Estimation of Rotation and Translation Simultaneously Using Dual Quaternion

本文使用雙四元數來取代表示物體轉移的旋轉矩陣 與移動向量 ，雙四元數最大的好處就是可以同時處理物體的轉動與移動，並且以雙四元數的連續乘積對一種特別的向量－雙向量做運算來表示一連串轉移。雙四元數的求解過程是將轉移前與轉移後的待測物上取出相對的雙向量，比對相對的雙向量來找出表示待測物轉移的雙四元數，又稱之為雙四元數的封閉式解法（Closed-Form Solution）。目前的標準作法是將雙向量與雙四元數之間的關係經過推導後以矩陣運算與SVD（Singular Value Decomposition）來解得[10]，本文中對此標準做法作改進使其運算更簡潔。 在電腦影像處理或者機械人的控制中，連結2-D觀察投影面與待測物實際在3-D空間中的資訊是很重要的，也普遍的被稱為手－眼問題（hand-eye problem）。在研究手－眼問題的過程中，我們發現雙四元數對於－2-D投影面上的資訊來反推待測物在3-D空間上的轉移－這個問題上有些特別的關連性，我們利用此關聯性推導出2-D投影面與3-D空間上待測物之間的雙四元數關係式，再經由模擬結果來證明此方式可以用來以2-D投影面上的資訊來正確估測出待測物在3-D空間上的轉動與移動。本文更利用雙四元數表示連續性轉移的方便性將以上的問題做實用範圍上的改進，最後以模擬實驗以驗證我們推導出的四元數關係式有更好的實用性與方便性。In this thesis, we use dual-quaternion to replace rotation matrix and translation vector which expressed object’s transformation in usual. 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 serious of rotation and translation.. The process of finding the dual-quaternion is to take out corresponding dual-vectors of object before transformation and past. Then we compare those corresponding dual-vectors to find the dual-quaternion expressing rotation and translation of object. We also called this process as closed-form solution of dual-quaternion. The standard method to find closed-form solution presently is to use the matrix relationship of dual-quaternion and dual-vector, and then apply SVD method to solve the equation. We improve this standard method and make the operation simpler. In the machine vision and robot control, connecting the information of image plane in 2-D observing and object transformation in 3-D world is very important and which is also called hand-eye problem in robot control. We find that dual-quaternion has special relationship in 3-D transformation and 2-D image plane, and use this relationship to estimate object’s transformation from 2-D observing image. Because of the benefit of dual-quaternion－easily to handle and express a serious of transformation, we make above estimation more practically and reality using dual-quaternion. At last, we design a simulation to prove our method has better practicality and is more convenient to estimate 3-D transformation.摘要.................................................. i Abstract.............................................. iii 目錄.................................................. v 圖表目錄.............................................. vii 第一章 序論......................................... 1 1.1 研究背景..................................... 1 1.2 章節介紹..................................... 3 1.3 符號介紹..................................... 3 第二章 四元數與雙四元數介紹......................... 5 2.1 四元數（quaternion）.............................. 5 2.2 雙向量（dual vector）............................. 11 2.3 雙四元數（dual quaternion）....................... 13 第三章 雙四元數封閉形式的估測....................... 19 3.1 封閉形式（closed form）的解法................. 19 3.1.1 封閉形式的估測模式....................... 19 3.1.2 封閉形式的估測解法....................... 21 3.2 封閉形式解法的估測模擬........................ 24 3.2.1 封閉形式解法的模擬參數設定............... 24 3.2.2 封閉形式解法的模擬結果................... 26 第四章 雙四元數遞迴型式的解法與其應用................. 31 4.1 遞迴式線性估測解法............................ 31 4.1.1 遞迴式線性估測的估測模式................. 31 4.1.2 遞迴式線性估測的估測解法................. 35 4.2遞迴式線性估測解法的模擬....................... 37 4.2.1 遞迴式線性估測的模擬參數設計............. 38 4.2.2 遞迴式線性估測的模擬結果................. 40 4.3遞迴式線性估測法的實際應用..................... 45 4.3.1 遞迴式線性估測的實際應用模式............. 45 4.3.2 遞迴式線性估測的實際應用解法............. 49 4.4遞迴式線性估測解法實際應用的模擬............... 51 4.4.1 遞迴式線性估測的實際應用模擬參數設計..... 51 4.4.2 遞迴式線性估測實際應用的模擬結果......... 51 第五章 結論與未來工作................................. 55 5.1 .............................................. 55 5.2 未來工作...................................... 56 參考文獻.............................................. 5