This paper brings together line geometry, kinematic geometry of line-symmetric motions, and computer aided geometric design to develop a method for geometric design of rational Be´zier line-symmetric motions. By taking advantage of the kinematic geometry of a line-symmetric motion, the problem of synthesizing a rational Be´zier line-symmetric motion is reduced to that of designing a rational Be´zier ruled surface. In this way, a recently developed de Casteljau algorithm for line-geometric design of ruled surfaces can be applied. An example is presented in which the Bennet motion is represented as a rational Be´zier line-symmetric motion whose basic surface is a hyperboloid.
1.
Ge, Q. J., and Ravani, B., 1991, “Computer Aided Geometric Design of Motion Interpolants,” ASME Publ. DE-Vol. 3202:33-41, Advances in Design Automation 1991; ASME J. Mech. Des., 116, pp. 756–762.
2.
Ge
, Q. J.
, and Ravani
, B.
, 1994a
, “Geometric Construction of Be´zier Motions
,” ASME J. Mech. Des.
, 116
, pp. 756
–762
.3.
Barr
, A.
, Curin
, B.
, Gabriel
, S.
, and Hughes
, J.
, 1992
, “Smooth Interpolation of Orientations With Angular Velocity Constraints Using Quaternions
,” Comput. Graphics
, 26
, pp. 313
–320
.4.
Park
, F. C.
, and Ravani
, B.
, 1995
, “Be´zier Curves on Riemannian Manifolds and Lie Groups With Kinematic Applications
,” ASME J. Mech. Des.
, 117
, pp. 35
–40
.5.
Kim
, K.-S.
, and Nam
, K.-W.
, 1995
, “Interpolating Solid Orientations With Circular Blending Quaternion Curves
,” Comput.-Aided Des.
, 27
, pp. 385
–398
.6.
Etzel, K., and McCarthy, J. M., 1996, “Spatial Motion Interpolation in an Image Space of SO(4),” Proc. 1996 ASME Mechanisms Conference, Paper Number 96-DETC/MECH-1164.
7.
Ju¨ttler
, B.
, and Wagner
, M.
, 1996
, “Computer Aided Design With Spatial Rational B-Spline Motions
,” ASME J. Mech. Des.
, 118
, pp. 193
–201
.8.
Ju¨ttler
, B.
, and Wagner
, M.
, 1999
, “Rational Motion-Based Surface Generation
,” Comput.-Aided Des.
, 31
, pp. 203
–213
.9.
Zefran
, M.
, and Kumar
, V.
, 1998
, “Rigid Body Motion Interpolation
,” Comput.-Aided Des.
, 30
, pp. 179
–189
.10.
Srinivasan
, L. N.
, and Ge
, Q. J.
, 1996
, “Parametrically Continuous and Smooth Motion Interpolation
,” ASME J. Mech. Des.
, 118
, pp. 494
–498
.11.
Srinivasan
, L.
, and Ge
, Q. J.
, 1998
, “Fine Tuning of Rational B-Spline Motions
,” ASME J. Mech. Des.
, 120
, pp. 46
–51
.12.
Ro¨schel
, O.
, 1998
, “Rational Motion Design—A Survey
,” Comput.-Aided Des.
, 30
, pp. 169
–178
.13.
Ge
, Q. J.
, and Sirchia
, M.
, 1999
, “Computer Aided Geometric Design of Two-Parameter Freeform Motions
,” ASME J. Mech. Des.
, 121
, pp. 502
–506
.14.
Ge
, Q. J.
, and Larochelle
, P.
, 1999
, “Algebraic Motion Approximation With NURBS Motions and Its Application to Spherical Mechanism Synthesis
,” ASME J. Mech. Des.
, 121
, pp. 529
–532
.15.
Xia
, J.
, and Ge
, Q. J.
, 2001
, “On the Exact Representation of the Boundary Surfaces of the Swept Volume of a Cylinder Undergoing Rational Bezier and B-Spline Motions
,” ASME J. Mech. Des.
, 123
, pp. 261
–265
.16.
Study, E., 1903, Die Geometrie der Dynamen, Leipzig.
17.
Klein, F., 1908, Elementary Mathematics From an Advanced Standpoint: Geometry. Translated by E. R. Hedrick and C. A. Noble (1939), Dover, New York.
18.
Krames
, J.
, 1937
, “Zur Aufrechten Ellipsenbewegung des Raumes (U¨ber Symmetrische Schotungen III)
,” Monatshefte Math. Phys.
, 46
, pp. 38
–50
.19.
Yang, A. T., 1964, “Application of Quaternion Algebra and Dual Numbers to the Analysis of Spatial Mechanisms,” Ph.D. dissertation, Columbia University, UMI-No. 64-2803.
20.
Woo
, L.
, and Freudenstein
, F.
, 1970
, “Application of Line Geometry to Theoretical Kinematics and the Kinematic Analysis of Mechanical Systems
,” J. Mech.
, 5
, pp. 417
–460
.21.
Bottema, O., and Roth, B., 1979, Theoretical Kinematics, North Holland, Amsterdam.
22.
Ro¨schel
, O.
, 1985
, “Rationale Ra¨umliche Zwangla¨ufe Vierter Ordnung
,” Sitzungsber. Akad. Wiss. Wien, Math.-Naturwiss. Kl., Abt. 2A
, 194
, pp. 185
–202
.23.
Hunt, K., 1990, Kinematic Geometry of Mechanisms, Oxford University Press, New York.
24.
Ravani
, B.
, and Wang
, J.
, 1991
, “Computer Aided Geometric Design of Line Constructs
,” ASME J. Mech. Des.
, 113
, pp. 363
–371
.25.
Ge, Q. J., and Ravani, B., 1998, “Geometric Design of Rational Bezier Line Congruences and Ruled Surfaces Using Line Geometry,” Computing Supplement 13: Geometric Modeling, edited by G. Farin, H. Beiri, G. Brunnett, and T. DeRose, Springer, Berlin, pp. 101–120.
26.
Dooley, J. R., and Ravani, B., 1994, “Geometric Analysis of Spatial Rigid Body Dynamics With Multiple Frictional Contacts,” Advances in Robot Kinematics and Computational Geometry, edited by J. Lenarcic and B. Ravani, Kluwer Academic, Dordrechtt, pp. 71–80.
27.
Sprott, K., and Ravani, B., 1997, “Ruled Surfaces, Lie Groups, and Mesh Generation,” Proc. 1997 Design Automation Conference, Paper no. DETC97/DAC-3996, Sacramento, CA.
28.
Pottmann, H., and Wallner, J., 2001, Computational Line Geometry, Springer, Berlin.
29.
Ge, Q. J., 1996, “Kinematics-Driven Geometric Modeling: A Framework for Simultaneous Tool-Path Generation and Sculptured Surface Design,” Proc. 1996 IEEE Robotics and Automation Conference, Vol. 2, pp. 1819–1824, Minneapolis, MN, April 1996.
30.
Xia, J., and Ge, Q. J., 2000, “Kinematic Approximation of Ruled Surfaces Using NUEBS Motions of a Cylindrical Cutter,” Proc. 2000 ASME Design Automation Conference, Baltimore, MD, Paper No. DETC2000/DAC-14280.
31.
Piegl
, L. A.
, and Tiller
, W.
, 2002
, “Biarc Approximation of NURBS
,” Comput.-Aided Des.
, 34
, pp. 807
–814
.32.
McCarthy, J. M., 1990, Introduction to Theoretical Kinematics, MIT Press, Cambridge, MA.
33.
Ball, R., 1900, A Treatise on the Theory of Screws, Cambridge, University, Press Cambridge.
34.
McCarthy
, J. M.
, 1986
, “Dual Orthogonal Matrices in Manipulator Kinematics
,” Int. J. Robot. Res.
, 5
, pp. 43
–49
.35.
Ravani, B., 1982, “Kinematic Mappings as Applied to Motion Approximation and Mechanism Synthesis,” Ph.D. dissertation, Stanford University.
36.
Struik, D. J., 1961, Lectures On Classical Differential Geometry, 2nd ed., Addison-Wesley, Reading, MA.
37.
Ravani
, B.
, and Roth
, B.
, 1984
, “Mappings of Spatial Kinematics
,” ASME J. Mech., Trans., Automation, Des.
, 106
, pp. 341
–347
.Copyright © 2005
by ASME
You do not currently have access to this content.
