Cusps and nodes on plane sections of the singularity locus in the joint space of parallel manipulators play an important role in nonsingular assembly-mode changing motions. This paper analyses in detail such points, both in the joint space and in the workspace. It is shown that a cusp (resp. a node) defines a point of tangency (resp. a crossing point) in the workspace between the singular curves and the curves associated with the so-called characteristics surfaces. The study is conducted on a planar 3-RPR manipulator for illustrative purposes.
Laboratoire d’informatique de Paris（巴黎信息学实验室）；Institut de Recherche en Communications et Cybernétique de Nantes（南特通信与控制论研究所）
This paper investigates the cuspidal configurations of 3-RPR parallel manipulators that may appear on their singular surfaces in the joint space. Cusp points play an important role in the kinematic behavior of parallel manipulators since they make possible a non-singular change of assembly mode. In previous works, the cusp points were calculated in sections of the joint space by solving a 24th-degree polynomial without any proof that this polynomial was the only one that gives all solutions. The purpose of this study is to propose a rigorous methodology to determine the cusp points of 3-R\unde…
This paper investigates the existence conditions of cusp points in the design parameter space of the R\underline{P}R-2P\underline{R}R parallel manipulators. Cusp points make possible non-singular assembly-mode changing motion, which can possibly increase the size of the aspect, i.e. the maximum singularity free workspace. The method used is based on the notion of discriminant varieties and Cylindrical Algebraic Decomposition, and resorts to Gröbner bases for the solutions of systems of equations.
Institut de Robòtica i Informàtica Industrial, CSIC - UPC（工业机器人与信息学研究所，西班牙国家研究委员会-加泰罗尼亚理工大学）；INRIA Nancy-Grand Est（法国国家信息与自动化研究所南锡-大东部研究中心）；Institut de Recherche en Communications et Cybernétique de Nantes（南特通信与控制论研究所）；INRIA Paris-Rocquencourt（法国国家信息与自动化研究所巴黎-罗克库尔研究中心）；Université Pierre et Marie Curie Paris VI（巴黎第六大学皮埃尔和玛丽·居里大学）
This paper investigates the conditions in the design parameter space for the existence and distribution of the cusp locus for planar parallel manipulators. Cusp points make possible non-singular assembly-mode changing motion, which increases the maximum singularity-free workspace. An accurate algorithm for the determination is proposed amending some imprecisions done by previous existing algorithms. This is combined with methods of Cylindric Algebraic Decomposition, Gröbner bases and Discriminant Varieties in order to partition the parameter space into cells with constant number of cusp points…
Institut de Recherche Mathématique de Rennes（雷恩数学研究所）；Université de Rennes I（雷恩第一大学）；Institut de Recherche en Communications et Cybernétique de Nantes（南特通信与控制研究所）
We study in this paper a class of 3-RPR manipulators for which the direct kinematic problem (DKP) is split into a cubic problem followed by a quadratic one. These manipulators are geometrically characterized by the fact that the moving triangle is the image of the base triangle by an indirect isometry. We introduce a specific coordinate system adapted to this geometric feature and which is also well adapted to the splitting of the DKP. This allows us to obtain easily precise descriptions of the singularities and of the cusp edges. These latter second order singularities are important for nonsi…
Nantes Université（南特大学）；École Centrale Nantes（南特中央理工学院）；CNRS（法国国家科学研究中心）
Cuspidal robots can move from one inverse or direct kinematic solution to another without ever passing through a singularity. These robots have remained unknown because almost all industrial robots do not have this feature. However, in fact, industrial robots are the exceptions. Some robots appeared recently in the industrial market can be shown to be cuspidal but, surprisingly, almost nobody knows it and robot users meet difficulties in planning trajectories with these robots. This paper proposes a review on the fundamental and application aspects of cuspidal robots. It addresses the importan…
Laboratoire des Sciences du Numérique de Nantes（南特数字科学实验室）；University of Innsbruck（因斯布鲁克大学）
Cuspidal robots can travel from one inverse kinematic solution to another without meeting a singularity. The name cuspidal was coined based on the existence of a cusp point in the workspace of 3R serial robots. The existence of a cusp point was proved to be a necessary and sufficient condition for orthogonal robots to be cuspidal, but it was not possible to extend this condition to non-orthogonal robots. The goal of this paper is to prove that this condition stands for any generic 3R robot. This result would give the designer more flexibility. In the presented work, the geometrical interpretat…
IRMAR, Université de Rennes 1 – CNRS（法国国家科学研究中心、雷恩第一大学数学研究所（IRMAR））；IRCCyN, CNRS – École Centrale de Nantes（法国国家科学研究中心、南特中央理工学院南特通信与控制研究所（IRCCyN））
This paper investigates a situation pointed out in a recent paper, in which a non-singular change of assembly mode of a planar 2-RPR-PR parallel manipulator was realized by encircling a point of multiplicity 4. It is shown that this situation is, in fact, a non-generic one and gives rise to cusps under a small perturbation. Furthermore , we show that, for a large class of singularities of multiplicity 4, there are only two types of stable singularities occurring in a small perturbation: these two types are given by the complex square mapping and the quarto mapping. Incidentally , this paper co…
