Many texts present robot geometry in terms of homogeneous transforms, but without the underlying formal mathematical structure. This book provides a good mathematical introduction and insight into the general kinematics of manipulators based on open and closed kinematic chains. It successfully bridges the gap between the mathematical underpinnings and more practical robot geometry.

The first chapter introduces the homogeneous transform representation of displacements in three types of mechanism: planar (acting in one plane), spherical (the end of the mechanism moves over a sphere), and spatial (general displacement). McCarthy presents important characteristics of the displacements, such as the fixed point of a planar displacement, the spherical screw axis (the equivalent axis of rotation), and the characterization of spatial displacement as a rotation about a screw axis and a displacement along the axis. Chapter 2 uses homogeneous transforms to express motion classes as mathematical groups, including the well-known SO ( 3 ), the group of spatial rotations; H ( 3 ), the group of planar displacements; and H ( 4 ), the group of spatial displacements. The tangent operator is introduced as a generalized derivative of movement specified by transforms.

The third chapter elaborates screw theory, elegantly relating it to the standard Denavit-Hartenberg representation for a single rigid link with a joint at each end, which gives a screw around the joint axis, a displacement along the axis, a displacement along the link’s common normal, and finally a screw around the link twist. This chapter leads into the fourth as screw theory and Clifford algebra lead to a unified quaternion representation for all three cases of displacement. The quaternions arise from the fixed point or screw axis. Chapter 5 explains the number of degrees of freedom of various mechanisms. Chapter 6 develops the structure equations for open and closed chains, which describe the motion of the end of an open chain or a selected joint in a closed one. Joint movements are related to displacements in the world frame. A closed chain is split and the specifications of each half are equated, forming the structure equation. A general mechanism Jacobian relates the tangent operator S to joint velocities, with S = [ J ] { θ } . The final chapter uses structure equations in quaternion form to express the hypersurfaces on which a mechanism may move. A closed chain manifold is the intersection of its two component open chain manifolds.

The book fulfills its implied purpose: to introduce the mathematical foundations of articulated link geometry for manipulators and mobile robots. Its conciseness is refreshing; the book has only 122 pages, yet it is clear and covers the material well. The diagrams are excellent, and the chapter notes are helpful. The mathematical background required is somewhat above that of most practically oriented computer science students, but the book is still a useful and important reference for graduates continuing in robotics and figure animation. The index is satisfactory.