Simon J. Julier

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.

Recent advances in augmented reality

In 1997, Azuma published a survey on augmented reality (AR). Our goal is to complement, rather than replace, the original survey by presenting representative examples of the new advances. We refer one to the original survey for descriptions of potential applications (such as medical visualization, maintenance and repair of complex equipment, annotation, and path planning); summaries of AR system characteristics (such as the advantages and disadvantages of optical and video approaches to blending virtual and real, problems in display focus and contrast, and system portability); and an introduction to the crucial problem of registration, including sources of registration error and error-reduction strategies.

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.

The scaled unscented transformation

This paper describes a generalisation of the unscented transformation (UT) which allows sigma points to be scaled to an arbitrary dimension. The UT is a method for predicting means and covariances in nonlinear systems. A set of samples are deterministically chosen which match the mean and covariance of a (not necessarily Gaussian-distributed) probability distribution. These samples can be scaled by an arbitrary constant. The method guarantees that the mean and covariance second order accuracy in mean and covariance, giving the same performance as a second order truncated filter but without the need to calculate any Jacobians or Hessians. The impacts of scaling issues are illustrated by considering conversions from polar to Cartesian coordinates with large angular uncertainties.

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.

The spherical simplex unscented transformation

This paper describes a new and better-behaved sigma point selection strategy for the unscented transformation (UT). The 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 have the same known statistics as the given estimate. This paper describes a sigma point selection strategy that requires, for n dimensions, n+2 sigma points; and n+1 of these points lie on a hypersphere whose radius is proportional to /spl radic/n. The weights on each point are proportional to 1/n. We illustrate the algorithm through an example which uses simultaneous localisation and map building.

On Kalman Filtering With Nonlinear Equality Constraints

The state space description of some physical systems possess nonlinear equality constraints between some state variables. In this paper, we consider the problem of applying a Kalman filter-type estimator in the presence of such constraints. We categorize previous approaches into pseudo-observation and projection methods and identify two types of constraints-those that act on the entire distribution and those that act on the mean of the distribution. We argue that the pseudo-observation approach enforces neither type of constraint and that the projection method enforces the first type of constraint only. We propose a new method that utilizes the projection method twice-once to constrain the entire distribution and once to constrain the statistics of the distribution. We illustrate these algorithms in a tracking system that uses unit quaternions to encode orientation