Synthesis of three-revolute spatial chains for body guidance

For a given mechanism type, the solution set of a body guidance synthesis problem comprises all mechanisms whose end-effector can reach a set of prescribed poses (position and orientation). For three-revolute spatial chains, five general poses will yield a synthesis problem having only finitely many solutions, while specifying fewer than five poses leads to higher-dimensional solution sets. We use numerical algebraic geometry to compute solution sets for two to five general poses, and in particular, we find, for the first time, that the five-pose synthesis problem generically has 456 solutions. We also show how our results agree with and extend results in the literature.