Further Reading

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.