Abstract:
This research deals with a general kinematic synthesis problem of how to generate the motion of a spatial platform subject to one or more spherical con- straints. Each spherical constraint can be modeled by a SS dyad, which is a spatial two-link kinematic chain connected by two spherical (S) joints. It defines a five-degree-of-freedom motion of the end link such that a point stays on the surface of a sphere. A spherical constraint degenerates into a plane constraint when the radius of the sphere becomes infinite. In this way, a plane constraint is considered as a special case of a spherical constraint and a one-degree-of-freedom motion of a moving platform defined by a combination of five spherical and/or plane constraints can be modeled by a 5-SS platform linkage. In addition to a SS dyad, a spherical constraint can also be realized alterna- tively with a TS dyad, where T denotes a universal joint, or a three-link Revolute-Revolute-Spherical chain, or RRS chain, where R denotes a revolute joint. The plane constraint can be realized by RRS, RPS and PRS kinematic chains, where P denotes a prismatic joint. In this way, a SS dyad based formulation could lead to a unified synthesis of platform linkages formed by all these kinematic chains. Traditionally, the problem of mechanism synthesis separates into type syn- thesis and dimensional synthesis. Type synthesis deals with the selection of a mechanism type based on given task; dimensional synthesis deals with the deter- mination of link dimensions. While dimensional synthesis is highly amendable to mathematical treatment, type synthesis remains elusive and highly depen- dent on designer's prior experience, despite of recent theoretical advancement in graph theory or topology. Furthermore, recent research indicates that there are mechanism synthesis problems that one can not separate type and dimensional synthesis, i.e., slight variation of input data might lead to different mechanism types. This calls for a data driven simultaneous type and dimensional synthesis approach. The central idea to this approach is to synthesize geometric constraints for a given task. This includes the location of the moving point as well as the location and radius associated with the spherical constraint. Once the geometric constraints are determined from the given task, the next step is to figure out kinematic chains such as TS and RRS that can be used to generate the geometric constraints. In this dissertation, we first consider a simpler version of the problem, which is to design a planar mechanism such that its moving link is constrained by one or more circular constraints. Instead of separating type and dimensional synthesis by selecting joint types, either revolute (R) or prismatic (P) joint before dimensional synthesis, we present a unified design equation such that the selection of joint types and determining link lengths can be carried out simultaneously. In the process, we developed a linear representation of the design equations that can be extended to the synthesis of spherical four-bar linkages. The spherical constraints in 5-SS linkage synthesis problem are extensions of circular constraints from two to three dimensions. This leads to linear form of design equations for SS dyads that are similar to that of planar and spherical RR dyads but with additional variables and bilinear constraints. In contrast to the existing approach that leads to a system of polynomial design equations, which can be solved using homotopy method, this linear formulation leads to a gener- alized eigenvalue problem that can be more readily solved than the homotopy method. It turns out the SS dyad formulation can be applied to the synthesis of spatial RR dyads whose moving and fixed axes in general neither intersect nor are in parallel. Planar and spherical dyads can be viewed as special cases of a spatial RR dyad when the two axes either intersect or are in parallel. In all RR dyads, a point on the moving axis traces out a circle. Furthermore, as a circle can be obtained as intersection of two spheres or a plane and a sphere, the circular constraints of spatial RR dyads can be obtained as intersection of spherical and plane constraints. Thus, planar, spherical and Bennett 4R linkages, which are composed of different types of RR dyads, may be treated as special cases of spatial 5-SS linkages. This leads to a new and unified methodology for synthesizing planar, spherical, and spatial RR dyads without a priori knowledge whether the given input positions are planar, spherical or spatial in nature.