多軸循跡運動控制之主要目的為確保系統之實際運動軌跡能維持在規劃的命令路徑上。傳統作法大都是設計追蹤控制器(tracking controller)儘量減少系統各軸之實際運動路徑與各軸命令路徑間之追蹤誤差。然而若任何一軸有顯著之追蹤誤差，則極有可能系統之實際運動軌跡與規劃路徑間之輪廓誤差會超過容許的範圍。事實上在多軸循跡運動問題中，輪廓誤差遠比追蹤誤差重要。有鑑於此，近年來有不少學者致力於研究以建構速度場的方式來規劃循跡運動的命令路徑。由於速度場僅為位置函數，與時間無關，因此在控制器的設計上可著重在減少輪廓誤差而非追蹤誤差。在這些研究中，速度場由速度向量所組成，而速度向量則是由距離向量與其相互垂直之切線向量構成。這樣的作法對於直線或圓形循跡運動問題而言，因距離向量的計算十分容易，所以可順利建構出適當之速度場。但對於自由曲線之循跡運動問題來說，因其距離向量無法即時且準確地被計算出來，相關文獻並未提及如何規劃其速度場。為解決此一問題，本論文應用方向場理論，以格子點的方式來建構自由曲線循跡運動問題中之速度場。模擬結果顯示本論文所提出之方法確實可行。

The aim of multi-axis contour following control is to ensure that the system’s motion trajectory maintain on the desired contour. Conventional approaches to this control problem are to design tracking controllers for each axis so that the tracking error between the position output and the motion command for each axis is as small as possible. However, if there exists significant tracking error in any axis, it is likely that the contour error between the motion trajectory and the desired contour will exceed the tolerable limit. In fact, the contour error is more important than the tracking error in contour following tasks. Consequently, a number of studies have focused on encoding the desired contour by constructing the velocity field. Since the constructed velocity field is a function of position only (independent of time), the motion controller design for contour following tasks can emphasize on reducing contour error rather than tracking error. However, in these studies, the velocity field is composed of a distance vector and a tangent vector that is perpendicular to the distance vector. For line or circle contour following tasks, it is easy to find the corresponding distance vector, whereas it is not the case for the free form contour following task. This fact may explain why those previous studies did not explore the case of free form contour following task. In order to cope with this problem, this thesis exploits the idea of direction filed theory and constructs the velocity field at pre-selected grid points. Simulation results demonstrate the proposed approach indeed feasible.

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