This paper presents methods for the realization of 2 × 2 translational compliance matrices using serial mechanisms having three joints, each either revolute or prismatic and each with selectable compliance. The geometry of the mechanism and the location of the compliance frame relative to the mechanism base are each arbitrary but specified. Necessary and sufficient conditions for the realization of a given compliance with a given mechanism are obtained. We show that, for an appropriately constructed serial mechanism having at least one revolute joint, any single 2 × 2 compliance matrix can be realized by properly choosing the joint compliances and the mechanism configuration. For each type of three-joint combination, requirements on the redundant mechanism geometry are identified for the realization of every point planar elastic behavior at a given location, just by changing the mechanism configuration and the joint compliances.

References

References
1.
Albu-Schaffer
,
A.
,
Eiberger
,
O.
,
Grebenstein
,
M.
,
Haddadin
,
S.
,
Ott
,
C.
,
Wimbock
,
T.
,
Wolf
,
S.
, and
Hirzinge
,
G.
,
2008
, “
Soft Robotics: From Torque Feedback-Controlled Lightweight Robots to Intrinsically Compliant Systems
,”
IEEE Rob. Autom. Mag.
,
15
(
3
), pp.
20
30
.
2.
Ham
,
R. V.
,
Sugar
,
T. G.
,
Vanderborght
,
B.
,
Hollander
,
K. W.
, and
Lefeber
,
D.
,
2009
, “
Compliant Actuator Designs
,”
IEEE Rob. Autom. Mag.
,
16
(
3
), pp.
81
94
.
3.
Pratt
,
G.
, and
Williamson
,
M.
,
1995
, “
Series Elastic Actuators
,” 1995
IEEE/RSJ
International Conference on Intelligent Robots and Systems
, Aug. 5–9, Vol.
1
, pp.
399
406
.
4.
Ham
,
R. V.
,
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 Rob. Autom. Mag.
,
16
(
3
), pp.
81
94
.
5.
Ball
,
R. S.
,
1900
,
A Treatise on the Theory of Screws
,
Cambridge University Press
,
London, UK
.
6.
Dimentberg
,
F. M.
,
1965
, “
The Screw Calculus and Its Applications in Mechanics
,” Foreign Technology Division, Wright-Patterson Air Force Base, Dayton, OH,
Document No. FTD-HT-23-1632-67
.
7.
Griffis
,
M.
, and
Duffy
,
J.
,
1991
, “
Kinestatic Control: A Novel Theory for Simultaneously Regulating Force and Displacement
,”
ASME J. Mech. Des.
,
113
(
4
), pp.
508
515
.
8.
Patterson
,
T.
, and
Lipkin
,
H.
,
1993
, “
Structure of Robot Compliance
,”
ASME J. Mech. Des.
,
115
(
3
), pp.
576
580
.
9.
Patterson
,
T.
, and
Lipkin
,
H.
,
1993
, “
A Classification of Robot Compliance
,”
ASME J. Mech. Des.
,
115
(
3
), pp.
581
584
.
10.
Loncaric
,
J.
,
1987
, “
Normal Forms of Stiffness and Compliance Matrices
,”
IEEE J. Rob. Autom.
,
3
(
6
), pp.
567
572
.
11.
Huang
,
S.
, and
Schimmels
,
J. M.
,
1998
, “
The Bounds and Realization of Spatial Stiffnesses Achieved With Simple Springs Connected in Parallel
,”
IEEE Trans. Rob. Autom.
,
14
(
3
), pp.
466
475
.
12.
Huang
,
S.
, and
Schimmels
,
J. M.
,
2000
, “
The Bounds and Realization of Spatial Compliances Achieved With Simple Serial Elastic Mechanisms
,”
IEEE Trans. Rob. Autom.
,
16
(
1
), pp.
99
103
.
13.
Roberts
,
R. G.
,
1999
, “
Minimal Realization of a Spatial Stiffness Matrix With Simple Springs Connected in Parallel
,”
IEEE Trans. Rob. Autom.
,
15
(
5
), pp.
953
958
.
14.
Ciblak
,
N.
, and
Lipkin
,
H.
,
1999
, “
Synthesis of Cartesian Stiffness for Robotic Applications
,”
IEEE
International Conference on Robotics and Automation
, May 10–15, pp.
2147
2152
.
15.
Huang
,
S.
, and
Schimmels
,
J. M.
,
1998
, “
Achieving an Arbitrary Spatial Stiffness With Springs Connected in Parallel
,”
ASME J. Mech. Des.
,
120
(
4
), pp.
520
526
.
16.
Huang
,
S.
, and
Schimmels
,
J. M.
,
2000
, “
The Eigenscrew Decomposition of Spatial Stiffness Matrices
,”
IEEE Trans. Rob. Autom.
,
16
(
2
), pp.
146
156
.
17.
Huang
,
S.
, and
Schimmels
,
J. M.
,
2001
, “
A Classification of Spatial Stiffness Based on the Degree of Translational–Rotational Coupling
,”
ASME J. Mech. Des.
,
123
(
3
), pp.
353
358
.
18.
Roberts
,
R. G.
,
2000
, “
Minimal Realization of an Arbitrary Spatial Stiffness Matrix With a Parallel Connection of Simple Springs and Complex Springs
,”
IEEE Trans. Rob. Autom.
,
16
(
5
), pp.
603
608
.
19.
Choi
,
K.
,
Jiang
,
S.
, and
Li
,
Z.
,
2002
, “
Spatial Stiffness Realization With Parallel Springs Using Geometric Parameters
,”
IEEE Trans. Rob. Autom.
,
18
(
3
), pp.
264
284
.
20.
Petit
,
F. P.
,
2014
, “
Analysis and Control of Variable Stiffness Robots
,”
Ph.D. thesis
, ETH Zurich, Zurich, Switzerland.
21.
Belfiore
,
N. P.
,
Verotti
,
M.
,
Giamberardino
,
P. D.
, and
Rudas
,
I. J.
,
2012
, “
Active Joint Stiffness Regulation to Achieve Isotropic Compliance in the Euclidean Space
,”
ASME J. Mech. Rob.
,
4
(
4
), p.
041010
.
22.
Verotti
,
M.
, and
Belfiore
,
N. P.
,
2016
, “
Isotropic Compliance in E(3): Feasibility and Workspace Mapping
,”
ASME J. Mech. Rob.
,
8
(
6
), p.
061005
.
23.
Verotti
,
M.
,
Masarati
,
P.
,
Morandini
,
M.
, and
Belfiore
,
N.
,
2016
, “
Isotropic Compliance in the Special Euclidean Group SE(3)
,”
Mech. Mach. Theory
,
98
, pp.
263
281
.
24.
Huang
,
S.
, and
Schimmels
,
J. M.
,
2016
, “
Realization of Point Planar Elastic Behaviors Using Revolute Joint Serial Mechanisms Having Specified Link Lengths
,”
Mech. Mach. Theory
,
103
, pp.
1
20
.
25.
Albu-Schaffer
,
A.
,
Fischer
,
M.
,
Schreiber
,
G.
,
Schoeppe
,
F.
, and
Hirzinger
,
G.
,
2004
, “
Soft Robotics: What Cartesian Stiffness Can We Obtain With Passively Compliant, Uncoupled Joints?
,” 2004
IEEE/RSJ
International Conference on Intelligent Robots and Systems
, Sept. 28–Oct. 2, pp.
3295
3301
.
26.
Huang
,
S.
, and
Schimmels
,
J. M.
,
2002
, “
The Duality in Spatial Stiffness and Compliance as Realized in Parallel and Serial Elastic Mechanisms
,”
ASME J. Dyn. Syst., Meas., Control
,
124
(
1
), pp.
76
84
.
You do not currently have access to this content.