magistrsko delo
Abstract
V magistrskem delu obravnavamo konstrukcijo gibanj togih teles v prostoru dualnih kvaternionov. Predstavljene so nekatere klasične interpolacijske sheme na gladki mnogoterosti $SE(3)$. Pri tem si pomagamo s sredstvi iz diferencialne geometrije in teorije Liejevih grup. Ločeno obravnavamo konstrukcijo rotacijskega in translacijskega dela gibanja, kjer večji poudarek namenimo ravno izpeljavi interpolacijskih shem za rotacijski del. Translacijski del je obravnavan le bežno, saj za njegovo konstrukcijo zadoščajo že klasični interpolacijski postopki v ${\mathbb R}^3$. S pomočjo teorije Cliffordovih algeber konstruiramo algebro dualnih kvaternionov in na naraven način izpeljemo zvezo med evklidsko grupo $SE(3)$ in podmnogoterostjo imenovano Studyjeva kvadrika, kjer so ti elementi tudi reprezentirani. S pomočjo projekcij iz prostora ${\mathbb R}^8$ na Studyjevo kvadriko vpeljemo različne eksplicitne interpolacijske postopke, kjer lahko ob primerno izbranih začetnih točkah dosežemo interpolacijo pozicij in orientacij togega telesa, kakor tudi kotnih hitrosti in translacijskih hitrosti.
Keywords
togo telo;Studyjeva kvadrika;zlepki;kvaternioni;dualni kvaternioni;
Data
Language: |
Slovenian |
Year of publishing: |
2019 |
Typology: |
2.09 - Master's Thesis |
Organization: |
UL FMF - Faculty of Mathematics and Physics |
Publisher: |
[K. Bolčič] |
UDC: |
519.6 |
COBISS: |
18731097
|
Views: |
3098 |
Downloads: |
254 |
Average score: |
0 (0 votes) |
Metadata: |
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Other data
Secondary language: |
English |
Secondary title: |
Construction of rigid body motion in dual quaternion space |
Secondary abstract: |
In the master thesis we consider the construction of rigid body motion in dual quaternion space. We show some classical examples of interpolation procedures on the smooth manifold $SE(3)$, where we use several known methods from differential geometry and the theory of Lie groups. In the procedure we often split the construction of the motion in the rotational and translational part, where we put more effort into the construction of the rotational part since translational movement of the rigid body is almost trivial using standard interpolation procedures in ${\mathbb R}^3$. From the theory of Clifford algebra we construct the space of dual quaternions. We search for a submanifold of $\mathbb {DH}$ which is isomorphic to the Euclidean group $SE(3)$, where rigid body movement transformations are represented. Using special projections from the Euclidean space ${\mathbb R}^8$ onto the Study quadric, which is a special submanifold of Dual quaternions representing body transformations, we develop several interpolations schemes which enables us to interpolate rotations, translations and rigid body twists. Twists are objects representing the angular velocity and the velocity of the moving frame. |
Secondary keywords: |
rigid bodies;Study quadric;splines;quaternions;dual quaternions; |
Type (COBISS): |
Master's thesis/paper |
Study programme: |
0 |
Embargo end date (OpenAIRE): |
1970-01-01 |
Thesis comment: |
Univ. v Ljubljani, Fak. za matematiko in fiziko, Oddelek za matematiko, Matematika - 2. stopnja |
Pages: |
IX, 75 str. |
ID: |
11234415 |