本論文研究的對象為古典圓規式雙足機器人之被動步態，嘗試在各種地面坡度下，找出可能存在的極限循環的數目，並針對擺動腳接觸地面的彈性碰撞條件，進行探討；希望後續的研究可以藉此找出合適的步伐，達到節能的目的。
在數學模型建立，以Euler-Lagrange方法推導出被動式步伐的動態方程式。在推導擺動腳碰觸地面的模型方面，需考慮擺動腳接觸地面時會產生衝擊力與摩擦力，以及在實際路面狀況會有不同的恢復係數。利用碰撞前後，法線方向存在與恢復係數相關的線動量關係，以及接觸點處的角動量平衡與機構動態的拘束條件，推導出其碰撞前後速度改變方程式。為節省確切極限循環的搜尋時間，則是利用雙足機器人在碰撞前有一特定的角速度，在碰撞瞬間失去的部分動能，所造成碰撞後的角速度為下一起步的初始狀態；此時，兩腳動作互換，在下一次新的擺動腳碰撞瞬間前，位能所轉換的動能恰為之前碰撞瞬間所損失的動能。
在模擬中考慮有數值誤差，利用絕對誤差來判定在各碰撞角度有無極限循環；並探討坡度與步距大小的相互關係，以及改變恢復係數對極限循環的影響，結果顯示下坡坡度稍微變大時，符合極限循環的步距會隨之變大些；而恢復係數變小時，符合極限循環的步距則變大。

The objective of this thesis is to study the existence of limit cycles in the passive gaits of a classical compass biped robot under different slopes and road conditions. It is expected that suitable limit cycles of passive gaits may be found to save energy in the future.
In mathematical modeling, the dynamic of passive gaits is derived using Euler-Lagrange method, while the impact equation impact and the conservation of angular momentum at the contact point. All the velocities are calculated based on the kinematic constraints. To make the search for limit cycles more efficient, we make use of the phenomenon that the kinetic energy loss during impact equals the kinetic energy converted from the potential energy in one downhill pace.
Acknowledging the fact that numerical error might exist in simulation, some absolute error must be tolerated in judging the existence of limit cycles. The effect of slope and the restoring coefficient on the limit cycles increases as the slope increases or the restitution coefficient decreases.

摘要 iv
Abstract v
誌謝 vi
目次 vii
表目錄 ix
圖目錄 x
第一章 緒論 1
1.1 前言 1
1.2 研究動機與目的 2
1.3 文獻回顧 2
1.4 論文架構 5
第二章 雙足機器人數學模型建立 7
2.1 模型定義 7
2.2 數學模型推導 8
2.2.1 單腳擺動期動態方程式 8
2.2.2 碰撞前後速度改變方程式 13
第三章 被動式步態的確切極限循環 18
3.1 被動式步態的穩定性 18
3.2 確切之極限循環 19
3.3 黃金分割搜尋法 25
第四章 模擬結果與討論 30
4.1 坡度與碰撞角度相互關係 30
4.2 改變恢復係數e值 34
第五章 結論 42
參考文獻 43
附錄A 47
A.1 坡度γ為0.004、恢復係數e為0.9 47
A.2 坡度γ為0.0045、恢復係數e為0.9 49
A.3 坡度γ為0.005、恢復係數e為0.9 51
A.4 坡度γ為0.0055、恢復係數e為0.9 53
A.5 坡度γ為0.006、恢復係數e為0.9 54
附錄B 57
B.1 證明det(H+)≠0 57

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