CONFIGURATION SPACE FORMULATION

The underlying idea of configuration space is to represent the robot as a point in an appropriate space - the robot's configuration space - and to map the obstacles in this space.This mapping transforms the problem of motion planning into the problem of planning the motion of a point.It makes constraints on the motion of robot more explicit.

Let a frame F_A be embedded in A and F_W in W. As A is rigid, every point a of a has a fixed position with respect to F_A. But a's position in W depends on the position and orientation of F_A relative to F_W.

DEFINITION

A configuration of an arbitrary object is a specification of the position of every point in this object relative to a fixed reference frame[Arnold,1978]. Configuration q of A is a specification of the position T and orientation of F_A with respect to F_W.

The configuration space of A is the space C of all configurations of A.

The subset of W occupied by A at configuration q is denoted by A(q). The point a on A at configuration q is denoted by a(q) in W. Obstacles in Configuration Space

The obstacles are mapped in configuration space and the subset of C that is made of contact-free configuration is considered. Every obstacle B_i i = 1...q, in the workspace W maps in C to a region

which is called a C-obstacle.The union of all the C-obstacles: is called the C-obstacle region, and the set:

is called the free spacei.Any configuration inC_free is called a free configuration.

FigureThe robot A (a square) can translate freely in the plane at a fixed orientation.The C-obstacle CB_i is obtained by "growing the corresponding workspace obstacle B_i(a triangle) by the shape of A.Planning a motion of a relative to B_i is equivalent to planning a motion of the vertex 1 of A relative to CB_i.
 

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