The paper presents the position analysis of a spatial structure composed of
two platforms mutually connected by one RRP and three SS serial kinematic
chains, where R, P, and S stand for revolute, prismatic, and spherical
kinematic pair respectively. A set of three compatibility equations is laid
down that, following algebraic elimination, results in a 28th-order univariate
algebraic equation, which in turn provides the addressed problem with 28
solutions in the complex domain.

It was shown recently that parallel manipulators with several inverse
kinematic solutions have the ability to avoid parallel singularities [Chablat
1998a] and self-collisions [Chablat 1998b] by choosing appropriate joint
configurations for the legs. In effect, depending on the joint configurations
of the legs, a given configuration of the end-effector may or may not be free
of singularity and collision.

This paper introduces a methodology to analyze geometrically the
singularities of manipulators, of which legs apply both actuation forces and
constraint moments to their moving platform. Lower-mobility parallel
manipulators and parallel manipulators, of which some legs do not have any
spherical joint, are such manipulators. The geometric conditions associated
with the dependency of six Pl\"ucker vectors of finite lines or lines at
infinity constituting the rows of the inverse Jacobian matrix are formulated
using Grassmann-Cayley Algebra.

The paper proposes a new calibration method for parallel manipulators that
allows efficient identification of the joint offsets using observations of the
manipulator leg parallelism with respect to the base surface. The method
employs a simple and low-cost measuring system, which evaluates deviation of
the leg location during motions that are assumed to preserve the leg
parallelism for the nominal values of the manipulator parameters. Using the
measured deviations, the developed algorithm estimates the joint offsets that
are treated as the most essential parameters to be identified.