Spatial Generalization of the Planar Path Generation Problem

Abstract

This paper deals with the dimensional synthesis of a 4C spatial mechanism. Several positions of a line in space are specified, and the goal is to design a 4C mechanism whose coupler can guide the line to pass through these positions. This problem is a spatial generalization of the planar 4R path generation problem. The maximum number of positions of lines that can be specified is found to be nine, which is identical to the maximum number of design points in the planar path generation problem. In order to avoid the complexity of obtaining numerical solutions, we use the screw triangle formulation for one CC dyad and matrix formulations for the other CC dyad to derive the design equations.We then use the Newton-Raphson method to solve the design equations, and a numerical example is provided. In this paper, the similarities between the planar and spatial path generation problems are established. A point in the planar path generation problem corresponds to a line in space, while revolute joints are replaced by cylindrical joints in the spatial path generation problem. Furthermore, the maximum numbers of allowable design positions of lines and points are both nine.