Genericity Conditions for Serial Manipulators

Applied Mathematics and Computation, 2010

Let X be a smooth quadric of dimension 2m in P 2mþ1 C and let Y; Z & X be subvarieties both of dimension m which intersect transversely. In this paper we give an algorithm for computing the intersection points of Y \ Z based on a homotopy method. The homotopy is constructed using a C Ã -action on X whose fixed points are isolated, which induces Bialynicki-Birula decompositions of X into locally closed invariant subsets. As an application we present a new solution to the inverse kinematics problem of a general six-revolute serial-link manipulator.

Mechanisms and Machine Science, 2013

We present a theoretical and algorithmic method for describing the singularity locus for the endpoint map of any serial manipulator with revolute joints. As a surface of revolution around the first joint, the singularity locus is determined by its intersection with a fixed plane through the first joint. The resulting plane curve is part of an algebraic curve called the singularity curve. Its degree can be computed from the specialized case of all pairs of consecutive joints coplanar, when the singularity curve is a union of circles, counted with multiplicity two. Knowledge of the degree and a simple iterative procedure for obtaining sample points on the singularity curve lead to the precise equation of the curve.

2010

Let X be a smooth quadric of dimension 2m in P 2mþ1 C and let Y; Z & X be subvarieties both of dimension m which intersect transversely. In this paper we give an algorithm for computing the intersection points of Y \ Z based on a homotopy method. The homotopy is constructed using a C Ã -action on X whose fixed points are isolated, which induces Bialynicki-Birula decompositions of X into locally closed invariant subsets. As an application we present a new solution to the inverse kinematics problem of a general six-revolute serial-link manipulator.

This article deals with the kinematics of serial manipulators. The serial manipulators are assumed to be rigid and are modeled using the well-known Denavit-Hartenberg parameters. Two well-known problems in serial manipulator kinematics, namely the direct and inverse problems, are discussed and several examples are presented. The important concept of the workspace of a serial manipulator and the approaches to determine the workspace are also discussed.

Applied Mathematics and Computation, 2014

The global analysis of the singularities of regional manipulators is addressed in this paper. The problem is approached from the point of view of computational algebraic geometry. The main novelty is to compute the syzygy module of the differential of the constraint map. Composing this with the differential of the forward kinematic map and studying the associated Fitting ideals allows for a complete stratification of the configuration space according to the corank of singularities. Moreover using this idea we can also compute the boundary of the image of the forward kinematic map. Obviously this gives us also a description of the image itself, i.e. the manipulator workspace. The approach is feasible in practice because generators of syzygy modules can be computed in a similar way as Gröbner bases of ideals.

Robotica, 2007

The significance of singularities in the design and control of robot manipulators is well known and there is an extensive literature on the determination and analysis of singularities for a wide variety of serial and parallel manipulators-indeed such an analysis is an essential part of manipulator design. Singularity theory provides methodologies for a deeper analysis with the aim of classifying singularities, providing local models and local and global invariants. This paper surveys applications of singularity-theoretic methods in robot kinematics and presents some new results.

Journal of Robotic Systems, 1988

This article is the first of three companion papers which introduce some novel concepts useful in the kinetmatic design, analysis, motion planning, control, and performance evaluation of serial manipulators. Here an efficient methodology is developed to classify the special configurations of an N DOF manipulator. This classification yields hypersurfaces in the joint space on each of which the manipulator loses a certain number of its degrees of freedom. The efficiency of the methodology is based on a theorem proved in Appendix B and the utilization of the minimal reference frames. These frames are obtained and tabulated for all possible types of manipulators. Furthermore the adaptability of a manipulator to a common control algorithm is quantified via an index. The numerical values of these indices are listed for all types of manipulators.

Proceedings 1992 IEEE International Conference on Robotics and Automation, 1992

A s serial robots parallel manipulator may be in a singular configuration. In these configurations the inverse jacobian matrix is singular and the end-effector may move although the articular velocities are equal to zero. The determination of the loci of these singular configurations is an important problem because in such configuration the articular forces may go to infinity and yield important mechanical damages. In a preceding paper we have proposed a geometrical approach for finding the singular configurations loci. We consider here a specific parallel manipulator and find what are the features of the infinitesimal motions associated to each of the singular configurations.

2017

The kinematics of parallel mechanisms are defined by means of a kinematic constraint map (KCM) that captures the constraints imposed on its links by the joints. The KCM incorporates both pose parameters describing the configuration of every link and the design parameters inherent in the mechanism architecture. This provides a coherent approach to determining C-space singularities and generalised Grashof conditions on the design parameters under which these can occur.Copyright © 2017 by ASME

2000

Engineers have for some time known that singularities play a significant role in the design and control of robot manipulators. Singularities of the kinematic mapping, which determines the position of the end-eector in terms of the manipu- lator's joint variables, may impede control algorithms, lead to large joint velocities, forces and torques and reduce instantaneous mobility. However they can also

Proceedings. 1998 IEEE International Conference on Robotics and Automation (Cat. No.98CH36146), 1998

We present a normal form approach to solving the singular inverse kinematic problem for non-redundant robot kinematics. A standing assumption is made that singular configurations of the kinematics have corank 1 and that the kinematics are equivalent to the quadratic normal form. The approach has been illustrated with a simulation case study of 2R kinematics. A comparison with the Singularity-Robust Inverse technique and the null-space based approach has demonstrated advantages of the normal for approach.

The International Journal of Robotics Research, 1994

Singularities, table Surfaces, and the Repeatable Behavior of Kinematically Redundant Manipulators which relates the Cartesian coordinates of the end effector, described by the m-vector x, to the n-dimensional vector () of joint values. For kinematically redundant manipulators n is, of course, larger than m. Because (1) is, in general, very nonlinear, one typically works with the Jacobian equation, which, for the positional component, can be found by differentiating (1) to obtain of pseudoinverse control of a planar three-link manipulator. Subsequently, Klein and Kee (1989) did a numerical study of the drift in joint space for the same manipulator performing the cyclic task of repeatedly drawing a square in the workspace, showing that the drift had a numerically stable limit in some situations. These findings motivated further research into the repeatability of kinematically redundant manipulators (Bay

ArXiv, 2019

We develop an algorithm that solves the inverse kinematics of general serial 2RP3R, 2R2P2R, 3RP2R and 6R manipulators based on the HuPf algorithm. We identify the workspaces of the 3-subchains of the manipulator with a quasi-projective variety in $\mathbb{P}^7$ via dual quaternions. This allows us to compute linear forms that describe linear spaces containing the workspaces of these 3-subchains. We present numerical examples that illustrate the algorithm and show the real solutions.

Advances in Robot Manipulators, 2010