Geometric constraint programming (GCP) is an approach to synthesizing planar mechanisms in the sketching mode of commercial parametric computer-aided design software by imposing geometric constraints using the software's existing graphical user interface. GCP complements the accuracy of analytical methods with the intuition developed from graphical methods. Its applicability to motion generation, function generation, and path generation for finitely separated positions has been previously reported. By implementing existing, well-known theory, this technical brief demonstrates how GCP can be applied to kinematic synthesis for motion generation involving infinitesimally and multiply separated positions. For these cases, the graphically imposed geometric constraints alone will in general not provide a solution, so the designer must parametrically relate dimensions of entities within the graphical construction to achieve designs that automatically update when a defining parameter is altered. For three infinitesimally separated positions, the designer constructs an acceleration polygon to locate the inflection circle defined by the desired motion state. With the inflection circle in place, the designer can rapidly explore the design space using the graphical second Bobillier construction. For multiply separated position problems in which only two infinitesimally separated positions are considered, the designer constrains the instant center of the mechanism to be in the desired location. For example, four-bar linkages are designed using these techniques with three infinitesimally separated positions and two different combinations of four multiply separated positions. The ease of implementing the techniques may make synthesis for infinitesimally and multiply separated positions more accessible to mechanism designers and undergraduate students.

References

References
1.
Beyer
,
R.
,
1963
,
The Kinematic Synthesis of Mechanisms, H. Kuenzel, trans.
,
McGraw-Hill
,
New York
.
2.
Hirschhorn
,
J.
,
1962
,
Kinematics and Dynamics of Plane Mechanisms
,
McGraw-Hill
,
New York
.
3.
Uicker
,
J. J.
, Jr.
,
Pennock
,
G. R.
, and
Shigley
,
J. E.
,
2011
,
Theory of Machines and Mechanisms
,
4th ed.
,
Oxford University Press, Inc.
,
New York
.
4.
Dijksman
,
E. A.
,
1976
,
Motion Geometry of Mechanism
,
Cambridge University Press
,
Cambridge, UK
.
5.
Erdman
,
A. G.
,
Sandor
,
G. N.
, and
Kota
,
S.
,
2001
,
Mechanism Design
,
4th ed.
, Vol.
1
,
Prentice-Hall
,
Upper Saddle River, NJ
.
6.
Hain
,
K.
,
1967
,
Applied Kinematics
,
2nd ed.
,
McGraw-Hill
,
New York
.
7.
Tesar
,
D.
, and
Eschenbach
,
P. W.
,
1967
, “
Four Multiply Separated Positions in Coplanar Motion
,”
ASME J. Eng. Ind.
,
89
(2), pp.
231
234
.10.1115/1.3610033
8.
Tesar
,
D.
,
1968
, “
The Generalized Concept of Four Multiply Separated Positions in Coplanar Motion
,”
J. Mech.
,
3
(
1
), pp.
11
23
.10.1016/0022-2569(68)90026-8
9.
Dijksman
,
E. A.
,
1969
, “
Geometrical Treatment of the PPP-P Case in Coplanar Motion: Three Infinitesimally and One Finitely Separated Position in a Plane
,”
J. Mech.
,
4
(
4
), pp.
375
389
.10.1016/0022-2569(69)90017-2
10.
Tesar
,
D.
, and
Sparks
,
J. W.
,
1968
, “
The Generalized Concept of Five Multiply Separated Positions in Coplanar Motion
,”
ASME J. Mech.
,
3
(
1
), pp.
25
33
.
11.
Chen
,
P.
, and
Roth
,
B.
,
1969
, “
A Unified Theory for the Finitely and Infinitesimally Separated Position Problems in Kinematic Synthesis
,”
ASME J. Eng. Ind.
,
91
(1), pp.
203
208
.10.1115/1.3591522
12.
Kinzel
,
E. C.
,
Schmiedeler
,
J. P.
, and
Pennock
,
G. R.
,
2006
, “
Kinematic Synthesis for Finitely Separated Positions Using Geometric Constraint Programming
,”
ASME J. Mech. Des.
,
128
(5), pp.
1070
1079
.10.1115/1.2216735
13.
Kinzel
,
E. C.
,
Schmiedeler
,
J. P.
, and
Pennock
,
G. R.
,
2007
, “
Function Generation With Finitely Separated Precision Points Using Geometric Constraint Programming
,”
ASME J. Mech. Des.
,
129
(11), pp.
1185
1190
.10.1115/1.2771575
14.
Hall
,
A. S.
, Jr.
,
1986
,
Kinematics and Linkage Design
,
Waveland Press, Inc.
, Prospect Heights, IL.
15.
Waldron
,
K. J.
, and
Kinzel
,
G. L.
,
2004
,
Kinematics, Dynamics, and Design of Machinery
,
2nd ed.
,
Wiley
,
New York
.
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