0
Research Papers

Compositional Submanifolds of Prismatic–Universal–Prismatic and Skewed Prismatic–Revolute– Prismatic Kinematic Chains and Their Derived Parallel Mechanisms

[+] Author and Article Information
Xinsheng Zhang

Key Lab for Mechanism Theory and Equipment
Design, International Centre for Advanced
Mechanisms and Robotics,
Tianjin University,
Centre for Robotics Research,
King's College London,
London WC2R 2 LS, UK
e-mail: xinsheng.zhang@kcl.ac.uk

Pablo López-Custodio

Centre for Robotics Research,
King's College London,
London WC2R 2 LS, UK
e-mail: pablo.lopez-custodio@kcl.ac.uk

Jian S. Dai

Chair of Mechanisms and Robotics International
Centre for Advanced Mechanisms and Robotics,
Tianjin University,
Centre for Robotics Research,
King's College London,
London WC2R 2 LS, UK
e-mail: jian.dai@kcl.ac.uk

Manuscript received October 12, 2016; final manuscript received August 23, 2017; published online March 1, 2018. Assoc. Editor: Venkat Krovi.

J. Mechanisms Robotics 10(3), 031001 (Mar 01, 2018) (9 pages) Paper No: JMR-16-1303; doi: 10.1115/1.4038218 History: Received October 12, 2016; Revised August 23, 2017

The kinematic chains that generate the planar motion group in which the prismatic-joint direction is always perpendicular to the revolute-joint axis have shown their effectiveness in type synthesis and mechanism analysis in parallel mechanisms. This paper extends the standard prismatic–revolute–prismatic (PRP) kinematic chain generating the planar motion group to a relatively generic case, in which one of the prismatic joint-directions is not necessarily perpendicular to the revolute-joint axis, leading to the discovery of a pseudo-helical motion with a variable pitch in a kinematic chain. The displacement of such a PRP chain generates a submanifold of the Schoenflies motion subgroup. This paper investigates for the first time this type of motion that is the variable-pitched pseudo-planar motion described by the above submanifold. Following the extraction of a helical motion from this skewed PRP kinematic chain, this paper investigates the bifurcated motion in a 3-prismatic–universal–prismatic (PUP) parallel mechanism by changing the active geometrical constraint in its configuration space. The method used in this contribution simplifies the analysis of such a parallel mechanism without resorting to an in-depth geometrical analysis and screw theory. Further, a parallel platform which can generate this skewed PRP type of motion is presented. An experimental test setup is based on a three-dimensional (3D) printed prototype of the 3-PUP parallel mechanism to detect the variable-pitched translation of the helical motion.

FIGURES IN THIS ARTICLE
<>
Copyright © 2018 by ASME
Your Session has timed out. Please sign back in to continue.

References

Wohlhart, K. , 1996, “ Kinematotropic Linkages,” Recent Advances in Robot Kinematics, Springer, Dordrecht, The Netherlands, pp. 359–368. [CrossRef]
Wei, G. , Chen, Y. , and Dai, J. S. , 2014, “ Synthesis, Mobility, and Multifurcation of Deployable Polyhedral Mechanisms With Radially Reciprocating Motion,” ASME J. Mech. Des., 136(9), p. 091003. [CrossRef]
Qin, Y. , Dai, J. S. , and Gogu, G. , 2014, “ Multi-Furcation in a Derivative Queer-Square Mechanism,” Mech. Mach. Theory, 81, pp. 36–53. [CrossRef]
Kong, X. , 2014, “ Reconfiguration Analysis of a 3-DOF Parallel Mechanism Using Euler Parameter Quaternions and Algebraic Geometry Method,” Mech. Mach. Theory, 74, pp. 188–201. [CrossRef]
Walter, D. R. , Husty, M. L. , and Pfurner, M. , 2009, “ A Complete Kinematic Analysis of the SNU 3-UPU Parallel Robot,” Contemp. Math., 496, pp. 331–346. [CrossRef]
Nurahmi, L. , Caro, S. , Wenger, P. , Schadlbauer, J. , and Husty, M. , 2016, “ Reconfiguration Analysis of a 4-RUU Parallel Manipulator,” Mech. Mach. Theory, 96(Pt. 2), pp. 269–289. [CrossRef]
López-Custodio, P. C. , Rico, J. M. , Cervantes-Sánchez, J. J. , and Pérez-Soto, G. , 2016, “ Reconfigurable Mechanisms From the Intersection of Surfaces,” ASME J. Mech. Rob., 8(2), p. 021029. [CrossRef]
Gan, D. , Dai, J. S. , Dias, J. , and Seneviratne, L. D. , 2013, “ Unified Kinematics and Singularity Analysis of a Metamorphic Parallel Mechanism With Bifurcated Motion,” ASME J. Mech. Rob., 5(3), p. 031004. [CrossRef]
Carbonari, L. , Callegari, M. , Palmieri, G. , and Palpacelli, M. C. , 2014, “ A New Class of Reconfigurable Parallel Kinematic Machines,” Mech. Mach. Theory, 79, pp. 173–183. [CrossRef]
Rodriguez-Leal, E. , Dai, J. S. , and Pennock, G. R. , 2009, “ Inverse Kinematics and Motion Simulation of a 2-DOF Parallel Manipulator With 3-PUP Legs,” 5th International Workshop on Computational Kinematics (CK), Duisburg, Germany, May 6–8, pp. 85–92.
Hervé, J. M. , 2004, “ Parallel Mechanisms With Pseudo-Planar Motion Generators,” On Advances in Robot Kinematics, J. Lenarčič and C. Galletti, eds., Springer, Dordrecht, The Netherlands.
Tu, L. , 2008, An Introduction to Manifolds, Vol. 200, Springer, New York.
Mourad, K. , and Hervé, J. M. , 2002, A Family of Novel Orientational 3-DOF Parallel Robots, Springer, Vienna, Austria.
Meng, J. , Liu, G. , and Li, Z. , 2007, “ A Geometric Theory for Analysis and Synthesis of Sub-6 DoF Parallel Manipulators,” IEEE Trans. Rob., 23(4), pp. 625–649. [CrossRef]
Lee, C. C. , and Hervé, J. M. , 2006, “ Translational Parallel Manipulators With Doubly Planar Limbs,” Mech. Mach. Theory, 41(4), pp. 433–455. [CrossRef]
Pérez-Soto, G. , and Tadeo, A. , 2006, “Sintesis de número de cadenas cinemáticas, un nuevo enfoque y nuevas herramientas matemáticas. (in Spanish),” M.Sc. thesis, Universidad de Guanajuato, Salamanca Gto., México.
Lee, C. C. , and Hervé, J. M. , 2007, “ Cartesian Parallel Manipulators With Pseudoplanar Limbs,” ASME J. Mech. Des., 129(12), pp. 1256–1264. [CrossRef]
Lee, C. C. , and Hervé, J. M. , 2009, “ Type Synthesis of Primitive Schoenflies-Motion Generators,” Mech. Mach. Theory, 44(10), pp. 1980–1997. [CrossRef]
Hervé, J. M. , 1978, “ Analyze Structurelle Des Mécanismes Par Groupe Des Déplacements (in French),” Mech. Mach. Theory, 13(4), pp. 437–450. [CrossRef]
Dai, J. S. , 2012, “ Finite Displacement Screw Operators With Embedded Chasles' Motion,” ASME J. Mech. Rob., 4(4), p. 041002. [CrossRef]
Dai, J. S. , 2015, “ Euler-Rodrigues Formula Variations, Quaternion Conjugation and Intrinsic Connections,” Mech. Mach. Theory, 92 , pp. 144–152. [CrossRef]
Dai, J. S. , 2006, “ An Historical Review of the Theoretical Development of Rigid Body Displacements From Rodrigues Parameters to the Finite Twist,” Mech. Mach. Theory, 41(1), pp. 41–52. [CrossRef]
Li, Q. C. , Huang, Z. , and Hervé, J. M. , 2004, “ Type Synthesis of 3R2T 5-DOF Parallel Mechanisms Using the Lie Group of Displacements,” IEEE Trans. Rob. Autom., 20(2), pp. 173–180. [CrossRef]
Kong, X. , and Gosselin, C. M. , 2007, Type Synthesis of Parallel Mechanisms, Springer, Berlin.
Gogu, G. , 2007, Structural Synthesis of Parallel Robots—Part I: Methodology, Springer-Verlag, Berlin.
Rico, J. M. , Cervantes-Sánchez, J. J. , Tadeo-Chávez, A. , Pérez-Soto, G. I. , and Rocha-Chavarría, J. , 2008, “ New Considerations on the Theory of Type Synthesis of Fully Parallel Platforms,” ASME J. Mech. Des., 130(11), p. 112302. [CrossRef]
Carricato, M. , and Rico, J. M. , 2010, “ Persistent Screw Systems,” Advances in Robot Kinematics: Motion in Man and Machine, J. Lenarčič and M. M. Stanišić , eds., Springer, Dordrecht, The Netherlands, pp. 185–194.
Tadeo-Chávez, A. , Rico, J. M. , Cervantes-Sánchez, J. J. , Pérez-Soto, G. , and Müller, A. , 2011, “Screw Systems Generated by Subalgebras: A Further Analysis,” ASME Paper No. DETC2011-48304.
Wu, Y. , Löwe, H. , Carricato, M. , and Li, Z. , 2016, “ Inversion Symmetry of the Euclidean Group: Theory and Application to Robot Kinematics,” IEEE Trans. Rob., 32(2), pp. 312–326. [CrossRef]
Zhang, K. , Dai, J. S. , and Fang, Y. , 2012, “ Constraint Analysis and Bifurcated Motion of the 3PUP Parallel Mechanism,” Mech. Mach. Theory, 49, pp. 256–269. [CrossRef]
Gan, D. , and Dai, J. S. , 2013, “ Geometry Constraint and Branch Motion Evolution of 3-PUP Parallel Mechanisms With Bifurcated Motion,” Mech. Mach. Theory, 61, pp. 168–183. [CrossRef]
Dai, J. S. , Huang, Z. , and Lipkin, H. , 2006, “ Mobility of Overconstrained Parallel Mechanisms,” ASME J. Mech. Des., 128(1), pp. 220–229. [CrossRef]
Dai, J. S. , 2014, Geometrical Foundations and Screw Algebra for Mechanisms and Robotics, Higher Education Press, Beijing, China.
Dai, J. S. , and Jones, J. R. , 2002, “ Null–Space Construction Using Cofactors From a Screw–Algebra Context,” Proc. R. Soc. London. Ser. A: Math., Phys. Eng. Sci., 458(2024), pp. 1845–1866. [CrossRef]
Dai, J. S. , and Rees Jones, J. , 2001, “ Interrelationship Between Screw Systems and Corresponding Reciprocal Systems and Applications,” Mech. Machine Theory, 36(5), pp. 633–651. [CrossRef]
Rico-Martínez, J. M. , and Ravani, B. , 2003, “ On Mobility Analysis of Linkages Using Group Theory,” ASME J. Mech. Des., 125(1), pp. 70–80. [CrossRef]
Fanghella, P. , and Galletti, C. , 1995, “ Metric Relations and Displacement Groups in Mechanism and Robot Kinematics,” ASME J. Mech. Des., 117(3), pp. 470–478. [CrossRef]
Lee, C.-C. , and Hervé, J. M. , 2010, “ Generators of the Product of Two Schoenflies Motion Groups,” Eur. J. Mech. – A/Solids, 29(1), pp. 97–108. [CrossRef]

Figures

Grahic Jump Location
Fig. 1

PRP chain and the related submanifolds

Grahic Jump Location
Fig. 2

Geometrical structures of the PRP chain with a tilting angle and the PRH chain: (a) PRP chain with a tilting angle α and (b) PRH kinematic chain

Grahic Jump Location
Fig. 3

The 3-DOF PRP-Schoenflies parallel mechanism

Grahic Jump Location
Fig. 4

The PRP chain extracted from pivoting limb that is limb 1 and its equivalent revolute–prismatic–revolute chain

Grahic Jump Location
Fig. 5

The Lie subgroups in limb 1

Grahic Jump Location
Fig. 6

The PRP chain with a tilting angle extracted from limb 2

Grahic Jump Location
Fig. 7

Feasible displacements in motion branch A: (a) home configuration (see Sec. 8); (b), (c) from the test setup in Sec. 7, the platform are performing pure rotation, without any translation along x- and y-axis

Grahic Jump Location
Fig. 8

Feasible displacements in motion branch B: (a) home configuration (see Sec. 8); (b), (c) the photos only illustrate the rotation of the platform, and the translation of the helical motion is to be detected in Sec. 7

Grahic Jump Location
Fig. 9

Bifurcated motion and constraint singularity in the 3-PUP

Grahic Jump Location
Fig. 10

The experimental environment for detecting the translation of the helical motion

Tables

Errata

Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In