The design of adjustable spherical mechanisms using plane-to-sphere and sphere-to-plane projections

The spherical mechanism is a particular type of spatial mechanism. Due to the orientation of its joint axes and the curvature of its links, the workspaces of spherical mechanisms (whether line segments, closed loops or area regions) are spherical in curvature. This characteristic of spherical mechanisms makes them quite effective and practical in motion path and function generation applications requiring spherical rigid body kinematics. Although there are design methods available for spherical mechanisms, most of these methods do not consider the design of a single adjustable spherical mechanism. With an adjustable spherical mechanism, the user could for example, relocate the fixed or moving pivots of the spherical mechanism to achieve a greater range of rigid body locations and orientations. Having adjustability would make a single mechanism effective for multiple design applications. Numerous methods have been published for the design of adjustable planar mechanisms. Unfortunately, the number of design methods for adjustable spherical mechanisms, in comparison, is extremely modest. This research bridges the gap between the need for adjustable spherical mechanism design methods and the design methods available for adjustable planar mechanisms. This research presents a new method for synthesizing adjustable spherical four and five-bar motion, path and function generators using planar motion, path and function generation methods respectively. The benefits of this method are twofold. One benefit is that the user can design spherical mechanisms to approximate multiple phases of prescribed rigid-body path points. Another benefit is that the user can design spherical path generators using synthesis methods for planar path generators. By projecting the coordinates of a given spherical mechanism on a plane or the coordinates of a given planar mechanism on a sphere using the method introduced in this work, the user can design both planar and spherical mechanisms respectively. This research introduces sphere-to-plane and plane-to-sphere projection methods with optimization methods to minimize the structural error between the prescribed performance of the adjustable spherical mechanism and the performance achieved by the synthesized adjustable spherical mechanism. This research considers two-phase moving pivot adjustment problems with constant crank and follower lengths for the spherical mechanism. The spherical mechanisms considered in this research are four-bar motion, path and function generators as well as five-bar motion and path generators. Codified models of the projection and optimization methodologies introduced are also included