Mainly drawing on screw theory and linear algebra, this paper presents an approach to determining the bases of three unknown twist and wrench subspaces of lower mobility serial kinematic chains, an essential step for kinematic and dynamic modeling of both serial and parallel manipulators. By taking the reciprocal product of a wrench on a twist as a linear functional, the underlying relationships among their subspaces are reviewed by means of the dual space and dual basis. Given the basis of a twist subspace of permissions, the causes of nonuniqueness in the bases of the other three subspaces are discussed in some depth. Driven by needs from engineering design, criteria, and a procedure are proposed that enable pragmatic, consistent bases of these subspaces to be determined in a meaningful, visualizable, and effective manner. Three typical examples are given to illustrate the entire process. Then, formulas are presented for the bases of the twist/wrench subspaces of a number of commonly used serial kinematic chains, which can readily be employed for the formulation of the generalized Jacobian of a variety of lower mobility parallel manipulators.
Issue Section:
Research Papers
References
1.
Whiteney
, D. E.
, 1972
, “The Mathematics of Coordinated Control of Prosthetic Arms and Manipulators
,” ASME J. Dyn. Syst., Meas., Control
, 94
(4
), pp. 303
–309
.10.1115/1.34266112.
Merlet
, J. P.
, 2006
, Parallel Robots, Solid Mechanics and Its Applications
, 2nd ed., Vol. 128
, Springer
, Dordrecht, The Netherlands
.3.
Angeles
, J.
, 2007
, Fundamentals of Robotic Mechanical Systems: Theory, Methods, and Algorithms, Mechanical Engineering Series
, 3rd ed., Vol. 1
, Springer Science + Business Media
, New York
.4.
Tsai
, L. W.
, 1999
, Robot Analysis: The Mechanics of Serial and Parallel Manipulators
, Wiley
, New York
.5.
Merlet
, J. P.
, 2006
, “Jacobian, Manipulability, Condition Number, and Accuracy of Parallel Robots
,” ASME J. Mech. Des.
, 128
(1
), pp. 199
–206
.10.1115/1.21217406.
Huang
, T.
, Liu
, H. T.
, and Chetwynd
, D. G.
, 2011
, “Generalized Jacobian Analysis of Lower Mobility Manipulators
,” Mech. Mach. Theory
, 46
(6
), pp. 831
–844
.10.1016/j.mechmachtheory.2011.01.0097.
Hunt
, K. H.
, 1978
, Kinematic Geometry of Mechanisms
, Oxford University Press
, Oxford, UK
.8.
Sugimoto
, K.
, and Duffy
, J.
, 1982
, “Applications of Linear Algebra to Screw Systems
,” Mech. Mach. Theory
, 17
(1
), pp. 73
–83
.10.1016/0094-114X(82)90025-89.
Kerr
, D.
R.
, and Sanger
, D. J.
, 1989
, “The Inner Product in the Evaluation of Reciprocal Screws
,” Mech. Mach. Theory
, 24
(2
), pp. 87
–92
.10.1016/0094-114X(89)90014-110.
Samuel
, A. E.
, McAree
, P. R.
, and Hunt
, K. H.
, 1991
, “Unifying Screw Geometry and Matrix Transformations
,” Int. J. Rob. Res.
, 10
(5
), pp. 454
–472
.10.1177/02783649910100050211.
Rico-Martínez
, J. M.
, and Duffy
, J.
, 1992
, “Orthogonal Spaces and Screw Systems
,” Mech. Mach. Theory
, 27
(4
), pp. 451
–458
.10.1016/0094-114X(92)90036-H12.
Joshi
, S. A.
, and Tsai
, L. W.
, 2002
, “Jacobian Analysis of Limited-DOF Parallel Manipulators
,” ASME J. Mech. Des.
, 124
(2
), pp. 254
–258
.10.1115/1.146954913.
Kim
, D.
, and Chung
, W. K.
, 2003
, “Analytic Formulation of Reciprocal Screws and Its Applications to Nonredundant Robot Manipulators
,” ASME J. Mech. Des.
, 125
(1
), pp. 158
–164
.10.1115/1.153950814.
Hunt
, K. H.
, 2004
, Robots and Screw Theory: Applications of Kinematics and Statics to Robotics
, Oxford University Press
, Oxford, UK
.15.
Fang
, Y. F.
, and Tsai
, L. W.
, 2002
, “Structure Synthesis of a Class of 4-DOF and 5-DOF Parallel Manipulators With Identical Limb Structures
,” Int. J. Rob. Res.
, 21
(9
), pp. 799
–810
.10.1177/027836490202100931416.
Fang
, Y. F.
, and Tsai
, L. W.
, 2004
, “Structure Synthesis of a Class of 3-DOF Rotational Parallel Manipulators
,” IEEE Trans. Rob. Autom.
, 20
(1
), pp. 117
–121
.10.1109/TRA.2003.81959717.
Bonev
, L. A.
, Zlatanov
, D.
, and Gosselin
, C. M.
, 2003
, “Singularity Analysis of 3-DOF Planar Parallel Mechanisms Via Screw Theory
,” ASME J. Mech. Des.
, 125
(3
), pp. 573
–581
.10.1115/1.158287818.
Carricato
, M.
, 2005
, “Fully Isotropic Four-Degrees-of-Freedom Parallel Mechanisms for Schoenflies Motion
,” Int. J. Rob. Res.
, 24
(5
), pp. 397
–414
.10.1177/027836490505368819.
Li
, Y. M.
, and Xu
, Q. S.
, 2007
, “Design and Development of a Medical Parallel Robot for Cardiopulmonary Resuscitation
,” IEEE/ASME Trans. Mechatron.
, 12
(3
), pp. 265
–273
.10.1109/TMECH.2007.89725720.
Gallardo-Alvarado
, J.
, Aguilar-Nájera
, C. R.
, Casique-Rosas
, L.
, Rico-Martínez
, J. M.
, and Islam
, M. N.
, 2008
, “Kinematics and Dynamics of 2(3-RPS) Manipulators by Means of Screw Theory and the Principle of Virtual Work
,” Mech. Mach. Theory
, 43
(10
), pp. 1281
–1294
.10.1016/j.mechmachtheory.2007.10.00921.
Li
, Y. M.
, and Xu
, Q. S.
, 2008
, “Stiffness Analysis for a 3-PUU Parallel Kinematic Machine
,” Mech. Mach. Theory
, 43
(2
), pp. 186
–200
.10.1016/j.mechmachtheory.2007.02.00222.
Liu
, H. T.
, Huang
, T.
, and Chetwynd
, D. G.
, 2011
, “A Method to Formulate a Dimensionally Homogeneous Jacobian of Parallel Manipulators
,” IEEE Trans. Rob.
, 27
(1
), pp. 150
–156
.10.1109/TRO.2010.208209123.
Liu
, H. T.
, Huang
, T.
, and Chetwynd
, D. G.
, 2011
, “An Approach for Acceleration Analysis of Lower Mobility Parallel Manipulators
,” ASME J. Mech. Rob.
, 3
(1
), p. 011013
.10.1115/1.400327124.
Liu
, H. T.
, Huang
, T.
, and Chetwynd
, D. G.
, 2011
, “A General Approach for Geometric Error Modeling of Lower Mobility Parallel Manipulators
,” ASME J. Mech. Rob.
, 3
(2
), p. 021013
.10.1115/1.400384525.
Liu
, H. T.
, Li
, Y. G.
, Huang
, T.
, and Chetwynd
, D. G.
, 2011
, “An Approach for Stiffness Modeling of Lower Mobility Parallel Manipulators Using the Generalized Jacobian
,” 13th World Congress in Mechanism and Machine Science
, Guanajuato
, Mexico
, June 19–25, p. A12_347-1
.26.
Dai
, J.
S.
, and Jones
, J. R.
, 2002
, “Null-Space Construction Using Cofactors From a Screw-Algebra Context
,” Proc. R. Soc. London, Ser. A
, 458
(2024
), pp. 1845
–1866
.10.1098/rspa.2001.094927.
Meyer
, C. D.
, 2000
, Matrix Analysis and Applied Linear Algebra
, Society for Industrial and Applied Mathematics
, Philadelphia, PA
.28.
Dai
, J. S.
, and Jones
, J. R.
, 2003
, “A Linear Algebraic Procedure in Obtaining Reciprocal Screw Systems
,” J. Rob. Syst.
, 20
(7
), pp. 401
–412
.10.1002/rob.1009429.
Angeles
, J.
, and Chen
, C.
, 2007
, “Generalized Transmission Index and Transmission Quality for Spatial Linkages
,” Mech. Mach. Theory
, 42
(9
), pp. 1225
–1237
.10.1016/j.mechmachtheory.2006.08.00130.
Arnold
, V.
, 1989
, Mathematical Methods of Classical Mechanics
, 2nd ed., Springer-Verlag
, Berlin
.31.
Abraham
, R.
, and Marsden
, J. E.
, 1994
, Foundations of Mechanics
, 2nd ed., Addison Wesley
, Menlo Park, CA
.32.
Selig
, J. M.
, and McAree
, P. R.
, 1996
, “A Simple Approach to Invariant Hybrid Control
,” 1996 IEEE International Conference on Robotics and Automation (ICAR’ 96)
, Minneapolis
, MN
, Apr. 22–28, pp. 2238
–2245
.33.
Featherstone
, R.
, 2000
, “On the Limits to Invariance in the Twist/Wrench and Motor Representations of Motion and Force Vectors
,” Symposium Commemorating the Legacy, Works, and Life of Sir Robert Stawell Ball Upon the 100th Anniversary of A Treatise on the Theory of Screws
, Trinity College, University of Cambridge
, Cambridge, UK
, July 9–11.34.
Featherstone
, R.
, 2003
, “A Dynamic Model of Contact Between a Robot and an Environment With Unknown Dynamics
,” Robotics Research, Springer Tracts in Advanced Robotics
, Vol. 6
, A. J.
Raymond
and Z.
Alexandar
, eds., Springer-Verlag
, Berlin
, pp. 433
–446
.35.
Lax
, P. D.
, 2007
, Linear Algebra and Its Applications
, 2nd ed., Wiley
, New York
, pp. 1
–31
.36.
Angeles
, J.
, 1994
, “On Twist and Wrench Generators and Annihilators
,” Computer-Aided Analysis of Rigid and Flexible Mechanical Systems
, M.
Seabra Pereira
and J. C.
Ambrósio
, eds., Kluwer Academic Publishers B.V., Dordrecht
, The Netherlands
, pp. 379
–411
.37.
Huang
, Z.
, and Li
, Q. C.
, 2002
, “General Methodology for Type Synthesis of Symmetrical Lower-Mobility Parallel Manipulators and Several Novel Manipulators
,” Int. J. Rob. Res.
, 21
(2
), pp. 131
–145
.10.1177/02783640276047534238.
Huang
, Z.
, and Li
, Q. C.
, 2003
, “Type Synthesis of Symmetrical Lower-Mobility Parallel Mechanisms Using the Constraint-Synthesis Method
,” Int. J. Rob. Res.
, 22
(1
), pp. 59
–79
.10.1177/027836490302200100539.
Kong
, X. W.
, and Gosselin
, C. M.
, 2007
, Type Synthesis of Parallel Mechanisms Springer Tracts in Advanced Robotics
, Vol. 33
, Springer-Verlag
, Berlin
, pp. 19
–83
.40.
Hervé
, J. M.
, and Sparacino
, F.
, 1991
, “Structural Synthesis of ‘Parallel’ Robots Generating Spatial Translation
,” 5th International Conference on Advanced Robotics (ICAR’ 91)
, Pisa
, Italy
, June 19–22, pp. 808
–813
.41.
Hervé
, J. M.
, 1999
, “The Lie Group of Rigid Body Displacements, a Fundamental Tool for Mechanism Design
,” Mech. Mach. Theory
, 34
(5
), pp. 719
–730
.10.1016/S0094-114X(98)00051-2Copyright © 2015 by ASME
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