Singular curves and cusp points in the joint space of 3-RPR parallel manipulators

Abstract

This paper investigates the singular curves in twodimensional slices of the joint space of a family of planar parallel manipulators. It focuses on special points, referred to as cusp points, which may appear on these curves. Cusp points play an important role in the kinematic behavior of parallel manipulators since they make possible a nonsingular change of assembly mode. The purpose of this study is twofold. First, it reviews an important previous work, which, to the authors' knowledge, has never been exploited yet. Second, it determines the cusp points in any twodimensional slice of the joint space. First results show that the number of cusp points may vary from zero to eight. This work finds applications in both design and trajectory planning.

Perturbation of Symmetric 3-RPR Manipulators and Asymptotic Singularities

“…(Figure 9, second row). In both cases an inner region with no solution of the DKP appears; all cusps disappear in the annulus case, thus ruling out the possibility of non-singular assembly mode change for large values of r 1 (see [12]). The slices in Figure 9 are taken for r 1 = 3, 10, 30.…”

“…Planar parallel manipulators have been extensively studied in the past, see [1,4,5,6,7,8,9,11,12] for example. Symmetric 3-RPR manipulators (those for which the platform triangle is congruent to the base triangle by an orientation-reversing isometry) have special kinematic properties (see [3]).…”

Singularities of non-redundant manipulators: A short account and a method for their computation in the planar case

“…Similarly, S can be projected to the input space V, as done e.g. in [20,21,31], where the forward singularities delimit the motion range that should be reachable by the actuators. Both the V and U spaces get partitioned into several regions after such projections, and it is possible to decide which regions correspond to feasible configurations of the manipulator by selecting a point in each region, and solving Eqs.…”

“…Previous methods for studying the set are effective, but restrict their attention to narrowly-defined classes of manipulators [14][15][16][17][18][19][20][21], or to particular singularity types [22], and a general approach able to isolate all possible singularities on a large class of manipulators is still lacking. Aiming to cover such gap, this paper pro-poses a numerical method for computing the whole singularity set of a planar non-redundant manipulator of general architecture, i.e., encompassing either open or closed kinematic chains interconnected in any possible way, by means of revolute or prismatic pairs.…”

Optimised assembly mode reconfiguration of the 5-DOF Gantry-Tau using mixed-integer programming

“…[5] the authors presented an analysis based on cusp points to help determine if singularity-free reconfiguration is possible. Cusp points are special coordinates on singularity curves where triple direct kinematic solutions meet in the joint-space for parallel manipulators.…”