Synthesis of multi-mode single-loop Bennett-based mechanisms using factorization of motion polynomials

This paper systematically deals with the synthesis of multi-mode single-loop 6R, 7R and 8R Bennett-based mechanisms from an algebraic viewpoint. Based on the factorization of motion polynomials over dual quaternions, an algebraic method is proposed to synthesize multi-mode single-loop 6R, 7R and 8R Bennett-based mechanisms. Using this method, several multi-mode single-loop Bennett-based mechanisms with different number of joints are constructed depending on explicit poses of joint axes. Then motion mode analysis of the 7R mechanism is carried out by formulating and solving a set of kinematic loop equations using tools from algebraic geometry. The analysis demonstrates that this multi-mode 7R mechanism has four motion modes, including a two degree-of-freedom (DOF) double Bennett mode, a 2-DOF hybrid mode, a 1-DOF rotation mode and a 1-DOF spatial 7R mode. Meanwhile, multimode characteristics of the single-loop 6R and 8R mechanisms also are concisely demonstrated in light of reconfiguration analysis. This work provides an algebraic representation framework for further investigation on multi-mode mechanisms that composed of two or more single-loop overconstraint mechanisms.