A.G. Saybel 1

1 North-West Regional Center VKO of Almaz-Antey Concern – Obukhov Plant JSC (St. Petersburg, Russia)
1 saybel_ag@mail.ru

High demands are placed on the accuracy of modern coordinate measurement systems, the implementation of which is impossible without reducing both random and systematic measurement errors, including those associated with the properties of the model of the system being created. In coordinate measurement, the principle of passive location is widely used to determine the position of objects, in which the location parameters are determined based on measurements of the differences in the reception times of radio signals. Hyperbolic systems have become widespread, however, the solution to the problem of spatial positioning, which consists in finding the intersection points of the position surfaces corresponding to the measured values of the navigation parameters, is carried out based on numerical methods for finding the extremum of a function of several arguments without taking into account the features of the analytical models used.

A single value of the measured difference in the reception times of a signal at two spaced points, without taking into account the mutual motion and inhomogeneity of the radiation propagation medium, corresponds to a position surface in the form of a two-sheeted hyperboloid of revolution, described by a nonlinear equation of the second degree. When describing hyperbolic navigation systems, the problem of finding intersection points of plane sections of second-order surfaces is traditionally considered, which gives grounds to consider difference-range systems hyperbolic. At the same time, the question of the form of intersection of hyperbolic surfaces in space has not been sufficiently studied.

In the three-point case, the system of three nonlinear equations connects the unknown values of the coordinates of the located point with the known coordinates of the reference points and the measured values of the range differences. However, due to the presence of a functional relationship between the equations, this system will have a set of solutions, which include the vectors of the coordinates of all intersection points of the surfaces of the point's position.

Since it is known that a spatial line containing all points whose coordinates are the roots of the considered system of equations is a second-order curve, a study of the spatial characteristics of this curve was carried out, as a result of which expressions were obtained that describe the type and conditions for the existence of position lines of each type. Based on the obtained expressions, a simulation was carried out, the results of which are presented as an example of the dependence of the hyperbola shape for various options for placing the located and reference points on the plane, as well as graphic illustrations demonstrating the nature of the evolution of the spatial boundaries of changes in the type of position line. In conclusion, directions for using the properties of hyper-hyperbolic sections in the interests of solving applied problems of location, navigation and radio monitoring are given.

Saybel A.G. Study of nonlinear solutions of the analytical model of the difference-range measuring system. Nonlinear World. 2026. V. 24. № 2. P. 31–41. DOI: https:// doi.org/10.18127/ j20700970-202602-04 (In Russian)

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