This paper reformulates and extends the new, group theoretic, mobility criterion recently developed by the authors, (Rico, J. M., and Ravani, B., 2003, ASME J. Mech. Des., 125, pp. 70–80). In contrast to the Kutzbach-Grübler criterion, the new mobility criterion and the approach presented apply to a large class of single-loop overconstrained linkages. The criterion is reformulated, in terms of the well-known Jacobian matrices, for exceptional linkages and extended to linkages with partitioned mobility and trivial linkages. Several examples are included.
Issue Section:
Research Papers
1.
Grübler
, M.
, 1883, “Allgemeine Eigenschaften der Zwangläufigen Ebenen Kinematische Kette: I
,” Civilingenieur
, 29
, pp. 167
–200
.2.
Grübler
, M.
, 1885, “Allgemeine Eigenschaften der Zwangläufigen Ebenen Kinematische Kette: II
,” Verein zur Beforderung des Gewerbefleisses. Verhandlungen
, 64
, pp. 179
–223
.3.
Kutzbach
, K.
, 1933, “Einzelfragen aus dem Gebiet der Maschinenteile
,” Zeitschrift der Verein Deutscher Ingenieur
, 77
, pp. 1168
–1169
.4.
Chebyshev
, P. L.
, 1869, “On Parallelograms
,” Collected Works
(in Russian), Moscow, 1955, Chelsea Publishing
, New York
, 1973, pp. 664
–669
.5.
Sylvester
, J. J.
, 1870, “On Recent Discoveries in the Mechanical Conversion of Motion
,” Collected Mathematical Papers
, 3
, pp. 7
–25
.6.
Somov
, P. O.
, 1887, “On the degree of Freedom of the Motion of Kinematic Chains
,” J. Russ. Phys.-Chem. Soc.
0372-9877, 9
, pp. 7
–25
.7.
Bottema
, O.
, 1950, “On Gruebler’s Formulae for Mechanisms
,” Appl. Sci. Res., Sect. A
0365-7132, A2
, pp. 162
–164
.8.
Freudenstein
, F.
, and Alizade
, R.
, 1975, “On the Degree of Freedom of Mechanisms With Variable General Constraint
,” Proc. of 4th World Congress on the Theory of Machines and Mechanisms
, Institution of Mechanical Engineers
, London
, pp. 51
–55
.9.
Bagci
, C.
, 1988, “Determining General and Overclosing Constraints in Mechanism Mobility Using Structural Finite Element Joint Freedoms
,” Proc. of ASME 20th Mechanisms Conference, Vol. 15-1, Trends and Developments in Mechanisms, Machines and Robotics
, ASME
, New York
, pp. 27
–36
.10.
Hervé
, J. M.
, 1978, “Analyse Structurelle des Mécanismes par Groupe des Déplacements
,” Mech. Mach. Theory
0094-114X, 13
, pp. 437
–450
.11.
Hervé
, J. M.
, 1997, “Conjugacy and Invariance in the Displacement Group as a Tool for Mechanism Design
,” Technical Report, Mechanical Research Team, Ecole Centrale Paris
, France.12.
Rico
, J. M.
, and Ravani
, B.
, 2003, “On Mobility Analysis of Linkages Using Group Theory
,” ASME J. Mech. Des.
1050-0472, 125
, pp. 70
–80
.13.
Rico
, J. M.
, Gallardo
, J.
, and Ravani
, B.
, 2003, “Lie Algebra and the Mobility of Kinematic Chains
,” J. Rob. Syst.
0741-2223, 20
, pp. 477
–499
.14.
Rico
, J. M.
, Aguilera
, L. D.
, Gallardo
, J.
, Rodríguez
, R.
, Orozco
, H.
, and Barrera
, J. M.
, 2005, “A More General Mobility Criterion for Parallel Platforms
,” ASME J. Mech. Des.
1050-0472, 128
, pp. 207
–219
.15.
Bentley
, D. L.
, and Cooke
, K. L.
, 1973, Linear Algebra with Differential Equations
, Holt, Rinehart, and Winston
, New York
.16.
Rico
, J. M.
, and Ravani
, B.
, 2005, “Mobility Determination of Paradoxical Linkages
,” Proc. of ASME 2005 Design Engineering Technical Conferences
, ASME
, New York
, ASME Paper No. DETC2005–84936.17.
Marghitu
, D. B.
, and Crocker
, M. J.
, 2001, Analytical Elements of Mechanisms
, Cambridge University Press
, Cambridge, England
.18.
Rico
, J. M.
, Aguilera
, D.
, and Gallardo
, J.
, 2002, “Computer Implementation of an Improved Kutzbach-Grübler Mobility Criterion
,” Proc. of ASME 2002 Design Engineering Technical Conferences
, ASME
, New York
, Paper No. DETC02∕DAC-34093.19.
Ball
, R. S.
, 1900, A Treatise on the Theory of Screws
, Cambridge University Press
, Cambridge, England
(reprinted in 1998).20.
Hunt
, K. H.
, 1978, Kinematic Geometry of Mechanisms
, Oxford University Press
, London
.Copyright © 2007
by American Society of Mechanical Engineers
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