應用雙四元數進行相對定位的估測

Abstract

本論文主要為處理相對定位的問題。不同於以往使用共平面條件配合最小平方差的作法，我們使用雙四元數來處理該問題。使用雙四元數最方便的地方在於，它同時處理了移動與轉動，並且使用一種特殊的向量表式方法 – 雙向量，來表達空間中的一條直線。 本文提供了兩種方法來解決相對定位的問題，分別為封閉式解法(Closed Form Solution) 與線性化遞迴式解法(Linearized Iteration Method)。在封閉式解法中，我們使用兩個鏡頭來觀察目標；並利用特徴值來求得連結兩作標系的雙四元數。 在遞迴式解法中，我們使用單一鏡頭觀察。利用目標的投影以及相片，我們可以比較其中的差異進而解得連結兩座標系的雙四元數。由於從三維度的目標投影到二維度的照片會造成一些資訊流失，所以我們必須使用遞迴的方法得到答案。 在論文的最後，我們實地設計了一個實驗來驗證我們的演算法。藉由實際的拍照以及利用全球定位系統(GPS)測量目標點的位置，我們証實了演算法的可行性。

In this thesis, we use dual quaternion to solve relative orientation in close form, which replaces the traditional way of coplanarity condition and least square solution. The best benefit of dual-quaternion is that it is able to handle rotation and translation simultaneously and apply continuous product of dual quaternion operating with a kind of special vector－dual vector to express a series of rotation and translation. We provide two kinds of method to solve the problem – Closed Form Solution and Linearized Iteration Method. In Closed Form Solution, we use two cameras to observe the target lines, and find the lines respect to RIGHT coordinate system. And find the relative orientation in close form by finding eigenvectors. Because of the benefit of dual quaternion－easily to handle and express a series of transformation, we make above solution more practically and reality using dual-quaternion. While in the second method – Linearized Iteration Method, we use only one camera to deal with the same problem. We find the relative orientation by comparing the projection of lines respect to LEFT and RIGHT coordinate systems. The projection of lines respect to RIGHT coordinate system is the photo we take. Since there must be lost of information in the 3-D (lines in space) to 2-D (photo), we use linearized iteration to make up the lost.

In the end, we make an experiment to clarify our algorithm by taking a photo and measure the position of targets by GPS practically.