The Synthesis of Spherical Motion Generators in the Presence of an Incomplete Set of Attitudes

Proposed in this paper is a general methodology applicable to the synthesis of spherical motion generators in the presence of an incomplete set of finitely separated attitudes. The spherical rigid-body guidance problem in the realm of four-bar linkage synthesis can be solved exactly for up to five prescribed attitudes of the coupler link, and hence, any number of attitudes smaller than five is considered incomplete in this paper. The attitudes completing the set are determined to produce a linkage whose performance is robust against variations in the unprescribed attitudes. Robustness is needed in this context to overcome the presence of uncertainty due to the selection of the unspecified attitudes, that many a time are specified implicitly by the designer upon choosing, for example, the location of the fixed joints of the dyads. A theoretical framework for model-based robust engineering design is thus, recalled, and a methodology for the robust synthesis of spherical four-bar linkages is laid down. An example is included here to concretize the concepts and illustrate the application of the proposed methodology.

A Robust Solution to the Spherical Rigid-Body Guidance Problem

In this paper we introduce a robust algorithm to solve the five-attitude spherical Burmester problem associated with exact linkage synthesis. The proposed algorithm solves for the unit vectors determining the four joint centers of the linkage while taking into account all redundant equations available, which enhances the robustness of the algorithm. In order to show the applicability of the proposed algorithm and to validate its robustness, two examples are included.

Synthesis of Spherical Linkages With Use of the Displacement Matrix

Equations of synthesis are derived which are based upon rigid body motion expressed as a displacement matrix. Applications are shown in spherical rigid body guidance, function generation, and path generation.