This paper is dedicated to the relationship between the external force applied on a point of a robot end-effector and its consequent displacement in static conditions. Both the force and the displacement are herein considered in the Euclidean space E(3). This fact represents a significant simplification of the approach, since it avoids some problems related to the absence of a natural positive definite metric on the Special Euclidean Group SE(3). On the other hand, such restriction allows the method to find closed-form solutions to a large class of problems in robot statics. The peculiar goal of this investigation consists of setting up a procedure which guarantees at least one pose at which any force applied (in E(3)) to an end-effector point is always parallel to its consequent displacement (also in E(3)). This property, which will be referred to as isotropic compliance in E(3), makes the robot tip static behavior uniform with respect to all directions, namely, isotropic, although not homogeneous, since it holds only in some poses. Achieving isotropic compliance in E(3) is a task more general than the classical problem of finding a pose with unit condition number, which does not include the case of different elements in the diagonal joint stiffness matrix. For this reason, the object of the present investigation could not be furtherer simplified to the classical kinetostatic problem in terms of the jacobian matrix alone. The paper reveals how the force–displacement parallelism can be achieved by using a method based on a simple proportional-derivative (PD) controller strategy. The method can be applied when the passive and active stiffness act, on the joints, either in parallel or in series, and the magnitude of the displacement response can be chosen by imposing appropriate values for the overall joints compliance. Results show that for the three analyzed examples, namely, the RR, RRP, and RRR manipulators, with arbitrary lengths of the links, there is, at least, one pose for which the sought property is achieved.

References

References
1.
Ball
,
S.
, 1998,
A Treatise on the Theory of Screws
,
Cambridge University Press
,
Cambridge, UK
.
2.
McCarthy
,
J. M.
, 1990,
Introduction to Theoretical Kinematics
,
MIT Press
,
Cambridge, MA
.
3.
Lipkin
,
H.
, 1985, “
Geometry and Mapping of Screws With Application of the Control of Robot Manipulators
,” Ph.D. thesis, University of Florida, Gainesville, FL.
4.
Patterson
,
T.
, and
Lipkin
,
H. A.
, 1993, “
Structure of Robot Compliance
,”
Trans. ASME J. Mech. Des.
,
115
(
5
), pp.
576
580
.
5.
Patterson
,
T.
, and
Lipkin
,
H. A.
, 1993, “
A Classification of Robot Compliance
,”
Trans. ASME J. Mech. Des.
,
115
(
5
), pp.
581
584
.
6.
Ciblak
,
N.
, and
Lipkin
,
H.
, 1999, “
Synthesis of Cartesian Stiffness for Robotic Applications
,” IEEE International Conference on Robotics and Automation, Vol.
3
, IEEE, Detroit, MI, pp.
2147
2152
.
7.
Uchiyama
,
M.
,
Bayo
,
E.
, and
Palma-Villalon
,
E.
, 1993, “
A Systematic Design Procedure to Minimize a Performance Index for Robot Force Sensors
,”
J. Dyn. Syst., Meas., Control
,
113
(
3
), pp.
388
395
.
8.
Pottmann
,
H.
,
Peternell
,
M.
, and
Ravani
,
B.
, 1999, “
An Introduction to Line Geometry With Applications
,”
Comput.-Aided Des.
,
31
(
1
), pp.
3
16
.
9.
Wolf
,
A.
, and
Shoham
,
M.
, 2003, “
Investigation of Parallel Manipulators Using Linear Complex Approximation
,”
Mech. Des.
,
125
(
3
), pp.
564
572
.
10.
Vinogradov
,
I. B.
,
Kobrinski
,
A. E.
,
Stepanenko
,
Y. E.
, and
Tives
,
L. T.
, 1971, “
Details of Kinematics of Manipulators With the Method of Volumes
,”
Mekhanika Mashin
,
1
(
5
), pp.
5
16
(in Russian).
11.
Yang
,
D. C.
, and
Lai
,
Z. C.
, 1985, “
On the Conditioning of Robotic Manipulators—Service Angle
,”
ASME J. Mech., Transm., Autom. Des.
,
107
(
5
), pp.
262
270
.
12.
Kumar
,
A.
, and
Waldron
,
K. J.
, 1981, “
The Workspace of a Mechanical Manipulator
,”
ASME J. Mech. Des.
,
103
, pp.
665
672
.
13.
Yoshikawa
,
T.
, 1984, “
Analysis and Control of Robot Manipulators With Redundancy
,” Robotics Research: The First International Symposium, pp.
735
747
.
14.
Yoshikawa
,
T.
, 1985, “
Manipulability of Robotic Mechanisms
,”
Int. J. Rob. Res.
,
4
(
2
), pp.
3
9
.
15.
Paul
,
R. P.
, and
Stevenson
,
C. N.
, 1983, “
Kinematics of Robot Wrists
,”
Int. J. Rob. Res.
,
2
(
1
), p.
31
38
.
16.
Forsythe
,
G. E.
, and
Moler
,
C. B.
, 1967,
Computer Solution of Linear Algebraic Systems
, Vol.
7
,
Prentice-Hall
,
New Jersey
.
17.
Salisbury
,
J.
, and
Craig
,
J.
, 1982, “
Articulated Hands: Force Control and Kinematic Issues
,”
Int. J. Rob. Res.
,
1
(
1
), pp.
4
17
.
18.
Angeles
,
J.
, and
Rojas
,
A.
, 1987, “
Manipulator Inverse Kinematics via Condition-Number Minimization and Continuation
,”
Int. J. Rob. Autom.
,
2
(
2
), pp.
61
69
.
19.
Angeles
,
J.
, and
López-Cajún
,
C. S.
, 1992, “
Kinematic Isotropy and the Conditioning Index of Serial Robotic Manipulators
,”
Int. J. Rob. Res.
,
11
(
6
), pp.
560
571
.
20.
Klein
,
C. A.
, and
Blaho
,
B. E.
, 1987, “
Dexterity Measures for the Design and Control of Kinematically Redundant Manipulators
,”
Int. J. Rob. Res.
,
6
(
2
), p.
72
83
.
21.
Gosselin
,
C.
, 1990, “
Dexterity Indices for Planar and Spatial Robotic Manipulators
,” IEEE International Conference on Robotics and Automation, IEEE, Cincinnati, OH, pp.
650
655
.
22.
Gosselin
,
C.
, and
Angeles
,
J.
, 1991, “
A Global Performance Index for the Kinematic Optimization of Robotic Manipulators
,”
J. Mech. Des.
,
113
, pp.
220
226
.
23.
Klein
,
C. A.
, and
Miklos
,
T. A.
, 1991, “
Spatial Robotic Isotropy
,”
Int. J. Rob. Res.
,
10
(
4
), pp.
426
437
.
24.
Angeles
,
J.
, 1992, “
The Design of Isotropic Manipulator Architectures in the Presence of Redundancies
,”
Int. J. Rob. Res.
,
11
(
3
), pp.
196
201
.
25.
Park
,
F. C.
, and
Brockett
,
R. W.
, 1994, “
Kinematic Dexterity of Robotic Mechanisms
,”
Int. J. Rob. Res.
,
13
(
1
), pp.
1
15
.
26.
Park
,
F. C.
, 1995, “
Optimal Robot Design and Differential Geometry
,”
J. Mech. Des.
,
117
, p.
87
92
.
27.
Kim
,
J. O.
, and
Khosla
,
K.
, 1991, “
Dexterity Measures for Design and Control of Manipulators
,”IEEE/RSJ International Workshop on Intelligent Robots and Systems, IEEE, Osaka, Japan, pp.
758
763
.
28.
Stocco
,
L.
,
Salcudean
,
S. E.
, and
Sassani
,
F.
, 1997, “
Mechanism Design for Global Isotropy With Applications to Haptic Interfaces
,” Proceedings of ASME Winter Annual Meeting, Vol.
61
, pp.
115
122
.
29.
Stocco
,
L.
,
Salcudean
,
S. E.
, and
Sassani
,
F.
, 1998, “
Fast Constrained Global Minimax Optimization of Robot Parameters
,”
Robotica
,
16
(
06
), pp.
595
605
.
30.
Merlet
,
J. P.
, 2006, “
Jacobian, Manipulability, Condition Number, and Accuracy of Parallel Robots
,”
J. Mech. Des.
,
128
, p.
199
206
.
31.
Cardou
,
P.
,
Bouchard
,
S.
, and
Gosselin
,
C.
, 2010, “
Kinematic-Sensitivity Indices for Dimensionally Nonhomogeneous Jacobian Matrices
,”
IEEE Trans. Rob.
,
26
(
1
), pp.
166
173
.
32.
Gogu
,
G.
, 2004, “
Structural Synthesis of Fully-Isotropic Translational Parallel Robots via Theory of Linear Transformation
,”
Eur. J. Mech. A/Solids
,
23
(
2
), pp.
1021
1039
.
33.
Asbeck
,
A.
, and
Cutkosky
,
M.
, 2012, “
Designing Compliant Spine Mechanisms for Climbing
,”
ASME J. Mech. Rob.
,
4
(
031007
), pp.
3
8
.
34.
Salisbury
,
J. K.
, 1980, “
Active Stiffness Control of a Manipulator in Cartesian Coordinates
,” 19th IEEE Conference on Decision and Control including the Symposium on Adaptive Processes, Vol.
19
, IEEE, Albuquerque, NM, pp.
95
100
.
35.
De Luca
,
A.
, and
Book
,
W.
, 2008, “
Robots With Flexible Elements
,”
Handbook of Robotics
,
B.
Siciliano
and
O.
Khatib
, eds.,
Springer
,
New York
, pp.
287
287
.
36.
Verotti
,
M.
,
Cappa
,
F.
,
Crescenzi
,
R.
,
Belfiore
,
N. P.
, and
Balucani
,
M.
, 2011, “
Construction and Experimental Simulation of a Tethered MEMS-Based 3 DOF Silicon Micro Robot
,” Proceedings of the RAAD 2011, 20th International Workshop on Robotics in Alpe-Adria-Danube Region,
E. Z.
Kolibal
, ed.
37.
Balucani
,
M.
,
Belfiore
,
N. P.
,
Crescenzi
,
R.
, and
Verotti
,
M.
, 2011, “
The Development of a MEMS/NEMS-Based 3 DOF Compliant Micro Robot
,”
Int. J. Mech. Control
,
12
(
1
), pp.
3
10
.
38.
Belfiore
,
N. P.
,
Faglia
,
R.
, and
Resconi
,
G.
, 1993, “
Revisiting Some Static and Kinematic Features of a SCARA Robot by Means of GSLT and the Lie Product
,” 3rd National Conference on Applied Mechanisms and Robotics, Vol.
8
.
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