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**A mathematical introduction to robotic manipulation.**
*(English)*
Zbl 0858.70001

Boca Raton, FL: CRC Press. xix, 456 p. (1994).

A slightly more abstract (mathematical) formulation of the kinematics, dynamics, and control of robot manipulators is presented, not only for a single robot but also for multifingered robot hands, involving multiple cooperating robots. As prerequisites, a good course in linear algebra at the undergraduate level and some familiarity with signals and systems are assumed.

The book is organized into nine chapters and two appendices. After a general introduction in chapter 1, chapter 2 is an introduction to rigid body motion. A modern treatment of the theory of screws based on linear algebra and matrix groups is presented. The fundamental tools are the use of homogeneous coordinates to describe rigid motions, and the matrix exponential which maps a twist into the corresponding screw motion. The chapter concludes with a discussion of reciprocal screws. Appendix A gives a more abstract version, using matrix Lie groups and Lie algebras. Appendix B contains a brief description of a Mathematica package for screw calculus.

The rest of the material may be subdivided into three parts: an introduction to manipulation for single robots (chapters 3 and 4), coordinated manipulation using multifingered robot hands (chapters 5 and 6), and nonholonomic motion planning (chapters 7 and 8). This is a selection of topics representing an evolution from the more basic concepts to the frontiers of the research in the field.

In chapter 3, a description of the kinematics for a general \(n\) degree-of-freedom, open-chain robot manipulator using the tools presented in chapter 2 is given, including an elegant formulation of a set of canonical problems for solving the inverse kinematics problem, and an introduction of the manipulator Jacobian using the product of exponentials formula. Finally, some of the main results of this paper are extended to redundant manipulators and parallel mechanisms.

Chapter 4 presents an introduction to the dynamics and control of single robots. Using Lagrange’s equations and making use of twists for representing the kinematics, the equations of motion for a general open-chain manipulator are derived. The control laws in the sense of workspace control for asymptotic tracking of a desired trajectory are constructed.

Chapter 5 is an introduction to the kinematics of grasping. The kinematics of a multifingered robot hand grasping an object is studied, not only for the usual fixed contact case but also for the case when the fingers are allowed to roll or slide along the object. The relationships between finger and object velocities and forces are derived, and the conditions under which a grasp can be used to manipulate an object are studied.

Chapter 6 is a derivation of the dynamics and control for multifingered robot hands. The dynamic formulation presented in chapter 4 is extended to include robotic systems with contact strains, redundant robots, and nonmanipulable robots. Kinematics and statics of tendon actuation and the control of robot hands are considered.

Nonholonomic constraints, arising in systems such as multifingered robot hands and wheeled mobile robots, where rolling contact is involved, as well as in systems where angular momentum is conserved, are considered in chapter 7. The effects of these constraints on the behavior of robotic systems are studied. Basic tools needed to analyze nonholonomic systems are drawn both from the basic theorems in differential geometry and from nonlinear control theory. The application of these tools in robotic manipulation is considered.

Chapter 8 contains an introduction to some methods of motion planning for systems with nonholonomic constraints. After a consideration of sinusoids in generating Lie bracket motions in nonholonomic systems and a model class referred to as chained form systems, more general methods for steering nonholonomic systems using piecewise constant controls and Ritz basis functions are discussed.

A number of exercises after chapters 2-8 are provided.

The short last chapter 9 gives future prospects describing some of the growth areas in robotics from a technological point of view.

The book is based upon courses which have been taught at several universities and institutes to a hybrid audience of electrical engineers, computer scientists, mechanical engineers, and mathematicians. The material is suitable for advanced courses in robotics.

The book is organized into nine chapters and two appendices. After a general introduction in chapter 1, chapter 2 is an introduction to rigid body motion. A modern treatment of the theory of screws based on linear algebra and matrix groups is presented. The fundamental tools are the use of homogeneous coordinates to describe rigid motions, and the matrix exponential which maps a twist into the corresponding screw motion. The chapter concludes with a discussion of reciprocal screws. Appendix A gives a more abstract version, using matrix Lie groups and Lie algebras. Appendix B contains a brief description of a Mathematica package for screw calculus.

The rest of the material may be subdivided into three parts: an introduction to manipulation for single robots (chapters 3 and 4), coordinated manipulation using multifingered robot hands (chapters 5 and 6), and nonholonomic motion planning (chapters 7 and 8). This is a selection of topics representing an evolution from the more basic concepts to the frontiers of the research in the field.

In chapter 3, a description of the kinematics for a general \(n\) degree-of-freedom, open-chain robot manipulator using the tools presented in chapter 2 is given, including an elegant formulation of a set of canonical problems for solving the inverse kinematics problem, and an introduction of the manipulator Jacobian using the product of exponentials formula. Finally, some of the main results of this paper are extended to redundant manipulators and parallel mechanisms.

Chapter 4 presents an introduction to the dynamics and control of single robots. Using Lagrange’s equations and making use of twists for representing the kinematics, the equations of motion for a general open-chain manipulator are derived. The control laws in the sense of workspace control for asymptotic tracking of a desired trajectory are constructed.

Chapter 5 is an introduction to the kinematics of grasping. The kinematics of a multifingered robot hand grasping an object is studied, not only for the usual fixed contact case but also for the case when the fingers are allowed to roll or slide along the object. The relationships between finger and object velocities and forces are derived, and the conditions under which a grasp can be used to manipulate an object are studied.

Chapter 6 is a derivation of the dynamics and control for multifingered robot hands. The dynamic formulation presented in chapter 4 is extended to include robotic systems with contact strains, redundant robots, and nonmanipulable robots. Kinematics and statics of tendon actuation and the control of robot hands are considered.

Nonholonomic constraints, arising in systems such as multifingered robot hands and wheeled mobile robots, where rolling contact is involved, as well as in systems where angular momentum is conserved, are considered in chapter 7. The effects of these constraints on the behavior of robotic systems are studied. Basic tools needed to analyze nonholonomic systems are drawn both from the basic theorems in differential geometry and from nonlinear control theory. The application of these tools in robotic manipulation is considered.

Chapter 8 contains an introduction to some methods of motion planning for systems with nonholonomic constraints. After a consideration of sinusoids in generating Lie bracket motions in nonholonomic systems and a model class referred to as chained form systems, more general methods for steering nonholonomic systems using piecewise constant controls and Ritz basis functions are discussed.

A number of exercises after chapters 2-8 are provided.

The short last chapter 9 gives future prospects describing some of the growth areas in robotics from a technological point of view.

The book is based upon courses which have been taught at several universities and institutes to a hybrid audience of electrical engineers, computer scientists, mechanical engineers, and mathematicians. The material is suitable for advanced courses in robotics.

Reviewer: L.Sperling (Magdeburg)

### MSC:

70-02 | Research exposition (monographs, survey articles) pertaining to mechanics of particles and systems |

70B15 | Kinematics of mechanisms and robots |

70Q05 | Control of mechanical systems |

93C85 | Automated systems (robots, etc.) in control theory |