On the geometry of rolling and interpolation curves on S n, SO n, and Grassmann manifolds

Journal of Dynamical and Control Systems

Volumn 13, Issue 4, 2007, Pages 467-502

  Hüper, Knut ^a,b   Silva Leite, F ^c,d  

^a CSIRO   (Australia)

^b AUSTRALIAN NATIONAL UNIVERSITY   (Australia)

^c UNIVERSITY OF COIMBRA   (Portugal)

^d UNIVERSITY OF COIMBRA   (Portugal)

Author keywords

Constrained variational problems; Geodesics; Geometric splines; Grassmann manifold; Interpolation; Kinematic equation; Orthogonal group; Parallel transport; Rolling mapping; Sphere

Indexed keywords

CONSTRAINED VARIATIONAL PROBLEMS; GRASSMANN MANIFOLD; ORTHOGONAL GROUP; PARALLEL TRANSPORT; ROLLING MAPPING;

CONSTRAINT THEORY; GEOMETRY; KINEMATICS; POLYNOMIALS; VECTORS;

INTERPOLATION;

EID: 35548981394     PISSN: 10792724     EISSN: 15738698     Source Type: Journal    
DOI: 10.1007/s10883-007-9027-3     Document Type: Article

References (30)

1. A. A. Agrachev and Yu. L. Sachkov, Control theory from the geometric viewpoint. Springer-Verlag, Berlin (2004).
2. A. M. Bloch, Nonholonomic mechanics and control. Springer-Verlag, New York (2003).
3. M. Camarinha, The geometry of cubic polynomials on Riemannian manifolds. Ph.D. thesis, University of Coimbra, Portugal (1996).
4. k on non-Eulidean spaces. J. Math. Control Inform. 12 (1995), 399-410.
5. P. Crouch, G. Kun, and F. Silva Leite, The De Casteljau algorithm on Lie groups and spheres. J. Dynam. Control Systems 5 (1999), No. 3, 397-429.
6. P. Crouch and F. Silva Leite, Geometry and the dynamic interpolation problem. In: Proc. American Control Conference, Boston (1991), pp. 1131-1136.
7. P. Crouch and F. Silva Leite, The dynamic interpolation problem on Riemannian manifolds, Lie groups, and symmetric spaces. J. Dynam. Control Systems 1 (1995), No. 2, 177-202.
8. P. De Casteljau, Outillages méthodes calcul. Technical report, A. Citroen, Paris (1959).
9. G. Farin, Curves and surfaces for CAGD: A practical guide. Morgan Kaufmann, San Francisco (2002).
10. R. Giambó, F. Giannoni, and P. Piccione, An analytical theory for Riemannian cubic polynomials. IMA J. Math. Control Inform. 19 (2002), 445-460.
11. R. Gilmore, Lie groups, Lie algebras, and some of their applications. Wiley, New York (1974).
12. U. Helmke, K. Hüper, and J. Trumpf, Newton's method on Grassmann manifolds. (in press).
13. U. Helmke and J. B. Moore, Optimization and dynamical systems. CCES, Springer-Verlag, London (1994).
14. K. Hüper, M. Kleinsteuber, and F. Silva Leite, Rolling Stiefel manifolds. Int. J. Systems Sci. (to appear).
15. K. Hüper and F. Silva Leite, Smooth interpolating curves with applications to path planning. In: 10th IEEE Mediterranean Conference on Control and Automation, Instituto Superior Técnico, Lisboa, Portugal, July, 9-12, 2002 (on CDROM).
16. n, and Grassmann manifolds. Technical Report SISSA 56/2005/M, Int. School for Adv. Stud., Trieste, Italy (2005).
17. P. E. Jupp and J. T. Kent, Fitting smooth paths to spherical data. Appl. Statist. 36 (1987), No. 1, 34-46.
18. V. Jurdjevic, Geometric control theory. Cambridge Univ. Press, Cambridge (1997).
19. S. Kobayashi and K. Nomizu, Foundations of differential geometry, Vol. I. Wiley, New York (1996).
20. J. M. Lee, Riemannian manifolds: An introduction to curvature. Springer-Verlag, New York (1997).
21. J. W. Milnor, Morse theory. Based on lecture notes by M. Spivak and R. Wells. Ann. Math. Stud. 51, Princeton Univ. Press, Princeton, New Jersey (1963).
22. L. Noakes, G. Heinzinger, and B. Paden, Cubic splines on curved spaces. IMA J. Math. Control Inform. 6 (1989), 465-473.
23. F. C. Park and B. Ravani, Bézier curves on Riemannian manifolds and Lie groups with kinematics applications. ASME J. Mech. Design 117 (1995), 36-40.
24. R. W. Sharpe, Differential geometry. Springer-Verlag, New York (1996).
25. Y. Shen and K. Hüper, Optimal joint trajectory planning for manipulator robot performing constrained motion tasks. In: Australasian Conf. on Robotics and Automation, Canberra, December 2004 (on CDROM).
26. Y. Shen and K. Hüper, Optimal joint trajectory planning of manipulators subject to motion constraints. In: Int. Conf. on Advanced Robotics (ICAR 2005), Seattle, July 2005 (on CDROM).).
27. Y. Shen, K. Hüper, and F. Silva Leite, Smooth interpolation of orientation by rolling and wrapping for robot motion planning. In: IEEE Int. Conf. on Robotics and Automation (ICRA 2006), Orlando, Florida, USA (2006), pp. 113-118.
28. S. T. Smith, Geometric optimization methods for adaptive filtering. Ph.D. thesis, Harvard University, Cambridge (1993).
29. S. T. Smith, Optimization techniques on Riemannian manifolds. In: Hamiltonian and gradient flows, algorithms and control (A. Bloch, ed.), Fields Inst. Commun. Amer. Math. Soc., Providence (1994), pp. 113-136.
30. n. Math. Control Signals Systems 17 (2005), 14-37.