The concepts of the previous chapter enable the construction and
transformation geometric models, which can be used to formulate path
planning problems in the world, 
. For the development and
analysis of algorithms, however, it will be convenient to view the
path planning as a search in the space of possible transformations
that can be applied to the robot, 
. This space of transformations
is termed the configuration space, which represents a powerful
representational tool that unifies a broad class of path planning
problems in a single mathematical framework. This facilitates the
development of path planning techniques that can be applied or adapted
to a wide variety of robots and models.
Section 3.1 introduces some preliminary mathematical concepts from topology. Section 3.2 defines the configuration space for a given robot and its set of possible transformations. Section 3.2 characterizes the portion of the configuration space in which the robot is in collision with an obstacle. Section 3.4 presents an overview of the collision detection problem. Section 3.5 represents configuration space concepts for the complicated case in which closed kinematic chains exist.
