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On the geometry of rolling and interpolation curves on $$S^n$$, $$\mathrm{SO}_n$$, and Grassmann manifolds. (English) Zbl 1140.58005
Summary: We present a procedure to generate smooth interpolating curves on submanifolds, which are given in closed form in terms of the coordinates of the embedding space. In contrast to other existing methods, this approach makes the corresponding algorithm easy to implement. The idea is to project the prescribed data on the manifold onto the affine tangent space at a particular point, solve the interpolation problem on this affine subspace, and then project the resulting curve back on the manifold. One of the novelties of this approach is the use of rolling mappings. The manifold is required to roll on the affine subspace like a rigid body, so that the motion is described by the action of the Euclidean group on the embedding space. The interpolation problem requires a combination of a pullback/push forward with rolling and unrolling. The rolling procedure by itself highlights interesting properties and gives rise to a new, but simple, concept of geometric polynomial curves on manifolds.
This paper is an extension of our previous work, where mainly the 2-sphere case was studied in detail. The present paper includes results for the $$n$$-sphere, orthogonal group $$\text{SO}_{n}$$, and real Grassmann manifolds. In particular, we present the kinematic equations for rolling these manifolds along curves without slip or twist, and derive from them formulas for the parallel transport of vectors along curves on the manifold.

##### MSC:
 58E40 Variational aspects of group actions in infinite-dimensional spaces 53A17 Differential geometric aspects in kinematics 53B05 Linear and affine connections 58E50 Applications of variational problems in infinite-dimensional spaces to the sciences 93C15 Control/observation systems governed by ordinary differential equations
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##### References:
 [1] A. A. Agrachev and Yu. L. Sachkov, Control theory from the geometric viewpoint. Springer-Verlag, Berlin (2004). · Zbl 1062.93001 [2] A. M. Bloch, Nonholonomic mechanics and control. Springer-Verlag, New York (2003). · Zbl 1045.70001 [3] M. Camarinha, The geometry of cubic polynomials on Riemannian manifolds. Ph.D. thesis, University of Coimbra, Portugal (1996). [4] M. Camarinha, F. Silva Leite, and P. Crouch, Splines of class $$\mathcal{C}^{k}$$ on non-Eulidean spaces. J. Math. Control Inform. 12 (1995), 399–410. · Zbl 0860.58013 [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. · Zbl 0961.53027 [6] P. Crouch and F. Silva Leite, Geometry and the dynamic interpolation problem. In: Proc. American Control Conference, Boston (1991), pp. 1131–1136. · Zbl 0736.93006 [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. · Zbl 0946.58018 [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. · Zbl 1138.58307 [11] R. Gilmore, Lie groups, Lie algebras, and some of their applications. Wiley, New York (1974). · Zbl 0279.22001 [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). · Zbl 0984.49001 [14] K. Hüper, M. Kleinsteuber, and F. Silva Leite, Rolling Stiefel manifolds. Int. J. Systems Sci. (to appear). · Zbl 1168.53007 [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] K. Hüper and F. Silva Leite, On the geometry of rolling and interpolation curves on S n , SO n , and Grassmann manifolds. Technical Report SISSA 56/2005/M, Int. School for Adv. Stud., Trieste, Italy (2005). · Zbl 1140.58005 [17] P. E. Jupp and J. T. Kent, Fitting smooth paths to spherical data. Appl. Statist. 36 (1987), No. 1, 34–46. · Zbl 0613.62086 [18] V. Jurdjevic, Geometric control theory. Cambridge Univ. Press, Cambridge (1997). · Zbl 0940.93005 [19] S. Kobayashi and K. Nomizu, Foundations of differential geometry, Vol. I. Wiley, New York (1996). · Zbl 0119.37502 [20] J. M. Lee, Riemannian manifolds: An introduction to curvature. Springer-Verlag, New York (1997). · Zbl 0905.53001 [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. · Zbl 0698.58018 [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. · Zbl 0816.49032 [30] J. A. Zimmerman, Optimal control of the sphere S n rolling on E n . Math. Control Signals Systems 17 (2005), 14–37. · Zbl 1064.49021
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