This paper deals with the characterization of the operation modes of the 4-RUU parallel manipulator with an algebraic approach, namely the Study's kinematic mapping of the Euclidean group SE(3). As the 4-RUU parallel manipulator is an over-constrained manipulator, it can be decomposed into two 2-RUU parallel manipulators. The manipulators are described by a set of constraint equations and the primary decomposition is computed. By combining the results of primary decomposition from two 2-RUU parallel manipulators, it reveals that the 4-RUU parallel manipulator has two Schönflies modes (4-dof) and one lower dimension operation mode (2-dof). The singularity conditions are obtained by deriving the determinant of the Jacobian matrix of the constraint equations with respect to the Study parameters in each operation mode. It is shown that there exist singular configurations in which the manipulators may switch from one operation mode to another operation mode. All the singular configurations are mapped onto the joint space, i.e., the actuated joint angles, and are geometrically interpreted. Eventually, the 4-RUU parallel manipulator may switch from the 1st Schönflies mode to the 2nd Schönflies mode, or vice versa, via the 2-dof third mode that contains self-motions.

The 3-RPS cube parallel manipulator, a three-degree-of-freedom parallel manipulator initially proposed by Huang and Fang (1995, “Motion Characteristics and Rotational Axis Analysis of Three DOF Parallel Robot Mechanisms,” IEEE International Conference on Systems, Man and Cybernetics. Intelligent Systems for the 21st Century, Vancouver, BC, Canada, Oct. 22–25, pp. 67–71) is analyzed in this paper with an algebraic approach, namely, Study kinematic mapping of the Euclidean group SE(3) and is described by a set of eight constraint equations. A primary decomposition is computed over the set of eight constraint equations and reveals that the manipulator has only one operation mode. Inside this operation mode, it turns out that the direct kinematics of the manipulator with arbitrary values of design parameters and joint variables, has 16 solutions in the complex space. A geometric interpretation of the real solutions is given. The singularity conditions are obtained by deriving the determinant of the Jacobian matrix of the eight constraint equations. All the singular poses are mapped onto the joint space and are geometrically interpreted. By parametrizing the set of constraint equations under the singularity conditions, it is shown that the manipulator is in actuation singularity. The uncontrolled motion gained by the moving platform is also provided. The motion of the moving platform is essentially determined by the fact that three vertices in the moving platform move in three mutually orthogonal planes. The workspace of each point of the moving platform (with exception of the three vertices) is bounded by a Steiner surface. This type of motion has been studied by Darboux in 1897. Moreover, the 3DOF motion of the 3-RPS cube parallel manipulator contains a special one-degree-of-freedom motion, called the vertical Darboux motion (VDM). In this motion, the moving platform can rotate and translate about and along the same axis simultaneously. The surface generated by a line in the moving platform turns out to be a right-conoid surface.

This paper discusses the notion of operation mode in parallel manipulators with less than six dof. This notion has been reported recently in several papers but the physical meaning of an operation mode is not always clear. Indeed, even if in some cases an operation mode can be associated with an understandable motion type e.g., three pure translations or a spherical motion, in some other cases, such an association is not straightforward. Therefore, the axodes are used in this paper to characterize any operation mode of lower-mobility parallel manipulators. A 3-RPS manipulator is used as an illustrative example. This manipulator is special because one can parameterize its operation modes.

Abstract. This paper deals with the comaprison of 3-RPS and 3-SPR parallel manipulators based on their operation modes and singularity-free workspace. The operation modes of the 3-SPR manipulator are identified by using Algebraic Geometry. Those operation modes amount to the operation modes of the 3-RPS parallel manipulator, which has already been studied in the literature [1]. Then, the parallel singularities of the 3-SPR and 3-RPS parallel manipulators are analyzed in order to trace their singularity loci in the orientation workspace. An index, named Maximum Inscribed Circle Radius (MICR), is defined to compare the two manipulators under study. It is based on their maximum singularity-free workspace and the ratio between their circum-radius of the movingplatform to that of the base.

The subject of this paper is about the study of the operation modes and the singularity conditions of the 3-PRS parallel manipulator with different arrangements of prismatic joints. The three prismatic joints of the PRS legs are attached to the base with an angle α between the horizontal plane of the base and their directions. By using an algebraic approach, namely the Study kinematic mapping of the Euclidean group SE(3), the mechanisms are described by a set of eight constraint equations. A primary decomposition is computed over a set of eight constraint equations and reveals that the 3-PRS manipulators with different arrangements of prismatic joints have identical operation modes, namely x0 = 0 and x3 = 0. Both operation modes are analysed. The singularity conditions are obtained by deriving the determinant of the Jacobian matrix of the eight constraint equations. All the singular configurations are mapped onto the joint space and are geometrically interpreted. The singularity loci of the 3-PRS parallel manipulators are also traced in its orientation workspace for different values of angle α.