Robots and Screw Theory: Applications of Kinematics and Statics to Robotics, by Joseph K. Davidson and Kenneth H. Hunt, Oxford University Press, 2004, Great Clarendon Street, Oxford, England. (ISBN 0-19-856245-4).
REVIEWED BY GORDON R. PENNOCK1
As stated in the preface, the goal of this book is two-fold: (i) to explore the underlying principles of kinematic geometry which are so important for an understanding of rigid body displacements and velocities in a robotic manipulator; and (ii) to explore the principles of the geometry of force systems in as much as they relate to the understanding of the kinematics. The book emphasizes important and long-established principles which provide the reader with a basis for a deeper understanding of the capabilities and the limitations of robot motion. The authors believe that this knowledge can be used effectively to design and control robotic manipulators.
The key to the treatment on robotics presented in this book is the Screw, that geometric entity which underlies the mechanics of statics and first-order kinematics. The earliest references to screws can be traced back to the beginning of the 19th century when Poinsot (1806) established the concept that any system of forces applied to a rigid body can be reduced to a single force and a couple. Then Chasles (1832) stated that any rigid body displacement can be conceived as a rotation about a line accompanied by a translation along that line. This type of representation is canonical in form; moreover, it is subject to easy geometrical interpretation. Ball (1900) established a firm physical base for the mathematics of screws when he wrote his treatise on the small oscillations of a rigid body. Although the basis for screw theory in spatial kinematics was introduced almost two hundred years ago, the treatment presented in this text is well-worth reading. The two fundamental concepts in the theory of screws—that the instantaneous motion of a rigid body is a twisting motion about the instantaneous screw axis, and that a system of forces acting upon the rigid body is a wrench acting about a particular screw axis—are the underlying basis for the work in this book.
This text builds upon the idea that the kinetostatics of serial and parallel robots is a valuable discipline. This concept was introduced for planar robot manipulators several years ago by Joseph Duffy in his book Statics and Kinematics with Applications to Robotics, published by Cambridge University Press (1996). The book was based solely on the concepts of classical geometry. The author was lamenting that the great developments in geometry of the last century and their application in mechanics have, for the most part, been forgotten or ignored by many researchers in the field of robotics. He attempted to correct this error with a rigorous study of instantaneous kinematics and statics applied to the field of robotics. The Davidson and Hunt text makes a significant step forward in applying geometry to the study of rigid body mechanics, in general, and robot manipulators, in particular. The text expands kinetostatics to spatial robots and integrates into two charts the detailed relationships between instantaneous kinematically equivalent serial and parallel manipulators. The charts are also adapted to parallel manipulators of reduced freedom (for example, those with translating platforms) in a way that leads to simple representations of their behavior.
Since the authors are primarily concerned with the geometry of mechanical motion, the most general of which is spatial, it is natural that they employ the well-known theory of screws. The instantaneous screw is fundamental to rigid body motion. An infinitesimal displacement of a rigid body can be reduced to a single twist about a unique screw that has a certain pitch. Similarly, any system of forces and moments that act on the rigid body can be reduced to a single wrench which lies on a unique screw that has a certain pitch. The concept of reciprocal screws is central to the treatment presented in this book. Two screws are said to be reciprocal if their scalar product, which can be defined in such a way that it corresponds exactly with the ordinary scalar product of a pair of single vectors, is zero. The reciprocity of screws directly relates the kinematics of a robot manipulator on the one hand to the statics of the manipulator on the other hand. The emphasis on reciprocity arises naturally because the authors focus on the kinematic joints of a robot, throughout the book, and the screws that are associated with them, as opposed to the links of the robot. The important principle of reciprocity links together the two fields of instantaneous kinematics and statics.
The first eight chapters of the book are to be regarded as a sequence; however, the final three chapters need not be treated sequentially since they depend only on Chapters 1 through 8. There are some worked examples included in the text and with the exception of Chapters 2, 3 and 4, the examples are drawn from robotic devices. The book contains over 300 exercises which can be divided into three categories: (i) exercises confirming analytical statements that are made in the book; (ii) exercises devoted to numerical manipulations of formulae and principles presented in the book, and (iii) exercises stimulating the curiosity and enquiring reader to extend further the developments in the book into tangential or more elaborate realms. The reader is introduced to the screw and its application in robotics by starting with the familiar and simple example of planar motion. No prior knowledge of screw theory and robotics is assumed by the authors. Also, it is not essential for the reader to learn any revolutionary techniques or procedures. All that is required by way of background is elementary mechanics, vector calculus, and linear algebra. The ability of the reader, however, to visualize motion in three-dimensions is a significant advantage since most robotic devices are spatial in some aspect of their motion.
A very brief summary of the eleven chapters in the book follows. Chapter 1: The Planar Serial Robot-Arm. This chapter is a brief review of the degrees of freedom of a robot end-effector, instantaneous centers of zero velocity (instant centers), and a solution to the inverse velocity problem using instant centers. Static equilbrium and torques at the actuated joints are also discussed and the reader is prepared for forthcoming chapters on spatial serial robot-arms. Chapter 2: Describing the Screw. This chapter introduces the role of the screw in mechanics. The analogy between the screw for a rigid body in static equilibrium and the screw for the instantaneous displacement of the rigid body is presented. The classical work on the finite twist and the freedom and constraint of a rigid body is reviewed. Chapter 3: Analysing the Screw. This chapter is a discussion of the various lines in a moving rigid body. In particular, the concept of right lines in a moving rigid body is explored. The conclusion is that these ensembles of lines figure largely and continuously in the theory of freedom and constraint. Chapter 4: Transformation for Coordinates that Locate a Rigid Body. The chapter is a brief discussion of enumerative geometry, concerned mainly with lines in Euclidean space. Chapter 5: Linear Dependence, Reciprocity of Screws: Linear and Non-Linear Screw Systems. This chapter is a discussion based on elementary principles of the instantaneous screw axis for the motion of a rigid body. The linear two-system of screws which leads to introduction of the cylindroid is presented here. Clifford (1878) assigned the fundamental theorem of the composition of twists and wrenches by the cylindroid an important position at the basis of mechanics. Mannheim (1868) also presented a theorem about the screws of zero pitch on a cylindroid. In addition to screw systems for the instantaneous kinematics of a robot, systems of finite screws are presented in book form for the first time.
Chapter 6: Spatial Serial Robot-Arms. This chapter deals with the systems of motion screws at a generalized joint and with the nature of possible joints in serial robot manipulators. It is a little unfortunate that much of the original literature for the robot geometries used in this chapter, and the following chapter, are related to commercial models of six-actuated robot arms that are no longer commercially available. Aside from the names, however, popular modern configurations serial robots are included. Chapter 7: The Assembly-Configurations of Serial Robot-Arms. This chapter covers the computation of actuator coordinates for robot manipulators. This material is included in the book because any specific computations for the instantaneous kinematics or statics of a robot, simply require values for the actuator coordinates. The chapter also includes robot workspace which relies heavily on reciprocal systems of screws. This is one of the few text books on robotics that covers dialytic elimination and continuation methods for polynomials. These analytical techniques are required for computing actuator coordinates in several popular robot models. Chapter 8: In-Parallel Actuation I: Simple and Direct. This chapter includes the 6-6 fully in-parallel manipulator and the octahedral manipulator. The authors develop, in this chapter, the two adjoining charts that bring together many cross-relationships between any six-actuator fully in-series robot and the equivalent fully in-parallel robot. There is also a very good discussion of assembly configurations. Chapter 9: In-Parallel Actuation II: Combinations with Serial Devices. This chapter contains a discussion of mobility, connectivity, and over-constraint. There are interesting sections on the adjustable tripod as a manipulator, generalized reciprocal connections, and applications of the charts to several limited-freedom manipulators. Chapter 10: Redundant Robotic Systems. This chapter focuses on the important issue of kinematic redundancy in robotic systems. Pseudoinverse control and the control of a four-axis spherical robot wrist and a study of statically redundant robots are included in the chapter. Chapter 11: Static Stability in Legged Vehicles. This chapter presents wheeled and legged vehicles. There is a discussion of the margin for static stability in these vehicles.
The book also contains four interesting appendices. Appendix A: Some Useful Expressions for Lines. This appendix contains material that is helpful for the exercises and is not covered elsewhere in the text. Appendix B: The Screw as a Point in Projective Five-Space. This is a brief account of some of the screw systems that were presented a decade ago by Gibson and Hunt (1990). Appendix C: The Finite Twist and Eduard Study’s Coordinates. This appendix compares different coordinate systems and presents Study’s coordinates of the finite twist axis for a general rigid body displacement. Appendix D: Computer File for Chapter 10. This appendix contains the listing of a computer file that is used by the authors to develop figures for Chapter 10. The appendix also provides a basis for some of the exercises. It is very good to see such an extensive list of references at the end of the book. Some of these references are source material for the text and others as sources for the reader to pursue more fully some of the topics that are only outlined in the book.
In conclusion, I have no doubt that this book will be welcomed and read with much interest by those in the kinematics, robotics and pure mechanics communities. The book is most suitable for a first-year graduate student. However, there are many in industry, and in academic institutions, already immersed in robotic technology (for example, in the field of feedback control systems or in computer science), who should find this book very helpful in their technical work or in classroom instruction. The book is believed to be an important addition to the literature on the theory of screws and robotics. The main thrust, which is unashamedly geometrical, is a stimulating reference for all serious students of spatial kinematics. Overall, this book is rich with content and examples, and instructors looking for a graduate text on the subject should seriously consider the adoption of this text.
School of Mechanical Engineering, Purdue University, West Lafayette, IN 47907-2088