A quadratic parallel manipulator refers to a parallel manipulator with a quadratic characteristic polynomial. This paper revisits the forward displacement analysis (FDA) of a quadratic parallel manipulator: $3-RP̱R$ planar parallel manipulator with similar triangular platforms. Although it has been revealed numerically elsewhere that for this parallel manipulator, the four solutions to the FDA fall, respectively, into its four singularity-free regions (in its workspace), it is unclear if there exists a one-to-one correspondence between the four formulas, each producing one solution to the FDA, and the four singularity-free regions. Using an algebraic approach, this paper will prove that such a one-to-one correspondence exists. Therefore, a unique solution to the FDA can be obtained in a straightforward way for such a parallel manipulator if the singularity-free region in which it works is specified.

1.
Kong
,
X.
, and
Gosselin
,
C. M.
, 2002, “
Kinematics and Singularity Analysis of a Novel Type of 3-C̱RR 3-DOF Translational Parallel Manipulator
,”
Int. J. Robot. Res.
0278-3649,
21
(
9
), pp.
791
798
.
2.
Richard
,
P. -L.
,
Gosselin
,
C. M.
, and
Kong
,
X.
, 2007, “
Kinematic Analysis and Prototyping of a Partially Decoupled 4-DOF 3T1R Parallel Manipulator
,”
ASME J. Mech. Des.
0161-8458,
129
(
6
), pp.
611
616
.
3.
Innocenti
,
C.
, and
Parenti-Castelli
,
V.
, 1998, “
Singularity-Free Evolution From One Configuration to Another in Serial and Fully-Parallel Manipulators
,”
ASME J. Mech. Des.
0161-8458,
120
(
1
), pp.
73
79
.
4.
Wenger
,
P.
, and
Chablat
,
D.
, 1998,
Advances in Robot Kinematics: Analysis and Control
,
J.
Lenarčič
and
M.
Husty
, eds.,
,
Dordrecht
, pp.
117
126
.
5.
Merlet
,
J. P.
, 2000, “
On the Separability of the Solutions of the Direct Kinematics of a Special Class of Planar 3-RPR Parallel Manipulator
,” ASME Paper No. DETC2000/MECH-14103.
6.
Kong
,
X.
, and
Gosselin
,
C. M.
, 2000, “
Classification of 6-SPS Parallel Manipulators According to Their Components
,” ASME Paper No. DETC2000/MECH-14105.
7.
Kong
X.
, 1995, “
Forward Displacement and Singularity Analysis a Class of Analytic 3-RPR Planar Parallel Manipulator
,”
Proceedings of Advanced Manufacturing Technology
, Beijing, Oct., pp.
103
104
.
8.
Kong
,
X.
, and
Gosselin
,
C. M.
, 2000, “
Determination of the Uniqueness Domains of 3-RPR Planar Parallel Manipulators With Similar Platforms
,” ASME Paper No. DETC2000/MECH-14094.
9.
Chablat
,
D.
,
Wenger
,
P.
, and
Bonev
,
I. A.
, 2006,
Advances in Robot Kinematics
,
J.
Lenarčič
and
B.
Roth
, eds.,
Springer
, New York, pp.
221
228
.
10.
Gosselin
,
C. M.
, and
Merlet
,
J. P.
, 1994, “
The Direct Kinematics of Planar Parallel Manipulators: Special Architectures and Number of Solutions
,”
Mech. Mach. Theory
0094-114X,
29
(
8
), pp.
1083
1097
.
11.
Collins
,
C. L.
, and
McCarthy
,
J. M.
, 1998, “
The Quartic Singularity Surfaces of Planar Platforms in the Clifford Algebra of the Projective Plane
,”
Mech. Mach. Theory
0094-114X,
33
(
7
), pp.
931
944
.
12.
Gosselin
,
C.
, and
Angeles
,
J.
, 1990, “
Singularity Analysis of Closed-Loop Kinematic Chains
,”
IEEE Trans. Robot. Autom.
,
6
(
3
), pp.
281
290
. 1042-296X
13.
Sefrioui
,
J.
, and
Gosselin
,
C.
, 1995, “
On the Quadratic Nature of the Singularity Curves of Planar Three-Degree-of-Freedom Parallel Manipulators
,”
Mech. Mach. Theory
0094-114X,
30
(
4
), pp.
533
551
.
14.
Kong
,
X.
, and
Gosselin
,
C. M.
, 2001, “
Forward Displacement Analysis of Third-Class Analytic 3-RP̱R Parallel Manipulators
,”
Mech. Mach. Theory
0094-114X,
36
(
9
), pp.
1009
1018
.
15.
Wenger
,
P.
,
Chablat
,
D.
, and
Zein
,
M.
, 2007, “
Degeneracy Study of the Forward Kinematics of Planar 3-RPR Parallel Manipulators
,”
ASME J. Mech. Des.
0161-8458,
129
(
12
), pp.
1265
1268
.
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