A visual, interactive approach to synthesis of spherical 6R closed chains for rational motions via constraint manifold modification

In this paper, we present an interactive, visual design approach for the dimensional synthesis of spherical 6R closed chains for a given rational motion using constraint manifold modification. This work is an extension of our previous work on the dimensional synthesis of planar 6R closed chains using the aforementioned approach. The theoretical underpinning of this work is based on representation of spherical displacements by quaternions and the kinematic mapping approach. In this way, motion of the coupler of a spherical 6R chain motion maps to a rational curve and the kinematic constraints of a spherical 6R closed chain map to hyper-dimensional algebraic surfaces in the image space of spherical displacements. Thus, the problem of determining dimensional parameters of the chain is reduced to finding the surface parameters that satisfy the geometry of the curve. This, in turn, resolves into satisfying the kinematic constraints of the chain. We provide designers an interactive, user friendly graphical tool that allows them to visually contain the image curve by simple geometric manipulation of the size, orientation, and the location of the constraint surfaces. This simple and straightforward design process lends designers an understanding of the mechanism design methodology.