現今有很多機器手臂在各個產業裡使用，有工廠的組裝、銲接等，以及服務業調酒、烹飪，但只要在機器手臂的工作空間內，不管是自己連桿本身，或是對周遭障礙物，都有可能會發生碰撞而會造成手臂損毀。本文的研究是以Lowner–John橢圓內嵌在手臂連桿與障礙物裡，藉由計算連桿與障礙物所代表兩個橢圓間的最短距離，用來估算連桿與障礙物的實際最短距離，本文的連桿與障礙物的最短距離以多邊形間的距離來呈現，但因為是以橢圓內嵌的方式，以致於會有連桿外露在橢圓外的情況。所以本文設計的演算法就是以補償為主，首先將一組橢圓固定，另一組本身作自轉且繞著前一組作公轉，產生的數據以圖表呈現，最後低估與高估的部分分別用高斯函數來補償，再搭配內差法求得其他相對位置的距離，既不會有低估的現象也能抑制高估的部分，使機器手臂能在不碰撞的前提下，貼近其他機器手臂或是障礙物，在整個工作空間能更有效率的被使用。最後，以內嵌與外接橢圓作位移、旋轉的方式及機器手臂的避撞環境模擬實驗來驗證所提方法的實用性，並由模擬的結果證明，此方法的準確率有明顯提升。

There are many manipulators used in various industries such as assembly, welding, bartending, cooking etc. However, during the working of the manipulator, there may be a collision with itself or obstacles will cause break to the arm. This paper presents a research on Lowner–John ellipsoid embedded in the links and obstacles. By calculating the minimum distance between two ellipses represented by the links and the obstacles, and it`s used to estimate the real minimum distance between links and obstacles. In this paper, the minimum distance between links and obstacles is represented by the distance between polygons. Because the link is embedded inside the ellipse, the corner of the link will be exposed. Therefore, the algorithm designed in this paper is based on compensation. First, one ellipse revolves both about the ellipse and on its own axis, and the result of data is presented graphically. Finally, the underestimate and overestimate parts are respectively compensated by the Gaussian function, and then the distance between the other relative positions is obtained by the internal difference method. The manipulators can be more close to other robot arm or obstacles on the premise that the manipulators has no collisions. It will be more efficient to be used on the workspace.
The result of simulation with ellipse and manipulators demonstrate the performance of the proposed approach.

論文審定書 i
中文摘要 ii
Abstract iii
目錄 iv
圖目錄 v
表目錄 vii
第一章 緒論 1
1-1 研究動機與目的 1
1-1-1 研究目的 1
1-1-2 研究動機 2
1-2 論文架構 2
第二章 研究背景 3
2-1 LOWNER –JOHN 橢圓 3
2-2 高斯函數 4
2-3 機器人運動學 6
第三章 研究方法 9
3-2 利用LOWNER - JOHN 橢圓建立三維資料表 10
3-3 使用高斯函數作補償 14
3-4 線性內差法 56
第四章 實驗設計與分析 58
第五章 結論與未來展望 68
5-1 結論與未來展望 68
參考文獻 69

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