On the Accuracy of Determining Incident Parameters by Angular Observations: Recurrent Hyperbola Estimate

Abstract

The article discusses the possibility of developing a device that determines the point and moment of fall based on angular observations, and whether it is possible to determine the fall location and time with sufficient accuracy for some time before the fall. There is an opinion that angular measurements (in which there is no range information) are unsuitable for obtaining accuracies that are of practical interest. However, if one assumes that the orbit is such that the fall location is in the vicinity of the observer, then such possibility arises.

It is assumed that the body moves at a speed less than the first cosmic speed, at an altitude of tens and a few hundred kilometers. Since angular measurements bear information about the “close” location of fall only during the descent stage, simplifying assumptions are made for the motion-observation model: the acceleration of gravity does not depend on height and, as a consequence, the angular velocity (relative to the center of the Earth) is constant; only the elevation angle is measured (azimuth measurements can be neglected), and the observer’s movement (due to the rotation of the Earth) is transferred to the orbital plane with an insignificant error. The motion model is simplified in comparison with the Kepler motion. Random errors with a given variance are superimposed on discrete elevation angle measurements. Under the accepted conditions, the tangent of the measured angle is expressed by the ratio of a linear function of time (with two unknown parameters) and equations for a hyperbola (with two parameters). The linear function parameters are estimated without problems based on the results of the initial observation section; the whole difficulty is in statistically estimating the hyperbola parameters corresponding to the fall time and location.

A statistical estimation procedure is constructed, which uses the Kalman filter approach with linearization elements. Simulation results on 300 tests for each trajectory are presented. It is shown that it is possible to obtain accuracies of practical interest, and for a time when there is about a minute left before the fall.

The analysis of the above computation scheme depends on six input data. The problem is reduced to a scheme with three conditional parameters. The calculation results on it are presented in graphic form. By using them, one can obtain an estimate of the accuracy for arbitrary input data.