Performing Nonsingular Transitions Between Assembly Modes in Analytic Parallel Manipulators by Enclosing Quadruple Solutions

Abstract

This paper analyzes the multiplicity of the solutions to forward kinematics of two classes of analytic robots: 2RPR-PR robots with a passive leg and 3-RPR robots with nonsimilar flat platform and base. Since their characteristic polynomials cannot have more than two valid roots, one may think that triple solutions, and hence nonsingular transitions between different assembly modes, are impossible for them. However, the authors show that the forward kinematic problems of these robots always admit quadruple solutions and obtain analytically the loci of points of the joint space where these solutions occur. Then, it is shown that performing trajectories in the joint space that enclose these points can produce nonsingular transitions, demonstrating that it is possible to design simple analytic parallel robots with two and three degrees-of-freedom (DOF) and the ability to execute these transitions.

Hidden Cusps

“…We begin with the example given in [9]. It is a 2RPR-PR planar manipulator with architecture described in Figure 1.…”

“…We compute the singularities in the workspace and in the joint space for an example with a 1 = 3, a 2 = 7, b 1 = 6, b 2 = 5, which is the same as the one considered in [9]. In this case we can check that the only real solutions of J = 0 and the equations (3) are (ϕ, y) = (0 mod π, 0).…”

“…The numbers in the zones delimited by these curves indicate the corresponding images by the inverse kinematic mapping in the joint space. The joint space is represented in Figure 2 (b) (the same figure appears in [9]). One can see the image of the singularity curve in blue.…”

“…a fold curve intersecting itself transversally) [6,7], and that the presence of cusps is not necessary for the existence of nonsingular assembly mode changes [8]. Michel Coste IRMAR, Université de Rennes 1 -CNRS, e-mail: michel.coste@univ-rennes1.fr Philippe Wenger IRCCyN, CNRS -École Centrale de Nantes, e-mail: philippe.wenger@irccyn.ec-nantes.fr Damien Chablat IRCCyN, CNRS -École Centrale de Nantes, e-mail: damien.chablat@irccyn.ec-nantes.fr Authors partially supported by ANR-14-CE34-0008-01 Kapamat A recent paper [9] exhibits an example of a 2-dof parallel manipulator with an isolated singularity of multiplicity 4 of the inverse kinematics mapping, such that circling around the image of this singularity in the joint space results in a nonsingular assembly mode change; moreover, after a second loop around the singularity, one is back in the same assembly mode. There is no cusp in the picture, but we intend to explain in the present paper that actually the cusps are hidden.…”

Análisis de estabilidad de singularidades aisladas en robots paralelos mediante desarrollos de Taylor de segundo orden

“…Considérese un robot paralelo de 2 grados de libertad (GDL), como es habitual en el estudio de transiciones no-singulares [7], para poder visualizar las singularidades del robot en el plano. Si el robot tuviera más de 2 GDL, simplemente se bloquean todos sus actuadores salvo dos, o bien se analiza un sub-mecanismo independiente de 2 GDL del robot [13,1].…”