On linearization of perturbed motion model in the problem of ballistic trajectories scattering probabilistic analysis

Аuthors

Vasil’eva S. N.^*, Kan Y. S.^**

Moscow Aviation Institute (National Research University), 4, Volokolamskoe shosse, Moscow, А-80, GSP-3, 125993, Russia

*e-mail: sofia_mai@mail.ru
**e-mail: yu_kan@mail.ru

Abstract

The article considers the probabilistic analysis problem of ballistic impacts scattering at spherical Earth surface. The starting point of the trajectory is also located on the earth’s surface. Trajectories scattering is caused by a random perturbation of the initial velocity vector. The circular probable deviation is employed as a scattering characteristic. It represents the circle radius length, the probability of hitting which equals to 1/2, i.e. it concurs with quantile of 1/2 level for the norm of a random vector, characterizing the trajectory impact deflection from the nominal one. These vector components are computed using the equations of Keplerian theory of elliptic motion in the central gravitational field. They depend nonlinearly on the velocity perturbation vector. In this respect, the analytical obtaining of the quantile from the norm of this vector is impossible.

The velocity perturbations are assumed to be small compared to its nominal absolute value. The vector of small random parameters of velocity perturbation is modeled as an componentwise product of a small deterministic parameter and a vector of random parameters. It is assumed that the random parameters vector has a standard normal distribution with independent components. The article suggests the linearization method consisting in approximation of the above said non-linear dependence by the linear model, which is obtained by deviation vector Tailor expansion along the velocity perturbation vector. It is being proved, that the error occurring while such substitution is of the order of a small deterministic parameter.

As a model example, The authors consider the flight of a material point to a specified distance at the known values of nominal initial velocity, departure angle and azimuth. The computation results of the circular probable error by the Monte Carlo and linearization methods for a wide range of initial values are presented. Calculations of linearization method relative error compared to Monte Carlo method are also given. This error does not exceed 1.5%, which was confirmed by theoretical accuracy estimates of the linearization method, proposed in the article.

Keywords:

linearization method, probabilistic analysis, circular probable deviation, ballistic trajectories scattering

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