Singularity Analysis of Limited-dof Parallel Manipulators using Grassmann-Cayley Algebra

This paper characterizes geometrically the singularities of limited DOF parallel manipulators. The geometric conditions associated with the dependency of six Pl\"ucker vector of lines (finite and infinite) constituting the rows of the inverse Jacobian matrix are formulated using Grassmann-Cayley algebra. Manipulators under consideration do not need to have a passive spherical joint somewhere in each leg. This study is illustrated with three example robot

SINGULAB - A Graphical user Interface for the Singularity Analysis of Parallel Robots based on Grassmann-Cayley Algebra

This paper presents SinguLab, a graphical user interface for the singularity analysis of parallel robots. The algorithm is based on Grassmann-Cayley algebra. The proposed tool is interactive and introduces the designer to the singularity analysis performed by this method, showing all the stages along the procedure and eventually showing the solution algebraically and graphically, allowing as well the singularity verification of different robot poses.Comment: Advances in Robot Kinematics, Batz sur Mer : France (2008

Singularity Analysis of Lower-Mobility Parallel Manipulators Using Grassmann-Cayley Algebra

This paper introduces a methodology to analyze geometrically the singularities of manipulators, of which legs apply both actuation forces and constraint moments to their moving platform. Lower-mobility parallel manipulators and parallel manipulators, of which some legs do not have any spherical joint, are such manipulators. The geometric conditions associated with the dependency of six Pl\"ucker vectors of finite lines or lines at infinity constituting the rows of the inverse Jacobian matrix are formulated using Grassmann-Cayley Algebra. Accordingly, the singularity conditions are obtained in vector form. This study is illustrated with the singularity analysis of four manipulators

Singularity Analysis of the 4-RUU Parallel Manipulator using Grassmann-Cayley Algebra

International audienceThis paper deals with the singularity analysis of 4-DOF parallel manipulators with identical limb structures performing Schönflies motions, namely, three independent translations and one rotation about an axis of fixed direction. The 6x6 Jacobian matrix of such manipulators contains two lines at infinity among its six Plücker lines. Some points at infinity are thus introduced to formulate the superbracket of Grassmann-Cayley algebra, which corresponds to the determinant of the Jacobian matrix. By exploring this superbracket, all the singularity conditions of such manipulators can be enumerated. The study is illustrated through the singularity analysis of the 4-\underline RUU parallel manipulator

Kinematics and workspace analysis of a 3ppps parallel robot with u-shaped base

This paper presents the kinematic analysis of the 3-PPPS parallel robot with an equilateral mobile platform and a U-shape base. The proposed design and appropriate selection of parameters allow to formulate simpler direct and inverse kinematics for the manipulator under study. The parallel singularities associated with the manipulator depend only on the orientation of the end-effector, and thus depend only on the orientation of the end effector. The quaternion parameters are used to represent the aspects, i.e. the singularity free regions of the workspace. A cylindrical algebraic decomposition is used to characterize the workspace and joint space with a low number of cells. The dis-criminant variety is obtained to describe the boundaries of each cell. With these simplifications, the 3-PPPS parallel robot with proposed design can be claimed as the simplest 6 DOF robot, which further makes it useful for the industrial applications

Singularities of serial robots: identification and distance computation using geometric algebra

The singularities of serial robotic manipulators are those configurations in which the robot loses the ability to move in at least one direction. Hence, their identification is fundamental to enhance the performance of current control and motion planning strategies. While classical approaches entail the computation of the determinant of either a 6×n or n×n matrix for an n-degrees-of-freedom serial robot, this work addresses a novel singularity identification method based on modelling the twists defined by the joint axes of the robot as vectors of the six-dimensional and three-dimensional geometric algebras. In particular, it consists of identifying which configurations cause the exterior product of these twists to vanish. In addition, since rotors represent rotations in geometric algebra, once these singularities have been identified, a distance function is defined in the configuration space C , such that its restriction to the set of singular configurations S allows us to compute the distance of any configuration to a given singularity. This distance function is used to enhance how the singularities are handled in three different scenarios, namely, motion planning, motion control and bilateral teleoperation.Peer ReviewedPostprint (published version

Using Singularities of Parallel Manipulators for Enhancing the Rigid-body Replacement Design Method of Compliant Mechanisms

International audienceThe rigid-body replacement method is often used when designing a compliant mechanism. The stiffness of the compliant mechanism, one of its main properties, is then highly dependent on the initial choice of a rigid-body architecture. In this paper, we propose to enhance the efficiency of the synthesis method by focusing on the architecture selection. This selection is done by considering the required mobilities and parallel manipulators in singularity to achieve them. Kinematic singularities of parallel structures are indeed advantageously used to propose compliant mechanisms with interesting stiffness properties. The approach is first illustrated by an example, the design of a one degree of freedom compliant architecture. Then the method is used to design a medical device where a compliant mechanism with three degrees of freedom is needed. The interest of the approach is outlined after application of the method

A general method for the numerical computation of manipulator singularity sets

The analysis of singularities is central to the development and control of a manipulator. However, existing methods for singularity set computation still concentrate on specific classes of manipulators. The absence of general methods able to perform such computation on a large class of manipulators is problematic because it hinders the analysis of unconventional manipulators and the development of new robot topologies. The purpose of this paper is to provide such a method for nonredundant mechanisms with algebraic lower pairs and designated input and output speeds. We formulate systems of equations that describe the whole singularity set and each one of the singularity types independently, and show how to compute the configurations in each type using a numerical technique based on linear relaxations. The method can be used to analyze manipulators with arbitrary geometry, and it isolates the singularities with the desired accuracy. We illustrate the formulation of the conditions and their numerical solution with examples, and use 3-D projections to visualize the complex partitions of the configuration space induced by the singularities.Preprin

Increase of Singularity-Free Zones in the Workspace of Parallel Manipulators Using Mechanisms of Variable Structure

International audienceThis paper is focused on the study of singularity of planar parallel manipulators taking into account the force transmission, i.e. study of singularity of planar manipulator by introducing the force transmission factor. Thus the singularity zones in the workspace of the manipulator are defined not only by kinematic criterions from the theoretical perfect model of the manipulator but also by the quality of force transmission. For this purpose, the pressure angle is used as an indicator of force transmission. The optimal control of the pressure angle for a given trajectory of the manipulator is realized by means of legs with variable structure. The suggested procedure to determination of the optimal structure of the planar parallel manipulator 3-RPR is illustrated by two numerical simulations

A general method for the numerical computation of manipulator singularity sets

The analysis of singularities is central to the development and control of a manipulator. However, existing methods for singularity set computation still concentrate on specific classes of manipulators. The absence of general methods able to perform such computation on a large class of manipulators is problematic because it hinders the analysis of unconventional manipulators and the development of new robot topologies. The purpose of this paper is to provide such a method for nonredundant mechanisms with algebraic lower pairs and designated input and output speeds. We formulate systems of equations that describe the whole singularity set and each one of the singularity types independently, and show how to compute the configurations in each type using a numerical technique based on linear relaxations. The method can be used to analyze manipulators with arbitrary geometry, and it isolates the singularities with the desired accuracy. We illustrate the formulation of the conditions and their numerical solution with examples, and use 3-D projections to visualize the complex partitions of the configuration space induced by the singularities.This work has been partially supported by the Spanish Ministry of Economy and Competitiveness under contract DPI2010-18449, and by a Juan de la Cierva contract supporting the fourth author.Peer Reviewe