The Kalman filter is the minimum-variance state estimator for linear dynamic systems with Gaussian noise. Even if the noise is non-Gaussian, the Kalman filter is the best linear estimator. For nonlinear systems it is not possible, in general, to derive the optimal state estimator in closed form, but various modifications of the Kalman filter can be used to estimate the state. These modifications include the extended Kalman filter, the unscented Kalman filter, and the particle filter. Although the Kalman filter and its modifications are powerful tools for state estimation, we might have information about a system that the Kalman filter does not incorporate. For example, we may know that the states satisfy equality or inequality constraints. In this case we can modify the Kalman filter to exploit this additional information and get better filtering performance than the Kalman filter provides. This paper provides an overview of various ways to incorporate state constraints in the Kalman filter and its nonlinear modifications. If both the system and state constraints are linear, then all of these different approaches result in the same state estimate, which is the optimal constrained linear state estimate. If either the system or constraints are nonlinear, then constrained filtering is, in general, not optimal, and different approaches give different results.

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Kalman filtering with state constraints: a survey of linear and nonlinear algorithms

Kalman filters are commonly used to estimate the states of a dynamic system. However, in the application of Kalman filters there is often known model or signal information that is either ignored or dealt with heuristically. For instance, constraints on state values (which may be based on physical considerations) are often neglected because they do not fit easily into the structure of the Kalman filter. A rigorous analytic method of incorporating state equality constraints in the Kalman filter is developed. The constraints may be time varying. At each time step the unconstrained Kalman filter solution is projected onto the state constraint surface. This significantly improves the prediction accuracy of the filter. The use of this algorithm is demonstrated on a simple nonlinear vehicle tracking problem.

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Kalman filtering with state 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

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On Kalman Filtering With Nonlinear Equality Constraints

This paper presents a generalization of the Kalman filter for linear and nonlinear fractional order discrete state-space systems. Linear and nonlinear discrete fractional order state-space systems are also introduced. The simplified kalman filter for the linear case is called the fractional Kalman filter and its nonlinear extension is named the extended fractional Kalman filter. The background and motivations for using such techniques are given, and some algorithms are discussed. The paper also shows a simple numerical example of linear state estimation. Finally, as an example of nonlinear estimation, the paper discusses the possibility of using these algorithms for parameters and fractional order estimation for fractional order systems. Numerical examples of the use of these algorithms in a general nonlinear case are presented.

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Fractional Kalman filter algorithm for the states, parameters and order of fractional system estimation

A procedure is developed to incorporate equality constraints in Kalman filters, including the ensemble Kalman filter (EnKF), and is referred to as the constrained ensemble Kalman filter (CEnKF). The constraint is carried out as a two-step filtering approach, with the first step being the standard (ensemble) Kalman filter. The second step is the constraint step carried out by another Kalman filter that optimally redistributes any imbalance from the first step. The CEnKF is implemented over a 75 000 km2 domain in the southern Great Plains region of the United States, using the terrestrial water balance as the constraint. The observations, consisting of gridded fields of the upper two soil moisture layers from the Oklahoma Mesonet system, Atmospheric Radiation Measurement Program Cloud and Radiation Testbed (ARM-CART) energy balance Bowen ratio (EBBR) latent heat estimates, and U.S. Geological Survey (USGS) streamflow from unregulated basins, are assimilated into the Variable Infiltration Capacity (...

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Data Assimilation for Estimating the Terrestrial Water Budget Using a Constrained Ensemble Kalman Filter

This paper presents the Smoothly Constrained Kalman Filter (SCKF) for nonlinear constraints A constraint is any relation that exists between the state variables Constraints can be treated as perfect observations But, linearization errors can prevent the estimate from converging to the true value Therefore, the SCKF iteratively applies nonlinear constraints as nearly perfect observations, or, equivalently, weakened constraints Integration of new measurements is interlaced with these iterations, which reduces linearization errors and, hence, improves convergence compared to other iterative methods The weakening is achieved by artificially increasing the variance of the nonlinear constraint The paper explains how to choose the weakening values, and when to start and stop the iterative application of the constraint

A smoothly constrained Kalman filter

A Kalman filter is a stochastic, recursive estimator, which estimates the state of a system based on the knowledge of the system input, the measurement of the system output, and a model of the relation between input and output. The Kalman filter equations are well known, but often little effort is spent to explain or understand how the Kalman filter really works, and what its assets and limitations are. This paper introduces the Kalman filter in a very intuitive way, and looks at its properties: what information can be extracted from it, and what are its limitations. This intuitive understanding should help to develop successful practical applications. The paper also points at further reading.

Kalman Filters : A Tutorial

This paper presents a general approach for the identification of contact and grasping uncertainties, and the monitoring of contact situation changes in force controlled assembly operations. The identification problem is solved using virtual contact manipulators and Kalman filter techniques. Monitoring is solved by carrying out a statistical test on the sum of normalized and squared innovations of the Kalman filter, within a moving window, identification and monitoring are verified by experimental results. The paper explains how the error covariance matrix of the Kalman filter is interpreted to analyse the observability of the Kalman filter's states. Preliminary simulation results are presented for an ad hoc active sensing strategy to achieve complete state observability.

Estimating contact and grasping uncertainties using Kalman filters in force controlled assembly

The choice of 'where to look next' is a special case of an optimal experiment design. This paper proposes the tolerance-optimal experiment design, which is a special instance of the well-known L-optimal design, that minimizes the weighted trace of the covariance matrix of the estimated state under Gaussian assumptions. The weighting matrix is chosen such that the design is invariant to transformations with non-singular Jacobians, and such that the emerging sensing sequence reflects the information needs of the task. This tolerance-optimal design does not require more calculations than existing optimal experiment designs. Existing optimal experiment designs do not reflect the information needs of the task. In addition, some of them physically do not make sense if the estimated state has inconsistent units.

Tolerance-weighted L-optimal experiment design: a new approach to task-directed sensing

This paper presents a recognition and location method for objects, composed of primitives with a simple analytical description, using local sensors such as an ultrasonic and an infra-red sensor. As these sensors only return local data, several measurements are needed to obtain global information. To allow effective exploration, on-line estimation is needed. Estimation is based on a constrained Kalman filter, the constraints defining relations between the primitives in the object model. Linearisation errors are reduced with a careful choice of the parametrisation of the primitives and by using a novel version of the constrained Kalman filter. Special features of the presented approach are the easy extendibility to objects composed of a different number and type of primitives, the fast convergence due to the effective use of geometric constraints, the ability to handle incomplete models, and the robustness to linearisation errors. The performance of the estimator has been thoroughly evaluated with simulations and experiments, showing that 10 measurements with an infrared sensor on the edges of a 800/spl times/550 mm/sup 2/ rectangular plate are sufficient to locate a point on the plate with standard deviation /spl sigma/=3 mm.

J. De Geeter

Papers

A smoothly constrained Kalman filter

Kalman Filters : A Tutorial

Estimating contact and grasping uncertainties using Kalman filters in force controlled assembly

Tolerance-weighted L-optimal experiment design: a new approach to task-directed sensing

Recognising and locating objects with local sensors