From the fundamental theorem of algebra to Kempe’s universality theorem. (English) Zbl 1397.70004

Summary: 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, it is already clear how beneficial it has been to both fields.
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.


70B15 Kinematics of mechanisms and robots
12D05 Polynomials in real and complex fields: factorization
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