VyacheslavV. Pechurin – Ph.D.(Eng.), Scientific and Pedagogical Worker

Maxim V. Pechurin – Ph.D.(Eng.), Scientific and Pedagogical Worker

Ivan V. Kolbasko – Ph.D.(Eng.), Dr.Sc. Candidate

Increasing requirements for positioning systems primarily relate to the exact measurement of the coordinates of a mobile object at any point of a coverage area of the systems. The positioning systems accuracy depending on their geometry and range and defines their coverage. Borders of a coverage area are calculated as product of an user equivalent error and the geometrical factor of the positioning system, on the basis of the set position accuracy. In Russian literature, expressions for calculate of the geometrical factor are presented either only for certain ground-based positioning systems with a minimum number of receiving points, or not in full. It does not allow to calculate the geometrical factor of space positioning systems with a considerable quantity of receiving points.

For this reason, the article considers a method for plotting the coverage area of various positioning systems, based on calculation of their geometrical factor. In the theory of a scalar field it express through a field gradient, which on algebraic value is equal derivative of function in a direction to a normal. In the mathematician the Jacobi matrix is made of such all possible private derivatives. Therefore, calculation of the geometrical factor is fulfilled on the basis of Jacobi matrixes of various positioning systems. Further, as it is known from probability theory and the mathematical statistics, dispersions and covariances of a random variable by means of a covariance matrix are defined. In a covariance matrix of a dispersion of errors of space coordinates are on its principal diagonal. The geometrical factor is calculated, as a trace of this matrix. It is calculated in all given points of the coverage area of positioning systems. Points are selected with some constant step which value depends on demanded resolution and available computing resources. This method is universal and makes it possible to calculate the geometric dilution of precision for ground-based and space positioning systems with an unlimited number of receiving points. A special feature of this method is the use of a single expression for calculating the geometric dilution of precision of any positioning system. In this expression, only the geometrical matrix changes when the positioning system parameters change. Expressions are given to calculate the various geometrical matrices. The code in the Python programming language for calculating the horizontal dilution of precision for the angle of arrival positioning system is presented. The horizontal and position dilution of precision are calculated and plotted for various ground-based and space positioning systems.

Pechurin V.V., Pechurin M.V., Kolbasko I.V. Dilution of precision computation for positioning systems. Electromagnetic waves and electronic systems. 2020. V. 25. № 6. P. 5−12. DOI: 10.18127/j15604128-202006-01. (in Russian)

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