High-Precision localization method for spheres with a theodolite

Abstract

Spherical objects are widely used in target localization applications, and the existing sphere localization methods with cameras or total stations both have some limitations. A new high-precision sphere localization method with a theodolite is proposed in this paper. From the view point of the theodolite, the contour points of a sphere with known radius are measured as latitude-longitude coordinates. It is observed that the center of the target sphere is located on a cylindrical surface constructed with the latitude-longitude coordinates, and therefore the latitude-longitude coordinates of at least three contour points can be used to construct a set of ternary quadratic equations. The Gröbner basis method is used to compute at most four real solutions of the sphere center coordinates. In order to distinguish the only meaningful solution from the other possible real solutions, a pre-processing of the measured longitude values is also proposed. The factors affecting the positioning accuracy of the sphere center are evaluated in simulation experiments, which are used to obtain an empirical estimation model of the positioning error. Real data experiments are also performed and the results show that the proposed method can achieve high localization precision.