The synthesis of an RPS serial chain to reach a given set of task positions

This paper examines the synthesis of the five degree-of-freedom robot formed by links connected by a revolute, prismatic and spherical joint to form an RPS serial chain. The reachable workspace of this robot is a right circular hyperboloid that defines a constraint equation with 10 dimensional parameters. Evaluating this equation on 10 arbitrary spatial positions yields a polynomial system of total degree 262,144. Polynomial continuation yields as many as 1024 RPS chains for a general 10 position task. The number of task positions and constraint equations can be reduced by specifying some of the dimensional parameters. For the cases of six through eight task positions, analytical solutions are provided using both a resultant elimination strategy as well as a generalized eigenvalue elimination procedure. The generalized eigenvalue method has been implemented using Java and integrated into our design software. A numerical example of the solution for eight design positions is presented.