On the Spin Surface of RSSR Mechanisms With Parallel Rotary Axes

Due to Cayley’s theorem the line $s∊Σ$ (⁠$=moving$ system) spanned by the centers of the spherical joints of a revolute-spherical-spherical-revolute linkage generates a surface of degree 8. In the special case of parallel rotary axes of the R-joints the corresponding ruled surface is only of degree 6. Now the point locus of any point $X∊Σ\{s}$ is a surface of order 16 (general case) or of order 12 (special case). Hunt (1978, Kinematic Geometry of Mechanisms, Clarendon, Oxford) suggested that the circularity of this so called spin-surface for the general case is 8 and this was later proved. We demonstrate that the circularity of the spin-surface for the special case is 4 instead of 6 as given in the literature (1994, “The (True) Stewart Platform Has 12 Configurations,” Proceedings of the IEEE International Conference on Robotics and Automation, pp. 2160–2165). As a consequence generalized triangular symmetric simplified manipulators (the three rotary axes need not be coplanar) with two parallel rotary joints can have up to 16 solutions instead of 12 (2006, Parallel Robots, 2nd ed., Springer, New York). We show that this upper bound cannot be improved by constructing an example for which the maximal number of assembly modes is reached. Moreover, we list all parallel manipulators of this type where more than $4×2=8$ points are located on the imaginary spherical circle.