利用數種電荷分佈所形成的場來模擬二維的工作空間

Abstract

解決路徑規劃的方法之一是使用位能場函數來代表自由空間的結構。這個 方法的主要優點包括簡化自由空間的表示式及利用推斥力來引導物體避碰 。過去的研究是用牛頓位能場函數來代表多邊形物體與障礙物,這些多邊 形是由線段組成,並且帶有均勻的電荷分佈。這篇論文把電荷分佈的情形 推廣到線性分佈或二次分佈,並且證明相關的運算式都可以用封閉式來表 示。根據重疊定理,多邊形物體與障礙物之間因推斥力所引起的相關量,可 由分別來自物體與障礙物的兩條線段因推斥力所引起的相關量求得。因為 每一條線段可能的電荷分佈有三種,分別為均勻,線性或二次分佈,故兩條 線段電荷分佈的組合共九種,本篇論文主要是推導這九種電荷分佈所引起 的相關量並用封閉式來表示。這些相關量與解析解可能的應用包括在自由 空間裡當作複雜形狀物體的模型及在路徑規劃問題中反應出非同質的物體 周界特性。 ne of the existing approaches to path planning problems uses a otential field function to represent the topological structuref free space. The main advantages of this approach include theimplicity of the representation of free space, the guidancerovided by the negative gradient of the potential filed functionor obstacle avoidance in the form of repulsive force. Newtonianotential function was used to represent polygonal objects andbstacles in 2D workspace where their boundaries are assumed toe uniformly charged. In this thesis, the source distributions arextended to more general cases which can be linear or quadratic.he expression of such potential field as well as some otherradient-related quantities can also be derived in closed form. Byuperposition principle, the gradient- related quantities due tohe repulsion between a pairs of polygonal object and obstalcesan be obtained from that due to line segments each pair has oneine segment from the object and the other from the obstacles.ecause the source distribution of a line segment can be uniform,inear or quadratic, there are nine types of source distributions he two line segments. The closed forms of these formulations arelso derived in this thesis. Possible applications of thesenalytic results include the modeling of free space of complexhape, and the representaion for objects and obstacles havingroperties of interest which are not homogeneous along theiroundary.