This paper investigates two situations in which the forward kinematics of planar 3-RP̱R parallel manipulators degenerates. These situations have not been addressed before. The first degeneracy arises when the three input joint variables ρ1, ρ2, and ρ3 satisfy a certain relationship. This degeneracy yields a double root of the characteristic polynomial in t=tan(φ2), which could be erroneously interpreted as two coalesce assembly modes. However, unlike what arises in nondegenerate cases, this double root yields two sets of solutions for the position coordinates (x,y) of the platform. In the second situation, we show that the forward kinematics degenerates over the whole joint space if the base and platform triangles are congruent and the platform triangle is rotated by 180deg about one of its sides. For these “degenerate” manipulators, which are defined here for the first time, the forward kinematics is reduced to the solution of a third-degree polynomial and a quadratic in sequence. Such manipulators constitute, in turn, a new family of analytic planar manipulators that would be more suitable for industrial applications.

1.
Hunt
,
K. H.
, 1978,
Kinematic Geometry of Mechanisms
,
Oxford University Press
,
Cambridge
.
2.
Hunt
,
K. H.
, 1983, “
Structural Kinematics of In-Parallel Actuated Robot Arms
,”
ASME J. Mech., Transm., Autom. Des.
0738-0666,
105
(
4
), pp.
705
712
.
3.
Gosselin
,
C.
,
Sefrioui
,
J.
, and
Richard
,
M. J.
, 1992, “
Solutions polynomiales au problème de la cinématique des manipulateurs parallèles plans à trois degrés de libertè
,”
Mech. Mach. Theory
0094-114X,
27
, pp.
107
119
.
4.
Pennock
,
G. R.
, and
Kassner
,
D. J.
, 1990, “
Kinematic Analysis of a Planar Eight-Bar Linkage: Application to a Platform-Type Robot
,”
ASME Proceedings of the 21th Biennial Mechanisms Conference
,
Chicago
, Sep., pp.
37
43
.
5.
Gosselin
,
C. M.
, and
Merlet
,
J.-P.
, 1994, “
On the Direct Kinematics of Planar Parallel Manipulators: Special Architectures and Number of Solutions
,”
Mech. Mach. Theory
0094-114X,
29
(
8
), pp.
1083
1097
.
6.
Kong
,
X.
, and
Gosselin
,
C. M.
, 2001, “
Forward Displacement Analysis of Third-Class Analytic 3-RPR Planar Parallel Manipulators
,”
Mech. Mach. Theory
0094-114X,
36
, pp.
1009
1018
.
7.
Merlet
,
J.-P.
, 2000,
Parallel Robots
,
Kluwer Academic
,
Dordrecht
.
8.
Mcaree
,
P. R.
, and
Daniel
,
R. W.
, 1999, “
An Explanation of Never-Special Assembly Changing Motions for 3-3 Parallel Manipulators
,”
Int. J. Robot. Res.
0278-3649,
18
(
6
), pp.
556
574
.
9.
Kong
,
X.
, and
Gosselin
,
C. M.
, 2000, “
Determination of the Uniqueness Domains of 3-RPR Planar Parallel Manipulators With Similar Platforms
,”
Proceedings of ASME 26th Biennial Mechanisms and Robotics Conference
,
Baltimore
, Sept. 2000, Paper No. MECH-14094.
10.
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.
1050-0472,
120
, pp.
73
99
.
You do not currently have access to this content.