Underweighting Nonlinear Measurements

Abstract

T HE extended Kalman filter [1] (EKF) is a nonlinear approximation of the optimal linear Kalman filter [2,3]. In the presence of measurements that are nonlinear functions of the state, the EKF algorithm expands the filter’s residual (difference between the actual measurement and the estimated measurement) in a Taylor series centered at the a priori state estimate. The EKF truncates the series to the first order, but second-order filters also exist [4,5]. It is well known that, in the presence of highly accurate measurements, the contribution of the second-order terms is essential when the a priori estimation error covariance is large [5,6]. Possible solutions include implementing a second-order Gaussian filter [5] or an unscented Kalman filter (UKF) [7]. The UKF is a nonlinear extension to the Kalman filter, capable of retaining the secondmoments (or higher) of the estimation error distribution. Even when retaining the secondorder terms of the Taylor series, the methods still rely on an approximation; therefore, good filtering results may not always be achievable. Historically, second-order filters are not used because of their computational cost. The space shuttle, for example, uses an ad hoc technique known as underweighting [8,9]. The commonly implemented method for the underweighting of measurements for human space navigation was introduced by Lear [10] for the space shuttle navigation system. In 1966, Denham and Pines showed the possible inadequacy of the linearization approximationwhen the effect ofmeasurement nonlinearity is comparable to the measurement error [11]. To compensate for the nonlinearity, Denham and Pines proposed to increase the measurement noise covariance by a constant amount. In the early 1970s, in anticipation of shuttle flights, Lear [8] and others developed relationships that accounted for the second-order effects in the measurements. It was noted that, in situations involving large state errors and very precise measurements, application of the standard EKF mechanization lead to conditions inwhich the state estimation error covariance decreased more rapidly than the actual state errors. Consequently, the EKF began to ignore new measurements, even when the measurement residualwas relatively large.Underweightingwas introduced to slow down the convergence of the state estimation error covariance, thereby addressing the situation in which the error covariance became overly optimistic with respect to the actual state errors. The original work on the application of second-order correction terms led to the determination of the underweighting method by trial and error [10]. More recently, studies on the effects of nonlinearity in sensor fusion problems with application to relative navigation have produced a so-called bump-up factor [12–15]. While Ferguson and How [12] initiated the use of the bump-up factor, the problem of mitigating filter divergence was more fully studied by Plinval [13] and, subsequently, by Mandic [14]. Mandic generalized Plinval’s [13] bump-up factor to allow flexibility and notes that the value selected influences the steady-state convergence of the filter. In essence, it was found that a larger factor keeps the filter from converging to the level that a lower factor would permit. This finding prompted Mandic [14] to propose a two-step algorithm in which the bump-up factor was only applied for a certain number of measurements, upon which the factor was completely turned off. Finally, Perea, et al. [15] summarized the findings of the previous works and introduced several ways of computing the applied factor. In all of the cases, the bump-up factor amounts, in application, to the underweighting factor introduced in Lear [10]. Save for the two-step procedure of Mandic [14], the bump-up factor was allowed to persistently affect the Kalman gain, which directly influenced the obtainable steady-state covariance. Effectively, the ability to remove the underweighting factor autonomously and under some convergence condition was not introduced. While of great historical importance, the work of Lear is not well known, as it is only documented in internal NASA memos [8,10]. Kriegsman and Tau [9] mention underweighting in their 1975 shuttle navigation paper, without a detailed explanation of the technique. The purpose of this Note is to review the motivations behind underweighting and to document its historical introduction. Lear’s [10] schemeuses a single scalar coefficient, and tuning is necessary in order to achieve good performance. A new method for determining the underweighting factor is introduced together with an automated method for deciding when the underweighting factor should and should not be applied. By using the Gaussian approximation and bounding the second-order contributions, suggested values for the coefficient are easily obtained. The proposed technique has the advantage of Lear’s scheme’s simplicity combined with the theoretical foundation of the Gaussian second-order filter. The result yields a simple algorithm to aid the design of the underweightedEKF.