本論文分析機器人腳與地面的彈性接觸，探討可能造成雙足機器人行走時發生傾倒及滑動的外力大小。若機器人腳所受外力矩較大，可能使腳與地面的接觸面積僅剩最後微小的區域，這就是發生傾倒不穩定的臨界狀態；而機器人在運動過程中，亦可能同時在地面上發生滑動。本研究針對滑動不穩定狀態進行分析，找出所有造成滑動臨界狀態的力及力矩(Fx, Fy, Mz)組合，並且亦找出數個造成傾倒臨界狀態的力及力矩組合(Mx, My, Fz)。分析的方法分成兩部分，第一部分是進行接觸分析，求出外力(矩)作用之下接觸面積及壓力分布。此部分結果可找出造成傾倒臨界狀態之(Mx, My, Fz)組合。第二部分是在接觸區域及正向壓力求出之後，利用平面運動之瞬時中心的觀念，找出滑動臨界狀態之(Fx, Fy, Mz)組合。

In this thesis the elastic contact between a robot foot and ground is analyzed to investigate the forces to cause falling or sliding of a biped robot in walking. In cases with large applied moments, the foot-to-ground contact area may reduce to a very small region, and this is the critical condition for falling. Gross sliding may also occur during robot walking, which is also a state of instability. In this thesis the size of foot-to contact region and pressure distribution in the region is determined first, from which the tangential stress distribution may also be found by using the technique of instantaneous center of zero velocity. With these stress distributions the resultant moments in gross sliding or falling may be determined. In this thesis we show all combinations of forces (moments) (Fx, Fy, Mz) to cause sliding, and a few combinations of forces (Mx, My, Fz) to cause falling. These forces are compared with results obtained by treating the foot as a rigid body.

目  錄
中文摘要	I
英文摘要	II
目錄	III
圖目錄	Ⅳ
表目錄	Ⅶ
第一章  緒論	1
1.1前言及研究動機	1
1.2 文獻回顧	1
1.3 雙足機器人腳之模型	3
第二章  雙足機器人之穩定性分析	4
2.1簡介	4
2.2邊界積分方程式	5
2.3穩定性分析	9
第三章  數值分析	12
3.1導角上任一點高度	12
3.2接觸區域網格化	14
第四章  結果與討論	17
第五章  結論及未來研究	23
參考文獻 	24
 附錄A  f(x,y,s,t)的積分	26

圖目錄
圖1  雙足機器人之示意圖...................................30
圖2  機器人腳及受力狀態...................................31
圖3  機器人腳對地面之相對位移圖...........................32
圖4  接觸區域及網格，虛線是腳的原始接觸區域...............33
圖5  機器人腳尺寸參數圖...................................34
圖6  將腳形劃分區域，針對不同區域導角上的點分別分析.......35
圖7  (a) 腳前方圓弧導角上某點P之xy面圖，(b)C1P之截面......36
圖8  腳左、右側之xz面圖...................................37
圖9  腳與地面的接觸區域網格化(a)沿y方向分割n個strip，再把每個strip分割成有限個網格(cell)，(b)第i個網格示意圖.........38
圖10 將腳形變為圓柱體後與赫氏應力分佈之比較...............39
圖11 theta_x=theta_y=0時delta與Fz之關係.................. 40
圖12 delta=0.002，theta_x=theta_y=0，在y/K=0.1390的應力分佈........................................................41
圖13 fz=0.0033時，theta_x與Fz和theta_y與Fz之關係..........42
圖14 fz=0.0033時，theta_x與Mx之關係.......................43
圖15 fz=0.0033時，theta_y與My之關係.......................44
圖16 fz=0.0033時，theta_x與對Q之力矩的關係............... 45
圖17 delta=0.002、theta_x=0、theta_y=0時mpz之等高線圖.....46
圖18 delta=0.002、theta_x=0.003、theta_y=0.005時mpz之等高線圖........................................................47
圖19 delta=0.002、theta_x=0、theta_y=0時，腳與地面之接觸區域
..........................................................48
圖20 delta=-0.77、theta_x=-0.0024、theta_y=-1.5時，腳與地面之接觸區域................................................49
圖21 delta=-0.77、theta_x=-0.0024、theta_y=-1.5時mpz之等高線圖........................................................50
圖22 delta=-0.77、theta_x=-0.0024、theta_y=-1.5時，Tz沿y方向之分佈....................................................51
圖23 機器人腳與地面接觸區域僅剩腳前方圓弧的頂端時之臨界數值；直線為最近似之線性解..................................52
圖24 機器人腳與地面接觸區域僅剩腳後端直線時之臨界數值；直線為最近似之線性解..........................................53
圖25 機器人腳為剛體且與地面接觸區域僅剩腳前方圓弧之頂端時，腳上之受力狀態................ ...........................54
圖26 機器人腳為剛體且與地面接觸區域僅剩腳後側一直線時，腳上之受力狀態................................................55

表目錄
表1  機器人腳與地面接觸區域僅剩腳前方圓弧的頂端時之臨界數值........................................................28
表2  機器人腳與地面接觸區域僅剩腳後端直線時之臨界數值........................................................28
表3  機器人腳與地面接觸區域僅剩腳左側直線時之臨界數值........................................................29

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