Lie Group Applications on Space Kinematics
Tez Türü: Doktora
Tezin Yürütüldüğü Kurum: Dokuz Eylül Üniversitesi, Fen Bilimleri Enstitüsü, Türkiye
Tez Danışmanı: İlhan Karakılıç
Tezin Onay Tarihi: 2023
Tezin Dili: İngilizce
Açık Arşiv Koleksiyonu: AVESİS Açık Erişim Koleksiyonu
Desteklendiği Program: Öğretim Üyesi Yetiştirme Programı (ÖYP)
Özet:
This thesis deals with the kinematics of the rigid body motions. The product of two group elements of Special Euclidean Group in 3-dimension is investigated. This problem corresponds to the Baker-Campell-Hausdorff formula and geometrically the screw triangle. The main focus of this work is to find the Baker-Campell-Hausdorff formula for space kinematics and investigate the screw triangle. Also, the same problem is studied for the group of rotations in 3-dimension and the group of planar displacements. From the geometric point of view, they correspond to the spherical triangle and the pole triangle, respectively. Then the Baker-Campell-Hausdorff formula is obtained for 2-systems of screws.
In the next study, the relations between the exponential map and Cayley maps are derived. Then the product of two group elements of Special Euclidean Group in 3-dimension is investigated by using the 4x4 and the 6x6 Cayley maps. The corresponding twists for the product of two group elements are written in terms of Lie brackets as in the Baker-Campell-Hausdorff formula.
Plane symmetric motions are studied in the rest of the thesis. The velocity twists, axodes, acceleration centres and inflection points of these motions, which are generated by reflections in the osculating, normal and rectifying planes of a space curve, are obtained. Finally, line-symmetric motions generated by reflections in the tangent, normal and binormal lines of a space curve are determined and their velocity twists and fixed axodes are obtained.