Overcoming Errors in Processing Nonlinear Measurements: A New Extension for the Kalman Filter

A. Draganov

Abstract:	The list of sensors that are used for navigation applications continues to expand. This brings a new challenge to the filtering algorithm: it must process a wide variety of measurements, including nonlinear and non-Gaussian ones. For example, a GPS pseudorange usually is an almost linear measurement, but ranging off a beacon or a peer user is nonlinear, if the position uncertainty is on the order of the distance to the signal source. Applying an Extended Kalman filter to such measurements would produce a bias in the state estimate. There are several known nonlinear extensions to the classic Kalman filter algorithm, which improve the result to some degree, but still may not meet requirements for a particular application. This paper introduces a new algorithm, which efficiently processes nonlinear measurements and achieves higher accuracy than previous alternatives. Nonlinear variants for the Kalman filter include the Extended Kalman Filter (which is nothing more than a linearization), the 2nd order filter, the Iterated Kalman Filter, the Unscented Kalman Filter, Gauss-Hermite Filter, etc. In a previous paper we showed that all these known extensions fall short of processing all types of nonlinear measurements. We also derived an analytical formulation, which is universally applicable for different types of nonlinearities. In this paper, we extend that idea and present a computer algorithm, which delivers robust, efficient and highly accurate results for processing nonlinear measurements. The problems with known algorithms and the gist of the new algorithm are as follows. A Kalman filter or its extension characterizes both the state and a measurement using only the mean and covariance, i.e. the first two moments of the statistical distribution. It has been long recognized that nonlinear measurements warp the joint distribution of measurement and state errors, making it non-Gaussian even in the case of Gaussian noise statistics. This gives rise to higher order cumulants, which must be truncated for processing by the Kalman filter. This is the reason for errors that a classic Extended Kalman filter produces when processing nonlinear measurements. In particular, a nonlinear measurement may produce a bias in the state estimate. A bias is especially dangerous as it is not removed by processing multiple measurements from the same source. For example, a range measurement from a beacon is nonlinear, if the distance to the beacon is on the order of the uncertainty in the user position. If a user repeatedly processes such ranging measurements, random measurement errors are averaged out by the filter; however the bias due to nonlinearity is not removed by the Extended Kalman filter and thus corrupts the solution. This bias increases with the increase in the uncertainty in the user solution. The popular Unscented Kalman Filter algorithm and its variations (such as the Gauss-Hermite or Quadrature algorithm) account for the biases induced by measurement or time update nonlinearity. They compute the mean and covariance for the distribution, which is “warped” by nonlinearity. This leads to a more accurate solution in the statistical sense. As a result, posteriori errors are not biased, but still can be substantial. In our algorithm we make use of the fact that nonlinear errors are a function of the user uncertainty, and this uncertainty is always smaller for the posteriori state than for a priori one. Hence, using the posteriori state for computing nonlinear corrections should produce more accurate result. In our algorithm, we apply a predictor-corrector scheme. At the predictor stage, the posteriori state and covariance are roughly estimated using the Unscented Kalman filter (or its variant). Corrector uses this rough estimate to compute and apply nonlinear corrections, again using the Unscented Kalman filter. The result is at least as good as for the standard UKF application; however if the Kalman gain is relatively high, the predictor-corrector algorithm is much more accurate. This reasoning is confirmed by computer simulations, which are described in the paper. The user determines its position on a plane by processing range measurements from two beacons, which are located on fixed poles above the plane. We compare results from the predictor-corrector algorithm with those from EKF, Iterated Extended Kalman filter, and the Unscented Kalman filter. The nonlinearity corrupts the EKF estimate by producing a position-dependent non-zero mean error, as expected. Depending on the measurement quality, IEKF may produce a much better estimate, but it can still be biased. UKF removes the bias, but the error (even though zero mean) persists. The predictor-corrector algorithm achieves the best of both results: the bias is removed and the magnitude of the error is reduced, in some cases by a large factor. The paper also discusses a possibility of applying the same logic to account for nonlinear effects in the time update (the propagation step) of a Kalman filter. If the time update is nonlinear, the transformation of the user state vector is not affine, i.e. the distribution is “warped”. Similarly to the case of measurement processing, known methods compute nonlinear effects on the mean and covariance for the time-updated distribution before processing new measurements. The paper presents a thought experiment, which shows that a predictor-corrector scheme would produce benefit for the time update step similarly to that for the measurement update. Again, the reason for this is the smaller state uncertainty for posteriori state. The full development of the nonlinear time update predictor-corrector algorithm is beyond the scope of this paper. The new algorithm is easy to implement and efficient to run. It retains the main attractiveness of the Kalman filter: the user state is described only by the mean and covariance. Thus, this algorithm can be applied in all applications that currently use a Kalman filter or one of its extensions. It delivers more accurate (in some cases, substantially more accurate) results than previously known comparable algorithms.
Published in:	Proceedings of the 26th International Technical Meeting of the Satellite Division of The Institute of Navigation (ION GNSS+ 2013) September 16 - 20, 2013 Nashville Convention Center, Nashville, Tennessee Nashville, TN
Pages:	1977 - 1987
Cite this article:	Draganov, A., "Overcoming Errors in Processing Nonlinear Measurements: A New Extension for the Kalman Filter," Proceedings of the 26th International Technical Meeting of the Satellite Division of The Institute of Navigation (ION GNSS+ 2013), Nashville, TN, September 2013, pp. 1977-1987.
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