Yaakov Oshman

Optimization of observer trajectories for bearings-only target localization

The problem of bearings-only target localization is to estimate the location of a fixed target from a sequence of noisy bearing measurements. Although, in theory, this process is observable even without an observer maneuver, estimation performance (i.e., accuracy, stability and convergence rate) can be greatly enhanced by properly exploiting observer motion to increase observability. This work addresses the optimization of observer trajectories for bearings-only fixed-target localization. The approach presented herein is based on maximizing the determinant of the Fisher information matrix (FIM), subject to state constraints imposed on the observer trajectory (e.g., by the target defense system). Direct optimal control numerical schemes, including the recently introduced differential inclusion (DI) method, are used to solve the resulting optimal control problem. Computer simulations, utilizing the familiar Stansfield and maximum likelihood (ML) estimators, demonstrate the enhancement to target position estimability using the optimal observer trajectories.

Attitude Determination from Vector Observations: Quaternion Estimation

Two recursive estimation algorithms, which use pairs of measured vectors to yield minimum variance estimates of the quaternion of rotation, are presented. The nonlinear relations between the direction cosine matrix and the quaternion are linearized, and a variant of the extended Kalman filter is used to estimate the difference between the quaternion and its estimate. With each measurement this estimate is updated and added to the whole quaternion estimate. This operation constitutes a full state reset in the estimation process. Filter tuning is needed to obtain a converging filter. The second algorithm presented uses the normality property of the quaternion of rotation to obtain, in a straightforward design, a filter which converges, with a smaller error, to a normal quaternion. This algorithm changes the state but not the covariance computation of the original algorithm and implies only a partial reset. Results of Monte-Carlo simulation runs are presented which demonstrate the superiority of the normalized quaternion.

Novel quaternion Kalman filter

This paper presents a novel Kalman filter (KF) for estimating the attitude-quaternion as well as gyro random drifts from vector measurements. Employing a special manipulation on the measurement equation results in a linear pseudo-measurement equation whose error is state-dependent. Because the quaternion kinematics equation is linear, the combination of the two yields a linear KF that eliminates the usual linearization procedure and is less sensitive to initial estimation errors. General accurate expressions for the covariance matrices of the system state-dependent noises are developed. In addition, an analysis shows how to compute these covariance matrices efficiently. An adaptive version of the filter is also developed to handle modeling errors of the dynamic system noise statistics. Monte-Carlo simulations are carried out that demonstrate the efficiency of both versions of the filter. In the particular case of high initial estimation errors, a typical extended Kalman filter (EKF) fails to converge whereas the proposed filter succeeds.

Optimal sensor selection strategy for discrete-time state estimators

A novel sensor selection strategy is introduced, which can be implemented on-line in time-varying discrete-time system. We consider a case in which several measurement subsystem are available, each of which may be used to drive a state estimation algorithm. However, due to practical implementation constraints (such as the ability of the on-board computer to process the acquired data), only one of these subsystems can actually by utilized at a measurement update. An algorithm is needed, by which the optimal measurement subsystem to be used is selected at each sensor selection epoch. The approach described is based on using the square root V-Lambda information filter as the underlying state estimation algorithm. This algorithm continuously provides its user with the spectral factors of the estimation error covariance matrix, which are used in this work as the basis for an on-line decision procedure by which the optimal measurement strategy is derived. At each sensor selection epoch, a measurement subsystem is selected, which contributes the largest amount of information along the principal state space direction associated with the largest current estimation error. A numerical example is presented, which demonstrates the performance of the new algorithm. The state estimation problem is solved for a third-order time-varying system equipped with three measurement subsystem, only one of which can be used at a measurement update. It is shown that the optimal measurement strategy algorithm enhances the estimator by substantially reducing the maximal estimation error. >

Optimal-REQUEST Algorithm for Attitude Determination

REQUEST is a recursive algorithm for least-squares estimation of the attitude quaternion of a rigid body using vector measurements. It uses a constant, empirically chosen gain and is, therefore, suboptimal when filtering propagation noises. The algorithm presented here is an optimized REQUEST procedure, which optimally filters measurement as well as propagation noises. The special case of zero-mean white noises is considered. The solution approach is based on state-space modeling of the K-matrix system and uses Kalman-filtering techniques to estimate the optimal K matrix. Then, the attitude quaternion is extracted from the estimated K matrix. A simulation study is used to demonstrate the performance of the algorithm.

Efficient multiple model adaptive estimation in ballistic missile interception scenarios

A novel efe cient algorithm, featuring a highly reduced computational load, is presented for multiple model adaptive estimation in a future real-life ballistic missile defense scenario, where the blind incoming target (having no information on the interceptor’ s state ) performs a bang ‐bang evasive maneuver characterized by a random switching time. The efe ciency of the algorithm derives mainly from its exploitation of the special structure of the hypothesis space in this problem to drastically reduce the number of concurrently active e lters in the bank without incurring any signie cant performance degradation. The proposed algorithm’ s efe ciency allows a substantial increase in the resolution of the discretized hypothesis space, thus enhancing considerably the attainable estimation performance. The effect of the new estimator’ s performance on guidance accuracy is examined. The homing performance of various perfect information guidance laws using this efe cient estimation method is compared, via Monte Carlo simulations, to the use of a Kalman e lter incorporating a shaping e lter representing the random target maneuver. The results demonstrate the superiority and viability of the proposed method.

Attitude Estimation from Vector Observations Using Genetic-Algorithm-Embedded Quaternion Particle Filter

A novel algorithm is presented for the estimation of spacecraft attitude quaternion from vector observations in gyro-equipped spacecraft. The new estimator is a particle filter that uses approximate numerical representation techniques for performing the otherwise exact time propagation and measurement update of potentially non-Gaussian probability density functions in inherently nonlinear systems. The new method can be applied using various kinds of vector observations. In this paper, the case of a low-Earth-orbit spacecraft, acquiring noisy geomagnetic field measurements via a three-axis magnetometer, is considered. A genetic algorithm is used to estimate the gyro bias parameters, avoiding the need to augment the particle filters state and rendering the estimator computationally efficient. Contrary to conventional filters, which address the quaternions unit norm constraint via special (mostly ad hoc) techniques, the new filter maintains this constraint naturally. An extensive simulation study is used to compare the new filter to three extended Kalman filters and to the unscented Kalman filter in Gaussian and non-Gaussian scenarios. The new algorithm is shown to be robust with respect to initial conditions and to possess a fast convergence rate. An evaluation of the Cramer-Rao estimation error lower bound demonstrates the filters asymptotic statistical efficiency and optimality.

Differential-game-based guidance law using target orientation observations

Modern 4th generation air-to-air missiles are quite capable of dealing with todays battlefield needs. Advanced aerodynamics, highly efficient warheads and smart target acquisition systems combine to yield higher missile lethality than ever. However, in order to intercept highly maneuverable targets, such as future unmanned combat air vehicles (UCAV), or to achieve higher tracking precision for missiles equipped with smaller warheads, further improvement in the missile guidance system is still needed. A new concept is presented here for deriving improved differential-game-based guidance laws that make use of information about the target orientation, which is acquired via an imaging seeker. The underlying idea is that of using measurements of the target attitude as a leading indicator of target acceleration. Knowledge of target attitude reduces the reachable set of target acceleration, facilitating the computation of an improved estimate of the zero-effort miss (ZEM) distance. In consequence, missile guidance accuracy is significantly improved. The new concept is applied in a horizontal interception scenario, where it is assumed that the target maneuver direction, constituting a partial attitude information, can be extracted via processing target images, acquired by an imaging sensor. The derivation results in a new guidance law that explicitly exploits the direction of the target acceleration. The performance of the new guidance law is studied via a computer simulation, which demonstrates its superiority over existing state-of-the-art differential-game-based guidance laws. It is demonstrated that a significant decrease in the miss distance can be expected via the use of partial target orientation information.

Integrated Estimation/Guidance Design Approach for Improved Homing Against Randomly Maneuvering Targets

Interceptor missiles, designed against aircraft, have substantial speed and maneuverability advantage over their targets. Thus, by exploiting the technological progress, even simple guidance concepts yielded satisfactory performance. For the interception of antisurface missiles, higher guidance precision is required. Using conventional guidance and estimation concepts, existing missile defense systems have demonstrated hit-to-kill accuracy against nonmaneuvering targets. Guaranteeing a similar performance against maneuvering targets can be achieved only if the estimation errors against such targets are minimized. This paper introduces a new, logic-based estimation/ guidance algorithm, that explicitly uses the time-to-go in the estimation process and modifies the guidance law to reduce the consequence of estimation errors. The successful outcome of the new approach is illustrated by an extensive Monte Carlo simulation study.

Eigenfactor solution of the matrix Riccati equation--A continuous square root algorithm

This paper introduces a new algorithm for solving the matrix Riccati equation. Differential equations for the eigenvalues and eigenvectors of the solution matrix are developed in which their derivatives are expressed in terms of the eigenvalues and eigenvectors themselves and not as functions of the solution matrix. The solution of these equations yields, then, the time behavior of the eigenvalues and eigenvectors of the solution matrix. A reconstruction of the matrix itself at any desired time is immediately obtained through a trivial similarity transformation. This algorithm serves two purposes. First, being a square-root solution, it entails all the advantages of square root algorithms such as nonnegativedefiniteness and accuracy. Secondly, it furnishes the eigenvalues and eigenvectors of the solution matrix continuously without resorting to the complicated route of solving the equation directly and then decomposing the solution matrix into its eigenvalues and eigenvectors. The accuracy and the stability of the new algorithm are demonstrated through an example. It is shown that the algorithm works in a case where the ordinary algorithm fails and the closed-form solution cannot be computed as a result of numerical difficulties.