Renato Zanetti

Norm-Constrained Kalman Filtering

The problem of estimating the state vector of a dynamical system from vector measurements when it is known that the state vector satisfies norm equality constraints is considered. The case of a linear dynamical system with linear measurements subject to a norm equality constraint is discussed with a review of existing solutions. The norm constraint introduces a nonlinearity in the system for which a new estimator structure is derived by minimizing a constrained cost function. It is shown that the constrained estimate is equivalent to the brute-force normalization of the unconstrained estimate. The obtained solution is extended to nonlinear measurement models and applied to the spacecraft attitude filtering problem.

Underweighting Nonlinear Measurements

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.

Recursive Update Filtering for Nonlinear Estimation

Nonlinear filters are often very computationally expensive and usually not suitable for real-time applications. Real-time navigation algorithms are typically based on linear estimators, such as the extended Kalman filter (EKF) and, to a much lesser extent, the unscented Kalman filter. This work proposes a novel nonlinear estimator whose additional computational cost is comparable to (N-1) EKF updates, where N is the number of recursions, a tuning parameter. The higher N the less the filter relies on the linearization assumption. A second algorithm is proposed with a differential update, which is equivalent to the recursive update as N tends to infinity.

Q-Method Extended Kalman Filter

A new algorithm is proposed that smoothly incorporates the nonlinear estimation of the attitude quaternion using Davenport’s q-method and the estimation of nonattitude states through an extended Kalman filter. The new algorithm is compared to an existing one and the various similarities and differences are discussed. The validity of the proposed approach is confirmed by numerical simulations.

Kalman Filters with Uncompensated Biases

A N UNDERLYING assumption of the Kalman filter is that the measurement and process disturbances can be accurately modeled as random white noise. Various mitigation strategies are available when this assumption is invalid. In practice, sensor errors are often modeled more accurately as the sum of a white-noise component and a strongly correlated component. The correlated components can, for example, be random constant biases. In this case, a standard technique is to augment theKalman filter state vector and estimate the random biases. In an attempt to decouple the bias estimation from the state estimation, Friedland [1] estimated the state as though the bias was not present and then added the contribution of the bias. Friedland showed that this approach is equivalent to augmenting the state vector. This technique, known as two-stage Kalman filtering or separate-bias Kalman estimation, was then extended to incorporate a walk in the bias forced by white noise [2]. To account for the bias walk, the process noise covariance was increased heuristically, and optimality conditions were derived [3,4]. In this work, the effects of the noise and biases are considered as sources of uncertainty and not as elements of the state vector. This approach is applicable in situations when the biases are not observable or when there is not enough information to discern the biases from the measurements. A common approach would be to tune the filter using process and measurement noise such that the sample covariance obtained through Monte Carlo analysis matches the filter state estimation error covariance. The technique presented here takes advantage of the structure of the biases to obtain a more precise representation of their contributions to the state estimation uncertainty. The resulting algorithms are useful in quantifying the uncertainty in a single simulation along the nominal state trajectory. This process can aid in tuning the filter as well as be employed onboard to obtain an accurate measure of the uncertainty of the state estimates. The approach taken is similar to that of the consider filter proposed by Schmidt [5]. The consider filter can be applied to the present problem and the two solution approaches, although different in form, are functionally the same. The goal of this paper is to introduce the uncompensated bias Kalman filter as presented by the authors in [6,7]. More recently Hough [8] independently derived a similar algorithm and applied it to orbit determination. This engineering note shows the relation between these two recent techniques as well as their equivalency to the Schmidt consider filter [5].

Quaternion Estimation and Norm Constrained Kalman Filtering

An analysis and comparison of two difierent strategies to implement the quaternion Kalman fllter is presented. Circumstances under which the two strategies are equivalent will be investigated, together with conditions for covariance singularity. Normalization will also be analyzed. Optimality will be shown in a stochastic sense and not only in a geometric sense as previously reported.

Attracting manifolds for attitude estimation in flatland and otherlands

Non-convex and non-affine parameterizations of uncertainty are intrinsic within every attitude estimation problem given the fact that minimal and/or nonsingular representations of the attitude matrix are invariably nonlinear functions of the unknown attitude variables. Of course, this fact remains true for rotation matrices both in the 2-D plane (flatland) and in higher dimensional spaces (otherlands). Therefore, estimation problems involving minimal nonsingular representations of unknown attitude matrices bring significant challenges to the adaptive estimation community. This paper develops a novel algorithm for attitude estimation. The proposed algorithm relies upon the design of an adaptive update law for the attitude estimate while preserving its inherent orthogonal structure. The underlying approach borrows from the classical Poisson differential equation in rigid-body rotational kinematics and endows certain manifold attractivity features within the adaptive estimation algorithm. Consequently, we are not only able to efficiently handle the non-affine and non-convex nature of the parameter uncertainty, but are also ensured of estimation algorithm stability and robustness under bounded measurement noise. In addition to a rigorous discussion on the overall methodology, the paper provides example simulations that help demonstrate the effectiveness of the attracting manifolds design.

Image Processing of Illuminated Ellipsoid

This study introduces novel algorithms and the underlying mathematics to process photographs of planetary illuminated bodies and use them for navigation purposes. The goal is to accurately estimate the observer-to-body relative position in inertial coordinates. The main motivation is to provide autonomous navigation capabilities to spacecrafts by observing a planet or a moon. This is needed, for example, in human-rated vehicles in order to provide navigation capabilities in a loss-of-communications scenario. The algorithm is derived for the general case of a triaxial ellipsoid that is observed bounded by an elliptical cone. The orientation of the elliptical cone reference frame is obtained by eigenanalysis, and the offset between the elliptical cone axis and the body center direction as well as the equation of the terminator are quantified. The main contribution of this paper is in the image-processing approach adopted to derive centroid and distance to the body. This is done by selecting a set of pixels ...

A Multiplicative Residual Approach To Attitude Kalman Filtering With Unit-Vector Measurements

Using direction vectors of unit length as measurements for attitude estimation in an extended Kalman filter inevitably results in a singular measurement covariance matrix. Singularity of the measurement covariance means no noise is present in one component of the measurement. Unit-vector measurements have no noise in the radial component. Singular measurement covariances can be dealt with by the classic Kalman filter formulation as long as the estimated measurement covariance is non-singular in the same direction. Unit-vector measurements violate this condition because both the true measurement and the estimated measurement have perfectly known lengths. Minimum variance estimation for the unit-vector attitude Kalman filter is studied in this work. An optimal multiplicative residual approach is presented. The proposed approach is compared with the classic additive residual attitude Kalman filter.

Autonomous Midcourse Navigation for Lunar Return

Autonomous navigation systems provide the vehicle with estimates of its states without ground support. This work develops an autonomous navigation architecture for lunar transfer using optical sensors and celestial navigation. Measurement and errormodels are developed for two classes of celestial measurements, the elevation of known stars from the Earth’s or moon’s limb, and the apparent radius of the Earth or moon. Monte Carlo methods are used to support the development of measurement error models. The proposed architecture is tested with linear covariance techniques; navigation errors and trajectory dispersions are obtained to confirm the feasibility of the approach. The navigation system is required to provide 0.5 deg flight-path angle accuracy at entry interface for mission safety. The simulation results show that the proposed autonomous navigation systemmeets the reentry safety requirement.