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**A survey on the theory of bonds.**
*(English)*
Zbl 1404.53009

The article discusses a special kind of mechanical linkages called “paradoxical” linkages. The word “paradoxical” is in my opinion a misnomer. The mobility \(m\) of a linkage is defined to be the dimension of its configuration space. Roughly speaking, a paradoxical linkage is a linkage which has mobility \(m \geq 1\), such that a generic linkage of the same type is not expected to have mobility (in other words, a generic linkage of the same type has mobility \(m=0\)).

Rare objects are interesting. It is therefore an interesting problem to classify paradoxical linkages for a small number of links and joints. Can one find necessary and sufficient conditions for a given linkage to be paradoxical? It is with this aim in mind that the theory of bonds was created by G. Hegedüs and the three authors of this article in a joint article in 2015. The article under review is a nice geometric introduction to the subject of paradoxical mechanical linkages and the related theory of bonds.

The authors use dual quaternions to represent Euclidean motions in \(3\)-space. A dual quaternion is a formal object of the form \(q + \epsilon p\), where \(q\) and \(p\) are quaternions, and \(\epsilon\) is a non-zero symbol which commutes with \(i\), \(j\) and \(k\), and such that \(\epsilon^2 = 0\). The norm of a dual quaternion is defined by \[ N(q + \epsilon p) = (q + \epsilon p) (\bar{q} + \epsilon \bar{p}), \] which takes values in \(\mathbb{R} \oplus \epsilon \mathbb{R}\) (the ring of dual numbers). It turns out that the group of motions of Euclidean \(3\)-space is isomorphic to the multiplicative group \(G\), which is the quotient of the group of dual quaternions of non-zero real norm by the normal subgroup \(\mathbb{R}^*\) of non-zero real numbers.

Using the language of dual quaternions, the authors describe linkages in geometric terms. Roughly speaking, bond diagrams are combinatorial diagrams associated to linkages of mobility \(1\) in this article (though they can be defined for more general linkages, as the authors point out), which encode information about the configuration space of a linkage. The authors end the article with a list of interesting open problems for \(6R\) linkages, which are linkages with \(6\) joints of a rotational type.

The reviewer strongly recommends this article to mathematically inclined engineers interested in linkages (which are for instance heavily used in robotics), as well as geometers interested in applying geometry to more concrete problems.

Rare objects are interesting. It is therefore an interesting problem to classify paradoxical linkages for a small number of links and joints. Can one find necessary and sufficient conditions for a given linkage to be paradoxical? It is with this aim in mind that the theory of bonds was created by G. Hegedüs and the three authors of this article in a joint article in 2015. The article under review is a nice geometric introduction to the subject of paradoxical mechanical linkages and the related theory of bonds.

The authors use dual quaternions to represent Euclidean motions in \(3\)-space. A dual quaternion is a formal object of the form \(q + \epsilon p\), where \(q\) and \(p\) are quaternions, and \(\epsilon\) is a non-zero symbol which commutes with \(i\), \(j\) and \(k\), and such that \(\epsilon^2 = 0\). The norm of a dual quaternion is defined by \[ N(q + \epsilon p) = (q + \epsilon p) (\bar{q} + \epsilon \bar{p}), \] which takes values in \(\mathbb{R} \oplus \epsilon \mathbb{R}\) (the ring of dual numbers). It turns out that the group of motions of Euclidean \(3\)-space is isomorphic to the multiplicative group \(G\), which is the quotient of the group of dual quaternions of non-zero real norm by the normal subgroup \(\mathbb{R}^*\) of non-zero real numbers.

Using the language of dual quaternions, the authors describe linkages in geometric terms. Roughly speaking, bond diagrams are combinatorial diagrams associated to linkages of mobility \(1\) in this article (though they can be defined for more general linkages, as the authors point out), which encode information about the configuration space of a linkage. The authors end the article with a list of interesting open problems for \(6R\) linkages, which are linkages with \(6\) joints of a rotational type.

The reviewer strongly recommends this article to mathematically inclined engineers interested in linkages (which are for instance heavily used in robotics), as well as geometers interested in applying geometry to more concrete problems.

Reviewer: Joseph Malkoun (Hazmieh)

### MSC:

53A17 | Differential geometric aspects in kinematics |