New approach to isometric transformations in oblique local coordinate systems of reference

Abstract

Abstract:The research article describes a method of isometric transformation and determining an exterior orientation of a measurement instrument. The method is based on a designation of a "virtual" translation of two relative oblique orthogonal systems to a common, known in the both systems, point. The relative angle orientation of the systems does not change as each of the systems is moved along its axis. The next step is the designation of the three rotation angles (e.g. Tait-Bryan or Euler angles), transformation of the system convoluted at the calculated angles and moving the system to the initial position where the primary coordinate system was. This way eliminates movements of the systems from the calculations and makes it possible to calculate angles of mutual rotation angles of two orthogonal systems primarily involved in the movement. The research article covers laboratory calculations for simulated data. The accuracy of the results is 10 -6 m (10 -3 regarding the accuracy of the input data). This confi rmed the correctness of the assumed calculation method. In the following step the method was verifi ed under fi eld conditions, where the accuracy of the method raised to 0.003 m. The proposed method enabled to make the measurements with the oblique and uncentered instrument, e.g. total station instrument set over an unknown point. This is the reason why the method was named by the authors as Total Free Station -TFS. The method may be also used for isometric transformations for photogrammetric purposes.

A New Geodetic Method of Examination of Geometrical Conditions of a Crane Bridge

“…Therefore, an isometric transformation of coordinates should be performed from the measurement level to the level of the object using two common points. Two common points should be chosen on the crane bridge, making it possible to perform an isometric transformation [11].…”

Method of the Determination of Exterior Orientation of Sensors in Hilbert Type Space

“…In case the scaling factor , then the transformation becomes the isometric transformation and is called Multi-Centroid Isometric Transformation (MCT). Both MCT and MCIT are development of the TFS transformation [ 15 ] to the case of at least three centroids. The next step is its translation and the last element is the rotation expressed in the A rotation matrix.…”

“… The transformation with the use of a centroid according to the relation (13)—development of TFS transformation [ 15 ]. …”

“…The proposed method, unlike the Moldensky-Badekas transformation or Bursa-Wolf transformation, can be applied for big rotation angles, and for close range photogrammetry purposes. The method is a further development of the Total Free Station (TFS) transformation [ 15 ] to several centroids in Hilbert type space. It can be applied for local, precise surveying as well as for geodynamic, close range photogrammetry and engineering purposes, especially for non-stable places (e.g., on floating vessels) where levelling of an instrument may not be possible and it is necessary to determine the exterior orientation of sensors.…”