We use the recently introduced factorization theory of motion polynomials over the dual quaternions and cubic interpolation on quadrics for the synthesis of closed kinematic loops with six revolute joints that visit four prescribed poses. The resulting 6R linkages are special in the sense that the relative motions of all links are rational. They exhibit certain elegant properties such as symmetry in the rotation angles and, at least in theory, full-cycle mobility. Our synthesis approach admits either no solution or two one-parametric families of solutions. We suggest strategies for picking good solutions from these families. They ensure a fair coupler motion and optimize other linkage characteristics such as total rotation angle or linkage size. A comprehensive synthesis example is provided.

References

References
1.
Brunnthaler
,
K.
,
Schröcker
,
H.-P.
, and
Husty
,
M.
,
2005
, “
A New Method for the Synthesis of Bennett Mechanisms
,” International Workshop on Computational Kinematics (CK2005), Cassino, Italy, May 4–6.
2.
Veldkamp
,
G. R.
,
1967
, “
Canonical Systems and Instantaneous Invariants in Spatial Kinematics
,”
J. Mech.
,
3
(
3
), pp.
329
388
.10.1016/0022-2569(67)90006-7
3.
Suh
,
C. H.
,
1969
, “
On the Duality in the Existence of R–R Links for Three Positions
,”
ASME J. Manuf. Sci. Eng.
,
91
(
1
), pp.
129
134
.10.1115/1.3591490
4.
Tsai
,
L. W.
, and
Roth
,
B.
,
1973
, “
A Note on the Design of Revolute–Revolute Cranks
,”
Mech. Mach. Theory
,
8
(
1
), pp.
23
31
.10.1016/0094-114X(73)90004-9
5.
Perez
,
A.
, and
McCarthy
,
J. M.
,
2004
, “
Dimensional Synthesis of Bennett Linkages
,”
ASME J. Mech. Des.
,
126
(
3
), pp.
425
435
.10.1115/1.1737378
6.
Hamann
,
M.
,
2011
, “
Line-Symmetric Motions With Respect to Reguli
,”
Mech. Mach. Theory
,
46
(
7
), pp.
960
974
.10.1016/j.mechmachtheory.2011.02.004
7.
Hegedüs
,
G.
,
Schicho
,
J.
, and
Schröcker
,
H.-P.
,
2013
, “
Factorization of Rational Curves in the Study Quadric and Revolute Linkages
,”
Mech. Mach. Theory
,
69
(
1
), pp.
142
152
.10.1016/j.mechmachtheory.2013.05.010
8.
Beyer
,
R.
,
1963
,
Technische Raumkinematik
,
Springer
,
Berlin, Germany
.
9.
Song
,
C.-Y.
,
Chen
,
Y.
, and
I-Ming
,
C.
,
2014
, “
Kinematic Study of the Original and Revised General Line-Symmetric Bricard 6R Linkages
,”
ASME J. Mech. Rob.
,
6
(
3
), p.
031002
.10.1115/1.4026339
10.
Perez
,
A.
, and
McCarthy
,
J. M.
,
2003
, “
Dual Quaternion Synthesis of Constrained Robotic Systems
,”
ASME J. Mech. Des.
,
126
(
3
), pp.
425
435
.10.1115/1.1737378
11.
McCarthy
,
J. M.
, and
Soh
,
G. S.
,
2011
, “
Geometric Design of Linkages
,”
Interdisciplinary Applied Mathematics
,
2nd ed.
, Vol.
11
,
Springer
,
New York
.
12.
Gfrerrer
,
A.
,
1999
, “
Rational Interpolation on a Hypersphere
,”
Comput. Aided Geom. Des.
,
16
(
1
), pp.
21
37
.10.1016/S0167-8396(98)00027-2
13.
Gfrerrer
,
A.
,
2000
, “
On the Construction of Rational Curves on Hyperquadrics
,”
Grazer Math. Ber.
,
340
, pp.
1
69
.
14.
Schröcker
,
H.-P.
, and
Jüttler
,
B.
,
2009
, “
Motion Interpolation With Bennett Biarcs
,”
Proceedings of the Computational Kinematics (CK 2009)
, Duisburg, Germany, May 6–8,
A.
Kecskeméthy
, and
A.
Müller
, eds.,
Springer
,
Berlin, Germany
, pp.
101
108
.
15.
Salmon
,
G.
,
1882
,
A Treatise on the Analytic Geometry of Three Dimensions
,
4th ed.
,
Hodges, Figgis & Co.
,
Dublin, Ireland
.
16.
Li
,
Z.
, and
Schicho
,
J.
,
2013
, “
Classification of Angle-Symmetric 6R Linkages
,”
Mech. Mach. Theory
,
70
, pp.
372
379
.10.1016/j.mechmachtheory.2013.08.002
17.
Fang
,
Y. C.
,
Hsieh
,
C. C.
,
Kim
,
M. J.
,
Chang
,
J. J.
, and
Woo
,
T. C.
,
1998
, “
Real Time Motion Fairing With Unit Quaternions
,”
CAD
,
30
(
3
), pp.
191
198
.10.1016/S0010-4485(97)00057-2
18.
Hsieh
,
C.-C.
, and
Chang
,
T.-Y.
,
2003
, “
Motion Fairing Using Genetic Algorithms
,”
CAD
,
35
(
8
), pp.
739
749
.10.1016/S0010-4485(03)00005-8
19.
Hofer
,
M.
,
Pottmann
,
H.
, and
Ravani
,
B.
,
2004
, “
From Curve Design Algorithms to the Design of Rigid Body Motions
,”
Visual Comput.
,
20
(
5
), pp.
279
297
.10.1007/s00371-003-0221-3
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