# Article

Full entry | PDF   (1.1 MB)
Keywords:
rolling; group of isometries; ellipsoid; kinematic equations; interpolation
Summary:
We present an algorithm to generate a smooth curve interpolating a set of data on an \$n\$-dimensional ellipsoid, which is given in closed form. This is inspired by an algorithm based on a rolling and wrapping technique, described in [11] for data on a general manifold embedded in Euclidean space. Since the ellipsoid can be embedded in an Euclidean space, this algorithm can be implemented, at least theoretically. However, one of the basic steps of that algorithm consists in rolling the ellipsoid, over its affine tangent space at a point, along a curve. This would allow to project data from the ellipsoid to a space where interpolation problems can be easily solved. However, even if one chooses to roll along a geodesic, the fact that explicit forms for Euclidean geodesics on the ellipsoid are not known, would be a major obstacle to implement the rolling part of the algorithm. To overcome this problem and achieve our goal, we embed the ellipsoid and its affine tangent space in \$\mathbb{R}^{n+1}\$ equipped with an appropriate Riemannian metric, so that geodesics are given in explicit form and, consequently, the kinematics of the rolling motion are easy to solve. By doing so, we can rewrite the algorithm to generate a smooth interpolating curve on the ellipsoid which is given in closed form.
References:
[1] Agrachev, A., Sachkov, Y.: Control Theory from the Geometric Viewpoint. In: Encyclopaedia of Mathematical Sciences 87 (2004), Springer-Verlag. MR 2062547 | Zbl 1062.93001
[2] Camarinha, M.: The Geometry of Cubic Polynomials on Riemannian Manifolds. PhD. Thesis, Departamento de Matemática, Universidade de Coimbra 1996.
[3] Crouch, P., Kun, G., Leite, F. S.: The De Casteljau algorithm on Lie groups and spheres. J. Dyn. Control Syst. 5 (1999), 3, 397-429. DOI 10.1023/A:1021770717822 | MR 1706785 | Zbl 0961.53027
[4] Crouch, P, Leite, F. S.: Geometry and the dynamic interpolation problem. In: Proc. American Control Conference Boston 1991, pp. 1131-1137.
[5] Crouch, P., Leite, F. S.: The dynamic interpolation problem: on Riemannian manifolds, Lie groups and symmetric spaces. J. Dyn. Control Syst. 1 (1995), 2, 177-202. DOI 10.1007/BF02254638 | MR 1333770 | Zbl 0946.58018
[6] Fedorov, Y. N., Jovanović, B.: Nonholonomic LR systems as generalized chaplygin systems with an invariant measure and flows on homogeneous spaces. J. Nonlinear Sci. 14 (2004), 4, 341-381. DOI 10.1007/s00332-004-0603-3 | MR 2076030 | Zbl 1125.37045
[7] Giambó, R., Giannoni, F., Piccione, P.: Fitting smooth paths to spherical data. IMA J. Math. Control Inform. 19 (2002), 445-460. MR 1949013
[8] Hüper, K., Kleinsteuber, M., Leite, F. S.: Rolling Stiefel manifolds. Int. J. Systems Sci. 39 (2008), 8, 881-887. MR 2437853 | Zbl 1168.53007
[9] Hüper, K., Krakowski, K. A., Leite, F. S.: Rolling Maps in a Riemannian Framework. In: Mathematical Papers in Honour of Fátima Silva Leite, Textos de Matemática 43, Department of Mathematics, University of Coimbra 2011, pp. 15-30. MR 2894254 | Zbl 1254.53018
[10] Hüper, K., Leite, F. S.: Smooth interpolating curves with applications to path planning. In: 10th IEEE Mediterranean Conference on Control and Automation (MED 2002), Lisbon 2002.
[11] Hüper, K., Leite, F. S.: On the geometry of rolling and interpolation curves on \$S^n\$, \$SO_n\$ and Graßmann manifolds. J. Dyn. Control Syst. 13 (2007), 4, 467-502. DOI 10.1007/s10883-007-9027-3 | MR 2350231
[12] Jupp, P., Kent, J.: Fitting smooth paths to spherical data. Appl. Statist. 36 (1987), 34-46. DOI 10.2307/2347843 | MR 0887825 | Zbl 0613.62086
[13] Jurdjevic, V., Zimmerman, J.: Rolling problems on spaces of constant curvature. In: Lagrangian and Hamiltonian methods for nonlinear control 2006, Proc. 3rd IFAC Workshop 2006 (F. Bullo and K. Fujimoto, eds.), Nagoya 2007, Lect. Notes Control Inform. Sciences, Springer, pp. 221-231. MR 2376942 | Zbl 1136.49028
[14] Krakowski, K., Leite, F. S.: Smooth interpolation on ellipsoids via rolling motions. In: PhysCon 2013, San Luis Potosí, Mexico 2013.
[15] Krakowski, K. A., Leite, F. S.: Why controllability of rolling may fail: a few illustrative examples. In: Pré-Publicações do Departamento de Matemática, No. 12-26, University of Coimbra 2012, pp. 1-30.
[16] Lee, J. M.: Riemannian Manifolds: An Introduction to Curvature. In? Graduate Texts in Mathematics No. 176, Springer-Verlag, New York 1997. MR 1468735 | Zbl 0905.53001
[17] Machado, L., Leite, F. S., Krakowski, K.: Higher-order smoothing splines versus least squares problems on riemannian manifolds. J. Dyn. Control Syst. 16 (2010), 1, 121-148. DOI 10.1007/s10883-010-9080-1 | MR 2580471 | Zbl 1203.65028
[18] Noakes, L., Heinzinger, G., Paden, B.: Cubic splines on curved spaces. IMA J. Math. Control Inform. 6 (1989), 465-473. DOI 10.1093/imamci/6.4.465 | MR 1036158 | Zbl 0698.58018
[19] Nomizu, K.: Kinematics and differential geometry of submanifolds. Tôhoku Math. J. 30 (1978), 623-637. DOI 10.2748/tmj/1178229921 | MR 0516894 | Zbl 0395.53005
[20] Park, F., Ravani, B.: Optimal control of the sphere \${S^n}\$ rolling on \${E^n}\$. ASME J. Mech. Design 117 (1995), 36-40.
[21] Samir, C., Absil, P.-A., Srivastava, A., Klassen, E.: A gradient-descent method for curve fitting on Riemannian manifolds. Found. Comput. Math. 12 (2012), 49-73. DOI 10.1007/s10208-011-9091-7 | MR 2886156 | Zbl 1245.65017
[22] Sharpe, R. W.: Differential Geometry: Cartan's Generalization of Klein's Erlangen Program. In: Graduate Texts in Mathematics, No. 166. Springer-Verlag, New York 1997. MR 1453120 | Zbl 0876.53001
[23] Zimmerman, J.: Optimal control of the sphere \${S^n}\$ rolling on \${E^n}\$. Math. Control Signals Systems 17 (2005), 1, 14-37. DOI 10.1007/s00498-004-0143-2 | MR 2121282 | Zbl 1064.49021

Partner of