本研究是針對雙足機器人在臨界狀態時腳與地面的反作用力，分別探討反作用力分布為單點力、線分布、與面分布三種情況，求出造成此臨界狀態的外力組合。本研究所針對的臨界狀態，主要是機器人在地面上滑動，但也探討機器人的傾倒。在線分布及面分布的情況皆是利用赫氏接觸理論，假設接觸區域內之應力為赫氏分布，並利用平面運動學中之瞬時中心原理求出臨界滑動狀態之切線應力。論文結果包括造成機器人滑動的力量(力矩)Fx，Fy，Mz的組合，以及可能造成傾倒之Mx，My的條件。

In this thesis we discuss ground reaction forces and moments on a foot of a biped robot under a critical stable condition. We treated the following three types of reaction force distribution on the foot: point force, line contact force, and area contact force, and determine force combinations to bring the foot into critical conditions. The emphasis of this study is on the critical stable condition of gross sliding, but the condition of falling is also discussed. We assume Hertz contact stress in line and area contact regions. The theory of instantaneous center of zero velocity in planar kinematics is utilized to determine tangential stress distribution. Results include all combinations of forces Fx, Fy, Mz to cause gross sliding and the possibility for moments Mx, My to cause falling.

目錄
中文摘要	I
英文摘要	II
目錄	III
圖目錄	IV
第一章  緒論	1
1.1 前言及研究動機	1
1.2文獻回顧	2
1.3研究內容	3
 第二章  臨界狀態之接觸形式	5
2.1簡介	5
2.2點接觸的臨界狀態	5
2.3線接觸的臨界狀態	7
2.4面接觸的臨界狀態	9
第三章  分析過程與數值結果	11
3.1點接觸	11
3.2線接觸	12
3.3面接觸	14
3.4點接觸與面接觸的比較	18
第四章  結論及未來研究方向	20
參考文獻	21
附錄A  接觸點B位置座標	23



圖目錄
圖一  (a)雙足機器人之示意圖；(b)機器人腳示意圖	24
圖二  機器人腳及受力狀態，i及j為x及y方向上之單位向量	25
圖三  腳繞直線L3旋轉時在地面上的壓力分佈	26
圖四  地面上線接觸區域內一點p(x,y)之切線應力	27
圖五  在圖二中B點附近的接觸區域	28
圖六  機器人腳之尺寸示意圖	29
圖七  當腳形大小w=1.0，L1=1.5，L2=1.0，a=0.2w，R1=w時，fx、
      fy及mz的關係；mBzmax=0.1862；(xB,yB)=(-0.8,-0.25) 30
圖八  當腳形大小w=1.0，L1=2.0，L2=2.0，a=0.2w，R1=w時，fx、
      fy及mz的關係；mBzmax=0.7194；(xB,yB)=(-0.8,-0.5)  31
圖九  當腳形大小w=2.0，L1=1.5，L2=1.0，a=0.2w，R1=w時，fx、
      fy及mz的關係；mBzmax=0.411；(xB,yB)=(-1.6,-0.75)  32
圖十  當腳形大小w=1.0，L1=1.5，L2=1.0，a=0.1w，R1=w時，fx、
      fy及mz的關係；mBzmax=0.0898；(xB,yB)=(-0.9,-0.25) 33
圖十一  當腳形大小w=2.0，L1=2.0，L2=3.0，a=0.4w，R1=時，
        fx、fy及mz的關係；mBzmax=3.2878；(xB,yB)=
        (-1.2,-1.5)    34
圖十二  當腳形大小w=1.0，L1=1.5，L2=1.0，a=w/100，R1=
        時，fx、fy及mz的關係；mBzmax=0.0089；(xB,yB)=
        (-0.9,-0.25)   35
圖十三  當腳形大小w=0.5，L1=1.2，L2=0.8，a=0.0585w，R1=w
        時，fx、fy及mz的關係；mBzmax=0.2078；(xB,yB)=
        (-0.9415,-0.1)   36
圖十四  線接觸情況即將滑動下的(mx,my)的關係；圖中顯示的是無
        因次化之xi及eta數值	   37
圖十五  R3為四周導角半徑   38
圖十六  當a=1.9834，b=0.4408，theta=pi/4時，fx、fy及mz的關
        係，mBzmax=1.4435   39
圖十七  當a=1.8410，b=0.6370，theta=pi/4時，fx、fy及mz的關
        係，mBzmax=1.8999   40
圖十八  當a=1.7272，b=0.8336，theta=pi/4時，fx、fy及mz的關
        係，mBzmax=2.3443   41
圖十九  當a=1.9834，b=0.4408，theta=pi/2時，fx、fy及mz的關
        係，mBzmax=1.4435   42
圖二十  當a=1.8410，b=0.6370，theta=pi/2時，fx、fy及mz的關
        係，mBzmax=1.8999   43
圖二十一  當a=1.7272，b=0.8336，theta=pi/2時，fx、fy及mz的
          關係，mBzmax=2.3443   44
圖二十二  當a=0.7577，b=0.4778，theta=7*pi/6時，fx、fy及mz
          的關係，mBzmax=0.2795   45
圖二十三  當a=0.7577，b=0.4778，theta=5*pi/4時，fx、fy及mz
          的關係，mBzmax=0.2795   46
圖二十四  當a=0.7577，b=0.4778，theta=4*pi/3時，fx、fy及mz 
          的關係，mBzmax=0.2795  47
圖二十五  面接觸情況，B點位在theta=pi/4時即將滑動下的 
          (mx,my)的關係；圖中顯示的是無因次化之xi及eta數值	 48
圖二十六  面接觸情況，B點位在theta=pi/2 時即將滑動下的
          (mx,my)的關係；圖中顯示的是無因次化之xi及eta數值	 49

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