This paper investigates the screw surface of the RPRP linkage, an overconstrained linkage with mobility one. The screw surface is a ruled surface containing screws for displacing the coupler of the RPRP linkage from one reference position to all reachable positions. In this paper, two types of RPRP linkages, in folded and unfolded forms, are constructed by using hexahedrons in accordance with Delassus’ criteria. It is shown that the screw surfaces of both types of RPRP linkages can be represented by screw systems of the second order. These novel finite screw systems are obtained by intersecting two 3-systems corresponding to the finite displacements of the RP and PR dyads. The intersection of finite screw systems is conducted by employing the intersection operation of vector subspaces of the six-dimensional vector space.

1.
Delassus
,
E.
, 1922, “
Les Chaînes Articulées Fermées et Déformables à Quatre Membres
,”
Bull. Sci. Math.
0007-4497,
46
, pp.
283
304
.
2.
Bennett
,
G. T.
, 1903, “
A New Mechanism
,”
Engineering (London)
0013-7782,
76
, pp.
777
778
.
3.
Hernandez-Gutierrez
,
I.
, 1980, “
Screw Surfaces in the Analysis and Synthesis of Mechanisms
,” Ph.D. thesis, Stanford University, Stanford, CA.
4.
Lohse
,
P.
, 1975,
Getriebesynthese
,
Springer-Verlag
,
Berlin
.
5.
Veldkamp
,
G. R.
, 1967, “
Rotation Curves
,”
J. Mech.
0022-2569,
2
, pp.
147
156
.
6.
Huang
,
C.
, 1997, “
The Cylindroid Associated With Finite Motions of the Bennett Mechanism
,”
ASME J. Mech. Des.
1050-0472,
119
, pp.
521
524
.
7.
Huang
,
C.
, 1994, “
On the Finite Screw System of the Third Order Associated With a Revolute-Revolute Chain
,”
ASME J. Mech. Des.
1050-0472,
116
, pp.
875
883
.
8.
Huang
,
C.
, and
Liu
,
T. E.
, 1997, “
On the Linear Finite Motions of the Goldberg Overconstrained Linkages
,”
Proc. the Fifth Applied Mechanisms and Robotics Conference
,
Cincinnati, Ohio
.
9.
Huang
,
C.
, and
Sun
,
C. C.
, 2000, “
An Investigation of Screw Systems in the Finite Displacements of Bennett-Based 6R Linkages
,”
ASME J. Mech. Des.
1050-0472,
122
, pp.
426
430
.
10.
Perez
,
A.
, and
McCarthy
,
J. M.
, 2003, “
Dimensional Synthesis of Bennett Linkages
,”
ASME J. Mech. Des.
1050-0472,
125
, pp.
98
104
.
11.
Brunnthaler
,
K.
,
Schröcker
,
H.-P.
, and
Husty
,
M.
, 2005, “
A New Method for the Synthesis of Bennett Mechanisms
,”
Proc. the 2005 International Workshop on Computational Kinematics
,
Cassino, Italy
.
12.
Huang
,
C.
, and
Tu
,
H. T.
, 2004, “
The Finite Screw System Associated With the Spatial RPRP Overconstrained Linkage
,”
Proc. of the 28th ASME Biennial Mechanisms and Robotics Conference
,
Salt Lake City, Utah
.
13.
Waldron
,
K. J.
, 1973, “
A Study of Overconstrained Linkage Geometry by Solution of Closure Equations—Part II. Four-bar Linkages With Low Pair Joints Other Than Screw Joints
,”
Mech. Mach. Theory
0094-114X,
8
, pp.
233
247
.
14.
Baker
,
E.
, 1975, “
Mobility Analysis of Spatial Linkages
,” Doctoral Dissertation, the University of New South Wales, Sydney, Australia.
15.
Hunt
,
K. H.
, 1978,
Kinematic Geometry of Mechanisms
,
Clarendon Press
,
Oxford
.
16.
Parkin
,
I. A.
, 1992, “
A Third Conformation With the Screw Systems: Finite Twist Displacements of a Directed Line and Point
,”
Mech. Mach. Theory
0094-114X,
27
, pp.
177
188
.
17.
Huang
,
C.
,
Sugimoto
,
K.
, and
Parkin
,
I.
, 2005, “
The Correspondence Between Finite Screw Systems and Projective Spaces
,”
Proc. the 2005 International Workshop on Computational Kinematics
,
Cassino, Italy
, May 4–5.
19.
Berger
,
M.
, 1987,
Geometry
,
Springer-Verlag
,
New York
.
20.
Shephard
,
G. C.
, 1966,
Vector Spaces of Finite Dimension
,
Interscience Publishers
,
New York
.
21.
Huang
,
C.
, 1994, “
The Finite Screw Systems Associated With a Prismatic-Revolute Dyad and Screw Displacement of a Point
,”
Mech. Mach. Theory
0094-114X,
29
(
28
), pp.
1131
1142
.
22.
Dimentberg
,
F. M.
, 1965,
The Screw Calculus and Its Application in Mechanics (in Russian)
, (English Translation: US Dept. of Commerce Translation No. AD680993).
23.
Yang
,
A. T.
, 1963, “
Application of Quaternion Algebra and Dual Numbers to the Analysis of Spatial Mechanisms
,” Ph.D. thesis, Columbia University, New York.
24.
Huang
,
C.
, 1997, “
Note on Screw Product Operations in the Formulations of Successive Finite Displacement
,”
ASME J. Mech. Des.
1050-0472,
119
, pp.
434
439
.
You do not currently have access to this content.