Abstract
The position synthesis of planar linkages is to locate the center point of the moving joint on a rigid link, whose trajectory is a circle or a straight line. Utilizing the min-max optimization scheme, the fitting curve needs to minimize the maximum fitting error to acquire the dimension of a planar binary P-R link. Based on the saddle point programming, the fitting straight line is determined to the planar discrete point-path traced by the point of the rigid body in planar motion. The property and evolution of the defined saddle line error can be revealed from three given separate points. A quartic algebraic equation relating the fitting error and the coordinates is derived, which agrees with the classical theory. The effect of the fourth point is discussed in three cases through the constraint equations. The multi-position saddle line error is obtained by combination and comparison from the saddle point programming. Several examples are presented to illustrate the solution process for the saddle line error of the moving plane. The saddle line error surface and the contour map presented to show the variations of the fitting error in the fixed frame. The discrete kinematic geometry is then set up to disclose the relations of the separate positions of the rigid body, the location of the tracing point on the moving body, and the position and orientation of the saddle line to the point-path. This paper presents a new analytic geometry method for saddle line fitting and provides a theoretical foundation for position synthesis.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
BURMESTER L. Lehrbuch der Kinematik[M]. Felix, Leipzig, 1888.
BAI S X. Advanced mechanism[M]. Shanghai: Shanghai Scientific and Technical Publishers, 1988. (in Chinese)
ROTH B. On the screw axes and other special lines associated with spatial displacements of a rigid body[J]. Journal of Manufacturing Science and Engineering, 1967, 89(1): 102–110.
ROTH B. On the multiple generation of coupler-curves[J]. Journal of Manufacturing Science and Engineering, 1965, 87(2): 177–183.
SUH C H, RADCLIFFE C W. Kinematic and mechanisms design[M]. New York: John Wiley & Sons, 1978.
SANDOR G N, FREUDENSTEIN F. Higher-Order plane motion theories in kinematic synthesis[J]. Journal of Manufacturing Science and Engineering, 1967, 89(2): 223–230.
FREUDENSTEIN F, SANDOR G N. On the Burmester points of a plane[J]. Journal of Applied Mechanics, 1961, 28(1): 41–49.
FREUDENSTEIN F. On the variety of motions generated by mechanisms[J]. Journal of Manufacturing Science and Engineering, 1962, 84(1): 156–159.
FREUDENSTEIN F, PRIMROSE E J F. Geared five-bar motion: Part I—Gear ratio minus one[J]. Journal of Applied Mechanics, 1963, 30(2): 161–169.
FREUDENSTEIN F. Higher path-curvature analysis in plane kinematics[J]. Journal of Manufacturing Science and Engineering, 1965, 87(2): 184–190.
VELDKAMP G R. Some remarks on higher curvature theory[J]. Journal of Manufacturing Science and Engineering, 1967, 89(1): 84–86.
AKHRAS R, ANGELES J. Unconstrained nonlinear least-square optimization of planar linkages for rigid-body guidance[J]. Mechanism and Machine Theory, 1990, 25(1): 97–118.
ANGELES J, ALIVIZATOSS A, AKHRAS R. An unconstrained nonlinear least-square method of optimization of RRRR planar path generators[J]. Mechanism and Machine Theory, 1988, 23(5): 343–353.
LIU J, WANG X M. Saddle point programming and geometric error evaluation[M]. Dalian: Dalian University of Technology Press, 1996. (in Chinese)
WANG S F. New Approach for kinematic synthesis of mechanism by adaptive curve fitting[D]. Dalian University of Technology, 2005. (in Chinese)
WANG D L, WANG S F, Li T. Kinematic synthesis of the planar four-bar linkage by adaptive curve fitting[J]. Journal of Mechanical Engineering, 2001, 37(12): 21–26. (in Chinese)
WANG D L, XIAO D Z. Distribution of coupler curves for crank-rocker linkages[J]. Mechanism and Machine Theory, 1993, 28(5): 671–684.
WANG D L, XIAO D Z, LIU J. Ball’s curve and Burmester’s curve for four-bar linkage[J]. Journal of Dalian University of Technology, 1994, 34(4): 411–417. (In Chinese)
LAN Z H, ZOU H J. Parallel optimization of mechanisms based on the local characteristics of the coupler curves [J]. Journal of Mechanical Engineering, 1999, 35(5): 16–19. (in Chinese)
WANG D L, GAO Y. Mechanism and machine theory[M]. Beijing: China Machine Press, 2011. (in Chinese)
XUE S. Course of Matlab[M]. Beijing: Tsinghua University press, 2011. (in Chinese)
Author information
Authors and Affiliations
Corresponding author
Additional information
Supported by National Natural Science Foundation of China(Grant No. 51275067)
WU Yu, born in 1991, is currently a PhD candidate at Dalian University of Technology, China. He received his bachelor degree from Dalian University of Technology, China, in 2013. His research interests include mechanisms analysis and synthesis.
WANG Delun, born in 1958, is currently a professor and a PhD candidate supervisor at Dalian University of Technology, China. He received his PhD degree from Dalian University of Technology, China, in 1995.
WANG Wei, born in 1985, is currently a PhD candidate at Dalian University of Technology, China.
YU Shudong, born in 1962, is currently a professor of mechanical engineering at Ryerson University, Toronto, Canada. He received his PhD degree from University of Toronto in 1995.
XU Wenji, born in 1964, is currently a professor and a PhD candidate supervisor at Dalian University of Technology, China. He received his PhD degree from Dalian University of Technology, China, in 2000.
Rights and permissions
About this article
Cite this article
Wu, Y., Wang, D., Wang, W. et al. Kinematic geometry for the saddle line fitting of planar discrete positions. Chin. J. Mech. Eng. 28, 763–768 (2015). https://doi.org/10.3901/CJME.2015.0119.050
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.3901/CJME.2015.0119.050