Abstract

An open path synthesis method for a spatial revolute-revolute-spherical-spherical (RRSS) mechanism is presented in this paper. The mathematical model for the trajectory curve is established. The characteristics of an RRSS mechanism in a standard installation position are revealed: the projection points of the coupler curve on the Oxy plane rotate by the corresponding input angles around the z-axis, and the generated points lie on an ellipse. Based on this finding, a 17-dimensional path generation problem can be translated into two lower-dimensional matching recognition problems and one actual size and installation position calculation problem. The path generation can be achieved by three steps. First, a database of four dimensional rotation angle parameters is established. By comparing the similarities between the mechanism feature curve of the prescribed open curve and its corresponding mechanism feature ellipse (MFE), the angles of installation, the initial angle of the input link, and the elliptic feature parameters of the desired RRSS mechanism can be approximately determined. Then, a 13-dimensional dynamic self-adapting numerical atlas database is established, which contains six basic dimensional types (BDTs) and seven wavelet feature parameters, and the BDTs of the desired RRSS mechanism are obtained. Finally, based on the relationship between the MFE of the prescribed curve and the BDTs of the desired RRSS mechanism, the calculation models for the actual link lengths and installation positions of the desired RRSS mechanism were established. Three examples are presented in this paper.

References

References
1.
Pennock
,
G. R.
, and
Sankaranarayanan
,
H.
,
2003
, “
Path Curvature of a Geared Seven-bar Mechanism
,”
Mech. Mach. Theory
,
38
(
12
), pp.
1345
1361
. 10.1016/S0094-114X(03)00091-0
2.
Pennock
,
G. R.
, and
Raje
,
N. N.
,
2004
, “
Curvature Theory for the Double Flier Eight-bar Linkage
,”
Mech. Mach. Theory
,
39
(
7
), pp.
665
679
. 10.1016/j.mechmachtheory.2004.01.003
3.
Kafash
,
S. H.
, and
Nahvi
,
A.
,
2017
, “
Optimal Synthesis of Four-Bar Path Generator Linkages Using Circular Proximity Function
,”
Mech. Mach. Theory
,
115
, pp.
18
34
. 10.1016/j.mechmachtheory.2017.04.010
4.
Singh
,
R.
,
Chaudhary
,
H.
, and
Singh
,
A. K.
,
2017
, “
Defect-free Optimal Synthesis of Crank-Rocker Linkage Using Nature-Inspired Optimization Algorithms
,”
Mech. Mach. Theory
,
116
, pp.
105
122
. 10.1016/j.mechmachtheory.2017.05.018
5.
Bulatović
,
R. R.
,
Miodragović
,
G.
, and
Bošković
,
M. S.
,
2016
, “
Modified Krill Herd (MKH) Algorithm and Its Application in Dimensional Synthesis of a Four-Bar Linkage
,”
Mech. Mach. Theory
,
95
, pp.
1
21
. 10.1016/j.mechmachtheory.2015.08.004
6.
Freudenstein
,
F.
,
1955
, “
Approximate Synthesis of Four-bar Linkages
,”
Trans. ASME
,
77
(
6
), pp.
853
861
.
7.
Bai
,
S. P.
, and
Angeles
,
J.
,
2008
, “
A Unified Input-Output Analysis of Four-bar Linkages
,”
Mech. Mach. Theory
,
43
(
2
), pp.
240
251
. 10.1016/j.mechmachtheory.2007.01.002
8.
Bai
,
S. P.
, and
Angeles
,
J.
,
2015
, “
Coupler-Curve Synthesis of Four-Bar Linkages via a Novel Formulation
,”
Mech. Mach. Theory
,
94
, pp.
177
187
. 10.1016/j.mechmachtheory.2015.08.010
9.
Peñuñuri
,
F.
,
Peón-Escalante
,
R.
,
Villanueva
,
C.
, and
Cruz-Villar
,
C. A.
,
2012
, “
Synthesis of Spherical 4R Linkage for Path Generation Using Differential Evolution
,”
Mech. Mach. Theory
,
57
, pp.
62
70
. 10.1016/j.mechmachtheory.2012.07.003
10.
Marble
,
S. D.
, and
Pennock
,
G. R.
,
2000
, “
Algebraic-Geometric Properties of the Coupler Curves of the RCCC Spatial Four-Bar Mechanism
,”
Mech. Mach. Theory
,
35
(
5
), pp.
675
693
. 10.1016/S0094-114X(99)00039-7
11.
Pennock
,
G. R.
, and
Raghavan
,
M.
,
2006
, “
Spatial Mechanisms and Robot Manipulators
,”
ASME J. Mech. Des.
,
128
(
1
), pp.
149
150
. 10.1115/1.2135674
12.
Figliolini
,
G.
,
Rea
,
P.
, and
Angeles
,
J.
,
2015
, “
The Synthesis of the Axodes of RCCC Linkages
,”
ASME J. Mech. Rob.
,
8
(
2
), p.
021011
. 10.1115/1.4031950
13.
Hrones
,
J. A.
, and
Nelson
,
G. L.
,
1951
,
Analysis of the Four-Bar Linkage
,
MIT Press and Wiley
,
New York
.
14.
Lin
,
S.
,
Wang
,
H. C.
,
Liu
,
J.
, and
Zhang
,
Y.
,
2018
, “
Geometric Method of Spatial Linkages Synthesis for Function Generation with Three Finite Positions
,”
ASME J. Mech. Des.
,
140
(
8
), p.
082303
. 10.1115/1.4040171
15.
Sun
,
J. W.
,
Liu
,
W. R.
, and
Chu
,
J. K.
,
2018
, “
Synthesis of Spherical Four-bar Linkage for Open Path Generation Using Wavelet Feature Parameters
,”
Mech. Mach. Theory
,
128
, pp.
33
46
. 10.1016/j.mechmachtheory.2018.05.008
16.
Li
,
X.
,
Wu
,
J.
, and
Ge
,
Q. J.
,
2016
, “
A Fourier Descriptor-Based Approach to Design Space Decomposition for Planar Motion Approximation
,”
ASME J. Mech. Rob.
,
8
(
6
), pp.
699
708
.10.1115/1.4033528
17.
McGarva
,
J. R.
, and
Mulineux
,
G.
,
1993
, “
Harmonic Representation of Closed Curves
,”
Appl. Math. Model
,
17
(
4
), pp.
213
218
. 10.1016/0307-904X(93)90109-T
18.
Lee
,
W. T.
,
Cosme
,
J.
, and
Russell
,
K.
,
2019
, “
Revolute-Cylindrical-Cylindrical-Cylindrical Linkage Optimum Dimensional Synthesis with Static Structural Loading
,”
IMechE Part C: J. Mech. Eng. Sci.
,
233
(
1
), pp.
256
268
. 10.1177/0954406218757566
19.
Li
,
X. Y.
, and
Chen
,
P.
,
2017
, “
A Parametrization-Invariant Fourier Approach to Planar Linkage Synthesis for Path Generation
,”
Math. Prob. Eng
.,
2017
. doi:10.1155/2017/8458149
20.
Lee
,
W. T.
, and
Russell
,
K.
,
2018
, “
Developments in Quantitative Dimensional Synthesis (1970–Present): Four-bar Path and Function Generation
,”
Inverse Prob. Sci. Eng.
,
26
(
9
), pp.
1280
1304
. 10.1080/17415977.2017.1396328
21.
Hoeltzel
,
D. A.
, and
Chieng
,
W. H.
,
1990
, “
Pattern Matching Synthesis as an Automated Approach to Mechanism Design
,”
ASME J. Mech. Des.
,
112
(
6
), pp.
190
199
. 10.1115/1.2912592
22.
Watanabe
,
K.
,
1992
, “
Application of Natural Equations to the Synthesis of Curve Generating Mechanisms
,”
Mech. Mach. Theory
,
27
(
3
), pp.
261
273
. 10.1016/0094-114X(92)90016-B
23.
Lan
,
Z. H.
, and
Zou
,
H. J.
,
1999
, “
Concurrent Optimum Synthesis of Path Generating Mechanisms Based on the Local Characteristics
,”
Chin. J. Mech. Eng.
,
35
(
5
), pp.
16
19
.
24.
Lan
,
Z.
,
Zou
,
H.
, and
Lu
,
L.
,
2002
, “
Kinematic Decomposition of Coupler Plane and the Study on the Formation and Distribution of Coupler Curves
,”
Mech. Mach. Theory
,
37
(
1
), pp.
115
126
. 10.1016/S0094-114X(01)00054-4
25.
McGarva
,
J.
,
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
26.
Unruh
,
V.
, and
Krishnaswami
,
P.
,
1995
, “
A Computer-Aided Design Technique for Semi-Automated Infinite Point Coupler Curve Synthesis of Four-bar Linkages
,”
ASME J. Mech. Des.
,
117
(
1
), pp.
143
149
. 10.1115/1.2826099
27.
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
28.
Mullineux
,
G.
,
2011
, “
Atlas of Spherical Four-Bar Mechanisms
,”
Mech. Mach. Theory
,
46
(
11
), pp.
1811
1823
. 10.1016/j.mechmachtheory.2011.06.001
29.
Wu
,
J.
,
Ge
,
Q. J.
,
Gao
,
F.
, and
Guo
,
W. Z.
,
2011
, “
On the Extension of a Fourier Descriptor Based Method for Planar Four-bar Linkage Synthesis for Generation of Open and Closed Paths
,”
ASME J. Mech. Rob.
,
3
(
3
), p.
031002
. 10.1115/1.4004227
30.
Yue
,
C.
,
Su
,
H. J.
, and
Ge
,
Q. J.
,
2012
, “
A Hybrid Computer-Aided Linkage Design System for Tracing Open and Closed Planar Curves
,”
Comput.-Aided Des.
,
44
(
11
), pp.
1141
1150
. 10.1016/j.cad.2012.06.004
31.
Galán-Marín
,
G.
,
Alonso
,
F. J.
, and
Castillo
,
J. M. D.
,
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
32.
Chu
,
J. K.
, and
Sun
,
J. W.
,
2010
, “
A new Approach to Dimension Synthesis of Spatial Four-Bar Linkage Through Numerical Atlas Method
,”
ASME J. Mech. Rob.
,
2
(
4
), p.
041004
. 10.1115/1.4001774
33.
Sun
,
J.
, and
Chu
,
J.
,
2010
, “
Fourier Series Representation of the Coupler Curves of Spatial Linkages
,”
Appl. Math. Modell
,
34
(
5
), pp.
1396
1403
. 10.1016/j.apm.2009.08.030
34.
Sun
,
J. W.
,
Liu
,
W. R.
, and
Chu
,
J. K.
,
2015
, “
Dimensional Synthesis of Open Path Generator of Four-Bar Mechanisms Using the Haar Wavelet
,”
ASME J. Mech. Des.
,
137
(
8
), p.
082303
. 10.1115/1.4030651
35.
McGarva
,
J. R.
, and
Mullineux
,
G.
,
1992
, “
A New Methodology for Rapid Synthesis of Function Generators
,”
IMechE Part C: J. Mech. Eng. Sci.
,
206
(
6
), pp.
391
398
. 10.1243/PIME_PROC_1992_206_146_02
36.
Ullah
,
I.
, and
Kota
,
S.
,
1997
, “
Optimal Synthesis of Mechanisms for Path Generation Using Fourier Descriptors and Global Search Method
,”
ASME J. Mech. Des
,
119
(
4
), pp.
504
510
. 10.1115/1.2826396
37.
Nie
,
X. C.
, and
Krovl
,
V.
,
2005
, “
Fourier Methods for Kinematic Synthesis of Coupled Serial Chain Mechanisms
,”
ASME J. Mech. Des.
,
127
(
2
), pp.
232
241
. 10.1115/1.1829726
38.
Khan
,
N.
,
Ullah
,
I.
, and
Algrafi
,
M.
,
2015
, “
Dimensional Synthesis of Mechanical Linkages Using Artificial Neural Networks and Fourier Descriptors
,”
Mech. Sci.
,
6
(
1
), pp.
29
34
. 10.5194/ms-6-29-2015
You do not currently have access to this content.