To create motion plans for robots, we must be able to specify the position of the robot. More specifically, we must be able to give a specification of the location of every point on the robot, since we need to ensure that no point on the robot collides with an obstacle. This raises some fundamental questions: How much information is required to completely specify the position of every point on the robot? How should this information be represented? What are the mathematical properties of these representations? How can obstacles in the robot's world be taken into consideration while planning the path of a robot?

In this chapter, we begin to address these questions. We first discuss exactly what is meant by a configuration of a robot and introduce the concept of the configuration space, one of the most important concepts in robot motion planning. We then briefly discuss how obstacles in the robot's environment restrict the set of admissible paths. We then begin a more rigorous investigation of the properties of the configuration space, including its dimension, how it sometimes can be represented by a differentiable manifold, and how manifolds can be represented by embeddings and parameterizations. We conclude the chapter by discussing mappings between different representations of the configuration, and the Jacobian of these mappings, which relates velocities in the different representations.