Non-null Lie quadratics in E3

Journal of Mathematical Physics

Volumn 45, Issue 11, 2004, Pages 4334-4351

  Noakes, Lyle ^a  

^a UNIVERSITY OF WESTERN AUSTRALIA   (Australia)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 22544465566     PISSN: 00222488     EISSN: None     Source Type: Journal    
DOI: 10.1063/1.1803609     Document Type: Article

References (35)

1. Abrunheiro, L. and Camarinha, M., " Riemannian cubic polynomials, " preprint 04-03, January 2004, Department of Mathematics, University of Coimbra, Portugal.
2. Altafini, C., " Reduction by group symmetry of second order variational problems on a semidirect product of Lie groups with positive definite Riemannian metric, " ESAIM: Control, Optimisation and Calculus of Variations (in press).
3. Angeles, J. and Akras, R., " Cartesian trajectory planning for 3-DOF spherical wrists, " IEEE Conference on Robotics and Automation, Scottsdale, AZ, 1989, pp. 68-74.
4. Barr, A. H., Currin, B., Gabriel, S., and Hughes, J. F., " Smooth interpolation of orientations with angular velocity constraints using quaternions, " Comput. Graphics 26, 313-320 (1992).
5. Brady J.M., Hollerbach, J.M., Johnson, T.L., Lozano-Perez, T., and Masson, M.T., Robot Motion: Planning and Control (MIT Press, Cambridge, MA, 1982).
6. Buss, S. R. and Fillmore, J. " Spherical averages and applications to spherical splines and interpolation, " ACM Trans. Graphics 20, 95-126 (2001).
7. Camarinha, M., " The geometry of cubic polynomials on Riemannian manifolds, " Ph.D. thesis in Pure Mathematics, University of Coimbra, Portugal, 1996.
8. Camarinha, M., Silva Leite, F., and Crouch, P., " On the geometry of Riemannian cubic polynomials, " Diff. Geom. Applic. 15, 107-135 (2001).
9. k on non-Euclidean spaces, " IMA J. Math. Control Inf. 12, 399-410 (1995).
10. Chapman, P. B. and Noakes, L., " Singular perturbations and interpolation - a problem in robotics, " Nonlinear Anal. Theory, Methods Appl. 16, 849-859 (1991).
11. Crouch, P., Kun, G., and Silva Leite, F., " The De Castlejau algorithm on Lie groups and spheres, " J. Dyn. Control Syst. 5, 397-429 (1999).
12. Duff, T., "Quaternion splines for animating rotations, " Second Summer Graphics Workshop, Monterey, CA, 12-13 December 1985 (Usenix Association), pp. 54-62.
13. Gabriel, S.A. and Kajiya, J.T., Spline Interpolation in curved manifolds (unpublished).
14. Giambo, R., Giannoni, F., and Piccione, P., " An analytical theory for Riemannian cubic polynomials, " IMA J. Math. Control Inf. 19, 445-460 (2002).
15. Giambo, R., Giannoni, F., and Piccione, P., Higher order interpolation in Riemannian manifolds (unpublished).
16. Jost, J., Riemannian Geometry and Geometrical Analysis (Springer-Verlag, Berlin, 1995).
17. Kang, I. G. and Park, F. C., " Cubic spline algorithms for orientation interpolation, " Int. J. Numer. Methods Eng. 46, 45-64 (1999).
18. Krakowski, K., "Geometrical methods of inference," Ph.D. thesis, University of Western Australia, 2002.
19. Noakes, L., " Asymptotically smooth splines, " World Scientific Series in Approximations and Decompositions 4, 131-137 (1994).
20. Noakes, L., " Riemannian quadratics, " in Curves and Surfaces with Applications in CAGD, edited by, A. Le Méhauté, C. Rabut, L. Schumaker, (Vanderbilt University Press, Nashville, TN, 1997), Vol. 1, pp. 319-328.
21. Noakes, L., " Nonlinear corner-cutting, " Adv. Comput. Math. 8, 165-177 (1998).
22. Noakes, L., " Accelerations of Riemannian quadratics, " Proc. Am. Math. Soc. 127, 1827-1836 (1999).
23. Noakes, L., " Quadratic interpolation on spheres, " Adv. Comput. Math. 17, 385-395 (2002).
24. Noakes, L., " Null cubics and Lie quadratics, " J. Math. Phys. 44, 1436-1448 (2003).
25. Noakes, L., " Duality and Riemannian cubics, " Adv. Comput. Math. (in press).
26. Noakes, L., Heinzinger, G., and Paden, B., " Cubic splines on curved spaces, " IMA J. Math. Control Inf. 6, 465-473 (1989).
27. Paul, R. P., " Manipulator path control, " IEEE Trans. Syst. Man Cybern. SMC-9, 702-711 (1979).
28. Park, F. C., and Ravani, B., " Smooth invariant interpolation of rotations, " ACM Trans. Graphics 16, 277-295 (1997).
29. Shoemake, K., " Animating rotation with quaternion curves, " Comput.-Aided Des. 19, 245-254 (1985).
30. Silva Leite, F., Camarinha, M., and Crouch, P., " Elastic curves as solutions of Riemannian and sub-Riemannian control problems, " Math. Control, Signals, Syst. 13, 140-155 (2000).
31. Tan, H. H. and Potts, R. B., " A discrete path/trajectory planner for robotic arms, " J. Aust. Math. Soc. Ser. B, Appl. Math. 31, 1-28 (1989).
32. Taylor, R. H., " Planning and execution of straight-line manipulator trajectories, " IBM J. Res. Dev. 23, 424-436 (1979).
33. Zefran, M., Kumar, V., and Croke, C., " On the generation of smooth three-dimensional rigid body motions, " IEEE Trans. Rob. Autom. (1995).
34. Zefran, M. and Kumar, V., " Planning of smooth motions on SE (3), " Proceedings of the 1996 International Conference on robotics and automation.
35. Zefran, M. and Kumar, V., " Interpolation scheme for rigid body motions, " Comput.-Aided Des. 30, 179-189 (1998).