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Research Papers

A Line Geometric Approach to Kinematic Acquisition of Geometric Constraints of Planar Motion

[+] Author and Article Information
Jun Wu

DMG MORI Manufacturing USA, Inc.,
Davis, CA 95618

Xiangyun Li

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

Q. J. Ge

Computational Design Kinematics Laboratory,
Stony Brook University,
Stony Brook, NY 11794

Feng Gao

School of Mechanical Engineering,
Shanghai Jiao Tong University,
Shanghai 200240, China

Xueyin Liu

R&D Center,
Sichuan Provincial Machinery
Research and Design Institute,
Chengdu 610063, China

1Corresponding author.

Manuscript received September 9, 2016; final manuscript received March 3, 2017; published online April 27, 2017. Assoc. Editor: Xilun Ding.

J. Mechanisms Robotics 9(4), 041004 (Apr 27, 2017) (7 pages) Paper No: JMR-16-1267; doi: 10.1115/1.4036222 History: Received September 09, 2016; Revised March 03, 2017

This paper examines the problem of geometric constraints acquisition of planar motion through a line-geometric approach. In previous work, we have investigated the problem of identifying point-geometric constraints associated with a motion task which is given in a parametric or discrete form. In this paper, we seek to extend the point-centric approach to the line-centric approach. The extracted geometric constraints can be used directly for determining the type and dimensions of a physical device such as mechanical linkage that generates this constrained motion task.

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References

Wu, J. , Ge, Q. J. , Su, H. J. , and Gao, F. , 2013, “ Kinematic Acquisition of Geometric Constraints for Task-Oriented Design of Planar Mechanisms,” ASME J. Mech. Rob., 5(1), p. 011003. [CrossRef]
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McCarthy, J. M. , 1990, Introduction to Theoretical Kinematics, MIT, Cambridge, MA.
McCarthy, J. M. , 2000, Geometric Design of Linkages, Springer, New York.
Hayes, M. J. D. , and Zsombor-Murrary, P. J. , 2004, “ Towards Integrated Type and Dimensional Synthesis of Mechanisms for Rigid Body Guidance,” CSME Forum, pp. 53–61.
Hayes, M. , and Rucu, S. R. , 2011, “ Quadric Surface Fitting Applications to Approximate Dimensional Synthesis,” 13th World Congress in Mechanism and Machine Theory, Guanajuato, Mexico, June 19–25, pp. 10–25.
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]
Zhao, P. , Li, X. , Zhu, L. , Zi, B. , and Ge, Q. J. , 2016, “ A Novel Motion Synthesis Approach With Expandable Solution Space for Planar Linkages Based on Kinematic-Mapping,” Mech. Mach. Theory, 105, pp. 164–175. [CrossRef]
Li, X. , Zhao, P. , Ge, Q. J. , and Purwar, A. , 2013, “ A Task Driven Approach to Simultaneous Type Synthesis and Dimensional Optimization of Planar Parallel Manipulator Using Algebraic Fitting of a Family of Quadrics,” ASME Paper No. DETC2013-13197.
Bruce, J. W. , and Giblin, P. J. , 1984, Curves and Singularities, Cambridge University Press, Cambridge, London.
Pottmann, H. , and Wallner, J. , 2001, Computational Line Geometry, Springer, Berlin.
Ravani, B. , and Ku, T. S. , 1991, “ Bertrand Offsets of Ruled and Developable Surfaces,” Comput.-Aided Des., 23(2), pp. 145–152. [CrossRef]
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Ge, Q. J. , and Ravani, B. , 1994, “ On Representation and Interpolation of Line-Segments for Computer Aided Geometric Design,” Advances in Design Automation, 69(1), pp. 191–198.
Ge, Q. J. , and Ravani, B. , 1998, “ Geometric Design of Rational Bézier Line Congruences and Ruled Surfaces Using Line Geometry,” Computing Supplement 13: Geometric Modeling, G. Farin , H. Beiri , G. Brunnett , and T. DeRose , eds., Springer-Verlag, Vienna, Austria, pp. 101–120.
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Figures

Grahic Jump Location
Fig. 1

A planar displacement

Grahic Jump Location
Fig. 2

The standard line constraint g, represented by its envelope, is shown at the lower corner. The desired line constraint G, shown by its envelope as the solid curve, is equiformly transformed from g. A line l in the moving body traces out a family of lines L, shown as the dash envelope curve.

Grahic Jump Location
Fig. 3

(a) A 2DOF mechanism that envelopes a circle, (b) a 2DOF mechanism that envelopes an ellipse, (c) a 2DOF mechanism that envelopes a hyperbola, and (d) a 2DOF mechanism that envelopes a parabola

Grahic Jump Location
Fig. 4

The task is a rigid body motion with one line tangent to the ellipse shown in the figure. The discrete moving frames are some sample positions of the task motion, whose origin traces out the dash line.

Grahic Jump Location
Fig. 5

The ellipse is the constraint identified from the motion, such that it is tangentially touched by the moving line (0, 1, 0). The fine solid lines are some samples of the line motion that is tangent to the identified ellipse. The bold solid lines represent the mechanism that corresponds to the elliptical constraint and generates the task motion.

Grahic Jump Location
Fig. 6

The ellipse is the constraint identified from the motion, such that it is tangentially touched by the moving line (–0.4160, 0.9094, 0.7641). The fine solid lines are some samples of the line motion that is tangent to the identified ellipse. The bold solid lines represent the mechanism that corresponds to the elliptical constraint and generates the task motion.

Grahic Jump Location
Fig. 7

A 2DOF five link mechanism whose end-effector motion envelops an ellipse

Grahic Jump Location
Fig. 8

The ellipse is the constraint identified from the motion, such that it is tangentially touched by the moving line (–0.4688, 0.8833, 1.1235). The fine solid lines are some samples of the line motion that is tangent to the identified ellipse. The bold solid lines represent the mechanism that corresponds to the elliptical constraint and generates the task motion.

Grahic Jump Location
Fig. 9

The ellipse is the constraint identified from the motion, such that it is tangentially touched by the moving line (0.8385, 0.5449, 0.3279). The fine solid lines are some samples of the line motion that is tangent to the identified ellipse. The bold solid lines represent the mechanism that corresponds to the elliptical constraint and generates the task motion.

Grahic Jump Location
Fig. 10

A 1DOF closed-loop mechanism composed of two 2DOF mechanisms corresponding to two identified constraints as shown in Figs. 8 and 9

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