本篇論文針對工業機器手臂之軌跡規劃，提出平面上最短路徑且平滑之軌跡的演算法。本演算法首先利用橢圓罩住障礙物，找出在平面上之歐基里德最短路徑，此最短路徑必須避開障礙物，接著藉由限制關節(joint)的力矩與力矩改變率(Torque rate)，使得機器手臂在沿著此最短路徑工作時，能保持平滑的軌跡。由本篇論文所找出的歐基里德最短路徑，我們可以確保在有障礙物的情況下，機器手臂沿著此路徑所行進的距離總和為最短。在找到此最短路徑之後，為了要確保機器手臂能快速且平滑的沿著此一路徑執行指定工作，我們加入了兩個限制的條件，亦即是限制每一個關節的力矩與力矩改變率。為了要確保軌跡的平滑，機器手臂在執行工作中可能要犧牲一點時間。最後我們可以得到在有障礙物的環境下要達到最短路徑與軌跡平滑所需的工作時間。本篇論文最後將此演算法運用在SCORBOT ER VII機器手臂及NONHOLONOMIC MOBILE ROBOT的軌跡追蹤以彰顯本論文在此方面的成果。

In this thesis, we proposed an algorithm about the smooth trajectory planning for industrial manipulators along the shortest path with obstacles in the plane. First, an Euclidean shortest path, avoiding obstacles which were well covered by ellipses, was found by our algorithms. A smooth trajectory could be kept by limiting the torque and torque rate of the joints when the manipulator worked along the shortest path. From the Euclidean shortest path, determined by the thesis, we can be sure that the summation of distances is smallest for manipulators working along the Euclidean shortest path. After finding the shortest path, we added two conditions, i.e., the torque and torque rate limits, and a prespecified work could be executed for ensuring that the manipulator could worked quickly and smoothly along the shortest path. For ensuring the smoothness of the trajectory, the manipulator, executing an prespecified work, may spent extra time. Finally, we could get an executing time for the manipulator working smooth along the shortest path with obstacles. The models of the SCORBOT ER VII manipulator and nonholonomic mobile robot are used to illustrate the effectiveness of the proposed algorithm for achieving the tracking purpose.

CONTENTS
ABSTRACT (IN CHINESE) I
ABSTRACT (IN ENGLISH) II
1 INTRODUCTION 1
2 PROBLEM FORMULATION 3
3 APPROXIMATE EUCLIDEAN SHORTEST PATHS 5
3.1 The history and our esult . . . . . . . . . . . . . . 5
3.2 Approximate shortest paths . . . . . . . . . . . . . . 6
3.3 Examples . . . . . . . . . . . . . . . . . . . . . . 9
4 SMOOTH PATH-CONSTRAINED TIME OPTIMAL MOTIONS 13
4.1 Path constraints . . . . . . . . . . . . . . . . . . . 13
4.2 Torque limits . . . . . . . . . . . . . . . . . . . 15
4.3 Torque rate limits . . . . . . . . . . . . . . . . . . 16
4.4 Admissible states . . . . . . . . . . . . . . . . . . 18
4.5 System dynamics . . . . . . . . . . . . . . . . . . . 21
4.6 Solution of the SPCTOM . . . . . . . . . . . . . . . . 22
4.6.1 SPCTOM trajectory characteristics . . . . . . . 22
4.6.2 Approximating functions in Robotics . . . . . . 25
4.6.3 Planning method . . . . . . . . . . . . . . . . 26
5 SIMULATIONS 29
5.1 Planning performance . . . . . . . . . . . . . . . . . 29
5.2 Trajectory tracking . . . . . . . . . . . . . . . . . 37
6 CONCLUSIONS 43

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