Geometric properties of zero-torsion parallel kinematics machines

The advantages of tilt-and-torsion angles in analysis of zero-torsion parallel kinematics machines (PKM) have been reported by several literatures. However, geometric properties of tilt-and-torsion angles are not completely understood and fully utilized in synthesis of novel zero-torsion PKMs. In this paper, we study geometric properties of the so called zero-torsion motion types via differential geometry of Lie groups. We show that zero-torsion motion types admit simple representations under canonical coordinates of the first kind of the special Euclidean group SE(3). Using the proposed representation, we give a classification of zero-torsion PKMs. The synthesis condition for several well known zero-torsion PKMs are correctly identified. We will conduct type synthesis of zero-torsion PKMs in a separate paper.

[1]  Hieu Minh Trinh,et al.  Two-mode overconstrained three-DOFs rotational-translational linear-motor-based parallel-kinematics mechanism for machine tool applications , 2007, Robotica.

[2]  Richard M. Murray,et al.  A Mathematical Introduction to Robotic Manipulation , 1994 .

[3]  Kadri Buruncuk,et al.  On the kinematic of a 3-DOF Stewart platform , 1999, J. Field Robotics.

[4]  J. M. Hervé,et al.  Translational parallel manipulators with doubly planar limbs , 2006 .

[5]  J. M. Hervé.  The Lie group of rigid body displacements, a fundamental tool for mechanism design , 1999 .

[6]  J. M. Hervé,et al.  Asymmetrical three-DOFs rotational-translational parallel-kinematics mechanisms based on lie group theory , 2006 .

[7]  Sébastien Briot,et al.  Singularity analysis of zero-torsion parallel mechanisms , 2008, 2008 IEEE/RSJ International Conference on Intelligent Robots and Systems.

[8]  J. M. Hervé,et al.  Asymmetrical 3-dof spherical parallel mechanisms , 2005 .

[9]  A. W. Knapp.  Lie groups beyond an introduction , 1988 .

[10]  Mark Elling Rosheim,et al.  New high-angulation omni-directional sensor mount , 2002, SPIE Optics + Photonics.

[11]  Qinchuan Li,et al.  1T2R Parallel Mechanisms Without Parasitic Motion , 2010, IEEE Transactions on Robotics.

[12]  K. H. Hunt,et al.  Kinematic geometry of mechanisms , 1978 .

[13]  Clément Gosselin,et al.  Constraint singularities of parallel mechanisms , 2002, Proceedings 2002 IEEE International Conference on Robotics and Automation (Cat. No.02CH37292).

[14]  C. Gosselin,et al.  Advantages of the modified Euler angles in the design and control of PKMs , 2002 .

[15]  Z. Huang,et al.  Type Synthesis of Symmetrical Lower-Mobility Parallel Mechanisms Using the Constraint-Synthesis Method , 2003, Int. J. Robotics Res..

[16]  Zexiang Li,et al.  A Geometric Theory for Analysis and Synthesis of Sub-6 DoF Parallel Manipulators , 2007, IEEE Transactions on Robotics.

[17]  Clément Gosselin,et al.  Type synthesis of three-DOF up-equivalent parallel manipulators using a virtual-chain approach , 2006, ARK.

[18]  Hui Zhao,et al.  New kinematic structures for 2-, 3-, 4-, and 5-DOF parallel manipulator designs , 2002 .

[19]  Pavel Winternitz,et al.  Subgroups of the Euclidean group and symmetry breaking in nonrelativistic quantum mechanics , 1977 .

[20]  W. Boothby.  An introduction to differentiable manifolds and Riemannian geometry , 1975 .

[21]  Jeha Ryu,et al.  Orientation workspace analysis of 6-DOF parallel manipulators , 1999 .

[22]  R. Mrugala,et al.  Lie Groups Beyond an Introduction, 2nd Edition, Anthony W. Knapp, in: Birkhäuser Series: Progress in Mathematics, Vol. 140. Birkhäuser, Boston Basel Berlin (2002), xviii+812 pp., CHF 138.-/EUR 88 (hardcover)., ISBN: 0-8176-4259-5 , 2005 .

[23]  Janusz,et al.  Geometrical Methods in Robotics , 1996, Monographs in Computer Science.

[24]  Ilian A. Bonev.  Direct kinematics of zero-torsion parallel mechanisms , 2008, 2008 IEEE International Conference on Robotics and Automation.

[25]  Xin-Jun Liu,et al.  Some New Parallel Mechanisms Containing the Planar Four-Bar Parallelogram , 2003, Int. J. Robotics Res..

[26]  Xianwen Kong,et al.  Type Synthesis of Parallel Mechanisms , 2010, Springer Tracts in Advanced Robotics.