We prove an a priori error estimate for the hp-version of the boundaryelement method with hypersingular operators on piecewise plane open orclosed surfaces. The underlying meshes are supposed to be quasi-uniform.The solutions of problems on polyhedral or piecewise plane open surfaces exhibittypical singularities which limit the convergence rate of the boundary element method.On closed surfaces, and for sufficiently smooth given data, the solution isH 1-regular whereas, on open surfaces, edge singularities arestrong enough to prevent the solution from being in H 1.In this paper we cover both cases and, in particular, prove an a priorierror estimate for the h-version with quasi-uniform meshes.For open surfaces we prove a convergence like O(h1/2p-1),h being the mesh size and p denoting the polynomial degree.This result had been conjectured previously.

The 3-PS structure features one rigid body (platform) connected to another rigid body (base) by means of three kinematic chains (limbs) of type PS (P and S stand for prismatic pair and spherical pair, respectively). All the 3-degree-of-freedom parallel manipulators with three connectivity-5 limbs, each one constituted of one passive (i.e. not actuated) prismatic pair, one passive spherical pair and one actuated kinematic pair of any type, become 3-PS structures when the actuated pairs are locked. Direct kinematics of this class of manipulators is tied to the properties of the 3-PS structure. In particular, the direct position analysis is tied to the assembly modes of the 3-PS structure; whereas the determination of the singularities of the direct instantaneous problem is tied to the determination of the singular geometries of the 3-PS structure, where instantaneous relative motions between platform and base are possible. The solution of these two problems is necessary both for designing the manipulators and for controlling them during motion. This paper deal with the determination of the singular geometries of the 3-PS structure.