We study 4-universal-prismatic-universal (UPU) parallel manipulators performing Schoenflies motion and show that they can have extra modes of operation with three degrees of freedom (3DOF), depending on the geometric parameters of the manipulators. We show that the transition between the different modes occurs along self-motion of the manipulator in the Schoenflies mode.

Abstract. In this paper we present new proofs using real spectra of the finiteness theorem on Nash trivial simultaneous resolution and the finiteness theorem on Blow-Nash triviality for isolated real algebraic singularities. That is, we prove that a family of Nash sets in a Nash manifold indexed by a semialgebraic set always admits a Nash trivial simultaneous resolution after a partition of the parameter space into finitely many semialgebraic pieces and in the case of isolated singularities it admits a finite Blow-Nash trivialization. We also complement the finiteness results with recursive bounds.