Three-Position Synthesis of Spherically Constrained Planar 3R Chains

This paper presents a dimensional finite position synthesis procedure for a 8-bar linkage, that we call the spherically constrained planar 3R chain. The procedure aims at using the well-developed constraint-based synthesis equations of spherical RR chains in order to constrain a planar serial 3R guiding chain, synthesized before for a maximum number of three task poses. This maximum number of task positions results from the specific linkage topology, which requires to select specific axes of spherical RR chains. However, three-position synthesis allows it to apply a specific version of the dyad triangle equation of planar RR chains to the problem. A particular assumption forces that this version of the dyad triangle equation becomes nothing but the synthesis equation of a planar 3R chain, which is easily solved for the three prescribed task poses. An example is provided showing a synthesized spherically constrained planar 3R chain reaching three prescribed planar poses.

STAR-TRIANGLE EQUATIONS AND IDENTITIES IN HYPERGEOMETRIC SERIES

In this paper, we introduce the cyclic basic hypergeometric series p + 1Φp with q → ω where ωN = 1. This is a terminating series with N terms, whose summand has period N. We show how the Fourier transform of the weights of the integrable chiral Potts model are related to the 2Φ1, which is summable. We show that 3Φ2 satisfies certain transformation formulae. We then show that the Saalschützian 4Φ3 series is summable at argument z = ω. This then gives the simplest proof of the star-triangle relation in the chiral Potts model. Finally, we let N → ∞, where the star-triangle equation becomes a two-sided identity for the hypergeometric series.

The Synthesis of Planar RR and Spatial CC Chains and the Equation of a Triangle

This paper formulates the planar and spatial versions of an equation that determines one vertex of a triangle in terms of the other two vertices and their interior angles. The fact that a slight modification of Sandor and Erdman’s standard form equation for the design of RR chains yields this planar triangle equation is the basis for identifying the equivalent equation for a spatial triangle as the standard form equation for CC chains. The simultaneous solution of two of the planar equations yields an analytical expression of Burmester’s relationship between the fixed pivot of an RR chain and the relative position poles of its floating link. A similar solution of simultaneous spatial triangle equations yields Roth’s generalization of this insight, specifically, the fixed axis of a CC chain views two relative screw axes in one-half the dual crank rotation angle. These results provide the foundation for generalizing planar linkage synthesis techniques based on complex numbers to the synthesis of spatial linkages.

The Synthesis of Planar RR and Spatial CC Chains and the Equation of a Triangle

This paper formulates the planar and spatial versions of an equation that determines one vertex of a triangle in terms of the other two vertices and their interior angles. The fact that a slight modification of Sandor and Erdman’s standard form equation for the design of RR chains yields this planar triangle equation is the basis for identifying the equivalent equation for a spatial triangle as the standard form equation for CC chains. The simultaneous solution of two of the planar equations yields an analytical expression of Burmester’s relationship between the fixed pivot of an RR chain and the relative position poles of its floating link. A similar solution of simultaneous spatial triangle equations yields Roth’s generalization of this insight, specifically, the fixed axis of a CC chain views two relative screw axes in one-half the dual crank rotation angle. These results provide the foundation for generalizing planar linkage synthesis techniques based on complex numbers to the synthesis of spatial linkages.