##
**Geometric fundamentals of robotics.
2nd ed.**
*(English)*
Zbl 1062.93002

Monographs in Computer Science. New York, NY: Springer (ISBN 0-387-20874-7/hbk). xv, 398 p. (2005).

This is the second edition of a book initially published with a slightly different title [Geometrical methods in robotics (1996; Zbl 0861.93001)].

Two new chapters have been added. Chapter ten explains how to describe the motion of points, lines and planes in \(\mathbb{R}^3\) with a Clifford algebra. Operations on these objects are described with the exterior product and shuffle product. This helps establishing in a more straightforward way the results of D. L. Pieper [The kinematics of manipulators under computer control. PhD thesis, Standford University (1968)] on the solvability of the inverse kinematics of a six-joint manipulator. In the last chapter, some basics in Riemannian geometry are introduced and used as a way to bypass the need for the inverse dynamics in a tracking problem. If the curve to be tracked is a geodesic and if the controllers can be set so that the solution to the closed-loop dynamics is a geodesic, then tracking will be realized. A last section deals with hybrid position and force control.

The restrictions expressed in the review of the first edition still apply, even if control issues are now addressed in the last chapter.

One could add that even if the book is not so strong in tracing the latest developments in the control of robots from the differential-geometric point of view, it excels in embedding the topic in a historical context, showing how the works of Ball one hundred years ago, which predate developments in differential geometry and which remained a “backwater”, can now be expressed very elegantly and efficiently. The author excels also in showing the interplay between robotics and mathematics. In particular, the presence of differential and algebraic geometry is well outlined. Finally, there is no trace of pedantism, and clear language harmonizes with sound intuition.

Two new chapters have been added. Chapter ten explains how to describe the motion of points, lines and planes in \(\mathbb{R}^3\) with a Clifford algebra. Operations on these objects are described with the exterior product and shuffle product. This helps establishing in a more straightforward way the results of D. L. Pieper [The kinematics of manipulators under computer control. PhD thesis, Standford University (1968)] on the solvability of the inverse kinematics of a six-joint manipulator. In the last chapter, some basics in Riemannian geometry are introduced and used as a way to bypass the need for the inverse dynamics in a tracking problem. If the curve to be tracked is a geodesic and if the controllers can be set so that the solution to the closed-loop dynamics is a geodesic, then tracking will be realized. A last section deals with hybrid position and force control.

The restrictions expressed in the review of the first edition still apply, even if control issues are now addressed in the last chapter.

One could add that even if the book is not so strong in tracing the latest developments in the control of robots from the differential-geometric point of view, it excels in embedding the topic in a historical context, showing how the works of Ball one hundred years ago, which predate developments in differential geometry and which remained a “backwater”, can now be expressed very elegantly and efficiently. The author excels also in showing the interplay between robotics and mathematics. In particular, the presence of differential and algebraic geometry is well outlined. Finally, there is no trace of pedantism, and clear language harmonizes with sound intuition.

Reviewer: A. Akutowicz (Berlin)

### MSC:

93-02 | Research exposition (monographs, survey articles) pertaining to systems and control theory |

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

70B15 | Kinematics of mechanisms and robots |

93B29 | Differential-geometric methods in systems theory (MSC2000) |

53Z05 | Applications of differential geometry to physics |

70G65 | Symmetries, Lie group and Lie algebra methods for problems in mechanics |

70E60 | Robot dynamics and control of rigid bodies |

70B10 | Kinematics of a rigid body |

82C20 | Dynamic lattice systems (kinetic Ising, etc.) and systems on graphs in time-dependent statistical mechanics |

82C80 | Numerical methods of time-dependent statistical mechanics (MSC2010) |