Abstract

Compliant mechanisms are typically designed for varying stiffness from nearly zero to rigid. However, targeted design for fine-tuning within an application's sensitive range of stiffness remains more desirable for practical implementation in accurate loading or positioning systems. To achieve various competing objectives, a “generalized spiral spring” (GSS) is proposed which achieves small size and other objectives by using a reduced number of parameters as provided by the spiral shape description of the components. An analytical model based on virtual work and curved beam theory is developed for accurate prediction of the stiffness. Moreover, finite element (FE) models are also developed for verification of the proposed designs. Multiobjective design optimization (MDO) is conducted to maximize the linearity in the stiffness versus control parameter (CP) response and improve resolution. The proposed analytical model is validated experimentally and computationally. This approach may be used to achieve finesse by accurate positioning with force control for industrial robots and elegant prostheses.

References

1.
Vanderborght
,
B.
,
Van Ham
,
R.
,
Lefeber
,
D.
,
Sugar
,
T. G.
, and
Hollander
,
K.
,
2009
, “
Comparison of Mechanical Design and Energy Consumption of Adaptable, Passive-Compliant Actuators
,”
Int. J. Robot. Res.
,
28
(
1
), pp.
90
103
. 10.1177/0278364908095333
2.
Sugano
,
S.
,
Tsuto
,
S.
, and
Kato
,
I.
,
1992
, “
Force Control of the Robot Finger Joint Equipped With Mechanical Compliance Adjuster
,”
Proceedings of the IEEE/RSJ International Conference on Intelligent Robots and Systems
,
Raleigh, NC
,
July 7–10
, pp.
2005
2013
.
3.
Van Ham
,
R.
,
Vanderborght
,
B.
,
Van Damme
,
M.
,
Verrelst
,
B.
, and
Lefeber
,
D.
,
2007
, “
MACCEPA, the Mechanically Adjustable Compliance and Controllable Equilibrium Position Actuator: Design and Implementation in a Biped Robot
,”
Robot. Auton. Syst.
,
55
(
10
), pp.
761
768
. 10.1016/j.robot.2007.03.001
4.
Van Ham
,
R.
,
Sugar
,
T. G.
,
Vanderborght
,
B.
,
Hollander
,
K. W.
, and
Lefeber
,
D.
,
2009
, “
Compliant Actuator Designs Review of Actuators With Passive Adjustable Compliance/Controllable Stiffness for Robotic Applications
,”
IEEE Robot. Autom. Mag.
,
16
(
3
), pp.
81
94
. 10.1109/MRA.2009.933629
5.
Shepherd
,
M. K.
, and
Rouse
,
E. J.
,
2017
, “
The VSPA Foot: A Quasi-Passive Ankle-Foot Prosthesis With Continuously Variable Stiffness
,”
IEEE Trans. Neural Syst. Rehabil. Eng.
,
25
(
12
), pp.
2375
2386
. 10.1109/TNSRE.2017.2750113
6.
Groothuis
,
S. S.
,
Rusticelli
,
G.
,
Zucchelli
,
A.
,
Stramigioli
,
S.
, and
Carloni
,
R.
,
2014
, “
The Variable Stiffness Actuator vsaUT-II: Mechanical Design, Modeling, and Identification
,”
IEEE-ASME Trans. Mechatron.
,
19
(
2
), pp.
589
597
. 10.1109/TMECH.2013.2251894
7.
Jafari
,
A.
,
Tsagarakis
,
N. G.
,
Sardellitti
,
I.
, and
Caldwell
,
D. G.
,
2014
, “
A New Actuator With Adjustable Stiffness Based on a Variable Ratio Lever Mechanism
,”
IEEE-ASME Trans. Mechatron.
,
19
(
1
), pp.
55
63
. 10.1109/TMECH.2012.2218615
8.
Claros
,
M.
,
Soto
,
R.
,
Rodriguez
,
J. J.
,
Cantu
,
C.
, and
Contreras-Vidal
,
J. L.
,
2013
, “
Novel Compliant Actuator for Wearable Robotics Applications
,”
Annual International Conference of the IEEE Engineering in Medicine and Biology Society
,
Osaka, Japan
,
July 3–7
, pp.
2854
2857
.
9.
Galloway
,
K. C.
,
Clark
,
J. E.
, and
Koditschek
,
D. E.
,
2013
, “
Variable Stiffness Legs for Robust, Efficient, and Stable Dynamic Running
,”
ASME J. Mech. Robot.
,
5
(
1
), pp.
1
11
. 10.1115/1.4007843
10.
Jafari
,
A.
,
Vu
,
H. Q.
, and
Iida
,
F.
,
2016
, “
Determinants for Stiffness Adjustment Mechanisms
,”
J. Intell. Robot. Syst.
,
82
(
3–4
), pp.
435
454
. 10.1007/s10846-015-0253-8
11.
Hollander
,
K.
, and
Sugar
,
T.
, “
Concepts for Compliant Actuation in Wearable Robotic Systems
,”
Proceedings of the US-Korea Conference Science, Technology and Entrepreneurship (UKC 04)
, pp.
644
650
.
12.
Tsagarakis
,
N. G.
,
Sardellitti
,
I.
, and
Caldwell
,
D. G.
,
2011
, “
A New Variable Stiffness Actuator (CompAct-VSA): Design and Modelling
,”
IEEE/RSJ International Conference on Intelligent Robots and Systems
,
San Francisco, CA
,
Sept. 25–30
, pp.
378
383
.
13.
Beyl
,
P.
,
Knaepen
,
K.
,
Duerinck
,
S.
,
Van Damme
,
M.
,
Vanderborght
,
B.
,
Meeusen
,
R.
, and
Lefeber
,
D.
,
2011
, “
Safe and Compliant Guidance by a Powered Knee Exoskeleton for Robot-Assisted Rehabilitation of Gait
,”
Adv. Robot.
,
25
(
5
), pp.
513
535
. 10.1163/016918611X558225
14.
Hollander
,
K. W.
,
Ilg
,
R.
,
Sugar
,
T. G.
, and
Herring
,
D.
,
2006
, “
An Efficient Robotic Tendon for Gait Assistance
,”
J. Biomech. Eng.-Trans. ASME
,
128
(
5
), pp.
788
791
. 10.1115/1.2264391
15.
Sugar
,
T.
, and
Hollander
,
K.
,
2011
,
Adjustable Stiffness Jack Spring Actuator
,
SpringActive Inc.
,
Tempe, AZ
.
16.
Bharadwaj
,
K.
,
Sugar
,
T. G.
,
Koeneman
,
J. B.
, and
Koeneman
,
E. J.
,
2005
, “
Design of a Robotic Gait Trainer Using Spring Over Muscle Actuators for Ankle Stroke Rehabilitation
,”
J. Biomech. Eng. Trans. ASME
,
127
(
6
), pp.
1009
1013
. 10.1115/1.2049333
17.
Rodriguez
,
A. G.
,
Chacon
,
J. M.
,
Donoso
,
A.
, and
Rodriguez
,
A. G. G.
,
2011
, “
Design of an Adjustable-Stiffness Spring: Mathematical Modeling and Simulation, Fabrication and Experimental Validation
,”
Mech. Mach. Theory
,
46
(
12
), pp.
1970
1979
. 10.1016/j.mechmachtheory.2011.07.002
18.
Ghorbani
,
R.
, and
Wu
,
Q.
,
2009
, “
Adjustable Stiffness Artificial Tendons: Conceptual Design and Energetics Study in Bipedal Walking Robots
,”
Mech. Mach. Theory
,
44
(
1
), pp.
140
161
. 10.1016/j.mechmachtheory.2008.02.009
19.
Qaiser
,
Z.
,
Kang
,
L.
, and
Johnson
,
S.
,
2017
, “
Design of a Bioinspired Tunable Stiffness Robotic Foot
,”
Mech. Mach. Theory
,
110
, pp.
1
15
. 10.1016/j.mechmachtheory.2016.12.003
20.
Qaiser
,
Z.
,
Kang
,
L.
,
Ou
,
H.
, and
Johnson
,
S.
,
2018
, “
e-Spring: Circular Arch Mechanism for Large and Linear Tunable Stiffness Control Based on Tuning Deformation Mode Contributions
,”
Mech. Mach. Theory
,
128
, pp.
368
381
. 10.1016/j.mechmachtheory.2018.06.007
21.
Hurst
,
J. W.
,
Chestnutt
,
J. E.
, and
Rizzi
,
A. A.
,
2010
, “
The Actuator With Mechanically Adjustable Series Compliance
,”
IEEE Trans. Robot.
,
26
(
4
), pp.
597
606
. 10.1109/TRO.2010.2052398
22.
Hurst
,
J. W.
,
Chestnutt
,
J. E.
, and
Rizzi
,
A. A.
,
2004
, “
An Actuator With Physically Variable Stiffness for Highly Dynamic Legged Locomotion
,”
International Conference on Robotics and Automation
,
New Orleans, LA
,
Apr. 26–May 1
, pp.
4662
4667
.
23.
Alexander
,
R. M.
,
1990
, “
Three Uses for Springs in Legged Locomotion
,”
Int. J. Robot. Res.
,
9
(
2
), pp.
53
61
. 10.1177/027836499000900205
24.
Dobson
,
A. A.
,
Wei
,
G.
, and
Ren
,
L.
,
2019
, “
Biologically Inspired Design and Development of a Variable Stiffness Powered Ankle-Foot Prosthesis
,”
ASME J. Mech. Robot.
,
11
(
4
), pp.
1
24
.
25.
Stücheli
,
M.
,
Daners
,
M. S.
, and
Meboldt
,
M.
,
2017
, “
Benchmark of the Compactness Potential of Adjustable Stiffness Mechanisms
,”
ASME J. Mech. Robot.
,
9
(
5
), p.
051009
. 10.1115/1.4037114
26.
Smelser
,
G. K.
,
1964
, “
Atlas of Topographical and Applied Human Anatomy: Vol I. Head and Neck
,”
Arch. Ophthalmol.
,
71
(
2
), pp.
288
288
. 10.1001/archopht.1964.00970010304026
27.
Marinković
,
S.
,
Stanković
,
P.
,
Štrbac
,
M.
,
Tomić
,
I.
, and
Ćetković
,
M.
,
2012
, “
Cochlea and Other Spiral Forms in Nature and Art
,”
Am. J. Otolaryngol.
,
33
(
1
), pp.
80
87
. 10.1016/j.amjoto.2011.01.006
28.
Petrides
,
G. A.
, and
Peterson
,
R. T.
,
1973
,
A Field Guide to Trees and Shrubs: Northeastern and North-Central United States and Southeastern and South-Central Canada
, 2nd ed.,
Houghton Mifflin
,
Boston
.
29.
Gartner
,
L. P.
, and
Hiatt
,
J. L.
,
2006
,
Color Textbook of Histology E-Book
, 3rd ed.,
Saunders Imprint
,
Philadelphia
.
30.
Campbell
,
N. A.
,
Reece
,
J. B.
,
Mitchell
,
L. G.
, and
Taylor
,
M. R.
,
2003
,
Biology Concepts & Connections
, 4th ed.,
Benjamin-Cummings
,
San Francisco, CA
.
31.
Bering
,
C. L.
,
1989
, “
Biochemistry, Second Edition (Zubay, Geoffrey)
,”
J. Chem. Educ.
,
66
(
3
), p.
A102
. 10.1021/ed066pA102.2
32.
Haimes
,
Y. V.
,
Lasdon
,
L. S.
, and
Wismer
,
D. A.
,
1971
, “
On a Bicriterion Formation of the Problems of Integrated System Identification and System Optimization
,”
IEEE Trans. Syst. Man Cybernet.
,
1
(
3
), pp.
296
297
.
33.
Rattan
,
S. S.
,
2008
,
Strength of Materials
,
McGraw-Hill Education Pvt Limited
,
India
.
34.
Boresi
,
A. P.
, and
Schmidt
,
R. J.
,
2003
,
Advanced Mechanics of Materials
, 6th ed.,
John Wiley and Sons, Inc.
,
New York
.
35.
Timoshenko
,
S.
,
1934
,
Theory of Elasticity
, 1st ed,
McGraw-Hill Book Company
,
New York, NY
.
36.
Hibbeler
,
R. C.
,
2012
,
Structural Analysis
, 8th ed.,
Prentice Hall
,
Boston
.
37.
Renton
,
J. D.
,
1991
, “
Generalized Beam Theory Applied to Shear Stiffness
,”
Int. J. Solids Struct.
,
27
(
15
), pp.
1955
1967
. 10.1016/0020-7683(91)90188-L
38.
Arora
,
J.
,
2004
,
Introduction to Optimum Design
, 2nd ed,
Elsevier Academic Press
,
Cambridge, MA
.
39.
Mises
,
V.
,
1913
, “
Mechanik der Festen Körper im Plastisch Deformablen Zustand
,”
Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse
,
1913
, pp.
582
592
.
40.
Pilkey
,
W. D.
,
1994
,
Formulas for Stress, Strain, and Structural Matrices
,
John Wiley
,
New York
.
41.
Sanfilippo
,
F.
,
Hatledal
,
L. I.
,
Zhang
,
H.
,
Fago
,
M.
, and
Pettersen
,
K. Y.
,
2015
, “
Controlling Kuka Industrial Robots: Flexible Communication Interface JOpenShowVar
,”
IEEE Robot. Autom. Mag.
,
22
(
4
), pp.
96
109
. 10.1109/MRA.2015.2482839
42.
Yamazaki
,
Y.
,
Sugito
,
K.
, and
Tsuc
,
S.
,
2014
, “
Development of Flexible Manufacturing System
,”
J. Robot. Mechatron.
,
26
(
4
), pp.
426
433
. 10.20965/jrm.2014.p0426
43.
Beyl
,
P.
,
Van Damme
,
M.
,
Van Ham
,
R.
,
Vanderborght
,
B.
, and
Lefeber
,
D.
,
2009
, “
Design and Control of a Lower Limb Exoskeleton for Robot-Assisted Gait Training
,”
Appl. Bionics Biomech.
,
6
(
2
), pp.
229
243
. 10.1155/2009/580734
44.
Niiyama
,
R.
,
Nagakubo
,
A.
, and
Kuniyoshi
,
Y.
,
2007
, “
Mowgli: A Bipedal Jumping and Landing Robot With an Artificial Musculoskeletal System
,”
IEEE International Conference on Robotics and Automation
,
Roma, Italy
,
Apr. 10–14
, pp.
2546
2551
.
45.
Kuindersma
,
S.
,
Deits
,
R.
,
Fallon
,
M.
,
Valenzuela
,
A.
,
Dai
,
H.
,
Permenter
,
F.
,
Koolen
,
T.
,
Marion
,
P.
, and
Tedrake
,
R.
,
2016
, “
Optimization-Based Locomotion Planning, Estimation, and Control Design for the Atlas Humanoid Robot
,”
Auton. Rob.
,
40
(
3
), pp.
429
455
. 10.1007/s10514-015-9479-3
46.
Hurst
,
J.
,
Chestnutt
,
J.
, and
Rizzi
,
A.
,
2007
, “
Design and Philosophy of the BiMASC, a Highly Dynamic Biped
,”
Proceedings of the IEEE International Conference on Robotics and Automation
,
Roma, Italy
,
Apr. 10–14
, pp.
1863
1868
.
47.
In
,
H.
,
Kang
,
B. B.
,
Sin
,
M.
, and
Cho
,
K.
,
2015
, “
Exo-Glove: A Wearable Robot for the Hand With a Soft Tendon Routing System
,”
IEEE Robot. Autom. Mag.
,
22
(
1
), pp.
97
105
. 10.1109/MRA.2014.2362863
48.
Zhang
,
J.
, and
Collins
,
S. H.
,
2017
, “
The Passive Series Stiffness That Optimizes Torque Tracking for a Lower-Limb Exoskeleton in Human Walking
,”
Front. Neurorobot.
,
11
, pp.
68
. 10.3389/fnbot.2017.00068
49.
Rouse
,
E. J.
,
Mooney
,
L. M.
, and
Herr
,
H. M.
,
2014
, “
Clutchable Series-Elastic Actuator: Implications for Prosthetic Knee Design
,”
Int. J. Robot. Res.
,
33
(
13
), pp.
1611
1625
. 10.1177/0278364914545673
50.
Vanderborght
,
B.
,
Van Ham
,
R.
,
Verrelst
,
B.
,
Van Damme
,
M.
, and
Lefeber
,
D.
,
2008
, “
Overview of the Lucy Project: Dynamic Stabilization of a Biped Powered by Pneumatic Artificial Muscles
,”
Adv. Robot.
,
22
(
10
), pp.
1027
1051
. 10.1163/156855308X324749
51.
Grebenstein
,
M.
,
Albu-Schäffer
,
A.
,
Bahls
,
T.
,
Chalon
,
M.
,
Eiberger
,
O.
,
Friedl
,
W.
,
Gruber
,
R.
,
Haddadin
,
S.
,
Hagn
,
U.
,
Haslinger
,
R.
,
Höppner
,
H.
,
Jörg
,
S.
,
Nickl
,
M.
,
Nothhelfer
,
A.
,
Petit
,
F.
,
Reill
,
J.
,
Seitz
,
N.
,
Wimböck
,
T.
,
Wolf
,
S.
,
Wüsthoff
,
T.
, and
Hirzinger
,
G.
,
2011
, “
The DLR Hand Arm System
,”
IEEE International Conference on Robotics and Automation
,
Shanghai, China
,
May 9–13
, pp.
3175
3182
.
52.
Potkonjak
,
V.
,
Svetozarevic
,
B.
,
Jovanovic
,
K.
, and
Holland
,
O.
,
2011
, “
Anthropomimetic Robot With Passive Compliance—Contact Dynamics and Control
,”
19th Mediterranean Conference on Control & Automation (MED)
,
Corfu, Greece
,
June 20–23
, pp.
1059
1064
.
53.
Cao
,
J.
,
Xie
,
S. Q.
, and
Das
,
R.
,
2018
, “
MIMO Sliding Mode Controller for Gait Exoskeleton Driven by Pneumatic Muscles
,”
IEEE Trans. Control Syst. Technol.
,
26
(
1
), pp.
274
281
. 10.1109/TCST.2017.2654424
54.
Festo
,
2018
,
New Scope for Interaction Between Humans and Machines
,
Festo
,
Esslingen, Germany
.
55.
Park
,
H.-W.
, and
Kim
,
S.
,
2014
, “
The MIT Cheetah, an Electrically-Powered Quadrupedal Robot for High-Speed Running
,”
J. Robot. Soc. Jap.
,
32
(
4
), pp.
323
328
. 10.7210/jrsj.32.323
56.
Raibert
,
M.
,
Blankespoor
,
K.
,
Nelson
,
G.
, and
Playter
,
R.
,
2008
, “
BigDog, the Rough-Terrain Quadruped Robot
,”
IFAC Proceedings Volumes Seoul
,
South Korea
, pp.
10822
10825
.
57.
Rehman
,
T. U.
,
Qaiser
,
Z.
, and
Johnson
,
S.
,
2019
, “
Tuning Bifurcation Loads in Bistable Composites With Tunable Stiffness Mechanisms
,”
Mech. Mach. Theory
,
142
, pp.
1
12
. https://doi.org/10.1016/j.mechmachtheory.2019.103585
58.
Jafari
,
A.
, and
Johnson
,
S.
,
2019
, “
The Inherent Power Efficiency of Continuous Tunable Stiffness Mechanisms
,”
Mech. Mach. Theory
,
135
, pp.
208
224
. 10.1016/j.mechmachtheory.2019.02.002
59.
Scharff
,
R. B. N.
,
Wu
,
J.
,
Geraedts
,
J. M. P.
, and
Wang
,
C. C. L.
,
2019
, “
Reducing Out-of-Plane Deformation of Soft Robotic Actuators for Stable Grasping
,”
2nd IEEE International Conference on Soft Robotics
,
(RoboSoft)Seoul, Korea (South)
, pp.
265
270
.
60.
Howell
,
L. L.
,
Midha
,
A.
, and
Norton
,
T. W.
,
1996
, “
Evaluation of Equivalent Spring Stiffness for Use in a Pseudo-Rigid-Body Model of Large-Deflection Compliant Mechanisms
,”
ASME J. Mech. Des.
,
118
(
1
), pp.
126
131
. 10.1115/1.2826843
61.
Awtar
,
S.
, and
Sen
,
S.
,
2010
, “
A Generalized Constraint Model for Two-Dimensional Beam Flexures: Nonlinear Load-Displacement Formulation
,”
ASME J. Mech. Des.
,
132
(
8
), p.
081008
.
62.
Awtar
,
S.
, and
Sen
,
S.
,
2010
, “
A Generalized Constraint Model for Two-Dimensional Beam Flexures: Nonlinear Strain Energy Formulation
,”
ASME J. Mech. Des.
,
132
(
8
), p.
081009
.
63.
Chen
,
G. M.
,
Ma
,
F. L.
,
Hao
,
G. B.
, and
Zhu
,
W. D.
,
2019
, “
Modeling Large Deflections of Initially Curved Beams in Compliant Mechanisms Using Chained Beam Constraint Model
,”
J. Mech. Robot. Trans. ASME
,
11
(
1
), p.
011002
.
64.
Zou
,
R.
,
Xia
,
Y.
,
Liu
,
S.
,
Hu
,
P.
,
Hou
,
W.
,
Hu
,
Q.
, and
Shan
,
C.
,
2016
, “
Isotropic and Anisotropic Elasticity and Yielding of 3D Printed Material
,”
Compos. Part B: Eng.
,
99
, pp.
506
513
. 10.1016/j.compositesb.2016.06.009
You do not currently have access to this content.