Application of the Statistically Linearized Filter to Re-entry Estimation

Abstract

THE purpose of this paper is to show that the Statistically Linearized Filter (SLF) is a competitive tool in estimating the of rather complex systems, and to outline its application to reconstructing re-entry trajectories from a variety of sensor data. The SLF is presented as a current-state squareroot filter of the continuous-discrete type. As a crucial test for full-sized application of this algorithm, the estimation of reentry trajectories of ballistic bodies from a variety of sensor data was considered. Both the Extended Kalman Filter (EKF) and SLF were applied to this estimation problem assuming a perfect error model. It was discovered that EKF diverges for this system, whereas SLF performs very well. The cost associated with the evaluation of statistical averages necessary in SLF can be dramatically reduced by computing as many averages as possible on the basis of closed-form expressions using the algebraic properties of the differential equations of motion. SLF was successfully applied to test-flight data employing a minimum error state approach to instrumenterror modeling. Contents Numerous attempts have been made over the years to find nonlinear filters which are workable on complex and realistically sized problems. The SLF has been a prominent candidate because of its formal similarity to the EKF. Examples have shown the power of this method; however, applications have been mostly limited to cases of modest dimension or well-isolated nonlinearities, because the effort in computing statistical averages over high-dimensional probability spaces proved formidable.J The basic formulation of the SLF has been re-examined to remove some of the numerical obstacles blocking the way to more substantial applications. This resulted in an algorithm, for the continuous-discrete case, which employs a square-root decomposition of the covariance matrix, a convenient modification of the Bierman UD-algorithm,2 and the selection of a fifth-order numerical integration scheme for the evaluation of statistical expectations. The SLF can be successfully and economically applied to problems of re-entry estimation. It is assumed that the reentry body carries strap-down instrumentation (bodymounted accelerometers and rate gyroscopes) as well as a magnetometer for attitude determination. The body is tracked by one or several radars and the impact point can be measured very accurately. The motion of the re-entry body involves both translational and rotational degrees of freedom and can