Further Reading

Section 4.1 introduced the basic definitions and concepts of topology. Further study of this fascinating subject can provide a much deeper understanding of configuration spaces. There are many books on topology, some of which may be intimidating, depending on your level of math training. For a heavily illustrated, gentle introduction to topology, see [535]. Another gentle introduction appears in [496]. An excellent text at the graduate level is available on-line: [439]. Other sources include [38,451]. To understand the motivation for many technical definitions in topology, [911] is helpful. The manifold coverage in Section 4.1.2 was simpler than that found in most sources because most sources introduce smooth manifolds, which are complicated by differentiability requirements (these were not needed in this chapter); see Section 8.3.2 for smooth manifolds. For the configuration spaces of points moving on a topological graph, see [5].

Section 4.2 provided basic C-space definitions. For further reading on matrix groups and their topological properties, see [63], which provides a transition into more advanced material on Lie group theory. For more about quaternions in engineering, see [210,563]. The remainder of Section 4.2 and most of Section 4.3 were inspired by the coverage in [588]. C-spaces are also covered in [220]. For further reading on computing representations of ${\cal C}_{obs}$, see [513,736] for bitmaps, and Chapter 6 and [865] for combinatorial approaches.

Much of the presentation in Section 4.4 was inspired by the nice introduction to algebraic varieties in [250], which even includes robotics examples; methods for determining the dimension of a variety are also covered. More algorithmic coverage appears in [704]. See [693] for detailed coverage of robots that are designed with closed kinematic chains.