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 , 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.

Steven M LaValle 2020-08-14