In this work, a method is presented to geometrically determine the dexterous workspace boundary of kinematically redundant n-PRRR (n-PRRR indicates that the manipulator consists of n serial kinematic chains that connect the base to the end-effector. Each chain is composed of two actuated (therefore underlined) joints and two passive revolute joints. P indicates a prismatic joint while R indicates a revolute joint.) planar parallel manipulators. The dexterous workspace of each nonredundant RRR kinematic chain is first determined using a four-bar mechanism analogy. The effect of the prismatic actuator is then considered to yield the workspace of each PRRR kinematic chain. The intersection of the dexterous workspaces of all the kinematic chains is then obtained to determine the dexterous workspace of the planar n-PRRR manipulator. The Gauss divergence theorem applied to planar surfaces is implemented to compute the total dexterous workspace area. Finally, two examples are shown to demonstrate applications of the method.

References

References
1.
Merlet
,
J.-P.
, 1997,
Les manipulateurs parallèles
,
2nd ed.
,
Éditions Hermes
,
Paris
.
2.
Gosselin
,
C.
, and
Angeles
,
J.
, 1988, “
The Optimum Kinematic Design of a Planar Three-Degree-of-Freedom Parallel Manipulator
,”
ASME J. Mech., Transm. Autom. Des.
,
110
(
1
), pp.
35
41
.
3.
Williams
II,
R. L.
, and
Reinholtz
,
C. F.
, 1988, “
Closed-Form Workspace Determination and Optimization for Parallel Robot Mechanisms
,”
ASME Design Technology Conferences 20th Biennial Mechanisms Conference
, pp.
341
351
.
4.
Kumar
,
V.
, 1992, “
Characterization of Workspaces of Parallel Manipulators
,”
ASME J. Mech. Des.
,
114
(
3
), pp.
368
375
.
5.
Pennock
,
G. R.
, and
Kassner
,
D. J.
, 1993, “
The Workspace of a General Geometry Planar Three-Degree-of-Freedom Platform-Type Manipulator
,”
ASME J. Mech. Des.
,
115
(
2
), pp.
269
276
.
6.
Merlet
,
J.-P.
,
Gosselin
,
C.
, and
Mouly
,
N.
, 1998, “
Workspaces of Planar Parallel Manipulators
,”
Mech. Mach. Theory
,
33
(
1–2
), pp.
7
20
.
7.
Zhaohui
,
L.
, and
Zhonghe
,
Y.
, 2004, “
Determination of Dexterous Workspace of Planar 3-DOF Parallel Manipulator by Auxiliary Linkages
,”
Chin. J. Mech. Eng. (English edition)
,
17
(
Suppl.
), pp.
76
78
.
8.
Gosselin
,
C.
, and
Jean
,
M.
, 1996, “
Determination of the Workspace of Planar Parallel Manipulators With Joint Limits
,”
Rob. Auton. Syst.
,
17
(
3
), pp.
129
138
.
9.
Hay
,
A. M.
, and
Snyman
,
J. A.
, 2002, “
The Chord Method for the Determination of Nonconvex Workspaces of Planar Parallel Manipulators
,”
Comput. Math. Appl.
,
43
, pp.
1135
1151
.
10.
Huang
,
M.
, and
Thebert
,
J.-L.
, 2010, “
A Study of the Workspace and Characteristics for Design of 3-DOF Planar Parallel Robots
,”
Int. J. Adv. Manuf. Technol.
,
51
, pp.
789
797
.
11.
Buck
,
R. C.
, 1965,
Advanced Calculus, International series in pure and applied mathematics
,
2nd ed.
,
McGraw-Hill Book Company
,
New York
.
12.
Gosselin
,
C.
, 1990, “
Determination of the Workspace of 6-DOF Parallel Manipulators
,”
ASME J. Mech. Des.
,
112
(
3
), pp.
331
336
.
13.
Lee
,
S.
, and
Kim
,
S.
, 1993, “
Kinematic Analysis of Generalized Parallel Manipulator Systems
,”
Proceedings of the IEEE Conference on Decision and Control
, Vol.
2
, pp.
1097
1102
.
14.
Zanganeh
,
K.
, and
Angeles
,
J.
, 1994, “
Instantaneous Kinematics and Design of a Novel Redundant Parallel Manipulator
,”
Proceedings of IEEE Conference on Robotics and Automation
, pp.
3043
3048
.
15.
Merlet
,
J.-P.
, 1996, “
Redundant Parallel Manipulators
,”
J. Lab. Rob. Autom.
,
8
(
1
), pp.
17
24
.
16.
Wang
,
J.
, and
Gosselin
,
C.
, 2004, “
Kinematic Analysis and Design of Kinematically Redundant Parallel Mechanisms
,”
ASME J. Mech. Des.
,
126
(
1
), pp.
109
118
.
17.
Dasgupta
,
B.
, and
Mruthyunjaya
,
T.
, 1998, “
Force Redundancy in Parallel Manipulators: Theoretical and Practical Issues
,”
Mech. Mach. Theory
,
33
(
6
), pp.
727
747
.
18.
Ebrahimi
,
I.
,
Carretero
,
J. A.
, and
Boudreau
,
R.
, 2008, “
A Family of Kinematically Redundant Planar Parallel Manipulators
,”
ASME J. Mech. Des.
,
130
(
6
), p.
062306
.
19.
Stamper
,
R. E.
,
Tsai
,
L. W.
, and
Walsh
,
G. C.
, 1997, “
Optimization of a Three DOF Translational Platform for Well-Conditioned Workspace
,”
Proceedings of IEEE Conference on Robotics and Automation
, pp.
3250
3255
.
20.
Gosselin
,
C.
, and
Angeles
,
J.
, 1991, “
A Global Performance Index for the Kinematic Optimization of Robotic Manipulators
,”
ASME J. Mech. Des.
,
113
(
3
), pp.
220
226
.
21.
Zarkandi
,
S.
,
Vafadar
,
A.
, and
Esmaili
,
M.
, 2011, “
PRRRRRP Redundant Planar Parallel Manipulator: Kinematics, Workspace and Singularity Analysis
,”
Proceedings of IEEE Comference on Robotics, Automation and Mechatronics (RAM)
, pp.
61
66
.
22.
Gallant
,
A.
,
Boudreau
,
R.
, and
Gallant
,
M.
, 2009, “
Dexterous Workspace of a 3-PRRR Kinematically Redundant Planar Parallel Manipulator
,”
Trans. Can. Soc. Mech. Eng.
,
33
(
4
), pp.
645
654
.
23.
Gallant
,
A.
,
Boudreau
,
R.
, and
Gallant
,
M.
, 2012, “
Geometric Determination of the Dexterous Workspace of n-RRRR and n-RRPR Manipulators
,”
Mech. Mach. Theory
,
52
, pp.
159
171
.
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