The mobility of a linkage is determined by the constraints imposed on its members. The geometric constraints define the configuration space (c-space) variety as the geometric entity in which the finite mobility of a linkage is encoded. The aim of a local kinematic analysis of a linkage is to deduce its finite mobility, in a given configuration, from the local c-space geometry. In this paper, a method for the local analysis is presented adopting the concept of the tangent cone to a variety. The latter is an algebraic variety approximating the c-space. It allows for investigating the mobility in regular as well as singular configurations. The instantaneous mobility is determined by the constraints, rather than by the c-space geometry. Shaky and underconstrained linkages are prominent examples that exhibit a permanently higher instantaneous than finite DOF even in regular configurations. Kinematic singularities, on the other hand, are reflected in a change of the instantaneous DOF. A c-space singularity as a kinematic singularity, but a kinematic singularity may be a regular point of the c-space. The presented method allows to identify c-space singularities. It also reveals the ith-order mobility and allows for a classification of linkages as overconstrained and underconstrained. The method is applicable to general multiloop linkages with lower pairs. It is computationally simple and only involves Lie brackets (screw products) of instantaneous joint screws. The paper also summarizes the relevant kinematic phenomena of linkages.

References

References
1.
Husty
,
M.
,
Pfurner
,
M.
,
Schröcker
,
H.-P.
, and
Brunnthaler
,
K.
,
2007
, “
Algebraic Methods in Mechanism Analysis and Synthesis
,”
Robotica
,
25
(
6
), pp.
661
675
.
2.
Gogu
,
G.
,
2008
, “
Constraint Singularities and the Structural Parameters of Parallel Robots
,”
Advances in Robot Kinematics: Analysis and Design
,
J.
Lenarcic
and
P.
Wenger
, eds.,
Springer
,
Amsterdam
.
3.
Dai
,
J.
,
Huang
,
Z.
, and
Lipkin
,
H.
,
2006
, “
Mobility of Overconstrained Parallel Mechanisms
,”
ASME J. Mech. Des.
,
128
(
1
), pp.
220
229
.
4.
Zhao
,
J.-S.
,
Feng
,
Z.-J.
, and
Dong
,
J.-X.
,
2006
, “
Computation of the Configuration Degree of Freedom of a Spatial Parallel Mechanism by Using Reciprocal Screw Theory
,”
Mech. Mach. Theory
,
41
(
12
), pp.
1486
1504
.
5.
Kong
,
X.
, and
Gosselin
,
C. M.
,
2007
,
Type Synthesis of Parallel Mechanisms
,
Springer
, Berlin.
6.
Baker
,
J. E.
,
1980
, “
An Analysis of the Bricard Linkages
,”
Mech. Mach. Theory
,
15
(
4
), pp.
267
286
.
7.
Waldron
,
K. J.
,
1973
, “
A Study of Overconstrained Linkage Geometry by Solution of Closure Equations—Part II. Four-Bar Linkages With Lower Pairs Other Than Screw Joints
,”
Mech. Mach. Theory
,
8
(
2
), pp.
233
247
.
8.
Mavroidis
,
C.
, and
Roth
,
B.
,
1995
, “
Analysis of Overconstrained Mechanisms
,”
ASME J. Mech. Des.
,
117
(
1
), pp.
69
74
.
9.
Gogu
,
G.
,
2005
, “
Mobility and Spatiality of Parallel Robots Revisited Via the Theory of Linear Transformations
,”
Eur. J. Mech. A/Solids
,
24
(
4
), pp.
690
711
.
10.
Kong
,
X.
,
2014
, “
Reconfiguration Analysis of a 3-DOF Parallel Mechanism Using Euler Parameter Quaternions and Algebraic Geometry Method
,”
Mech. Mach. Theory
,
74
, pp.
188
201
.
11.
Kong
,
X.
, and
Pfurner
,
M.
,
2015
, “
Type Synthesis and Reconfiguration Analysis of a Class of Variable-DOF Single-Loop Mechanisms
,”
Mech. Mach. Theory
,
85
, pp.
116
128
.
12.
Arponen
,
T.
,
Piipponen
,
S.
, and
Tuomela
,
J.
,
2009
, “
Kinematic Analysis of Bricard's Mechanism
,”
Nonlinear Dyn.
,
56
(
1
), pp.
85
99
.
13.
Arponen
,
T.
,
Müller
,
A.
,
Piipponen
,
S.
, and
Tuomela
,
J.
,
2014
, “
Kinematical Analysis of Overconstrained and Underconstrained Mechanisms by Means of Computational Algebraic Geometry
,”
Meccanica
,
49
(
4
), pp.
843
862
.
14.
Liu
,
R.
,
Serré
,
P.
, and
Rameau
,
J. F.
,
2013
, “
A Tool to Check Mobility Under Parameter Variations in Over-Constrained Mechanisms
,”
Mech. Mach. Theory
,
69
, pp.
44
61
.
15.
Wampler
,
C. W.
, and
Sommese
,
A. J.
,
2011
, “
Numerical Algebraic Geometry and Kinematics
,”
Acta Numer.
,
20
, pp.
469
567
.
16.
Wampler
,
C. W.
,
Hauenstein
,
J. D.
, and
Sommese
,
A. J.
,
2011
, “
Mechanism Mobility and a Local Dimension Test
,”
Mech. Mach. Theory
,
46
(9), pp.
1193
1206
.
17.
Rico
,
J. M.
,
Gallardo
,
J.
, and
Duffy
,
J.
,
1999
, “
Screw Theory and Higher Order Kinematic Analysis of Open Serial and Closed Chains
,”
Mech. Mach. Theory
,
34
(
4
), pp.
559
586
.
18.
Gallardo-Alvarado
,
J.
, and
Rico-Martinez
,
J. M.
,
2001
, “
Jerk Influence Coefficients, Via Screw Theory, of Closed Chains
,”
Meccanica
,
36
(
2
), pp.
213
228
.
19.
Cervantes-Sánchez
,
J. J.
,
Rico-Martínez
,
J. M.
,
González-Montiel
,
G.
, and
González-Galván
,
E. J.
,
2009
, “
The Differential Calculus of Screws: Theory, Geometrical Interpretation, and Applications
,”
Proc. Inst. Mech. E., Part C: J. Mech. Eng. Sci.
,
223
(
6
), pp.
1449
1468
.
20.
Müller
,
A.
,
2002
, “
Higher Order Local Analysis of Singularities in Parallel Mechanisms
,”
ASME
Paper No. DETC2002/MECH-34258.
21.
Müller
,
A.
,
2014
, “
Higher Derivatives of the Kinematic Mapping and Some Applications
,”
Mech. Mach. Theory
,
76
, pp.
70
85
.
22.
Müller
,
A.
, “
Higher-Order Constraints for Linkages With Lower Kinematic Pairs
,”
Mech. Mach. Theory
,
100
, pp. 33–43.
23.
Lerbet
,
J.
,
1999
, “
Analytic Geometry and Singularities of Mechanisms
,”
Z. Angew. Math. Mech.
,
78
(
10b
), pp.
687
694
.
24.
Whitney
,
H.
,
1965
, “Local Properties of Analytic Varieties, Differential and Combinatorial Topology,”
A Symposium in Honor of Marston Morse
(Princeton Mathematical Series 27), S. S. Cairns, ed.,
Princeton University Press
, Princeton, NJ.
25.
Müller
,
A.
, and
Rico
,
J. M.
,
2008
, “
Mobility and Higher Order Local Analysis of the Configuration Space of Single-Loop Mechanisms
,”
Advances in Robot Kinematics
,
J. J.
Lenarcic
and
P.
Wenger
, eds.,
Springer
, Amsterdam, pp.
215
224
.
26.
Chen
,
C.
,
2011
, “
The Order of Local Mobility of Mechanisms
,”
Mech. Mach. Theory
,
46
(
9
), pp.
1251
1264
.
27.
de Bustos
,
I. F.
,
Aguirrebeitia
,
J.
,
Avilés
,
R.
, and
Ansola
,
R.
,
2012
, “
Second Order Mobility Analysis of Mechanisms Using Closure Equations
,”
Meccanica
,
47
(
7
), pp.
1695
1704
.
28.
Karger
,
A.
,
1996
, “
Singularity Analysis of Serial Robot-Manipulators
,”
ASME J. Mech. Des.
,
118
(
4
), pp.
520
525
.
29.
Wohlhart
,
K.
,
1999
, “
Degrees of Shakiness
,”
Mech. Mach. Theory
,
34
(
7
), pp.
1103
1126
.
30.
Connelly
,
R.
, and
Servatius
,
H.
,
1994
, “
Higher-Order Rigidity–What is the Proper Definition?
,”
Discrete Comput. Geom.
,
11
(
2
), pp.
193
200
.
31.
Wohlhart
,
K.
,
2010
, “
From Higher Degrees of Shakiness to Mobility
,”
Mech. Mach. Theory
,
45
(
3
), pp.
467
476
.
32.
Müller
,
A.
,
2015
, “
Representation of the Kinematic Topology of Mechanisms for Kinematic Analysis
,”
Mech. Sci.
,
6
, pp.
1
10
.
33.
Müller
,
A.
,
2015
, “
Local Analysis of Closed-Loop Linkages: Mobility, Singularities, and Shakiness
,”
ASME
Paper No. DETC2015-47485.
34.
Chai
,
W. H.
, and
Chen
,
Y.
, “
The Line-Symmetric Octahedral Bricard Linkage and Its Structural Closure
,”
Mech. Mach. Theory
,
45
(
5
), pp.
772
779
.
35.
Wohlhart
,
K.
,
1996
, “
Kinematotropic Linkages
,”
Recent Advances in Robot Kinematics
,
J.
Lenarčič
and
V.
Parent-Castelli
, eds.,
Kluwer
, Alphen aan den Rijn, The Netherlands, pp.
359
368
.
36.
Golubitsky
,
M.
, and
Guillemin
,
V.
,
1973
,
Stable Mappings and Their Singularities
,
Springer
,
New York
.
37.
Zlatanov
,
D.
,
Bonev
,
I. A.
, and
Gosselin
,
C. M.
,
2002
, “
Constraint Singularities as C-Space Singularities
,”
8th International Symposium on Advances in Robot Kinematics
(
ARK 2002
), Caldes de Malavella, Spain, June 24–28.
38.
Kutznetsov
,
E. N.
,
1991
, “
Systems With Infinitesimal Mobility—Part I: Matrix Analysis and First-Order Infinitesimal Mobility
,”
ASME J. Appl. Mech.
,
58
(
2
), pp.
513
526
.
39.
Brockett
,
R. W.
,
1984
, “
Robotic Manipulators and the Product of Exponentials Formula, Mathematical Theory of Networks and Systems
,”
Lect. Notes Control Inf. Sci.
,
58
, pp.
120
129
.
40.
Selig
,
J.
,
2005
,
Geometric Fundamentals of Robotics
, Monographs in Computer Science Series,
Springer-Verlag
,
New York
.
41.
Müller
,
A.
,
2014
, “
Derivatives of Screw Systems in Body-Fixed Representation
,”
Advances in Robot Kinematics (ARK)
,
J.
Lenarcic
and
O.
Khatib
, eds.,
Springer
, Cham, Switzerland, pp.
123
130
.
42.
Rameau
,
J. F.
, and
Serré
,
P.
,
2015
, “
Computing Mobility Condition Using Groebner Basis
,”
Mech. Mach. Theory
,
91
, pp.
21
38
.
43.
Greuel
,
G. M.
, and
Pfister
,
G.
,
2012
,
A Singular Introduction to Commutative Algebra
,
Springer
, Berlin.
44.
Cox
,
D.
,
Little
,
J.
, and
O'Shea
,
D.
,
2007
,
Ideals, Varieties and Algorithms
,
3rd ed.
,
Springer
,
Berlin
.
45.
Goldberg
,
M
,
1943
, “
New Five-Bar and Six-Bar Linkages in Three Dimensions
,”
Trans. ASME
,
65
, pp.
649
661
.
46.
Baker
,
J. E.
,
1993
, “
A Comparative Survey of the Bennett-Based, 6-Revolute Kinematic Loops
,”
Mech. Mach. Theory
,
28
(1), pp.
83
96
.
47.
Servatius
,
B.
,
Shai
,
O.
, and
Whiteley
,
W.
,
2010
, “
Geometric Properties of Assur Graphs
,”
Eur. J. Combinatorics
,
31
(
4
), pp.
1105
1120
.
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