In this paper, we give a full classification of all pentapods with mobility 2, where neither all platform anchor points nor all base anchor points are located on a line. Therefore, this paper solves the famous Borel–Bricard problem for two-dimensional motions beside the excluded case of five collinear points with spherical trajectories. But even for this special case, we present three new types as a side-result. Based on our study of pentapods, we also give a complete list of all nonarchitecturally singular hexapods with two-dimensional self-motions.

References

References
1.
Borel
,
E.
,
1908
, “
Mémoire sur les Déplacements à Trajectoires Sphériques
,” Mémoire Présenteés par divers savants à l'Académie des Sciences de l'Institut National de France,
33
(
1
), pp.
1
128
.
2.
Bricard
,
R.
,
1906
, “
Mémoire sur les Déplacements à Trajectoires Sphériques
,”
J. Éc. Polytech.
,
11
(
2
), pp.
1
96
.
3.
Husty
,
M.
,
2000
, “
E. Borel's and R. Bricard's Papers on Displacements With Spherical Paths and Their Relevance to Self-Motions of Parallel Manipulators
,” International Symposium on History of Machines and Mechanisms (
HMM 2000
), Cassino, Italy, May 11–13, pp.
163
171
.10.1007/978-94-015-9554-4_19
4.
Gallet
,
M.
,
Nawratil
,
G.
, and
Schicho
,
J.
, “
Möbius Photogrammetry
,” e-print
arXiv:1408.6716v1
. http://arXiv:1408.6716
5.
Nawratil
,
G.
,
2014
, “
On Stewart Gough Manipulators With Multidimensional Self-Motions
,”
Comput. Aided Geom. Des.
,
31
(
7–8
), pp.
582
594
.http://arxiv.org/abs/1408.6716
6.
Nawratil
,
G.
, and
Schicho
,
J.
, “
Self-Motions of Pentapods With Linear Platform
,” e-print
arXiv:1407.6126v1
. http://arXiv:1407.6126
7.
Pottmann
,
H.
, and
Wallner
,
J.
,
2001
,
Computational Line Geometry
,
Springer
,
Berlin, Germany
.
8.
Karger
,
A.
,
2008
, “
Architecturally Singular Non-Planar Parallel Manipulators
,”
Mech. Mach. Theory
,
43
(
3
), pp.
335
346
.10.1016/j.mechmachtheory.2007.03.006
9.
Nawratil
,
G.
,
2012
, “
Comments on ‘Architectural Singularities of a Class of Pentapods'
,”
Mech. Mach. Theory
,
57
(
1
), pp.
139
.10.1016/j.mechmachtheory.2012.06.007
10.
Nawratil
,
G.
,
2012
, “
Self-Motions of Planar Projective Stewart Gough Platforms
,”
Latest Advances in Robot Kinematics
,
J.
Lenarcic
and
M.
Husty
, eds.,
Springer
,
Dordrecht, The Netherlands
, pp.
27
34
.
11.
Nawratil
,
G.
,
2014
, “
Introducing the Theory of Bonds for Stewart Gough Platforms With Self-Motions
,”
ASME J. Mech. Rob.
,
6
(
1
), p.
011004
.10.1115/1.4025623
12.
Hegedüs
,
G.
,
Schicho
,
J.
, and
Schröcker
,
H.-P.
,
2012
, “
Bond Theory and Closed 5R Linkages
,”
Latest Advances in Robot Kinematics
,
J.
Lenarcic
and
M.
Husty
, eds.,
Springer
,
Dordrecht, The Netherlands
, pp.
221
228
.
13.
Hegedüs
,
G.
,
Schicho
,
J.
, and
Schröcker
,
H.-P.
,
2013
, “
The Theory of Bonds: A New Method for the Analysis of Linkages
,”
Mech. Mach. Theory
,
70
, pp.
407
424
.10.1016/j.mechmachtheory.2013.08.004
14.
Husty
,
M. L.
,
1996
, “
An Algorithm for Solving the Direct Kinematics of General Stewart-Gough Platforms
,”
Mech. Mach. Theory
,
31
(
4
), pp.
365
380
.10.1016/0094-114X(95)00091-C
15.
Gallet
,
M.
,
Nawratil
,
G.
, and
Schicho
,
J.
,
2014
, “
Bond Theory for Pentapods and Hexapods
,”
J. Geom.
(in press).10.1007/s00022-014-0243-1
16.
Nawratil
,
G.
,
2014
, “
Congruent Stewart Gough Platforms With Non-Translational Self-Motions
,”
16th International Conference on Geometry and Graphics
, Innsbruck, Austria, Aug. 4–8,
H.-P.
Schröcker
and
M.
Husty
, eds.,
Innsbruck University
, Innsbruck Austria, pp.
204
215
.
17.
Nawratil
,
G.
,
2013
, “
On Equiform Stewart Gough Platforms With Self-Motions
,”
J. Geom. Graphics
,
17
(
2
), pp.
163
175
.
18.
Chasles
,
M.
,
1861
, “
Sur les Six Droites qui Peuvent étre les Directions de Six Forces en Équilibre
,”
C. R. Acad. Sci.
,
52
, pp.
1094
1104
.
19.
Duporcq
,
E.
,
1898
, “
Sur la Correspondance Quadratique et Rationnelle de Deux Figures Planes et sur un Déplacement Remarquable
,”
C. R. Acad. Sci.
,
126
, pp.
1405
1406
.
20.
Koenigs
,
G.
,
1897
, Leçons de Cinématique (avec notes par M. G. Darboux), A. Hermann, Paris.
21.
Mannheim
,
A.
,
1894
,
Principes et Développements de Géométrie Cinématique
,
Gauthier-Villars
, Paris.
22.
Duporcq
,
E.
,
1898
, “
Sur le Déplacement le Plus Général D'une Droite Dont Tous les Points Décrivent des Trajectoires Sphériques
,”
J. Math. Pures Appl.
,
4
(
5
), pp.
121
136
.
23.
Borras
,
J.
,
Thomas
,
F.
, and
Torras
,
C.
,
2010
, “
Singularity-Invariant Leg Rearrangements in Stewart-Gough Platforms
,”
Advances in Robot Kinematics: Motion in Man and Machine
,
J.
Lenarcic
and
M. M.
Stanisic
, eds.,
Springer
, Dordrecht, The Netherlands, pp.
421
428
.
You do not currently have access to this content.