Abstract

This paper studies the problem of geometric constraint acquisition from a given planar motion task using Fourier descriptor. In the previous work, we established a computational geometric framework for simultaneous type and dimensional synthesis of planar dyads by extracting line or circle constraints from a sequence of task poses. In cases where six or more poses are specified as the desired movement, the resulting optimal constraint may be nowhere in the accuracy neighborhood to be viewed as an approximate line or circle. The approach herein enhances the framework by exploiting Fourier transform to capture the feasible constraint of a continuous motion with a large set of poses. Theoretically, any arbitrary point trajectory on the task motion can be transformed to an array of harmonics and used as a constraint; on a practical level, only those with low number of harmonics could allow accurate realization by simple planar mechanisms suitable for real applications, e.g., four- and six-bar linkages, cams, and coupled serial chains. Therefore, the practical goal is to find the simple Fourier constraint defined with the least number of harmonics. Two examples of designing assistive mechanisms for upper- and lower-limb rehabilitation are provided in the end to illustrate the effectiveness of our approach.

References

1.
Sandor
,
G. N.
, and
Erdman
,
A. G.
,
2009
,
Advanced Mechanism Design: Analysis and Synthesis
,
Prentice Hall
,
Englewood Cliffs, NJ
.
2.
McCarthy
,
J. M.
, and
Soh
,
G. S.
,
2010
,
Geometric Design of Linkages
,
Springer
,
New York
.
3.
Lohse
,
P.
,
2013
,
Getriebesynthese: Bewegungsablufe Ebener Koppelmechanismen
,
Springer
,
Berlin
.
4.
Liu
,
Y.
, and
McPhee
,
J.
,
2007
, “
Automated Kinematic Synthesis of Planar Mechanisms With Revolute Joints
,”
Mech. Based Des. Struct. Mach.
,
35
(
4
), pp.
405
445
.
5.
Oliva
,
J.
, and
Goodman
,
E. D.
,
2010
, “
Simultaneous Type and Dimensional Synthesis of Planar 1DOF Mechanisms Using Evolutionary Search and Convertible Agents
,”
ASME J. Mech. Rob.
,
2
(
3
), p.
031001
.
6.
Shen
,
Z.
,
Allison
,
G.
, and
Cui
,
L.
,
2018
, “
An Integrated Type and Dimensional Synthesis Method to Design One Degree-of-Freedom Planar Linkages With Only Revolute Joints for Exoskeletons
,”
ASME J. Mech. Des.
,
140
(
9
), p.
092302
.
7.
Pucheta
,
M.
, and
Cardona
,
A.
,
2013
, “
Topological and Dimensional Synthesis of Planar Linkages for Multiple Kinematic Tasks
,”
Multibody Syst. Dyn.
,
29
(
2
), pp.
189
211
.
8.
Funke
,
L.
, and
Schmiedeler
,
J. P.
,
2017
, “
Simultaneous Topological and Dimensional Synthesis of Planar Morphing Mechanisms
,”
ASME J. Mech. Rob.
,
9
(
2
), p.
021009
.
9.
Santiago-Valentín
,
E.
,
Portilla-Flores
,
E. A.
,
Mezura-Montes
,
E.
,
Vega-Alvarado
,
E.
,
Calva-Yanez
,
M. B.
, and
Pedroza-Villalba
,
M.
,
2019
, “
A Graph-Theory-Based Method for Topological and Dimensional Representation of Planar Mechanisms as a Computational Tool for Engineering Design
,”
IEEE Access
,
7
, pp.
587
596
.
10.
Heo
,
J. C.
, and
Yoon
,
G. H.
,
2013
, “
Size and Configuration Syntheses of Rigid-Link Mechanisms With Multiple Rotary Actuators Using the Constraint Force Design Method
,”
Mech. Mach. Theory
,
64
, pp.
18
38
.
11.
Yu
,
J.
,
Han
,
M. S.
, and
Kim
,
Y. Y.
,
2020
, “
Simultaneous Shape and Topology Optimization of Planar Linkage Mechanisms Based on the Spring-Connected Rigid Block Model
,”
ASME J. Mech. Des.
,
142
(
1
), p.
011401
.
12.
Yim
,
N. H.
,
Lee
,
J.
,
Kim
,
J.
, and
Kim
,
Y. Y.
,
2021
, “
Big Data Approach for the Simultaneous Determination of the Topology and End-Effector Location of a Planar Linkage Mechanism
,”
Mech. Mach. Theory
,
163
, p.
104375
.
13.
Wu
,
J.
,
Ge
,
Q. J.
,
Su
,
H.-J.
, and
Gao
,
F.
,
2013
, “
Kinematic Acquisition of Geometric Constraints for Task-oriented Design of Planar Mechanisms
,”
ASME J. Mech. Rob.
,
5
(
1
), p.
011003
.
14.
Bai
,
S.
,
Wang
,
D.
, and
Dong
,
H.
,
2016
, “
A Unified Formulation for Dimensional Synthesis of Stephenson Linkages
,”
ASME J. Mech. Rob.
,
8
(
4
), p.
041009
.
15.
Lin
,
S.
,
Wang
,
H.
,
Zhang
,
Y.
, and
Jiang
,
J.
,
2020
, “
Kinematic Geometry Description of a Line With Four Positions and Its Application in Dimension Synthesis of Spatial Linkage
,”
ASME J. Mech. Rob.
,
12
(
3
), p.
031006
.
16.
Cao
,
Y.
, and
Han
,
J.
,
2020
, “
Solution Region-Based Synthesis Methodology for Spatial Hccc Linkages
,”
Mech. Mach. Theory
,
143
, p.
103619
.
17.
Cera
,
M.
,
Cirelli
,
M.
,
Pennestrì
,
E.
,
Salerno
,
R.
, and
Valentini
,
P.
,
2022
, “
Path-Constrained Points Synthesis of Symmetric Mechanisms for Prescribed Higher-Order Curvature Features
,”
Mech. Mach. Theory
,
167
, p.
104562
.
18.
Hayes
,
M. J. D.
,
Luu
,
T.
, and
Chang
,
X.-W.
,
2004
, “Kinematic Mapping Application to Approximate Type and Dimension Synthesis of Planar Mechanism,”
9th Advances in Robotic Kinematics
,
J.
Lenarčič
and
C.
Galletti
, eds.,
Kluwer Academic Publishers
,
Dordrecht
, pp.
41
48
.
19.
Hayes
,
M.
, and
Rucu
,
S. R.
,
2011
, “
Quadric Surface Fitting Applications to Approximate Dimensional Synthesis
,”
13th World Congress in Mechanisms and Machine Theory
,
Guanajuato, Mexico
,
June 19–23
, pp.
10
25
.
20.
McCarthy
,
J. M.
,
1990
,
Introduction to Theoretical Kinematics
,
MIT Press
,
Cambridge, MA
.
21.
Zhao
,
P.
,
Li
,
X.
,
Purwar
,
A.
, and
Ge
,
Q. J.
,
2016
, “
A Task-Driven Unified Synthesis of Planar Four-Bar and Six-Bar Linkages With R- and P-Joints for Five-Position Realization
,”
ASME J. Mech. Rob.
,
8
(
6
), p.
061003
.
22.
Zhao
,
P.
,
Li
,
X.
,
Zhu
,
L.
,
Zi
,
B.
, and
Ge
,
Q. J.
,
2016
, “
A Novel Motion Synthesis Approach With Expandable Solution Space for Planar Linkages Based on Kinematic-Mapping
,”
Mech. Mach. Theory
,
105
, pp.
164
175
.
23.
Ge
,
Q. J.
,
Purwar
,
A.
, and
Zhao
,
P.
,
2017
, “
A Task-Driven Approach to Unified Synthesis of Planar Four-Bar Linkages Using Algebraic Fitting of a Pencil of G-Manifolds
,”
ASME J. Comput. Inf. Sci. Eng.
,
17
(
3
), p.
031011
.
24.
McCarthy
,
J.
, “Want a Patent? Try a Six-Bar Linkage,” https://mechanicaldesign101.com/2016/07/26/want-a-patent-try-a-six-bar-linkage/, Accessed December 24, 2021.
25.
Boothroyd
,
G.
,
Dewhurst
,
P.
, and
Knight
,
W. A.
,
2010
,
Product Design for Manufacture and Assembly
,
CRC Press
,
Boca Raton, FL
.
26.
Deshpande
,
S.
, and
Purwar
,
A.
,
2019
, “
A Machine Learning Approach to Kinematic Synthesis of Defect-Free Planar Four-Bar Linkages
,”
ASME J. Comput. Inf. Sci. Eng.
,
19
(
2
), p.
021004
.
27.
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
.
28.
Wu
,
J.
,
Ge
,
Q. J.
,
Gao
,
F.
, and
Guo
,
W. Z.
,
2011
, “
On the Extension of a FD Based Method for Planar Four-Bar Linkage Synthesis for Generation of Open and Closed Paths
,”
ASME J. Mech. Rob.
,
3
(
3
), p.
031002
.
29.
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
), p.
064501
.
30.
Li
,
X.
,
Wei
,
S.
,
Liao
,
Q.
, and
Zhang
,
Y.
,
2020
, “
A Novel Analytical Method for Four-Bar Path Generation Synthesis Based on Fourier Series
,”
Mech. Mach. Theory
,
144
, p.
103671
.
31.
Sun
,
J.
,
Chen
,
L.
, and
Chu
,
J.
,
2016
, “
Motion Generation of Spherical Four-Bar Mechanism Using Harmonic Characteristic Parameters
,”
Mech. Mach. Theory
,
95
(
6
), pp.
76
92
.
32.
Chu
,
J.
, and
Sun
,
J.
,
2010
, “
A New Approach to Dimensional Synthesis of Spatial Four-Bar Linkage Through Numerical Atlas Method
,”
ASME J. Mech. Rob.
,
2
(
4
), p.
041004
.
33.
Krovi
,
V.
, and
Nie
,
X.
,
2008
, “
Design of Reconfigurable Coupled-Serial-Chain-Based Manipulation Assistive Aids
,”
Robot. Comput. Integr. Manuf.
,
24
(
3
), pp.
345
358
.
34.
Liu
,
Y.
, and
McCarthy
,
J. M.
,
2017
, “
Design of Mechanisms to Draw Trigonometric Plane Curves
,”
ASME J. Mech. Rob.
,
9
(
2
), p.
024503
.
35.
Liu
,
Y.
, and
Michael McCarthy
,
J.
,
2017
, “
Design of a Linkage System to Write in Cursive
,”
ASME J. Comput. Inform. Sci. Eng.
,
17
(
3
), p.
031015
.
36.
Wu
,
J.
,
2010
, “Variational Kinematic Geometry for Task Centered Design of Mechanisms and Robotic Systems,” Ph.D. thesis,
Department of Mechanical Engineering, Stony Brook University
,
Stony Brook, NY
.
37.
Li
,
X.
, and
Chen
,
P.
,
2017
, “
A Parametrization-Invariant Fourier Approach to Planar Linkage Synthesis for Path Generation
,”
Math. Probl. Eng.
,
2017
,
1
16
.
38.
Han
,
J.
,
Kamber
,
M.
, and
Pei
,
J.
,
2012
,
Data Mining: Concepts and Techniques
, 3rd ed.,
Morgan Kaufmann
,
Waltham, MA
.
39.
Maciejasz
,
P.
,
Eschweiler
,
J.
,
Gerlach-Hahn
,
K.
,
Jansen-Troy
,
A.
, and
Leonhardt
,
S.
,
2014
, “
A Survey on Robotic Devices for Upper Limb Rehabilitation
,”
J. Neuroeng. Rehabil.
,
11
(
1
), pp.
1
29
.
40.
Yang
,
N.
,
An
,
Q.
,
Kogami
,
H.
,
Yoshida
,
K.
,
Yamakawa
,
H.
,
Tamura
,
Y.
,
Shimoda
,
S.
, et al.
2020
, “
Temporal Muscle Synergy Features Estimate Effects of Short-Term Rehabilitation in Sit-to-Stand of Post-Stroke Patients
,”
IEEE Robot. Autom. Lett.
,
5
(
2
), pp.
1796
1802
.
41.
Veerubhotla
,
A.
,
Ehrenberg
,
N.
,
Ibironke
,
O.
, and
Pilkar
,
R.
,
2021
, “
Objective Evaluation of Risk of Falls in Individuals With Chronic Stroke: Feasibility Study
,”
Arch. Phys. Med. Rehabil.
,
102
(
10
), p.
e101
.
42.
An
,
Q.
,
Yang
,
N.
,
Yamakawa
,
H.
,
Kogami
,
H.
,
Yoshida
,
K.
,
Wang
,
R.
,
Yamashita
,
A.
, et al.
2021
, “
Classification of Motor Impairments of Post-Sstroke Patients Based on Force Applied to a Handrail
,”
IEEE Trans. Neural Syst. Rehabil. Eng.
,
29
, pp.
2399
2406
.
43.
Li
,
X.
,
Chen
,
P.
,
Yu
,
X.
, and
Jiang
,
N.
,
2022
, “
Analysis of the Relationship Between Motor Imagery and Age-Related Fatigue for CNN Classification of the EEG Data
,”
Front. Aging Neurosci.
,
14
, p.
909571
.
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