Duality and Riemannian cubics

Advances in Computational Mathematics

Volumn 25, Issue 1-3, 2006, Pages 195-209

  Noakes, Lyle ^a  

^a UNIVERSITY OF WESTERN AUSTRALIA   (Australia)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 33745662909     PISSN: 10197168     EISSN: None     Source Type: Journal    
DOI: 10.1007/s10444-004-7621-4     Document Type: Article

References (32)

1. J.H. Ahlberg, E.N. Nilson and J.H. Walsh, The Theory of Splines and Their Applications, Mathematics in Science and Engineering, Vol. 38 (Academic Press, New York, 1967).
2. J. Angeles and R. Akras, Cartesian trajectory planning for 3-DOF spherical wrists, in: IEEE Conf. on Robotics and Automation, Scottsdale, AZ, May 1989, pp. 68-74.
3. A.H. Barr, B. Currin, S. Gabriel and J.F. Hughes, Smooth interpolation of orientations with angular velocity constraints using quaternions, Comput. Graphics 26(2) (1992) 313-320.
4. J.M. Brady, J.M. Hollerbach, T.L. Johnson, T. Lozano-Perez and M.T. Masson, Robot Motion: Planning and Control (MIT Press, Cambridge, MA, 1982).
5. S. Buss, Accurate and efficient simulations of rigid body rotations, J. Comput. Phys. 164 (2000) 377-406.
6. M. Camarinha, F.S. Leite and P. Crouch, On the geometry of Riemannian cubic polynomials, Differential Geom. Appl. 15(2) (2001) 107-135.
7. k on non-Euclidean spaces, IMA J. Math. Control Inform. 12(4) (1995) 399-410.
8. P.B. Chapman and L. Noakes, Singular perturbations and interpolation - a problem in robotics, Nonlinear Analysis TMA 16(10) (1991) 849-859.
9. P. Crouch, G. Kun and F.S. Leite, The De Castlejau algorithm on Lie groups and spheres, J. Dynam. Control Systems 5(3) (1999) 397-429.
10. P. Crouch and F.S. Leite, The dynamic interpolation problem: On Riemannian manifolds, Lie groups, and symmetric spaces, J. Dynam. Control Systems 1(2) (1995) 177-202.
11. T. Duff, Quaternion splines for animating rotations, in: Second Summer Graphics Workshop, Monterey, CA, Usenix Association, 12-13 December 1985, pp. 54-62.
12. S.A. Gabriel and J.T. Kajiya, Spline interpolation in curved manifolds, unpublished manuscript (1985).
13. J. Jost, Riemannian Geometry and Geometrical Analysis (Springer, Berlin, 1995).
14. I.G. Kang and F.C. Park, Cubic spline algorithms for orientation interpolation, Internat. J. Numer. Methods Engrg. 46 (1999) 45-64.
15. K. Krakowski, Ph.D. thesis, University of Western Australia (2002) submitted.
16. F.S. Leite, M. Camarinha and P. Crouch, Elastic curves as solutions of Riemannian and sub-Riemannian control problems, Math. Control Signals Systems 13(2) (2000) 140-155.
17. L. Noakes, Asymptotically smooth splines, in: World Scientific Series in Approximations and Decompositions, Vol. 4 (1994) pp. 131-137.
18. L. Noakes, Riemannian quadratics, in: Curves and Surfaces with Applications in CAGD, eds. A. Le Méhauté, C. Rabut and L.L. Schumaker, Vol. 1 (Vanderbilt Univ. Press, Nashville, TN, 1997) pp. 319-328.
19. L. Noakes, Nonlinear corner-cutting, Adv. Comput. Math. 8 (1998) 165-177.
20. L. Noakes, Accelerations of Riemannian quadratics, Proc. Amer. Math. Soc. 127 (1999) 1827-1836.
21. L. Noakes, Quadratic interpolation on spheres, Adv. Comput. Math., in press.
22. L. Noakes, Null cubics and Lie quadratics, J. Math. Phys., in press.
23. L. Noakes, G. Heinzinger and B. Paden, Cubic splines on curved spaces, J. Math. Control Inform. 6 (1989) 465-473.
24. F.C. Park and B. Ravani, Smooth invariant interpolation of rotations, ACM Trans. Graphics 16(3) (1997) 277-295.
25. R.P. Paul, Manipulator path control, IEEE Trans. Systems Man Cybernet. 9 (1979) 702-711.
26. K. Shoemake, Animating rotation with quaternion curves, SIGGRAPH 19(3) (1985) 245-254.
27. H.H. Tan and R.B. Potts, A discrete path/trajectory planner for robotic arms, J. Austral. Math. Soc. Ser. B 31 (1989) 1-28.
28. R.H. Taylor, Planning and execution of straight-line manipulator trajectories, IBM J. Res. Develop. 23 (1979) 424-436.
29. M. Zefran and V. Kumar, Planning of smooth motions on SE(3), in: IEEE Internat. Conf. on Robotics and Automation, Minneapolis, MN, 1996.
30. M. Zefran and V. Kumar, Two methods for interpolating rigid body motions, in: IEEE Internat. Conf. on Robotics and Automation, Leuven, Belgium, 1996.
31. M. Zefran and V. Kumar, Interpolation schemes for rigid body motions, Comput. Aided Design 30(3) (1998) 179-189.
32. M. Zefran, V. Kumar and C. Croke, Choice of Riemannian metrics for rigid body dynamics, in: Proc. of ASM E Design Engineering Technical Conf. and Computers in Engineering Conf., Irvine, CA, 18-22 August 1996, pp. 1-11.