APPLICATION OF QUATERNIONS FOR REFLECTOR PARAMETER

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.

 1. Dzhabiev A.N., Konyakhin I.A., Pankov E.D. Autocollimation Angle Measuring Tools for Deformation Monitoring. St. Petersburg, SPbSIFMO(TU), 2000, 198 p.
2. Greim I.A. Zerkal'no-Prizmennye Sistemy [Mirrored-Prism Systems]. Moscow, Mashinostroenie Publ., 1981, 125 p.
3. Bronshtein Yu.L. Krupnogabaritnye Zerkal'nye Sistemy (Kontrol' Geometrii, Yustirovka) [Large Mirror Systems (Geometry Control, Adjusting)]. Moscow, DPK Press, 2013, 480 p.
4. Amel'kin N.I. Kinematics and Dynamics of Solid State Body (Quaternionic Statement). Moscow, MPTI Publ., 2000, 64 p.
5. Goldobin N.N., Goldobina L.A. Continuity in the development of scientific knowledge: practical application of quaternions in solving engineering problems. Technico-Tehnologicheskie Problemy Servisa, 2013, no. 2, pp. 59–62.
6. Sivtsov G.P. Prostranstvennye Opticheskie Sistemy [Spatial Optical System]. Novosibirsk, SGGA Publ., 2011, 331 p.
7. Konyakhin I.A., Pankov E.D. Three-Axis Optical and Optoelectronic Goniometers: Handbook. Moscow, Nedra Publ., 1991, 224 p. (In Russian)
8. Kork G.A., Korn T.M. Mathematical Handbook for Scientists and Engineers: Definitions, Theorems, and Formulas for Reference and Review. New York, Dover Publications, 2000, 1130 p.
9. Karataev E.A. Kvaterniony i Trekhmernye Povoroty. Prakticheskii Podkhod [Quaternions and 3D Rotations. Practical Approach]. Moscow, Aprel' Publ., 2000, 32 p.
10. Erdogdu M., Ozdemir M. On Eigenvalues of Split Quaternion Matrices. Advances in Applied Clifford Algebras, 2013, vol. 23, no. 3, pp. 615–623. doi: 10.1007/s00006-013-0391-7
11. Branets V.N., Shmyglevskii I.P. Primenenie Kvaternionov v Zadachakh Orientatsii Tverdogo Tela [Application of Quaternions in Problems of Solid Body Orientation]. Moscow, Nauka Publ., 1973, 320 p.
12. Chelnokov Yu.N. Kvaternionnye i Bikvaternionnye Modeli i Metody Mekhaniki Tverdogo Tela i ikh Prilozheniya. Geometriya i Kinematika Dvizheniya. Moscow, Fizmatlit Publ., 2006, 512 p.
13. Pobegailo A.P. Primenenie Kvaternion v Komp'yuternoi Geometrii i Grafike [Application of Quaternion in Computer Geometry and Graphics]. Minsk, BGU Publ., 2010, 216 p.
14. Konyakhin I.A., Kopylova T.V. Three-axis optical-electronic autocollimator with increased sensitivity of the roll angle measuring. Scientific and Technical Journal of Information Technologies, Mechanics and Optics, 2010, no. 6, pp. 9–12. (In Russian)
15. Konyakhin I.A., Turgalieva T.V. Three-coordinate digital autocollimator. Journal of Optical Technology, 2013, vol. 80, no. 12, pp. 772–777. doi: 10.1364/JOT.80.000772
16. Konyakhin I.A., Kopylova T.V., Konyakhin A.I., Smekhov A.A. Optic-electronic systems for measurement the three-dimension angular deformation of axles at the millimeter wave range radiotelescope. Proceedings of SPIE, 2013, vol. 8759, art. 87593E. doi: 10.1117/12.2014605
17. Angot A. Comlements de Mathematiques. Paris, 1957.
18. Vanderwerf D.F. Applied Prismatic and Reflective Optics. Bellingham, SPIE, 2010, 310 p. doi: 10.1117/3.867634
 

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