From the Fundamental Theorem of Algebra to Kempe's Universality Theorem

摘要

This article provides a gentle introduction for a general mathematical audience to the factorization theory of motion polynomials and its application in mechanism science. This theory connects in a rather unexpected way a seemingly abstract mathematical topic, the non-unique factorization of certain polynomials over the ring of dual quaternions, with engineering applications. Four years after its introduction [9, 10], it is already clear how beneficial it has been to both fields [6, 12, 17, 18, 19, 20, 21, 22, 23]. In Section 1 we introduce the notion of motion polynomials and discuss their decomposition into products of linear motion polynomials. It can be used to synthesize linkages following a prescribed motion and is related to a variant of Kempe's Universality Theorem. We explain the relation to mechanism science in more detail in Section 2. In Sections 3 and 4 we present examples from linkage synthesis and discuss exceptional factorizations.