Analytical Boundary of the Workspace for General3-DOF Mechanisms

An analytical method is presented to obtain all surfaces en veloping the workspace of a general 3-DOF mechanism. The method is applicable to kinematic chains that can be mod eled using the Denavit-Hartenberg representation for serial kinematic chains or its modification for closed-loop kinematic chains. The method developed is based upon analytical crite ria for determining singular behavior of the mechanism. By manipulating the Jacobian of the underlying mechanism, first- order singularities are computed. These singularities are then substituted into the constraint equation to parameterize singu lar surfaces representing barriers to motion. Singular surfaces are those resultihg from a singular behavior of a joint gen eralized coordinate, allowing the manipulator to lose one or more degrees of mobility. These surfaces are then intersected to determine singular curves, which represent the manipulator losing at least two degrees of mobility. Difficulties in sepa rating singular behaviors at points along singular curves are encountered. Also, difficulties in computing tangents at the intersections of singular curves are addressed. These difficul ties are resolved using an analysis of a quadratic form of the intersection of singular surfaces. An example is presented to validate the theory. Although the methods used are numerical, the main result of this work is the ability to analytically define boundary surfaces of the workspace.

[1]  Chia-Hsiang Menq,et al.  The dexterous workspace of simple manipulators , 1988, IEEE J. Robotics Autom..

[2]  Clément Gosselin,et al.  Singularity analysis of closed-loop kinematic chains , 1990, IEEE Trans. Robotics Autom..

[3]  Itzhak Wilf,et al.  Quadric-surface intersection curves: shape and structure , 1993, Comput. Aided Des..

[4]  J. Denavit,et al.  A kinematic notation for lower pair mechanisms based on matrices , 1955 .

[5]  W. Rheinboldt.  Numerical analysis of parametrized nonlinear equations , 1986 .

[6]  R. Walker,et al.  The Dexterous Workspace of Simple Manipulators , 1988 .

[7]  Gregor Müllenheim.  On determining start points for a surface/surface intersection algorithm , 1991, Comput. Aided Geom. Des..

[8]  J. K. Wu,et al.  Dextrous Workspaces of Manipulators, Part 2: Computational Methods∗ , 1993 .

[9]  E. Allgower,et al.  Numerical Continuation Methods , 1990 .

[10]  F. A. Adkins,et al.  Numerical Algorithms for Mapping Boundaries of Manipulator Workspaces , 1996 .

[11]  Phillip J. McKerrow,et al.  Introduction to robotics , 1991 .

[12]  Edward J. Haug,et al.  Dextrous Workspaces of Manipulators. I. Analytical Criteria , 1992 .

[13]  D. C. H. Yang,et al.  On the Dexterity of Robotic Manipulators—Service Angle , 1985 .

[14]  Ying-Chien Tsai,et al.  Accessible Region and Synthesis of Robot Arms , 1981 .

[15]  K. C. Gupta,et al.  Design Considerations for Manipulator Workspace , 1982 .

[16]  S. Agrawal.  Workspace boundaries of in-parallel manipulator systems , 1991, Fifth International Conference on Advanced Robotics 'Robots in Unstructured Environments.

[17]  J. Kleinfinger.  Modelisation dynamique de robots a chaine : cinematique simple, arborescente, ou fermee, en vue de leur commande , 1986 .

[18]  H. Keller.  Lectures on Numerical Methods in Bifurcation Problems , 1988 .

[19]  Eugene L. Allgower,et al.  Numerical continuation methods - an introduction , 1990, Springer series in computational mathematics.

[20]  D. M. Emiris.  Workspace analysis of realistic elbow and dual-elbow robot , 1993 .

[21]  B. J. Davies,et al.  Surface Engineering Geometry for Computer Aided Design and Manufacture , 1987 .

[22]  K. Abdel-Malek.  DEXTERITY OF MANIPULATOR ARMS AT AN OPERATING POINT , 1995 .

[23]  Marco Ceccarelli,et al.  On the Workspace of General 4R Manipulators , 1995, Int. J. Robotics Res..

[24]  D. C. H. Yang,et al.  On the Workspace of Mechanical Manipulators , 1983 .

[25]  Edward J. Haug,et al.  Dextrous Workspaces of Manipulators, Part III: Calculation of Continuation Curves at Bifurcation Points , 1995 .

[26]  G. R. Pennock,et al.  The Workspace of a General Geometry Planar Three-Degree-of-Freedom Platform-Type Manipulator , 1993 .