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1

Derivation of formulas in spherical trigonometry based on rotation matrix

Hsieh T. H., Wang S. Z., Liu W., Zhao J. S., Chen C. L.

TransNav : International Journal on Marine Navigation and Safety of Sea Transportation

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2019

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Vol. 13, no. 3

553--558

EN

The formulas of spherical triangle, which are widely used to solve various navigation problems, are the important basic knowledge of nautical mathematics. Because the sine rules and the cosine rules for the sides are the fundamental formulas to derive the other spherical triangle formulas, they are also called the genetic codes of the spherical triangle formulas. In the teaching process, teachers usually use the geometric method to derive and prove these fundamental formulas. However, the derivation of geometric methods is complicated and difficult to understand. To improve the teaching process, this paper proposes the three-dimensional rotation method, which is based on conversion of two cartesian coordinate frames using the rotation matrices. This method can easily and simultaneously derive the sine rules, the cosine rules for the sides, and the five-part formulas (I), and is also helpful to solve different kinds of spherical navigation problems.

2

A method for local approximation of a planar deformation field

Ligas Marcin, Banaś Marek, Szafarczyk Anna

Reports on Geodesy and Geoinformatics

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2019

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Vol. 108

1--8

EN

We present a method of approximation of a deformation eld based on the local ane transformations constructed based on n nearest neighbors with respect to points of adopted grid. The local ane transformations are weighted by means of inverse distance squared between each grid point and observed points (nearest neighbors). This work uses a deformation gradient, although it is possible to use a displacement gradient instead – the two approaches are equivalent. To decompose the deformation gradient into components related to rigid motions (rotations, translations are excluded from the deformation gradient through dierentiation process) and deformations, we used a polar decomposition and decomposition into a sum of symmetric and an anti-symmetric matrices (tensors). We discuss the results from both decompositions. Calibration of a local ane transformations model (i.e., number of nearest neighbors) is performed on observed points and is carried out in a cross-validation procedure. Verication of the method was conducted on simulated data-grids subjected to known (functionally generated) deformations, hence, known in every point of a study area.

3

Algorithms for detecting disorders of the BLDC motor with direct control

Chodnicki M., Kordowski P., Nowakowski M., Kowaleczko G.

Journal of KONES

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2016

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Vol. 23, No. 4

49--54

EN

Unmanned aerial vehicles are used to observe objects from the air in both kind of users – military and civilian. During the flight, UAV constantly is changing its orientation. This is due to weather conditions and commands coming from the ground control station. It has directly influence to the quality of the video stream transmitted from the cameras to the GCS. That disturbed image is hard to effective analyse. In order to eliminate the interference, UAVs are equipped with stabilized gimbals. The most of gimbals designed in Air Force Institute of Technology are using BLDC motors. Some of these systems use encoders for determining the absolute angle of motor rotation. The smaller one, which is up to 200 g, is not provided with these sensors. This angle is calculated without any feedback from the system. It is calculated by the BLDC motor`s control signals. However, lack of feedback can provide unstable work of gimbal, if it will be pushed by any force from environment. During the flight, it is unacceptable to head optoelectronic stopped working steadily. Therefore, there is a need to develop algorithms for securing the proper operation of the system stability. These algorithms can be used in other systems using stepper or BLDC motors.

4

Koncepcja rozpoznawania orientacji obiektów w obrazach 3D

Żorski W.

Biuletyn Instytutu Automatyki i Robotyki

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2010

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R. 16, nr 28

3-21

PL

Artykuł porusza zagadnienie rozpoznawania orientacji obiektów 3D. Przystosowano opracowaną przez Ballarda metodę rozpoznawania obiektów nieregularnych. Podstawą rozpoznawania orientacji jest wyznaczanie wartości macierzy akumulatora dla kątów Eulera. Wartości akumulatora dla zadanych wspłrzędnych uzyskiwane są metodą zliczania voxeli Wartości kątów Eulera, dla któych akumulator przyjmuje największą liczbę zliczeń, określają orientację badanego obiektu. Mechanizm zliczania voxeli został zaimplementowany i zbadany dla algorytmu bazującego na macierzy obrotu oraz algorytmu z zastosowaniem kwaternionów. Wykazano istnienie szczególnego rodzaju symetrii akumulatora, co pozwoliło na redukcję obliczeń o 50%.

EN

This paper considers the problem of 3D object orientation recognition. The Ballard method of arbitrary shapes detection is adopted. The basis of the orientation recognition is the mapping of an accumulator array for Euler angles. Accumulator values for given coordinates are calculated using the voxel counting method. An object orientation is determined by Euler angles with the maximum number of votes in the accumulator array. The voxel counting method was implemented and verified for an algorithm based on a rotation matrix as well as for an algorithm based on quaternions. A characteristic kind of accumulator symmetry was detected, which reduced computations by 50%.