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APPLICATION OF QUATERNIONS FOR REFLECTOR PARAMETER

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Subject of Research.  The paper deals with application of quaternions for optimization of reflector parameters at autocollimation measurements in comparison with a matrix method. Computer-based results on the quaternionic models are presented that have given the possibility to determine conditions of measurement error reduction in view of apriori information on the rotation axis position. The practical synthesis technique for tetrahedron reflector parameters using found ratios is considered. Method. Originally, received conditions for reduction of autocollimation system measurement error are determined with the use of a matrix method for definition of an angular object position as a set of three equivalent consecutive turns about coordinate axes. At realization of these conditions the numerous recalculation of orientation parameters between various systems of coordinates is necessary that increases complexity and reduces resulting accuracy of autocollimation system at practical measurements. The method of quaternions gives the possibility to analyze the change of an absolute angular position in space, thus, there are conditions of accuracy increase regardless of the used systems of coordinates. Main Results. Researches on the mathematical model have shown, that the orthogonal arrangement of two basic constant directions for autocollimator tetrahedron reflector is optimal with respect to criterion of measurement error reduction at bisection arrangement of actual turn axis against them. Practical Relevance. On the basis of the found ratios between tetrahedron reflector angles and angles of its initial orientation parameters we have developed a practical method of reflector synthesis for autocollimation measurements in case of apriori information on an actual turn axis at monitoring measurements of the shaft or pipelines deformations.

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