Variable degree-of-freedom spatial mechanisms composed of four circular translation joints

This paper deals with the construction and reconfiguration analysis of a spatial mechanism composed of four circular translation (G) joints. Two links connected by a G joint, which can be in different forms such as a planar parallelogram, translate along a circular trajectory with respect to each other. A spatial 4G mechanism, which is composed of four G joints, usually has one degree-of-freedom (DOF). First, a 2-DOF spatial 4G mechanism is constructed. Then, a novel variable-DOF spatial 4G mechanism is constructed starting from the 2-DOF 4G mechanism using the approach based on screw theory. Finally, the reconfiguration analysis is carried out in the configuration space using dual quaternions and tools from algebraic geometry. The analysis shows that the variable-DOF spatial 4G mechanism has one 2-DOF motion mode and one to two 1-DOF motion modes and reveals how the 4G mechanism can switch among these motion modes. By removing one link from two adjacent G joints each and two links from each of the remaining two G joints, we can obtain a queer-rectangle and a queer-parallelogram, which are the generalization of the queer-square or derivative queer-square in the literature. The approach in this paper can be extended to the analysis of other types of coupled mechanisms using cables and gears and multi-mode spatial mechanisms involving G joints.