4.2 Defining the Configuration Space

This section defines the manifolds that arise from the transformations of Chapter 3. If the robot has $n$ degrees of freedom, the set of transformations is usually a manifold of dimension $n$. This manifold is called the configuration space of the robot, and its name is often shortened to C-space. In this book, the C-space may be considered as a special state space. To solve a motion planning problem, algorithms must conduct a search in the C-space. The C-space provides a powerful abstraction that converts the complicated models and transformations of Chapter 3 into the general problem of computing a path that traverses a manifold. By developing algorithms directly for this purpose, they apply to a wide variety of different kinds of robots and transformations. In Section 4.3 the problem will be complicated by bringing obstacles into the configuration space, but in Section 4.2 there will be no obstacles.