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## Presentations

### On determining the parameters of the incidence from angular observations: estimation of the hyperbola by the Kalman filter

Goritskiy Y.A., Shevchenko O.V., Zaharova A.I.

Moscow Power Engeneering Insnitut, Moscow, Krasnokazarmennaya str., 14

The problem is related to the question of the possibility of constructing an angle-measuring system that determines the displacement (relative to the observer) of the point of incidence of a body (hereinafter DT) moving freely in the gravity field of the Earth. Is it possible to determine the place and time of the fall with sufficient accuracy? Among system developers, there is an idea of the unsuitability of angular measurements (in the absence of information about the range) to obtain practically interesting accuracy. However, if we make the assumption that the orbit is such that the crash site is in the vicinity of the observer, then the possibility appears.

Situations are analyzed when the DT moves in the gravity field at a speed of about one kilometer per second, heights – hundreds of kilometers. Since angular measurements carry information about the "close" point of incidence only at the last observation range, simplifying assumptions are made for the motion and observation model: the acceleration of gravity does not depend on height, the angular velocity is constant, only the angle of the place is measured (azimuth measurements are neglected), the observer's movement (due to the rotation of the Earth) is transferred to the plane orbits; the model is justified in [1] and [2]. Random errors with a given variance are superimposed on discrete measurements of the angle of the place. Under the accepted conditions, the tangent (or cotangent) of the measured angle is expressed by the ratio of two expressions: a linear function of time (with two unknown parameters) and an expression for a hyperbola (with two parameters). The parameters of the linear function are estimated without problems based on the results of the initial observation section, the whole difficulty is in the parameters of the hyperbola (time and place of fall). The least squares method leads to a nonlinear system and the complexity of the iterative process. The Kalman filter approach with linearization elements is used to evaluate the parameters.

The results of calculations are given. It is possible to obtain almost interesting accuracy at the moment before the fall of the order of a minute.

Literature

1. Goritsky Yu.A., Tigetov D.G., Anufriev A.M. A two-dimensional model for evaluating the effectiveness of angular measurements on elliptical orbits// "Izvestiya RAS. Theory and Control Systems", 2021, No. 2. pp.14-24. DOI: 10.31857/S0002338820060025

2. Goritsky Yu.A., Zakharova A.I. Estimation of the potential accuracy of some orbital parameters by angular measurements: a two-dimensional model//"Bulletin of the MEI", No.5, 2022. pp.133-144. DOI 10.24160/1993-6982-2022-5-133-1

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