In this paper, the analytic 3-RPR platform studied by Wenger and Chablat in 2009 will be analyzed regarding its cuspidality condition. Many investigations have paid great attention to the cuspidality phenomena that arise for some designs of the 3-RPR parallel manipulator. Nevertheless, most of them focus on obtaining the cusp points of direct kinematic singularity curves on a section of the joint space, meaning that one of the input variables must remain constant. The authors, in this paper, obtain the locus of cusp points of the manipulator under study in a complete analytic way and in 3-D joint space. This way, the three inputs can be actuated to perform a nonsingular transition that encircles one of the curves that belong to the locus of cusp points. It will be shown that the duplicity of one of the output variables of this manipulator causes the overlapping of two different cusp points in the joint space. In order to visualize this characteristic, initially, transitions in a section of the joint space will be analyzed using, additionally, the reduced configuration space. Then, transitions in the 3-D joint space will be performed, showing that, in order to ensure a feasible nonsingular transition, it is necessary to represent the direct kinematic singularity surface and assess the evolution of the three output variables along the trajectory.

It is known that there are parallel manipulators that can perform nonsingular transitions between different assembly modes. In particular, 3-degree-of-freedom (DOF) manipulators have received primary attention related to this phenomenon. In this paper, the conditions for the existence of special points in the projection of the direct-kinematic-problem-singularity locus onto the joint space for one constant input are obtained. From these conditions, the coordinates of all cusp points can be obtained analytically. Encircling one of these cusp points, it is possible to make a nonsingular transition between two assembly modes of a parallel manipulator. Utilizing these conditions, the range for the existence of cusp points of each input value can be also determined. An extension of the concept of cusp points to the complete joint space is also performed. The procedure is applied to an RPR-2-PRR parallel manipulator that can be solved analytically. Its dimensional variables are parametrized as a 1-D function, and all results are obtained in closed form, which is a benchmark example for other procedures.

This paper presents the application of the procedure known as the Geometrical-Iterative Method to the resolution of the direct position problem in spatial mechanisms, and in particular parallel platforms. This procedure is based on the successive application of geometric constraints used for modelling the mechanism. The application sequence of these is defined from a search algorithm directed at satisfying two basic convergence conditions. The method is a competitive alternative for the Newton-Raphson procedure in the kinematic control of parallel platforms.

The authors present a general method for the automated singularity analysis of any mechanism at a given configuration. The procedure uses a base of the motion space. This is obtained from a velocity equation characterized by a geometric matrix. This special form of Jacobian matrix has some advantages for automatic implementation. This approach provides the degree of freedom associated with the singularity, uncontrolled motion, and kinematic dependencies. It also facilitates the choice of actuators and redundant devices. The method has been implemented in a computer program for kinematic analysis.