A two-phase synthesis method is described, which is capable of solving quite challenging path generation problems. A combined discrete Fourier descriptor (FD) is proposed for shape optimization, and a geometric-based approach is used for the scale–rotation–translation synthesis. The combined discrete FD comprises three shape signatures, i.e., complex coordinates (CCs), centroid distance (CD), and triangular centroid area (TCA), which can capture greater similarity of shape. The genetic algorithm–differential evolution (GA–DE) optimization method is used to solve the optimization problem. The proposed two-phase synthesis method, based on the combined discrete FD, successfully solves the challenging path generation problems with a relatively small number of function evaluations. A more accurate path shape can be obtained using the combined FD than the one-phase synthesis method. The obtained coupler curves approximate the desired paths quite well.

References

References
1.
Murray
,
A. P.
,
Schmiedeler
,
J. P.
, and
Korte
,
B. M.
,
2008
, “
Kinematic Synthesis of Planar, Shape-Changing Rigid Body Mechanisms
,”
ASME J. Mech. Des.
,
130
(
3
), p.
032302
.10.1115/1.2829892
2.
Zhao
,
K.
,
Schmiedeler
,
J. P.
, and
Murray
,
A. P.
,
2012
, “
Design of Planar, Shape-Changing Rigid-Body Mechanisms for Morphing Aircraft Wings
,”
ASME J. Mech. Rob.
,
4
(
4
), p.
041007
.10.1115/1.4007449
3.
Funke
,
L. W.
,
Schmiedeler
,
J. P.
, and
Zhao
,
K.
,
2015
, “
Design of Planar Multi-Degree-of-Freedom Morphing Mechanisms
,”
ASME J. Mech. Rob.
,
7
(
1
), p.
011007
.10.1115/1.4029289
4.
Morgan
,
A. P.
, and
Wampler
,
C. W.
,
1990
, “
Solving a Planar Fourbar Design Problem Using Continuation
,”
ASME J. Mech. Des.
,
112
(
4
), pp.
544
550
.10.1115/1.2912644
5.
Krishnamurty
,
S.
, and
Turcic
,
D. A.
,
1992
, “
Optimal Synthesis of Mechanisms Using Nonlinear Goal Programming Techniques
,”
Mech. Mach. Theory
,
27
(
5
), pp.
599
612
.10.1016/0094-114X(92)90048-M
6.
Subbian
,
T.
, and
Flugrad
,
D. R.
,
1994
, “
Six and Seven Position Triad Synthesis Using Continuation Methods
,”
ASME J. Mech. Des.
,
116
(
2
), pp.
660
665
.10.1115/1.2919429
7.
Mariappan
,
J.
, and
Krishnamurty
,
S.
,
1996
, “
A Generalized Exact Gradient Method for Mechanism Synthesis
,”
Mech. Mach. Theory
,
31
(
4
), pp.
413
421
.10.1016/0094-114X(95)00077-C
8.
Roston
,
G. D.
, and
Sturges
,
R. H.
,
1996
, “
Genetic Algorithm Synthesis of Four-Bar Mechanisms
,”
Artif. Intell. Eng. Des. Anal. Manuf.
,
10
(
5
), pp.
371
390
.10.1017/S0890060400001700
9.
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
10.
Lin
,
W.-Y.
,
2010
, “
A GA-DE Hybrid Evolutionary Algorithm for Path Synthesis of Four-Bar Linkage
,”
Mech. Mach. Theory
,
45
(
8
), pp.
1096
1107
.10.1016/j.mechmachtheory.2010.03.011
11.
Bulatović
,
R. R.
,
Đorđević
,
S. R.
, and
Đorđević
,
V. S.
,
2013
, “
Cuckoo Search Algorithm: A Metaheuristic Approach to Solving the Problem of Optimum Synthesis of a Six-Bar Double Dwell Linkage
,”
Mech. Mach. Theory
,
61
(
1
), pp.
1
13
.10.1016/j.mechmachtheory.2012.10.010
12.
Schmiedeler
,
J. P.
,
Clark
,
B. C.
,
Kinzel
,
E. C.
, and
Pennock
,
G. R.
,
2014
, “
Kinematic Synthesis for Infinitesimally and Multiply Separated Positions Using Geometric Constraint Programming
,”
ASME J. Mech. Des.
,
136
(
3
), p.
034503
.10.1115/1.4026152
13.
Ullah
,
I.
, and
Kota
,
S.
,
1997
, “
Optimal Synthesis of Mechanisms for Path Generation Using Fourier Descriptor and Global Search Methods
,”
ASME J. Mech. Des.
,
119
(
4
), pp.
504
510
.10.1115/1.2826396
14.
Dibakar
,
S.
, and
Mruthyunjaya
,
T. S.
,
1999
, “
Synthesis of Workspaces of Planar Manipulators With Arbitrary Topology Using Shape Representation and Simulated Annealing
,”
Mech. Mach. Theory
,
34
(
3
), pp.
391
420
.10.1016/S0094-114X(98)00045-7
15.
Smaili
,
A.
, and
Diab
,
N.
,
2007
, “
A New Approach to Shape Optimization for Closed Path Synthesis of Planar Mechanisms
,”
ASME J. Mech. Des.
,
129
(
9
), pp.
941
948
.10.1115/1.2753164
16.
Buśkiewicz
,
J.
,
Starosta
,
R.
, and
Walczak
,
T.
,
2009
, “
On the Application of the Curve Curvature in Path Synthesis
,”
Mech. Mach. Theory
,
44
(
6
), pp.
1223
1239
.10.1016/j.mechmachtheory.2008.08.001
17.
Buśkiewicz
,
J.
,
2010
, “
Use of Shape Invariants in Optimal Synthesis of Geared Five-Bar Linkage
,”
Mech. Mach. Theory
,
45
(
2
), pp.
273
290
.10.1016/j.mechmachtheory.2009.09.004
18.
Zhang
,
C.
,
Norton
,
R. L.
, and
Hammond
,
T.
,
1984
, “
Optimization of Parameters for Specified Path Generation Using an Atlas of Coupler Curves of Geared Five-Bar Linkage
,”
Mech. Mach. Theory
,
19
(
6
), pp.
459
466
.10.1016/0094-114X(84)90052-1
19.
Hoeltzel
,
D. A.
, and
Chieng
,
W.-H.
,
1990
, “
Patten Matching Synthesis as an Automated Approach to Mechanical Design
,”
ASME J. Mech. Des.
,
112
(
2
), pp.
190
199
.10.1115/1.2912592
20.
McGarva
,
J. R.
,
1994
, “
Rapid Search and Selection of Path Generating Mechanisms From a Library
,”
Mech. Mach. Theory
,
29
(
2
), pp.
223
235
.10.1016/0094-114X(94)90032-9
21.
Vasiliu
,
A.
, and
Yannou
,
B.
,
2001
, “
Dimensional Synthesis of Planar Mechanisms Using Neural Networks: Application to Path Generator Linkage
,”
Mech. Mach. Theory
,
36
(
2
), pp.
299
310
.10.1016/S0094-114X(00)00037-9
22.
Yu
,
H.
,
Tang
,
D.
, and
Wang
,
Z.
,
2007
, “
Study on a New Computer Path Synthesis Method of a Four-Bar Linkage
,”
Mech. Mach. Theory
,
42
(
4
), pp.
383
392
.10.1016/j.mechmachtheory.2006.05.003
23.
Galán-Marín
,
G. F.
,
Alonso
,
J.
, and
Del Castillo
,
J. M.
,
2009
, “
Shape Optimization for Path Synthesis of Crank-Rocker Mechanisms Using a Wavelet-Based Neural Network
,”
Mech. Mach. Theory
,
44
(
6
), pp.
1132
1143
.10.1016/j.mechmachtheory.2008.09.006
24.
Sun
,
J. W.
, and
Chu
,
J. K.
,
2009
, “
Fourier Method to Function Synthesis of an RCCC Mechanism
,”
Proc. Inst. Mech. Eng., Part C
,
223
(
2
), pp.
503
513
.10.1243/09544062JMES1091
25.
Chu
,
J.
, and
Sun
,
J.
,
2010
, “
Numerical Atlas Method for Path Generation of Spherical Four-Bar Mechanism
,”
Mech. Mach. Theory
,
45
(
6
), pp.
867
879
.10.1016/j.mechmachtheory.2009.12.005
26.
Mullineux
,
G.
,
2011
, “
Atlas of Spherical Four-Bar Mechanisms
,”
Mech. Mach. Theory
,
46
(
11
), pp.
1811
1823
.10.1016/j.mechmachtheory.2011.06.001
27.
Li
,
X.
,
Ge
,
X.
,
Purwar
,
A.
, and
Ge
,
Q. J.
,
2015
, “
A Unified Algorithm for Analysis and Simulation of Planar Four-Bar Motions Defined With R- and P-Joints Mechanisms
,”
ASME J. Mech. Rob.
,
7
(
1
), p.
011014
.10.1115/1.4029295
28.
Zahn
,
T.
, and
Roskies
,
R. Z.
,
1972
, “
Fourier Descriptors for Plane Closed Curves
,”
IEEE Trans. Comput.
,
C-21
(
3
), pp.
269
281
.10.1109/TC.1972.5008949
29.
Lin
,
W.-Y.
,
2014
, “
Optimization of Scale-Rotation-Translation Synthesis After Shape Synthesis for Path Generation of Planar Mechanisms
,”
J. Chin. Inst. Eng.
,
37
(
4
), pp.
497
505
.10.1080/02533839.2013.815006
30.
Lin
,
W.-Y.
,
2013
, “
Optimum Path Synthesis of a Geared Five-Bar Mechanism
,”
Adv. Mech. Eng.
,
5
, p.
757935
.10.1155/2013/757935
31.
Kauppinen
,
H.
,
Seppänen
,
T.
, and
Pietikäinen
,
M.
,
1995
, “
An Experimental Comparison of Autoregressive and Fourier-Based Descriptors in 2D Shape Classification
,”
IEEE Trans. Pattern Anal. Mach. Intell.
,
17
(
2
), pp.
201
207
.10.1109/34.368168
32.
Zhang
,
D.
, and
Lu
,
G.
,
2001
, “
A Comparative Study on Shape Retrieval Using Fourier Descriptors With Different Shape Signatures
,”
International Conference on Multimedia and Distance Education (ICIMADE01)
, Fargo, ND, June 1–3.
33.
Zhang
,
D.
, and
Lu
,
G.
,
2005
, “
Study and Evaluation of Different Fourier Methods for Image Retrieval
,”
Image Vision Comput.
,
23
(
1
), pp.
33
49
.10.1016/j.imavis.2004.09.001
34.
Zhang
,
D.
, and
Lu
,
G.
,
2003
, “
A Comparative Study of Curvature Scale Space and Fourier Descriptors
,”
J. Visual Commun. Image Representation
,
14
(
1
), pp.
39
57
.10.1016/S1047-3203(03)00003-8
35.
El-Ghazal
,
A.
,
Basir
,
O.
, and
Belkasim
,
S.
,
2009
, “
Farthest Point Distance: A New Shape Signature for Fourier Descriptors
,”
Signal Process.: Image Commun.
,
24
(
7
), pp.
572
586
.10.1016/j.image.2009.04.001
36.
Mei
,
Y.
,
2010
, “
Robust Affine Invariant Shape Descriptors
,” Ph.D. thesis, Ryerson University, Toronto, ON, Canada.
37.
Hewitt
,
D.
,
2011
, “
Two Dimensional Shape Recognition Using Complex Fourier Analysis and Extension to Three Dimensional Shape Recognition
,” Master's thesis, The University of Texas at San Antonio, San Antonio, TX.
38.
Noklebya
,
S. B.
, and
Podhorodeski
,
R. P.
,
2001
, “
Optimization-Based Synthesis of Grashof Geared Five-Bar Mechanisms
,”
ASME J. Mech. Des.
,
123
(
4
), pp.
529
534
.10.1115/1.1401736
39.
Norton
,
R. L.
,
2004
,
Design of Machinery
,
McGraw-Hill
,
New York
.
40.
Bigun
,
J.
,
2006
,
Vision With Direction
,
Springer
,
Berlin
.
41.
Ting
,
K. L.
,
1994
, “
Mobility Criteria of Geared Five-Bar Linkages
,”
Mech. Mach. Theory
,
29
(
2
), pp.
251
264
.10.1016/0094-114X(94)90034-5
You do not currently have access to this content.