Research Papers

Simplified Kinematics for a Parallel Manipulator Generator of the Schönflies Motion

[+] Author and Article Information
Jaime Gallardo-Alvarado

Department of Mechanical Engineering,
Instituto Tecnologico de Celaya, TecNM,
Celaya 38010 GTO, Mexico
e-mail: jaime.gallardo@itcelaya.edu.mx

Mohammad H. Abedinnasab

Department of Biomedical Engineering,
Rowan University,
Glassboro, NJ 08028
e-mail: abedin@rowan.edu

Daniel Lichtblau

Wolfram Research,
100 Trade Center Drive,
Champaign, IL 61820
e-mail: danl@wolfram.com

1Corresponding author.

Manuscript received March 4, 2016; final manuscript received September 19, 2016; published online October 25, 2016. Assoc. Editor: Yuefa Fang.

J. Mechanisms Robotics 8(6), 061020 (Oct 25, 2016) (10 pages) Paper No: JMR-16-1055; doi: 10.1115/1.4034884 History: Received March 04, 2016; Revised September 19, 2016

This work is devoted to simplify the inverse–forward kinematics of a parallel manipulator generator of the 3T1R motion. The closure equations of the displacement analysis are formulated based on the coordinates of two points embedded in the moving platform. Afterward, five quadratic equations are solved by means of a novel method based on Gröbner bases endowed with first-order perturbation and local stability of parameters. Meanwhile, the input–output equations of velocity and acceleration are systematically obtained by resorting to reciprocal-screw theory. In that concern, the inclusion of pseudokinematic pairs connecting the limbs to the fixed platform and a passive kinematic chain to the robot manipulator allows to avoid the handling of rank-deficient Jacobian matrices. The workspace of the robot is determined by using a discretized method associated to its inverse–forward displacement analysis, whereas the singularity analysis is approached based on the input–output equation of velocity. Numerical examples are provided with the purpose to show the application of the method.

Copyright © 2016 by ASME
Your Session has timed out. Please sign back in to continue.


Lee, C.-C. , and Hervé, J. M. , 2005, “ On the Enumeration of Schönflies Motion Generators,” Ninth IFToMM International Symposium on Theory of Machines and Mechanisms, Bucharest, Romania, Sept. 1–4, Paper 16, pp. 673–678.
Lee, C.-C. , and Hervé, J. M. , 2011, “ Isoconstrained Parallel Generators of Schoenflies Motion,” ASME J. Mech. Rob., 3(2), p. 021006. [CrossRef]
Pierrot, F. , Nabat, V. , Company, O. , Krut, S. , and Poignet, P. , 2009, “ Optimal Design of a 4-DOF Parallel Manipulator: From Academia to Industry,” IEEE Trans. Rob., 25(2), pp. 212–224. [CrossRef]
Rosenzveig, V. , Briot, S. , and Martinet, P. , 2013, “ Minimal Representation for the Control of the Adept Quattro With Rigid Platform Via Leg Observation Considering a Hidden Robot Model,” 2013 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), Tokyo, Japan, Nov. 3–7, pp. 430–435.
Özgür, E. , Dahmouche, R. , Andreff, N. , and Martinet, P. , 2014, “ A Vision-Based Generic Dynamic Model of PKMs and Its Experimental Validation on the Quattro Parallel Robot,” IEEE/ASME International Conference on Advanced Intelligent Mechatronics (AIM), Besancon, France, July 8–11, pp. 937–942.
Clavel, R. , 1988, “ Delta: A Fast Robot With Parallel Geometry,” 18th International Symposium on Industrial Robots, IFS Publications, Lausanne, Switzerland, Apr. 26–28, pp. 91–100.
Huang, Z. , and Li, Q. C. , 2003, “ Symmetrical Lower-Mobility Parallel Mechanisms Using the Constraint-Synthesis Method,” Int. J. Rob. Res., 22(1), pp. 59–79.
Kong, X. , and Gosselin, C. M. , 2004, “ Type Synthesis of 3T1R 4-DOF Parallel Manipulators Based on Screw Theory,” IEEE Trans. Rob. Autom., 20(2), pp. 181–190. [CrossRef]
Li, Q. , and Hervé, J. M. , 2009, “ Parallel Mechanisms With Bifurcation of Schönflies Motion,” IEEE Trans. Rob., 25(1), pp. 158–163. [CrossRef]
Kong, X. , and Gosselin, C. , 2011, “ Forward Displacement Analysis of a Quadratic 4-DOF 3T1R Parallel Manipulator,” Meccanica, 46(1), pp. 147–154. [CrossRef]
Altuzarra, O. , Pinto, C. , Sandru, B. , and Hernandez, A. , 2011, “ Optimal Dimensioning for Parallel Manipulators: Workspace, Dexterity, and Energy,” ASME J. Mech. Des., 133(4), p. 041007. [CrossRef]
Amine, S. , Masouleh, M. T. , Caro, S. , and Wenger, P. , 2012, “ Singularity Conditions of 3T1R Parallel Manipulators With Identical Limb Structures,” ASME J. Mech. Rob., 4(1), p. 011011. [CrossRef]
Masouleh, M. , Walter, D. , Husty, M. , and Gosselin, C. , 2012, “ Solving the Forward Kinematic Problem of 4-DOF Parallel Mechanisms (3T1R) With Identical Limb Structures and Revolute Actuators Using the Linear Implicitization Algorithm,” ASME Paper No. DETC2011-47884.
Liu, S. , Huang, T. , Mei, J. , Zhao, X. , Wang, P. , and Chetwynd, D. G. , 2012, “ Optimal Design of a 4-DOF SCARA Type Parallel Robot Using Dynamic Performance Indices and Angular Constraints,” ASME J. Mech. Rob., 4(3), p. 031005. [CrossRef]
Cammarata, A. , and Rosario, S. , 2014, “ Elastodynamic Optimization of a 3T1R Parallel Manipulator,” Mech. Mach. Theory, 73, pp. 184–196. [CrossRef]
Gogu, G. , 2014, Structural Synthesis of Parallel Robots, Springer, New York.
Xie, F. , and Liu, X.-J. , 2015, “ Design and Development of a High-Speed and High-Rotation Robot With Four Identical Arms and a Single Platform,” ASME J. Mech. Rob., 7(4), p. 041015. [CrossRef]
Angeles, J. , Caro, S. , Khan, W. , and Morozov, A. , 2006, “ The Design and Prototyping of an Innovative Schoenflies Motion Generator,” Proc. Inst. Mech. Eng., Part C, 220(C7), pp. 935–944. [CrossRef]
Lee, P.-C. , and Lee, J.-J. , 2016, “ On the Kinematics of a New Parallel Mechanism With Schoenflies Motion,” Robotica, 34(9), pp. 2056–2070.
Nurahmi, L. , Caro, S. , and Wenger, P. , 2015, “ Operation Modes and Self-Motions of a 2-RUU Parallel Manipulator,” Recent Advances in Mechanism Design for Robotics, S. Bai and M. Ceccarelli , eds., Springer International Publishing, Aalborg, Denmark, pp. 417–426.
Nurahmi, L. , Caro, S. , Wenger, P. , Schadlbauer, J. , and Husty, M. , 2016, “ Reconfiguration Analysis of a 4-RUU Parallel Manipulator,” Mech. Mach. Theory, 96, pp. 269–289. [CrossRef]
Merlet, J.-P. , 2004, “ Solving the Forward Kinematics of a Gough-Type Parallel Manipulator With Interval Analysis,” Int. J. Rob. Res., 23(3), pp. 221–235. [CrossRef]
Gallardo-Alvarado, J. , 2014, “ A Simple Method to Solve the Forward Displacement Analysis of the General Six-Legged Parallel Manipulator,” Rob. Comput. Integr. Manuf., 30(1), pp. 55–61. [CrossRef]
Kong, X. , and Gosselin, C. M. , 2007, Type Synthesis of Parallel Mechanisms, Springer, New York.
Cervantes-Sánchez, J. J. , Rico-Martínez, J. M. , and Pérez-Muñoz, V. H. , 2016, “ An Integrated Study of the Workspace and Singularity for a Schönflies Parallel Manipulator,” J. Appl. Res. Technol., 14(1), pp. 9–37. [CrossRef]
Lichtblau, D. , 2000, “ Solving Finite Algebraic Systems Using Numeric Gröbner Bases and Eigenvalues,” World Conference on Systemics, Cybernetics, and Informatics, Vol. 10, pp. 555–560.
Lichtblau, D. , 2016, “ First Order Perturbation and Local Stability of Parametrized Systems,” Math. Comput. Sci., 10(1), pp. 143–163. [CrossRef]
Gallardo-Alvarado, J. , Orozco-Mendoza, H. , and Rodríguez-Castro, R. , 2008, “ Finding the Jerk Properties of Multibody Systems Using Helicoidal Vector Fields,” Proc. Inst. Mech. Eng., Part C, 222(11), pp. 2217–2229. [CrossRef]
Choi, H.-B. , and Ryu, J. , 2012, “ Singularity Analysis of a Four Degree-of-Freedom Parallel Manipulator Based on an Expanded 6 × 6 Jacobian Matrix,” Mech. Mach. Theory, 57, pp. 51–61. [CrossRef]
Rico, J. M. , Gallardo, J. , and Duffy, J. , 1995, “ A Determination of Singular Configurations of Serial Non-Redundant Manipulators, and Their Escapement From Singularities Using Lie Products,” Computational Kinematics'95 (Solid Mechanics and Its Applications Series), J.-P. Merlet and B. Ravani , eds., Vol. 40, Springer, Dordrecht, The Netherlands, pp. 143–152.
Waldron, K. J. , Wang, S. L. , and Bolin, S. J. , 1985, “ A Study of the Jacobian Matrix of Serial Manipulators,” ASME J. Mech. Trans. Autom. Des., 107(2), pp. 230–238. [CrossRef]
Hausner, M. , and Schwartz, J. T. , 1968, Lie Groups, Lie Algebras, Gordon & Breach, New York.
Johnson, A. , Kong, X. , and Ritchie, J. , 2015, “ Determination of the Workspace of a Three-Degrees-of-Freedom Parallel Manipulator Using a Three-Dimensional Computer-Aided-Design Software Package and the Concept of Virtual Chains,” ASME J. Mech. Rob., 8(2), p. 024501. [CrossRef]
Verschelde, J. , 1999, “ Algorithm 795: PHCpack: A General-Purpose Solver for Polynomial Systems by Homotopy Continuation,” ACM Trans. Math. Software, 25(2), pp. 251–276. [CrossRef]
Lichtblau, D. , 2013, “ Approximate Gröbner Bases, Overdetermined Polynomial Systems, and Approximate GCDs,” ISRN Comput. Math., 2013, p. 352806. [CrossRef]


Grahic Jump Location
Fig. 1

The layout of the parallel manipulator under study

Grahic Jump Location
Fig. 2

Two direct singularities, the lines $i(i=1,2,3,4) are (i) coplanar and (ii) parallel

Grahic Jump Location
Fig. 6

Workspace of the robot

Grahic Jump Location
Fig. 5

The two poses of example 1

Grahic Jump Location
Fig. 3

Example of inverse singularity (left) and escapement from the singularity (right)

Grahic Jump Location
Fig. 4

Example 1: Geometry of the moving platform

Grahic Jump Location
Fig. 7

Time history of the kinematics of the moving platform



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