In this work, the solution to certain geometric constraint problems are studied. The possible rigid displacements allowed by the constraints are shown to be intersections of the Study quadric of rigid-body displacements with quadratic hypersurfaces. The geometry of these constraint varieties is also studied and is found to be isomorphic to products of subgroups in many cases. This information is used to find extremely simple derivations for general solutions to some problems in kinematics. In particular, the number of assembly configurations for RRPS and RRRS mechanisms are found in this way. In order to treat planes and spheres on an equal footing, the Clifford algebra for the Möbius group is introduced.

1.
Wampler
,
C.
, 2006, “
On a Rigid Body Subject to Point-Plane Constraints
,”
ASME J. Mech. Des.
0161-8458,
128
(
1
), pp.
151
158
.
2.
Kramer
,
G.
, 1992,
Solving Geometric Constraint Systems: A Case Study in Kinematics
,
MIT
,
Boston, MA
.
3.
Selig
,
J.
, 2005, “
Geometric Fundamentals of Robotics
,”
Monographs in Computer Science
,
2nd ed.
,
Springer-Verlag
,
New York
.
4.
Zsombor-Murray
,
P.
, and
Gfrerrer
,
A.
, 2010, “
A Unified Approach to Direct Kinematics of Some Reduced Motion Parallel Manipulators
,”
ASME J. Mech. Rob.
1942-4302,
2
, p.
021006
.
5.
Hestenes
,
D.
, 2001,
Geometric Algebra: A Geometric Approach to Computer Vision, Quantum and Neural Computing, Robotics, and Engineering
,
E.
Bayro-Corrochano
and
G.
Sobczyk
, eds.,
Birkhäuser
,
Boston
, pp.
498
520
.
6.
Bayro-Corrochano
,
E.
,
Reyes-Lozano
,
L.
, and
Zamora-Esquivel
,
J.
, 2006, “
Geometrical Methods of Classical Mechanics
,”
J. Math. Imaging Vision
0924-9907,
24
(
1
), pp.
55
81
.
7.
Wareham
,
R.
,
Cameron
,
J.
, and
Lasenby
,
J.
, 2004, “
Applications of Conformal Geometric Algebra in Computer Vision and Graphics
,”
Proceedings of the Sixth International Workshop on Mathematics Mechanization; Computer Algebra and Geometric Algebra With Applications
, pp.
329
349
.
8.
Porteous
,
I.
, 1995, “
Clifford Algebras and the Classical Groups
,”
Cambridge Studies in Advanced Mathematics
,
Cambridge University Press
,
Cambridge
, Vol.
50
.
9.
Lounesto
,
P.
, 1992, “
Clifford Algebras and Spinors
,”
London Mathematical Society Lecture Notes
,
2nd ed.
,
Cambridge University Press
,
Cambridge
, Vol.
286
.
10.
Harris
,
J.
, 1992, “
Algebraic Geometry a First Course
,”
Graduate Texts in Mathematics
,
Springer-Verlag
,
New York
, Vol.
133
.
11.
Hervé
,
J.
, 2004,
On Advances in Robot Kinematics
,
J.
Lenarčič
and
C.
Galletti
, eds.,
Kluwer Academic
,
Dordrecht
, pp.
431
440
.
12.
Alizadeh
,
D.
,
Angeles
,
J.
, and
Nokleby
,
S.
, 2009, “
On the Computation of the Home Posture of the McGill Schönflies Motion Generator
,”
Computational Kinematics, Proceedings of the Fifth International Workshop
,
A.
Kecskeméthy
and
A.
Müller
, eds.,
Springer-Verlag
,
Berlin
, pp.
149
158
.
13.
Hervé
,
J.
, 1999, “
The Lie Group of Rigid Body Displacements, a Fundamental Tool for Mechanism Design
,”
Mech. Mach. Theory
0094-114X,
34
(
5
), pp.
719
730
.
14.
Wu
,
Y. H. D.
,
Meng
,
J.
, and
Li
,
Z.
, 2006, “
Finite Motion Validation for Parallel Manipulators: A Differential Geometry Approach
,”
Proceedings of the 2006 IEEE/RSJ International Conference on Intelligent Robots and Systems
, Beijing, P.R.China, pp.
502
507
.
15.
Parkin
,
I.
, 1993, “
Composition and Additive Combination of Finite Displacement Screws—Part I
,” School of Information Technologies, University of Sydney, Technical Report No. 454.
16.
Hunt
,
K.
, and
Parkin
,
I.
, 1995, “
Finite Displacements of Points, Planes, and Lines via Screw Theory
,”
Mech. Mach. Theory
0094-114X,
30
(
2
), pp.
177
192
.
17.
Huang
,
C.
, and
Roth
,
B.
, 1994, “
Analytic Expressions for the Finite Screw Systems
,”
Mech. Mach. Theory
0094-114X,
29
(
2
), pp.
207
222
.
18.
Huang
,
C.
, and
Wang
,
J. -C.
, 2003, “
The Finite Screw System Associated With the Displacement of a Line
,”
ASME J. Mech. Des.
0161-8458,
125
(
1
), pp.
105
109
.
You do not currently have access to this content.