四元數方法應用於二軸機構之姿態規劃

Abstract

描述三維空間中剛性物體的位移(Displacement)，通常是以該物體之質量中心的「點」作為基礎。依據這個點的位移，來理解剛體是如何改變位置。規劃物體位移的路徑，即承接著「點」的精神，巳經成熟地發展出相當多演算法，以達成平滑位移的目的。 然而，三維空間中剛性物體的姿態(Pose)，因為涉及轉動，故不像位移那般單純。因為姿態的本質涉及座標的旋轉，無法直觀地承接「點」的概念進行分析。 本文中我們以單位四元數(Unit quaternion)詮釋物體姿態。單位四元數在經過對數映射，可以成為三維的向量。在這個三維向量上，我們發展出姿態圖，使得與物體位移相關之發展成熟的平滑軌跡規劃，亦能實現於姿態的平滑軌跡規劃上。 本文將研究應用於生活中常見的二軸機構(Two-axis mechanism)姿態規劃上。在這個基本型式的姿態機構，探討諸如平滑軌跡、避障，以及短路徑軌跡等議題。

To describe the displacement of a rigid body in 3-dimensional space , the “point” associated to the center of the mass of the rigid body is considered. The displacement of this point will represent the displacement of the rigid body. There are many matured algorithms have been developed to obtain a smoothed translation trajectory. To describe the pose of a rigid body , on the other hand , is much more complicated compared to that of the displacement. Since the rotation of the coordinate frame is involved , the concept of “point” only is not enough , to deal with the rotational operations. We use the unit quaternion to describe the pose of a rigid body. By using the mapping of the log function , the unit quaternion can be transformed to a 3-dimensional vector. The map of pose , which is based on the transformed 3-dimensional vector , is proposed in this thesis. The matured algorithm of smoothed displacement trajectory planning then can be applied to the pose map. Finally , the smoothed pose trajectory is obtained via the mapping of exponential function. The pose trajectory planning of the popular two-axis mechanism in our daily life is investigated. The smoothed trajectory , the obstacle avoiding , and the short route are demonstrated in our work.