This paper presents a technique for determining the Burmester points of five finitely separated positions in a plane. A Burmester point is the location of the fixed pivot of an RR dyad that can reach each of the five positions. These Burmester points are obtained from a “compatibility platform” constructed from selected relative poles associated with the five specified displacements. We show that the relative position pole of two assembly configurations of this platform is a fixed pivot compatible with the five positions. An example originally presented by Sandor and Erdman is used to verify the calculations.

Issue Section:
Research Papers
Topics:
Manufacturing, Poles (Building)
1.
Beyer, R., 1953, Kinematische Getriebesynthese, Springer-Verlag, Berlin.
2.
Bottema, O., and Roth, B., 1979, Theoretical Kinematics, North-Holland Publishing Company, New York.
3.
Burmester, L., 1886, Lehrbuch der Kinematik, Verlag Von Arthur Felix, Leipzig, Germany.
4.
Chang, Ching Yu, Angeles, Jorge, and Gonza´lez-Palacios, Max Antonio, 1991, “A Semigraphical Method for the Solution of the Burmester Problem,” ASME Advanced in Design Automation, DE-Vol. 32-2.
5.
Freudenstein, F., and Sandor, G. N., 1961, “On the Burmester Points of a Plane,” ASME Journal of Applied Mechanics.
6.
Gosselin, C. M., Sefrioui, J., and Richard, M. J., 1992, “Polynomial Solutions to the Direct Kinematic Problem of Planar 3 Degree-of-Freedom Parallel Manipulators,” Mechanism and Machine Theory, Vol. 27, No. 2, March.
7.
Hall, A. S., Jr., 1961, Kinematics and Linkage Design, Prentice-Hall, Inc., Englewoods Cliffs, NJ.
8.
Hartenberg, R. S., and Denavit, J., 1964, Kinematic Synthesis of Linkages, McGraw-Hill, New York.
9.
Lin, Chuen-Sun, 1987, “Dimensional Synthesis of a Planar Triad for Six Precision Points,” Ph.D. thesis, University of Minnesota.
10.
Lin, Cheun-Sun, and Jia, Bao-Ping, 1992, “Use of Resultant in the Dimensional Synthesis of Linkage Components: Motion Generation with Prescribed Timing,” ASME Mechanical Design and Synthesis. DE-Vol. 46.
11.
McCarthy, J. M., 1990, “An Introduction to Theoretical Kinematics,” MIT Press, Cambridge, MA.
12.
McCarthy
J. M.
,
1993
a, “
The Opposite Pole Quadrilateral as a Compatibility Linkage for Parameterizing the Center Point Curve
,”
ASME JOURNAL OF MECHANICAL DESIGN
, Vol.
115
, pp.
332
–
336
.
13.
McCarthy, J. M., 1993b, “Dual Quaternions and the Pole Triangle,” Ch. 4, Modern Kinematics: Developments in the Last Forty Years, Erdman A. G., ed., John Wiley & Sons, Inc., New York.
14.
McCarthy, J. M., 1994, “The Synthesis of Planar RR and Spatial CC Chains and the Equation of a Triangle,” ASME JOURNAL OF MECHANICAL DESIGN. December, in press.
15.
Murray, A. P., and McCarthy, J. M., 1994, “Five Position Synthesis of Spatial CC Dyads,” Proceedings of the ASME Mechanics Conference, Minneapolis, MN, DE-Vol. 70, pp. 143–152.
16.
Pennock
G. R.
, and
Kassner
D. J.
,
1993
, “
The Workspace of a General Geometry Planar 3 Degree-of-Freedom Platform-Type Manipulator
,”
ASME JOURNAL OF MECHANICAL DESIGN
, Vol.
115
, No.
2
, pp.
269
–
276
.
17.
Sandor, G. N., and Erdman, A. G., 1984, Advanced Mechanism Design: Analysis and Synthesis, Vol. 2, Prentice-Hall, Inc., Englewoods Cliffs, NJ.
This content is only available via PDF.
Copyright © 1995
 by The American Society of Mechanical Engineers
You do not currently have access to this content.