This paper derives the expressions of an equivalent finite screw of two successive screw motions in a simplified form using purely vectorial analysis. This is achieved by tracing the trajectories of specific points on the moving body, which together with the known axis and angle of combined rotation, yield the expressions of the screw triangle. This paper also gives a short overview of different known expressions of the screw triangle and shows that the one given in this paper reduces the number of arithmetic operations by about a third compared with the most efficient algorithm in the literature.

1.
Yu
,
J.
,
Dai
,
J. S.
,
Zhao
,
T.
,
Bi
,
S.
, and
Zong
,
G.
, 2009, “
Mobility Analysis of Complex Joints by Means of Screw Theory
,”
Robotica
0263-5747,
27
, pp.
915
927
.
2.
Kong
,
X.
, and
Gosselin
,
C. M.
, 2004, “
Type Synthesis of 3-DOF Spherical Parallel Manipulators Based on Screw Theory’
,”
ASME J. Mech. Des.
0161-8458,
126
(
1
), pp.
101
109
.
3.
Gallardo
,
J.
,
Rico
,
J. M.
,
Frisoli
,
A.
,
Checcacci
,
D.
, and
Bergamasco
,
M.
, 2003, “
Dynamics of Parallel Manipulators by Means of Screw Theory
,”
Mech. Mach. Theory
0094-114X,
38
, pp.
1113
1131
.
4.
Wolf
,
A.
, and
Shoham
,
M.
, 2003, “
Investigation of Parallel Manipulators Using Linear Complex Approximation
,”
ASME J. Mech. Des.
0161-8458,
125
(
3
), pp.
564
572
.
5.
Gal
,
J. A.
,
Gallo
,
L. M.
,
Murray
,
G.
, and
Kllneberg
,
I.
, 2004, “
Analysis of Human Mandibular Machanics Based on Screw Theory and In Vivo Data
,”
J. Biomech.
0021-9290,
37
(
9
), pp.
1405
1412
.
6.
Selig
,
J. M.
, and
Ding
,
X.
, 2009, “
A Screw Theory of Timoshenko Beams
,”
ASME J. Appl. Mech.
0021-8936,
76
(
3
), p.
031003
.
7.
Roth
,
B.
, 1967, “
On the Screw Axes and Other Special Lines Associated With Spatial Displacements of a Rigid Body
,”
ASME J. Eng. Ind.
0022-0817,
89
(
1
), pp.
102
110
.
8.
Ames
,
J. S.
, and
Murnaghan
,
F. D.
, 1929,
Theoretical Mechanics
,
Ginn and Co.
,
Boston, MA
.
9.
Bottema
,
O.
, and
Roth
,
B.
, 1990,
Theoretical Kinematics
,
Dove
,
New York
.
10.
Tsai
,
L. W.
, and
Roth
,
B.
, 1972, “
Design of Dyads With Helical, Cylindrical, Spherical, Revolute and Prismatic Joints
,”
Mech. Mach. Theory
0094-114X,
7
, pp.
85
102
.
11.
Rodrigues
,
O.
, 1840, “
Des Lois Geometriques Qui Reagissent les Deplacements d’un Systeme Solide dans l’Espace
,”
Journal de Mathematique Pures et Appliquees de Liouville
,
5
, pp.
380
440
.
12.
Huang
,
C.
, 1994, “
On the Finite Screw System of the Third Order Associated With a Revolute-Revolute Chain
,”
Mech. Mach. Theory
0094-114X,
116
(
2
), pp.
875
883
.
13.
Huang
,
C.
, and
Chen
,
C. M.
, 1995, “
The Linear Representation of the Screw Triangle—A Unification of Finite and Infinitesimal Kinematics
,”
ASME J. Mech. Des.
0161-8458,
117
(
4
), pp.
554
561
.
14.
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
(
2
), pp.
177
188
.
15.
Parkin
,
I. A.
, 1997, “
Unifying the Geometry of Finite Displacement Screws and Orthogonal Matrix Transformations
,”
Mech. Mach. Theory
0094-114X,
32
(
8
), pp.
975
991
.
16.
Dimentberg
,
F. M.
, 1965, “
The Screw Calculus and Its Applications in Mechanics
,” Clearinghouse for Federal and Scientific Technical Information Report No. AD680993.
17.
McCarthy
,
J. M.
, 2000,
Geometric Design of Linkages
Springer-Verlag
,
New York
.
19.
Angeles
,
J.
, 1982,
Spatial Kinematic Chains: Analysis, Synthesis, Optimization
,
Springer-Verlag
,
New York
.
20.
Davidson
,
J. K.
, and
Hunt
,
K. H.
, 2004,
Robots and Screw Theory
,
Oxford University Press
,
New York
.
You do not currently have access to this content.