Tag Archives: Kalman Filtering

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.

Integration of the ICP algorithm with a Kalman filter to improve relative localization, with a good state-of-the-art of ICP algorithms

F. Aghili and C. Y. Su, “Robust Relative Navigation by Integration of ICP and Adaptive Kalman Filter Using Laser Scanner and IMU,” in IEEE/ASME Transactions on Mechatronics, vol. 21, no. 4, pp. 2015-2026, Aug. 2016.DOI: 10.1109/TMECH.2016.2547905.

This paper presents a robust six-degree-of-freedom relative navigation by combining the iterative closet point (ICP) registration algorithm and a noise-adaptive Kalman filter in a closed-loop configuration together with measurements from a laser scanner and an inertial measurement unit (IMU). In this approach, the fine-alignment phase of the registration is integrated with the filter innovation step for estimation correction, while the filter estimate propagation provides the coarse alignment needed to find the corresponding points at the beginning of ICP iteration cycle. The convergence of the ICP point matching is monitored by a fault-detection logic, and the covariance associated with the ICP alignment error is estimated by a recursive algorithm. This ICP enhancement has proven to improve robustness and accuracy of the pose-tracking performance and to automatically recover correct alignment whenever the tracking is lost. The Kalman filter estimator is designed so as to identify the required parameters such as IMU biases and location of the spacecraft center of mass. The robustness and accuracy of the relative navigation algorithm is demonstrated through a hardware-in-the loop simulation setting, in which actual vision data for the relative navigation are generated by a laser range finder scanning a spacecraft mockup attached to a robotic motion simulator.

Substituting the update step of a bayesian filter by a maximum likelihood optimisation in order to use non-linear observation models in a (linear-transition) Kalman framework

Damián Marelli, Minyue Fu, and Brett Ninness, Asymptotic Optimality of the Maximum-Likelihood Kalman Filter for Bayesian Tracking With Multiple Nonlinear Sensors, IEEE Transactions on signal processing, vol. 63, no. 17, DOI: 10.1109/TSP.2015.2440220.

Bayesian tracking is a general technique for state estimation of nonlinear dynamic systems, but it suffers from the drawback of computational complexity. This paper is concerned with a class of Wiener systems with multiple nonlinear sensors. Such a system consists of a linear dynamic system followed by a set of static nonlinear measurements. We study a maximum-likelihood Kalman filtering (MLKF) technique which involves maximum-like-lihood estimation of the nonlinear measurements followed by classical Kalman filtering. This technique permits a distributed implementation of the Bayesian tracker and guarantees the boundedness of the estimation error. The focus of this paper is to study the extent to which the MLKF technique approximates the theoretically optimal Bayesian tracker. We provide conditions to guarantee that this approximation becomes asymptotically exact as the number of sensors becomes large. Two case studies are analyzed in detail.