Abstract
During more than one century, while the results and the concepts of the theory of mechanisms were developed, the central problem of mechanism theory, that is to calculate the degree of mobility of any mechanism, remained unsolved in general. Several formula have been successively suggested for evaluating this number but particular mechanisms that were outside the scope of the suggested formula were exhibited; those mechanisms are called paradoxical or exceptional. Unfortunately, although paradoxical, those mechanisms may have interesting mechanical properties and may not be simply rejected. At the end of this chapter, the insights provided by the Lie group language in these extremely difficult problems will first give the correct framework to formalize them, second it will bring answers to a large part of these issues, third it will explains why some cases are out of the scope of any formula. Retrospectively, it allows to understand why the usual mechanism theory can fail to overcome the mathematical difficulties of those problems.
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