-Hausdorff formula with different numbers of terms. Algorithms are developed and numerical results are presented that are shown to generate the equivalent fused density with good accuracy

Kalman Filter as the extreme case of finite impulse response filters as the horizon increases in length

Shunyi Zhao, Biao Huang, Yuriy S. Shmaliy, Bayesian state estimation on finite horizons: The case of linear state–space model,Automatica, Volume 85, 2017, Pages 91-99, DOI: 10.1016/j.automatica.2017.07.043.

The finite impulse response (FIR) filter and infinite impulse response filter including the Kalman filter (KF) are generally considered as two different types of state estimation methods. In this paper, the sequential Bayesian philosophy is extended to a filter design using a fixed amount of most recent measurements, with the aim of exploiting the FIR structure and unifying some basic FIR filters with the KF. Specifically, the conditional mean and covariance of the posterior probability density functions are first derived to show the FIR counterpart of the KF. To remove the dependence on initial states, the corresponding likelihood is further maximized and realized iteratively. It shows that the maximum likelihood modification unifies the existing unbiased FIR filters by tuning a weighting matrix. Moreover, it converges to the Kalman estimate with the increase of horizon length, and can thus be considered as a link between the FIR filtering and the KF. Several important properties including stability and robustness against errors of noise statistics are illustrated. Finally, a moving target tracking example and an experiment with a three degrees-of-freedom helicopter system are introduced to demonstrate effectiveness.

Dealing with nonlinearities in Kalman filters through Monte Carlo modelling for minimizing divergence

S. Gultekin and J. Paisley, Nonlinear Kalman Filtering With Divergence Minimization, IEEE Transactions on Signal Processing, vol. 65, no. 23, pp. 6319-6331, DOI: 10.1109/TSP.2017.2752729.

We consider the nonlinear Kalman filtering problem using Kullback-Leibler (KL) and α-divergence measures as optimization criteria. Unlike linear Kalman filters, nonlinear Kalman filters do not have closed form Gaussian posteriors because of a lack of conjugacy due to the nonlinearity in the likelihood. In this paper, we propose novel algorithms to approximate this posterior by optimizing the forward and reverse forms of the KL divergence, as well as the α-divergence that contains these two as limiting cases. Unlike previous approaches, our algorithms do not make approximations to the divergences being optimized, but use Monte Carlo techniques to derive unbiased algorithms for direct optimization. We assess performance on radar and sensor tracking, and options pricing, showing general improvement over the extended, unscented, and ensemble Kalman filters, as well as competitive performance with particle filtering.

The problem of the interdependence among particles in PF after the resampling step, and an approach to solve it

R. Lamberti, Y. Petetin, F. Desbouvries and F. Septier, Independent Resampling Sequential Monte Carlo Algorithms, IEEE Transactions on Signal Processing, vol. 65, no. 20, pp. 5318-5333, DOI: 10.1109/TSP.2017.2726971.

Sequential Monte Carlo algorithms, or particle filters, are Bayesian filtering algorithms, which propagate in time a discrete and random approximation of the a posteriori distribution of interest. Such algorithms are based on importance sampling with a bootstrap resampling step, which aims at struggling against weight degeneracy. However, in some situations (informative measurements, high-dimensional model), the resampling step can prove inefficient. In this paper, we revisit the fundamental resampling mechanism, which leads us back to Rubin’s static resampling mechanism. We propose an alternative rejuvenation scheme in which the resampled particles share the same marginal distribution as in the classical setup, but are now independent. This set of independent particles provides a new alternative to compute a moment of the target distribution and the resulting estimate is analyzed through a CLT. We next adapt our results to the dynamic case and propose a particle filtering algorithm based on independent resampling. This algorithm can be seen as a particular auxiliary particle filter algorithm with a relevant choice of the first-stage weights and instrumental distributions. Finally, we validate our results via simulations, which carefully take into account the computational budget.

Varying the number of particles in a PF in order to improve the speed of convergence, with a short related work about adapting the number of particles for other goals

V. Elvira, J. Míguez and P. M. Djurić, “Adapting the Number of Particles in Sequential Monte Carlo Methods Through an Online Scheme for Convergence Assessment,” in IEEE Transactions on Signal Processing, vol. 65, no. 7, pp. 1781-1794, April1, 1 2017. DOI: 10.1109/TSP.2016.2637324.

Particle filters are broadly used to approximate posterior distributions of hidden states in state-space models by means of sets of weighted particles. While the convergence of the filter is guaranteed when the number of particles tends to infinity, the quality of the approximation is usually unknown but strongly dependent on the number of particles. In this paper, we propose a novel method for assessing the convergence of particle filters in an online manner, as well as a simple scheme for the online adaptation of the number of particles based on the convergence assessment. The method is based on a sequential comparison between the actual observations and their predictive probability distributions approximated by the filter. We provide a rigorous theoretical analysis of the proposed methodology and, as an example of its practical use, we present simulations of a simple algorithm for the dynamic and online adaptation of the number of particles during the operation of a particle filter on a stochastic version of the Lorenz 63 system.

A new Kalman Filter that is more robust under certain deviations of the gaussian hypothesis

Badong Chen, Xi Liu, Haiquan Zhao, Jose C. Principe, Maximum correntropy Kalman filter, Automatica, Volume 76, February 2017, Pages 70-77, ISSN 0005-1098, DOI: 10.1016/j.automatica.2016.10.004.

Traditional Kalman filter (KF) is derived under the well-known minimum mean square error (MMSE) criterion, which is optimal under Gaussian assumption. However, when the signals are non-Gaussian, especially when the system is disturbed by some heavy-tailed impulsive noises, the performance of KF will deteriorate seriously. To improve the robustness of KF against impulsive noises, we propose in this work a new Kalman filter, called the maximum correntropy Kalman filter (MCKF), which adopts the robust maximum correntropy criterion (MCC) as the optimality criterion, instead of using the MMSE. Similar to the traditional KF, the state mean vector and covariance matrix propagation equations are used to give prior estimations of the state and covariance matrix in MCKF. A novel fixed-point algorithm is then used to update the posterior estimations. A sufficient condition that guarantees the convergence of the fixed-point algorithm is also given. Illustration examples are presented to demonstrate the effectiveness and robustness of the new algorithm.