本文所探討的演算法，是姿態判定時所使用的方法之ㄧ。姿態判定可以描述載具在空間中運動之軌跡、位置及指向。人造衛星、太空梭、手寫儀器等之運動皆與姿態判定有關。然而，姿態判定的準確性影響載具的安全和效率，這時候演算法的演繹就顯得格外重要。

隨著新興產品的需要，近年來包括互動式電子筆、手術用之手持儀器等應用，都有相關的研究。大部分所使用的姿態判定演算法不外乎三元組演算法、最佳三元組演算法、四元數估測法、遞迴式四元數估測法、最佳遞迴式四元數估測法等，有些方法配合卡爾曼濾波器的使用，可以降低雜訊帶來的誤差。

然而，有些演算法的細節可以改進，使演算過程更加健全。本文便是根據最佳三元組演算法的想法，引入球形線性內插法及蒙地卡羅模擬法幫助運算，我們稱此演算法為「建基於球形線性內插法之最佳三元組演算法」。

在本演算法提出後，透過電腦模擬，驗證結果是合乎預期的。最後設計兩個實驗，以期將演算法的功能付諸實際應用。

本文將詳細介紹相關背景知識、演算法推導過程、模擬驗證方法、實驗結果分析等，在附錄中並有相關公式推導。

This article discusses an algorithm which is used for attitude determination. Attitude determination describes the trajectory, position, and orientation of vehicles in space. The movements of satellites, space shuttles, or writing instruments have strong relations to attitude determination. However, the accuracy of attitude determination affects the safety or efficiency of vehicles. Thus, the deduction of the algorithm is especially important.

With the need for new products recent years, there are a lot of applications and related works, including interactive electronic pen and hand-held instruments used in surgery. TRIAD algorithm, optimized TRIAD algorithm, QUEST, REQUEST, and optimal REQUEST are all in common use for attitude determination. Moreover, with the use of Kalman filter, some of these methods can reduce the error caused by noise.

However, some of the algorithm details can be improved to make the process more robust. This article is based on the idea of the optimized TRIAD algorithm. In addition, SLERP and Monte Carlo simulation are introduced for calculation. We name this algorithm as “SLERP-Based Optimal TRIAD Algorithm.”

After this algorithm is proposed, the verifications are in line with our expectations through computer simulation. Finally, two experiments are designed to realize the algorithm in reality.

This article will detail the relevant background knowledge, algorithm derivation, simulation verification, and experimental results analysis. Besides, there are related formulas in the appendixes.

口試委員審定書 i
致謝 ii
中文摘要 iii
英文摘要 iv
目錄 v
圖目錄 viii
表目錄 x

第一章 緒論 1
1-1 研究背景 1
1-2 章節介紹 2
1-3 符號介紹 3

第二章 背景介紹 5
2-1 座標系 5
2-1.1 向量與座標系 5
A. 附地座標系統 7
B. 當地水平座標系統 9
C. 附體座標系統 10
2-1.2 座標轉換 10
A. 旋轉矩陣 10
B. 尤拉角 12
C. 四元數 14
2-2 姿態判定 16
2-2.1 三元組演算法 16
2-2.2 四元數估測法 17
2-3 蒙地卡羅模擬法 20

第三章 最佳三元組演算法 23
3-1 球形線性內插法 23
3-1.1 單位四元數之距離 23
3-1.2 球形線性內插法 24
3-2 建基於球形線性內插法之最佳三元組演算法 25
3-2.1 演算法 25
3-2.2 引入蒙地卡羅模擬法 27
3-3 模擬結果 30
3-3.1 模擬環境建立及流程簡介 30
3-3.2 模擬結果與分析 32

第四章 實驗結果與分析 39
4-1 手寫筆姿態判定 39
4-1.1 慣性導航系統與電子羅盤簡介 39
4-1.2 實驗環境建立及流程簡介 43
4-1.3 實驗結果與分析 45
4-2 北方方位判定 47
4-2.1 磁偏角簡介 47
4-2.2 實驗環境建立及流程簡介 48
4-2.3 實驗結果與分析 50

第五章 結論與未來工作 57
5-1 結論 57
5-2 未來工作 58

附錄A：經緯高轉至附地座標系推導 59
附錄B：附地座標系轉至當地水平座標系推導 63
參考文獻 65

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