Jeffrey K. Uhlmann

Unscented filtering and nonlinear estimation

The extended Kalman filter (EKF) is probably the most widely used estimation algorithm for nonlinear systems. However, more than 35 years of experience in the estimation community has shown that is difficult to implement, difficult to tune, and only reliable for systems that are almost linear on the time scale of the updates. Many of these difficulties arise from its use of linearization. To overcome this limitation, the unscented transformation (UT) was developed as a method to propagate mean and covariance information through nonlinear transformations. It is more accurate, easier to implement, and uses the same order of calculations as linearization. This paper reviews the motivation, development, use, and implications of the UT.

A New Extension of the Kalman Filter to Nonlinear Systems

The Kalman Filter (KF) is one of the most widely used methods for tracking and estimation due to its simplicity, optimality, tractability and robustness. However, the application of the KF to nonlinear systems can be difficult. The most common approach is to use the Extended Kalman Filter (EKF) which simply linearizes all nonlinear models so that the traditional linear Kalman filter can be applied. Although the EKF (in its many forms) is a widely used filtering strategy, over thirty years of experience with it has led to a general consensus within the tracking and control community that it is difficult to implement, difficult to tune, and only reliable for systems which are almost linear on the time scale of the update intervals. In this paper a new linear estimator is developed and demonstrated. Using the principle that a set of discretely sampled points can be used to parameterize mean and covariance, the estimator yields performance equivalent to the KF for linear systems yet generalizes elegantly to nonlinear systems without the linearization steps required by the EKF. We show analytically that the expected performance of the new approach is superior to that of the EKF and, in fact, is directly comparable to that of the second order Gauss filter. The method is not restricted to assuming that the distributions of noise sources are Gaussian. We argue that the ease of implementation and more accurate estimation features of the new filter recommend its use over the EKF in virtually all applications.

A new method for the nonlinear transformation of means and covariances in filters and estimators

This paper describes a new approach for generalizing the Kalman filter to nonlinear systems. A set of samples are used to parametrize the mean and covariance of a (not necessarily Gaussian) probability distribution. The method yields a filter that is more accurate than an extended Kalman filter (EKF) and easier to implement than an EKF or a Gauss second-order filter. Its effectiveness is demonstrated using an example.

A new approach for filtering nonlinear systems

In this paper we describe a new recursive linear estimator for filtering systems with nonlinear process and observation models. This method uses a new parameterisation of the mean and covariance which can be transformed directly by the system equations to give predictions of the transformed mean and covariance. We show that this technique is more accurate and far easier to implement than an extended Kalman filter. Specifically, we present empirical results for the application of the new filter to the highly nonlinear kinematics of maneuvering vehicles.

Satisfying general proximity / similarity queries with metric trees

Abstract Divide-and-conquer search strategies are described for satisfying proximity queries involving arbitrary distance metrics.

A non-divergent estimation algorithm in the presence of unknown correlations

This paper addresses the problem of estimation when the cross-correlation in the errors between different random variables are unknown. A new data fusion algorithm, the covariance intersection algorithm (CI), is presented. It is proved that this algorithm yields consistent estimates irrespective of the actual correlations. This property is illustrated in an application of decentralised estimation where it is impossible to consistently use a Kalman filter.

Reduced sigma point filters for the propagation of means and covariances through nonlinear transformations

The Unscented Transform (UT) approximates the result of applying a specified nonlinear transformation to a given mean and covariance estimate. The UT works by constructing a set of points, referred to as sigma points, which has the same known statistics, e.g., first and second and possibly higher moments, as the given estimate. The given nonlinear transformation Is applied to the set, and the unscented estimate is obtained by computing the statistics of the transformed set of sigma points. For example, the mean and covariance of the transformed set approximates the nonlinear transformation of the original mean and covariance estimate. The computational efficiency of the UT therefore depends on the number of sigma points required to capture the known statistics of the original estimate. In this paper we examine methods for minimizing the number of sigma points for real-time control, estimation, and filtering applications. We demonstrate results in a 3D localization example.

Using covariance intersection for SLAM

One of the greatest obstacles to the use of Simultaneous Localization And Mapping (SLAM) in a real-world environment is the need to maintain the full correlation structure between the vehicle and all of the landmark estimates. This structure is computationally expensive to maintain and is not robust to linearization errors. In this tutorial we describe SLAM algorithms that attempt to circumvent these difficulties through the use of Covariance Intersection (CI). CI is the optimal algorithm for fusing estimates when the correlations among them are unknown. A feature of CI relative to techniques which exploit full correlation information is that it provides provable consistency with much less computational overhead. In practice, however, a tradeoff typically needs to be made between estimation accuracy and computational cost. We describe a number of techniques that span the range of tradeoffs from maximum computational efficiency with straight CI to maximum estimation efficiency with the maintenance of all correlation information. We present a set of examples illustrating benefits of CI-based SLAM.

Covariance consistency methods for fault-tolerant distributed data fusion

Abstract This paper presents a general, rigorous, and fault-tolerant framework for maintaining consistent mean and covariance estimates in an arbitrary, dynamic, distributed network of information processing nodes. In particular, a solution is provided that addresses the information deconfliction problem that arises when estimates from two or more different nodes are determined to be inconsistent with each other, e.g., when two high precision (small covariance) estimates place the position of a particular object at very different locations. The challenge is to be able to resolve such inconsistencies without having to access and exploit global information to determine which of the estimates is spurious. The solution proposed in this paper is called Covariance Union.

General data fusion for estimates with unknown cross covariances

In this paper we present a new theoretic framework for combining sensor measurements, state estimates, or any similar type of quantity given only their means and covariances. The key feature of the new framework is that it permits the optimal fusion of estimates that are correlated to an unknown degree. This framework yields a new filtering paradigm that avoids all of the restrictive independence assumptions required by the standard Kalman filter, though at the cost of reduced rates of convergence for cases in which independence can be established.