Abstract
Rolling motions describe how two manifolds both embedded in the same manifold roll on each other without slip and without twist. With these constraints on the velocities of motion, an interesting issue is that of knowing whether such rolling motions are controllable or not. In this paper, after embedding the symplectic group and its affine tangent space at a point in an appropriate pseudo-Riemannian space, we derive the kinematic equations for rolling and prove that the corresponding rolling motions are controllable.
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References
Crouch, P., Silva Leite, F.: Rolling maps for pseudo-Riemannian manifolds. In: Proc. IEEE-CDC 2012, Hawaii, USA, December 10-13 (2012)
Godoy, M., Grong, E., Markina, I., Silva Leite, F.: An intrinsic formulation of the rolling manifolds problem. Journal of Dynamical and Control Systems 18(2), 181–214 (2012)
Grong, E.: Controllability of rolling without twisting or slipping in higher dimensions. SIAM J. Control Optim. 50(4), 2462–2485 (2012)
Hüper, K., Silva Leite, F.: On the geometry of rolling and interpolation curves on S n, SO(n) and Grassmann manifolds. Journal of Dynamical and Control Systems 13(4), 467–502 (2007)
Hüper, K., Krakowski, K., Silva Leite, F.: Rolling maps in a Riemannian framework. Textos de Matemática 43, 15–30 (2011), (J. Cardoso, K. Hüper, P. Saraiva, Eds.) Department of Mathematics, Universisity of Coimbra
Jurdjevic, V., Sussmann, H.: Control systems on Lie groups. Journal of Differential Equations 12, 313–329 (1972)
Jurdjevic, V., Zimmerman, J.A.: Rolling sphere problems on spaces of constant curvature. Math. Proc. Camb. Phil. Soc. 144, 19–729 (2008)
Korolko, A., Silva Leite, F.: Kinematics for rolling a Lorentzian sphere. In: Proc. CDC-ECC, Orlando, USA, December 2-15, pp. 6522–6528 (2011)
Marques, A., Silva Leite, F.: Rolling a pseudohyperbolic space over the affine tangent space at a point. In: Proc. CONTROLO 2012, Paper # 36, Funchal, Portugal, pp. 16–18 (July 2012)
Mehl, C., Mehrmann, V., Ran, A., Rodman, L.: Perturbation analysis of Lagrangian invariant subspaces of symplectic matrices. Linear and Multilinear Algebra 57(2), 141–184 (2009)
O’Neill, B.: Semi-Riemannian geometry with applications to relativity. Academic Press, Inc., N.Y (1983)
Sharpe, R.W.: Differential geometry. Springer, New York (1996)
Tar, J.K., Kozlowski, K., Rudas, I.J., Ilkei, T.: Application of the symplectic group in a novel branch of soft computing for controlling of electro-mechanical devices. In: Proc. IEEE International Workshop on Robot Motion and Control, Dworek, Poland, October 18-20, pp. 71–77 (2001)
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Marques, A., Leite, F.S. (2015). Controllability for the Constrained Rolling Motion of Symplectic Groups. In: Moreira, A., Matos, A., Veiga, G. (eds) CONTROLO’2014 – Proceedings of the 11th Portuguese Conference on Automatic Control. Lecture Notes in Electrical Engineering, vol 321. Springer, Cham. https://doi.org/10.1007/978-3-319-10380-8_1
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DOI: https://doi.org/10.1007/978-3-319-10380-8_1
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-10379-2
Online ISBN: 978-3-319-10380-8
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