Technical Brief

A Computational Geometric Approach for Motion Generation of Spatial Linkages With Sphere and Plane Constraints

[+] Author and Article Information
Xiangyun Li

School of Mechanical Engineering,
Southwest Jiaotong University,
Chengdu 610031, China
e-mail: xiangyun.app@gmail.com

Q. J. Ge

Department of Mechanical Engineering,
Stony Brook University,
Stony Brook, NY 11794
e-mail: qiaode.ge@stonybrook.edu

Feng Gao

School of Mechanical Engineering,
Shanghai Jiao Tong University,
Shanghai 200240, China
e-mail: fengg@sjtu.edu.cn

1Corresponding author.

Contributed by the Mechanisms and Robotics Committee of ASME for publication in the JOURNAL OF MECHANISMS AND ROBOTICS. Manuscript received June 6, 2018; final manuscript received October 11, 2018; published online December 10, 2018. Assoc. Editor: Andrew P. Murray.

J. Mechanisms Robotics 11(1), 014504 (Dec 10, 2018) (7 pages) Paper No: JMR-18-1168; doi: 10.1115/1.4041788 History: Received June 06, 2018; Revised October 11, 2018

This paper studies the problem of spatial linkage synthesis for motion generation from the perspective of extracting geometric constraints from a set of specified spatial displacements. In previous work, we have developed a computational geometric framework for integrated type and dimensional synthesis of planar and spherical linkages, the main feature of which is to extract the mechanically realizable geometric constraints from task positions, and thus reduce the motion synthesis problem to that of identifying kinematic dyads and triads associated with the resulting geometric constraints. The proposed approach herein extends this data-driven paradigm to spatial cases, with the focus on acquiring the point-on-a-sphere and point-on-a-plane geometric constraints which are associated with those spatial kinematic chains commonly encountered in spatial mechanism design. Using the theory of kinematic mapping and dual quaternions, we develop a unified version of design equations that represents both types of geometric constraints, and present a simple and efficient algorithm for uncovering them from the given motion.

Copyright © 2019 by ASME
Your Session has timed out. Please sign back in to continue.


Soh, G. S. , and McCarthy, J. M. , 2008, “ The Synthesis of Six-Bar Linkages as Constrained Planar 3R Chains,” Mech. Mach. Theory, 43(2), pp. 160–170. [CrossRef]
Hayes, M. J. D. , and Rusu, S. R. , 2011, “ Quadric Surface Fitting Applications to Approximate Dimensional Synthesis,” 13th World Congress in Mechanism and Machine Science, Guanajuato, Mexico, June 19–23, Paper No. A7-561.
Lin, S. , Liu, J. , Wang, H. , and Zhang, Y. , 2018, “ A Novel Geometric Approach for Planar Motion Generation Based on Similarity Transformation of Pole Maps,” Mech. Mach. Theory, 122, pp. 97–112. [CrossRef]
Brunnthaler, K. , Schreocker, H.-P. , and Husty, M. , 2006, “ Synthesis of Spherical Four-Bar Mechanisms Using Spherical Kinematic Mapping,” Advances in Robot Kinematics, Springer, Dordrecht, The Netherlands, pp. 377–385.
Léger, J. , and Angeles, J. , 2015, “ A Solution to the Approximate Spherical Burmester,” Multibody Mechatronic Systems, Springer, Cham, Switzerland, pp. 521–529.
Su, H.-J. , Wampler, C. W. , and McCarthy, J. M. , 2004, “ Geometric Design of Cylindric PRS Serial Chains,” ASME J. Mech. Des., 126(2), pp. 269–277. [CrossRef]
Robson, N. P. , McCarthy, J. M. , and Tumer, I. Y. , 2008, “ The Algeraic Synthesis of a Spatial TS Chain for a Prescribed Acceleration Task,” Mech. Mach. Theory, 43(10), pp. 1268–1280. [CrossRef]
Rao, N. M. , and Rao, K. M. , 2009, “ Dimensional Synthesis of a Spatial 3-RPS Parallel Manipulator for a Prescribed Range of Motion of Spherical Joints,” Mech. Mach. Theory, 44(2), pp. 477–486. [CrossRef]
Erdman, A. G. , Sandor, G. N. , and Kota, S. , 2001, Mechanism Design: Analysis and Synthesis, 4th ed., Prentice Hall, Englewood Cliffs, NJ.
Lee, E. , and Mavroidis, C. , 2006, “ An Elimination Procedure for Solving the Geometric Design Problem of Spatial 3R Manipulators,” ASME J. Mech. Des., 128(1), pp. 142–145. [CrossRef]
Su, H.-J. , McCarthy, J. M. , and Watson, L. T. , 2004, “ Generalized Linear Product Homotopy Algorithms and the Computation of Reachable Surface,” ASME J. Comput. Inf. Sci. Eng., 4(3), pp. 226–234. [CrossRef]
Luu, T. , and Hayes, M. J. D. , 2012, “ Integrated Type and Dimensional Synthesis of Planar Four-Bar Mechanisms,” Latest Advances in Robot Kinematics, Springer, New York, pp. 317–324.
Eberhard, P. , Gaugele, T. , and Sedlaczek, K. , 2009, “ Topology Optimized Synthesis of Planar Kinematic Rigid Body Mechanisms,” Advanced Design of Mechanical Systems: From Analysis to Optimization, Springer, Vienna, Austria, pp. 287–302.
Zhao, P. , Li, X. , Purwar, A. , and Ge, Q. J. , 2016, “ A Task Driven Unified Synthesis of Planar Four-Bar and Six-Bar Linkages With R- and P-Joints for Five Position Realization,” ASME J. Mech. Rob., 8(6), p. 061003. [CrossRef]
Li, X. , Zhao, P. , Purwar, A. , and Ge, Q. J. , 2018, “ A Unified Approach to Exact and Approximate Motion Synthesis of Spherical Four-Bar Linkages Via Kinematic Mapping,” ASME J. Mech. Rob., 10(1), p. 011003. [CrossRef]
Ge, X. , Purwar, A. , and Ge, Q. , 2016, “ From 5-SS Platform Linkage to Four-Revolute Jointed Planar, Spherical and Bennett Mechanisms,” ASME Paper No. DETC2016-60574.
Ge, X. , 2016, “ A Data Driven Design Methodology for SS Dyads and Its Application to Unified Synthesis of Planar, Spherical and Spatial Linkages,” Ph.D. thesis, Stony Brook University, Stony Brook, NY.
Bottema, O. , and Roth, B. , 1979, Theoretical Kinematics, Dover Publications, Mineola, NY.
McCarthy, J. M. , 1990, Introduction to Theoretical Kinematics, The MIT Press, Cambridge, MA.
Ge, Q. J. , and Sirchia, M. , 1999, “ Computer Aided Geometric Design of Two-Parameter Freeform Motions,” ASME J. Mech. Des., 121(4), pp. 502–506. [CrossRef]
Husty, M. L. , 1996, “ An Algorithm for Solving the Direct Kinematics of General Stewart-Gough Platforms,” Mech. Mach. Theory, 31(4), pp. 365–380. [CrossRef]
Verschelde, J. , and Yu, X. , 2015, “ Polynomial Homotopy Continuation on GPUs,” ACM Commun. Comput. Algebra, 49(4), pp. 130–133. [CrossRef]


Grahic Jump Location
Fig. 1

A spatial displacement

Grahic Jump Location
Fig. 2

Spherical RR-S Leg

Grahic Jump Location
Fig. 8

Solution 1: a sphere with center at (11.2473, 5.3352, 2.4121) and radius of 14.1004

Grahic Jump Location
Fig. 9

Solution 2: a plane defined by the homogeneous equation 0.7736X1 – 0.4464a2X2 + 0.0006X3 + 0.0075X4 = 0

Grahic Jump Location
Fig. 10

Solution 3: a sphere with center at (2.6928, 2.1003, 2.7649) and radius of 3.9451

Grahic Jump Location
Fig. 11

Solution 4: a plane constraint defined by the homogeneous equation –0.7806X1 + 0.4528X2 + 0.0070 * X3 + 0.0794 X4 = 0

Grahic Jump Location
Fig. 12

Solution 5: a sphere with center at (–0.4067, 1.1529, 2.8930) and radius of 4.8645

Grahic Jump Location
Fig. 13

Solution 6: a sphere with center at (3.0556, –2.0322, 2.4240) and radius of 2.1526

Grahic Jump Location
Fig. 14

Solution 7: a plane defined by the homogeneous equation 0.7746X1 + 0.4472X2 = 0

Grahic Jump Location
Fig. 15

A spatial parallel manipulator constructed by a spherical RR-S leg associated with the sphere constraint given by solution 5, a SS leg with the sphere constraint by solution 3, and a RPS leg with the plane constraint by solution 4



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