New Method That Solves the Three-Point Resection Problem Using Straight Lines Intersection
Publication: Journal of Surveying Engineering
Volume 135, Issue 2
Abstract
The three-point resection problem, i.e., the problem of obtaining the position of an unknown point from relative angular measurements to three known stations is a basic operation in surveying engineering. Several approaches to solve this problem, graphically or analytically, have been developed in the last centuries. In this paper, a new analytical approach to solve this problem is presented. The method determines the coordinates of the unknown point by intersecting straight lines through the three stations. The required azimuths of these lines are obtained from the geometric relationship between two similar triangles. Numerical simulations that show the good performance and accuracy of this approach are also reported.
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References
Allan, A. L., Hollwey, J. R., and Maynes, J. H. B. (1968). Practical field surveying and computations, Elsevier, New York.
Batlle, J. A., Font-Llagunes, J. M., and Escoda, J. (2004). “Dynamic positioning of a mobile robot using a laser based goniometer.” Proc., Int. Symp. on Intelligent Autonomous Vehicles, IFAC, Lisbon, Portugal.
Betke, M., and Gurvits, L. (1997). “Mobile robot localization using landmarks.” IEEE Trans. Rob. Autom., 13(2), 251–263.
Bock, W. (1959). “Mathematische und geschichtliche Betrachtungen zum Einschneiden.” Rep. 9, Schriftenreihe Niedersaechsisches Landesvermessungsamt, Hanover, Germany.
Briechle, K., and Hanebeck, U. D. (2004). “Localization of a mobile robot using relative bearing measurements.” IEEE Trans. Rob. Autom., 20(1), 36–44.
Burtch, R. (2005). “Three point resection problem.” Surveying computations course notes 2005/2006, ⟨http://www.ferris.edu/faculty/burtchr/sure215/notes/resection/res-ection.pdf⟩ (Jun. 14, 2007).
Cohen, C., and Koss, F. (1992). “A comprehensive study of three object triangulation.” Proc., SPIE Conf. on Mobile Robots, Vol. 1831, SPIE, Boston, 95–106.
Danial, N. F. (1978). “Another solution of the three-point problem.” J. Surv. and Mapping Div., 38(4), 329–333.
Font-Llagunes, J. M., and Batlle, J. A. (2006). “Mobile robot localization. Revisiting the triangulation methods.” Proc., Int. Symp. on Robot Control, IFAC, Bologna, Italy.
Greulich, F. E. (1999). “The barycentric coordinates solution to the optimal road junction problem.” J. Forest Engineering, 10(1), 111–114.
Hu, W. C., and Kuang, J. S. (1997). “Proof of Tienstra’s formula by finite-element method.” J. Surv. Eng., 123(1), 1–10.
Hu, W. C., and Kuang, J. S. (1998). “Proof of Tienstra’s formula for an external observation point.” J. Surv. Eng., 124(1), 49–55.
Kelly, A. (2003). “Precision dilution in triangulation based mobile robot position estimation.” Proc., Int. Conf. on Intelligent Autonomous Systems, IAS, Amsterdam, The Netherlands.
Klinkenberg, H. (1955). “Coordinate systems and the three-point problem.” Can. Surveyor, 9, 508–518.
Möbius, A. F. (1827). “Der barycentrische calcul: ein neues Hülfsmittel zur analytischen Behandlung der Geometrie.” Barth Verlag, Leipzig, Germany.
Neuberg, J. B. J., and Gob, A. (1889). “Sur les foyers de Steiner d’un triangle.” Compte-Rendu de la 18ième Séance. Deuxième Partie: Notes et Mémoires, Association Française pour l’Avancement des Sciences, Paris, 179–196.
Piaggio, M., Sgorbissa, A., and Zaccaria, R. (2001). “Autonomous navigation and localization in service mobile robotics.” Proc., Int. Conf. on Intelligent Robots and Systems, IEEE, Maui, Hawaii, 2024–2029.
Information & Authors
Information
Published In
Journal of Surveying Engineering
Volume 135 • Issue 2 • May 2009
Pages: 39 - 45
Copyright
© 2009 ASCE.
History
Received: Dec 17, 2007
Accepted: Sep 26, 2008
Published online: May 1, 2009
Published in print: May 2009
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