Niclas Bergman

Particle filters for positioning, navigation, and tracking

A framework for positioning, navigation, and tracking problems using particle filters (sequential Monte Carlo methods) is developed. It consists of a class of motion models and a general nonlinear measurement equation in position. A general algorithm is presented, which is parsimonious with the particle dimension. It is based on marginalization, enabling a Kalman filter to estimate all position derivatives, and the particle filter becomes low dimensional. This is of utmost importance for high-performance real-time applications. Automotive and airborne applications illustrate numerically the advantage over classical Kalman filter-based algorithms. Here, the use of nonlinear models and non-Gaussian noise is the main explanation for the improvement in accuracy. More specifically, we describe how the technique of map matching is used to match an aircrafts elevation profile to a digital elevation map and a cars horizontal driven path to a street map. In both cases, real-time implementations are available, and tests have shown that the accuracy in both cases is comparable with satellite navigation (as GPS) but with higher integrity. Based on simulations, we also argue how the particle filter can be used for positioning based on cellular phone measurements, for integrated navigation in aircraft, and for target tracking in aircraft and cars. Finally, the particle filter enables a promising solution to the combined task of navigation and tracking, with possible application to airborne hunting and collision avoidance systems in cars.

Terrain navigation using Bayesian statistics

The performance of terrain-aided navigation of aircraft depends on the size of the terrain gradient in the area. The point-mass filter (PMF) described in this work yields an approximate Bayesian solution that is well suited for the unstructured nonlinear estimation problem in terrain navigation. It recursively propagates a density function of the aircraft position. The shape of the point-mass density reflects the estimate quality; this information is crucial in navigation applications, where estimates from different sources often are fused in a central filter. Monte Carlo simulations show that the approximation can reach the optimal performance, and realistic simulations show that the navigation performance is very high compared with other algorithms and that the point-mass filter solves the recursive estimation problem for all the types of terrain covered in the test. The main advantages of the PMF is that it works for many kinds of nonlinearities and many kinds of noise and prior distributions. The mesh support and resolution are automatically adjusted and controlled using a few intuitive design parameters. The main disadvantage is that it cannot solve estimation problems of very high dimension since the computational complexity of the algorithm increases drastically with the dimension of the state space. The implementation used in this work shows real-time performance for 2D and in some cases 3D models, but higher state dimensions are usually intractable.

Auxiliary particle filters for tracking a maneuvering target

We consider the recursive state estimation of a highly maneuverable target. We apply optimal recursive Bayesian filters directly to the nonlinear target model. We present novel sequential simulation based algorithms developed explicitly for the maneuvering target tracking problem. These Monte Carlo filters perform optimal inference by simulating a large number of tracks, or particles. Each particle is assigned a probability weight determined by its likelihood. The main advantage of our approach is that linearizations and Gaussian assumptions need not be considered. Instead, a nonlinear model is directly used during the prediction and likelihood update. Detailed nonlinear dynamics models and non-Gaussian sensors can therefore be utilized in an optimal manner resulting in high performance gains. In a simulation comparison with current state-of-the-art tracking algorithms we show that our approach yields performance improvements.

Optimal Estimation and Cramér-Rao Bounds for Partial Non-Gaussian State Space Models

Partial non-Gaussian state-space models include many models of interest while keeping a convenient analytical structure. In this paper, two problems related to partial non-Gaussian models are addressed. First, we present an efficient sequential Monte Carlo method to perform Bayesian inference. Second, we derive simple recursions to compute posterior Cramér-Rao bounds (PCRB). An application to jump Markov linear systems (JMLS) is given.

Markov chain Monte Carlo data association for target tracking

We consider the estimation of the state of a discrete-time Markov process using observations which are sets of measurements from a finite number of known linear models. The measurement to model association is unknown and false measurements that do not yield any information about the Markov process are contained in the measurement set. The objective is to perform data association between the detected measurements and the models and determine optimal estimates of the state of the Markov process. The application of this problem is found in over the horizon target tracking. We derive iterative deterministic and stochastic algorithms based on Gibbs sampling. Rao-Blackwellisation allows us to solve the problem efficiently, yielding methods with computational complexity linear in the number of received data sets. Contrary to recent approaches based on the EM algorithm, the novel procedures we propose do not require an introduction of a missing data set and consequently their range of applicability is wider. A simulation study shows that the new algorithms are superior to previously proposed methods.

Point-mass filter and Cramer-Rao bound for terrain-aided navigation

The nonlinear estimation problem in navigation using terrain height variations is studied. The optimal Bayesian solution to the problem is derived. The implementation is grid based, calculating the probability of a set of points on an adaptively dense mesh. The Cramer-Rao bound is derived. Monte Carlo simulations over a commercial map shows that the algorithm, after convergence, reaches the Cramer-Rao lower bound.

A Bayesian Approach to Terrain-Aided Navigation

The terrain-aided navigation problem is a highly nonlinear estimation problem with application to aircraft navigation and missile guidance. In this work the Bayesian approach is used to estimate the aircraft position. With a quantization of the state space an implementable algorithm is found. Problems with low excitation, rough terrain and parallel position hypothesis are handled in a reliable way. The algorithm is evaluated using simulations on real terrain databases.

Deterministic and stochastic Bayesian methods in terrain navigation

Terrain navigation is an application where inference between conceptually different sensors is performed recursively online. In this work the Bayesian framework of statistical inference is applied to this recursive estimation problem. Three algorithms for approximative Bayesian estimation are evaluated in simulations, one deterministic algorithm and two stochastic. The deterministic method solve the Bayesian inference problem by numerical integration while the stochastic methods simulate several candidate solutions and evaluates the integral by averaging between these candidates. Simulations show that all three algorithms are efficient and approximately reach the Cramer-Rao bound. However, the stochastic methods are sensitive to outliers and the deterministic method has the limitation of being hard to implement in higher dimensions.

Applications of Matlab

A guided tour 38 (Differentiation and integration of polynomials) One class of functions that can easily be differ- entiated and integrated symbolically is poly-nomials.