Section 3.1 barely scratches the surface of geometric modeling. Most literature focuses on parametric curves and surfaces [376,718,788]. These models are not as popular for motion planning because obtaining efficient collision detection is most important in practice, and processing implicit algebraic surfaces is most important in theoretical methods. A thorough coverage of solid and boundary representations, including semi-algebraic models, can be found in [454]. Theoretical algorithm issues regarding semi-algebraic models are covered in [704,705]. For a comparison of the doubly connected edge list to its variants, see [522].

The material of Section 3.2 appears in virtually any book
on robotics, computer vision, or computer graphics. Consulting linear
algebra texts may be helpful to gain more insight into rotations.
There are many ways to parameterize the set of all 3D rotation
matrices. The yaw-pitch-roll formulation was selected because it is
the easiest to understand. There are generally 12 different variants
of the yaw-pitch-roll formulation (also called *Euler angles*)
based on different rotation orderings and axis selections. This
formulation, however, is not well suited for the development of motion
planning algorithms. It is easy (and safe) to use for making quick 3D
animations of motion planning output, but it incorrectly captures the
structure of the state space for planning algorithms. Section
4.2 introduces the quaternion parameterization, which
correctly captures this state space; however, it is harder to
interpret when constructing examples. Therefore, it is helpful to
understand both. In addition to Euler angles and quaternions, there
is still motivation for using many other parameterizations of
rotations, such as spherical coordinates, Cayley-Rodrigues parameters,
and stereographic projection. Chapter 5 of [210] provides
extensive coverage of 3D rotations and different parameterizations.

The coverage in Section 3.3 of transformations of chains of bodies was heavily influenced by two classic robotics texts [252,775]. The DH parameters were introduced in [434] and later extended to trees and loops in [524]. An alternative to DH parameters is exponential coordinates [725], which simplify some computations; however, determining the parameters in the modeling stage may be less intuitive. A fascinating history of mechanisms appears in [435]. Other texts on kinematics include [29,310,531,689]. The standard approach in many robotics books [366,856,907,994] is to introduce the kinematic chain formulations and DH parameters in the first couple of chapters, and then move on to topics that are crucial for controlling robot manipulators, including dynamics modeling, singularities, manipulability, and control. Since this book is concerned instead with planning algorithms, we depart at the point where dynamics would usually be covered and move into a careful study of the configuration space in Chapter 4.

Steven M LaValle 2012-04-20