doi: 10.3934/jgm.2021033
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Pure rolling motion of hyperquadrics in pseudo-Euclidean spaces

1. 

Scientific Area of Mathematics, School of Technology and Management of Viseu, Polytechnic Institute of Viseu, Campus Politécnico de Repeses, 3504-510 Viseu, Portugal

2. 

Department of Mathematics, University of Coimbra, Largo D. Dinis, 3001-454 Coimbra, Portugal

3. 

Institute of Systems and Robotics, University of Coimbra - Pólo II, Pinhal de Marrocos, 3030-290 Coimbra, Portugal

* Corresponding author: André Marques

Received  June 2021 Revised  November 2021 Early access January 2022

This paper is devoted to rolling motions of one manifold over another of equal dimension, subject to the nonholonomic constraints of no-slip and no-twist, assuming that these motions occur inside a pseudo-Euclidean space. We first introduce a definition of rolling map adjusted to this situation, which generalizes the classical definition of Sharpe [26] for submanifolds of an Euclidean space. We also prove some important properties of these rolling maps. After presenting the general framework, we analyse the particular rolling of hyperquadrics embedded in pseudo-Euclidean spaces. The central topic is the rolling of a pseudo-hyperbolic space over the affine space associated with its tangent space at a point. We derive the kinematic equations, as well as the corresponding explicit solutions for two specific cases, and prove the existence of a rolling map along any curve in that rolling space. Rolling of a pseudo-hyperbolic space on another and rolling of pseudo-spheres are equally treated. Finally, for the central theme, we write the kinematic equations as a control system evolving on a certain Lie group and prove its controllability. The choice of the controls corresponds to the choice of a rolling curve.

Citation: André Marques, Fátima Silva Leite. Pure rolling motion of hyperquadrics in pseudo-Euclidean spaces. Journal of Geometric Mechanics, doi: 10.3934/jgm.2021033
References:
[1]

A. Agrachev and Y. Sachkov, Control Theory from the Geometric Viewpoint, Vol. 87 of Encyclopaedia of Mathematical Sciences, Springer-Verlag, 2004. doi: 10.1007/978-3-662-06404-7.  Google Scholar

[2]

A. BlochM. Camarinha and L. Colombo, Dynamic interpolation for obstacle avoidance on Riemannian manifolds, Internat. J. Control, 94 (2021), 588-600.  doi: 10.1080/00207179.2019.1603400.  Google Scholar

[3]

A. M. Bloch, Nonholonomic Mechanics and Control, with the collaboration of J. Baillieul, P. E. Crouch and J. Marsden, Interdisciplinary Applied Mathematics, Springer Verlag, 2003. doi: 10.1007/b97376.  Google Scholar

[4]

A. M. Bloch and P. E. Crouch, Nonholonomic control systems on Riemannian manifolds, SIAM J. Control Optim., 33 (1995), 126-148.  doi: 10.1137/S036301299223533X.  Google Scholar

[5]

A. M. Bloch and A. G. Rojo, Kinematics of the rolling sphere and quantum spin, Commun. Inf. Syst., 10 (2010), 221-238.  doi: 10.4310/CIS.2010.v10.n4.a4.  Google Scholar

[6]

Y. Chitour and P. Kokkonen, Rolling Manifolds: Intrinsic Formulation and Controllability, arXiv: 1011.2925v2, 2011. Google Scholar

[7]

P. Crouch and F. Silva Leite, Rolling Motions of Pseudo-Orthogonal Groups, Proc. 51st IEEE-CDC 2012, 10-13 December 2012, Hawaii, USA. Google Scholar

[8]

M. Godoy MolinaE. GrongI. Markina and F. Silva Leite, An intrinsic formulation of the rolling manifolds problem, J. Dyn. Control Syst., 18 (2012), 181-214.  doi: 10.1007/s10883-012-9139-2.  Google Scholar

[9]

K. Hüper, K. Krakowski and F. Silva Leite, Rolling Maps in a Riemannian Framework, Textos de Matemática, Vol. 43 (2011), p. 15–30 (J. Cardoso, K. Hüper, P. Saraiva, Eds.), Department of Mathematics, University of Coimbra.  Google Scholar

[10]

K. Hüper, K. Krakowski and F. Silva Leite, Rolling maps and nonlinear data, In Handbook of Variational Methods for Nonlinear Geometric Data (Chapter 21), P. Grohs, M. Holler, A. Weinmann (Eds.), Springer, 2020,577–610. doi: 10.1007/978-3-030-31351-7_21.  Google Scholar

[11]

K. Hüper and F. Silva Leite, On the geometry of rolling and interpolation curves on $S^n$, $SO_n$ and Grassmann manifolds, J. Dyn. Control Syst., 13 (2007), 467-502.  doi: 10.1007/s10883-007-9027-3.  Google Scholar

[12]

B. D. Johnson, The nonholonomy of the rolling sphere, Amer. Math. Monthly, 114 (2007), 500-508.  doi: 10.1080/00029890.2007.11920439.  Google Scholar

[13]

P. E. Jupp and J. T. Kent, Fitting smooth paths to spherical data, J. Roy. Statist. Soc. Ser. C, 36 (1987), 34-46.  doi: 10.2307/2347843.  Google Scholar

[14] V. Jurdjevic, Geometric Control Theory, Cambridge University Press, Cambridge, 1997.   Google Scholar
[15]

V. Jurdjevic and H. Sussmann, Control systems on Lie groups, J. Differential Equations, 12 (1972), 313-329.  doi: 10.1016/0022-0396(72)90035-6.  Google Scholar

[16]

V. Jurdjevic and J. Zimmerman, Rolling sphere problems on spaces of constant curvature, Math. Proc. Cambridge Philos. Soc., 144 (2008), 729-747.  doi: 10.1017/S0305004108001084.  Google Scholar

[17]

A. Korolko and F. Silva Leite, Kinematics for rolling a Lorentzian sphere, Proc. 50th IEEE CDC-ECC, 6522–6528, 12-15 December 2011, Orlando, USA. Google Scholar

[18]

K. KrakowskiL. Machado and F. Silva Leite, A unifying approach for rolling symmetric spaces, J. Geom. Mech., 13 (2021), 145-166.  doi: 10.3934/jgm.2020016.  Google Scholar

[19]

I. Markina and F. Silva Leite, Introduction to the intrinsic rolling with indefinite metric, Comm. Anal. Geom., 24 (2016), 1085-1106.  doi: 10.4310/CAG.2016.v24.n5.a7.  Google Scholar

[20]

A. Marques and F. Silva Leite, Rolling a pseudohyperbolic space over the affine tangent space at a point, In: Proc. CONTROLO'2012, Paper 36, Funchal, Portugal, 16-18 July, 2012. Google Scholar

[21]

A. Marques and F. Silva Leite, 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, 2015. Google Scholar

[22]

A. MortadaP. Kokkonen and Y. Chitour, Rolling manifolds of different dimensions, Acta Appl. Math., 139 (2015), 105-131.  doi: 10.1007/s10440-014-9972-2.  Google Scholar

[23] Ba rrett O'Neill, Semi-Riemannian Geometry with Applications to Relativity, Academic Press, Inc., N. Y., 1983.   Google Scholar
[24]

A. G. Rojo and A. M. Bloch, The rolling sphere, the quantum spin, and a simple view of the Landau-Zener problem, American Journal of Physics, 78 (2010), 1014-1022.   Google Scholar

[25]

Y. L. Sachkov, Control theory on Lie groups, J. Math. Sci., 156 (2009), 381-439.  doi: 10.1007/s10958-008-9275-0.  Google Scholar

[26]

R. W. Sharpe, Differential Geometry, Springer, N. Y., 1997.  Google Scholar

[27]

Y. Shen, K. Huper and F. Silva Leite, Smooth Interpolation of Orientation by Rolling and Wrapping for Robot Motion Planning, Proc. 2006 IEEE International Conference on Robotics and Automation (ICRA2006), Orlando, USA, May 2006. Google Scholar

[28]

F. Silva Leite and F. Louro, Sphere rolling on sphere: Alternative approach to kinematics and constructive proof of controllability, In: Bourguignon JP., Jeltsch R., Pinto A., Viana M. (eds) Dynamics, Games and Science, 341–356. CIM Series in Mathematical Sciences, vol 1. Springer, Cham, 2015.  Google Scholar

[29]

J. A. Zimmerman, Optimal control of the sphere $S^{n}$ rolling on $E^n$, Math. Control Signals Systems, 17 (2005), 14-37.  doi: 10.1007/s00498-004-0143-2.  Google Scholar

show all references

References:
[1]

A. Agrachev and Y. Sachkov, Control Theory from the Geometric Viewpoint, Vol. 87 of Encyclopaedia of Mathematical Sciences, Springer-Verlag, 2004. doi: 10.1007/978-3-662-06404-7.  Google Scholar

[2]

A. BlochM. Camarinha and L. Colombo, Dynamic interpolation for obstacle avoidance on Riemannian manifolds, Internat. J. Control, 94 (2021), 588-600.  doi: 10.1080/00207179.2019.1603400.  Google Scholar

[3]

A. M. Bloch, Nonholonomic Mechanics and Control, with the collaboration of J. Baillieul, P. E. Crouch and J. Marsden, Interdisciplinary Applied Mathematics, Springer Verlag, 2003. doi: 10.1007/b97376.  Google Scholar

[4]

A. M. Bloch and P. E. Crouch, Nonholonomic control systems on Riemannian manifolds, SIAM J. Control Optim., 33 (1995), 126-148.  doi: 10.1137/S036301299223533X.  Google Scholar

[5]

A. M. Bloch and A. G. Rojo, Kinematics of the rolling sphere and quantum spin, Commun. Inf. Syst., 10 (2010), 221-238.  doi: 10.4310/CIS.2010.v10.n4.a4.  Google Scholar

[6]

Y. Chitour and P. Kokkonen, Rolling Manifolds: Intrinsic Formulation and Controllability, arXiv: 1011.2925v2, 2011. Google Scholar

[7]

P. Crouch and F. Silva Leite, Rolling Motions of Pseudo-Orthogonal Groups, Proc. 51st IEEE-CDC 2012, 10-13 December 2012, Hawaii, USA. Google Scholar

[8]

M. Godoy MolinaE. GrongI. Markina and F. Silva Leite, An intrinsic formulation of the rolling manifolds problem, J. Dyn. Control Syst., 18 (2012), 181-214.  doi: 10.1007/s10883-012-9139-2.  Google Scholar

[9]

K. Hüper, K. Krakowski and F. Silva Leite, Rolling Maps in a Riemannian Framework, Textos de Matemática, Vol. 43 (2011), p. 15–30 (J. Cardoso, K. Hüper, P. Saraiva, Eds.), Department of Mathematics, University of Coimbra.  Google Scholar

[10]

K. Hüper, K. Krakowski and F. Silva Leite, Rolling maps and nonlinear data, In Handbook of Variational Methods for Nonlinear Geometric Data (Chapter 21), P. Grohs, M. Holler, A. Weinmann (Eds.), Springer, 2020,577–610. doi: 10.1007/978-3-030-31351-7_21.  Google Scholar

[11]

K. Hüper and F. Silva Leite, On the geometry of rolling and interpolation curves on $S^n$, $SO_n$ and Grassmann manifolds, J. Dyn. Control Syst., 13 (2007), 467-502.  doi: 10.1007/s10883-007-9027-3.  Google Scholar

[12]

B. D. Johnson, The nonholonomy of the rolling sphere, Amer. Math. Monthly, 114 (2007), 500-508.  doi: 10.1080/00029890.2007.11920439.  Google Scholar

[13]

P. E. Jupp and J. T. Kent, Fitting smooth paths to spherical data, J. Roy. Statist. Soc. Ser. C, 36 (1987), 34-46.  doi: 10.2307/2347843.  Google Scholar

[14] V. Jurdjevic, Geometric Control Theory, Cambridge University Press, Cambridge, 1997.   Google Scholar
[15]

V. Jurdjevic and H. Sussmann, Control systems on Lie groups, J. Differential Equations, 12 (1972), 313-329.  doi: 10.1016/0022-0396(72)90035-6.  Google Scholar

[16]

V. Jurdjevic and J. Zimmerman, Rolling sphere problems on spaces of constant curvature, Math. Proc. Cambridge Philos. Soc., 144 (2008), 729-747.  doi: 10.1017/S0305004108001084.  Google Scholar

[17]

A. Korolko and F. Silva Leite, Kinematics for rolling a Lorentzian sphere, Proc. 50th IEEE CDC-ECC, 6522–6528, 12-15 December 2011, Orlando, USA. Google Scholar

[18]

K. KrakowskiL. Machado and F. Silva Leite, A unifying approach for rolling symmetric spaces, J. Geom. Mech., 13 (2021), 145-166.  doi: 10.3934/jgm.2020016.  Google Scholar

[19]

I. Markina and F. Silva Leite, Introduction to the intrinsic rolling with indefinite metric, Comm. Anal. Geom., 24 (2016), 1085-1106.  doi: 10.4310/CAG.2016.v24.n5.a7.  Google Scholar

[20]

A. Marques and F. Silva Leite, Rolling a pseudohyperbolic space over the affine tangent space at a point, In: Proc. CONTROLO'2012, Paper 36, Funchal, Portugal, 16-18 July, 2012. Google Scholar

[21]

A. Marques and F. Silva Leite, 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, 2015. Google Scholar

[22]

A. MortadaP. Kokkonen and Y. Chitour, Rolling manifolds of different dimensions, Acta Appl. Math., 139 (2015), 105-131.  doi: 10.1007/s10440-014-9972-2.  Google Scholar

[23] Ba rrett O'Neill, Semi-Riemannian Geometry with Applications to Relativity, Academic Press, Inc., N. Y., 1983.   Google Scholar
[24]

A. G. Rojo and A. M. Bloch, The rolling sphere, the quantum spin, and a simple view of the Landau-Zener problem, American Journal of Physics, 78 (2010), 1014-1022.   Google Scholar

[25]

Y. L. Sachkov, Control theory on Lie groups, J. Math. Sci., 156 (2009), 381-439.  doi: 10.1007/s10958-008-9275-0.  Google Scholar

[26]

R. W. Sharpe, Differential Geometry, Springer, N. Y., 1997.  Google Scholar

[27]

Y. Shen, K. Huper and F. Silva Leite, Smooth Interpolation of Orientation by Rolling and Wrapping for Robot Motion Planning, Proc. 2006 IEEE International Conference on Robotics and Automation (ICRA2006), Orlando, USA, May 2006. Google Scholar

[28]

F. Silva Leite and F. Louro, Sphere rolling on sphere: Alternative approach to kinematics and constructive proof of controllability, In: Bourguignon JP., Jeltsch R., Pinto A., Viana M. (eds) Dynamics, Games and Science, 341–356. CIM Series in Mathematical Sciences, vol 1. Springer, Cham, 2015.  Google Scholar

[29]

J. A. Zimmerman, Optimal control of the sphere $S^{n}$ rolling on $E^n$, Math. Control Signals Systems, 17 (2005), 14-37.  doi: 10.1007/s00498-004-0143-2.  Google Scholar

Figure 1.  Illustration of the no-twist condition (tangential part)
Figure 2.  Rolling $ H_0^2(r) $ over $ T_{p_{0}}^{\mathrm{aff}}H^{2}_0(r) $, with $ p_0 = \left[ r \,\,\, 0 \,\, \, 0 \right]^\top $
Figure 3.  Rolling of $ H_0^2(r) $, $ H_1^2(r) $ and $ H_2^2(r) $, in case II
Figure 4.  Rolling $ H^{n}_\kappa(r_1) $ over $ H^{n}_\kappa(r_2,\eta) $, with $ n = 1 $ and $ \kappa = 0 $
Table 1.  Equations of the curves which contain $ s(t) $, on plane $ x_2ox_3 $
hyperquadric curve $t\mapsto s(t)$, when $r=1$, $c=\left[c_1 c_2\right]^\top$, $\overline{s}_0=0$
$H_0^2(r)$ $(x_2-c_1)^2+(x_3-c_2)^2=c_1^2+c_2^2$
$H_1^2(r)$ $x_3=\pm x_2+(c_2\mp c_1)$ if $c_1=c_2$ or $c_1=-c_2$
$(x_2-c_1)^2-(x_3-c_2)^2=c_1^2-c_2^2$ if $c_1\neq \pm c_2$
$H_2^2(r)$ $(x_2-c_1)^2+(x_3-c_2)^2=c_1^2+c_2^2$
hyperquadric curve $t\mapsto s(t)$, when $r=1$, $c=\left[c_1 c_2\right]^\top$, $\overline{s}_0=0$
$H_0^2(r)$ $(x_2-c_1)^2+(x_3-c_2)^2=c_1^2+c_2^2$
$H_1^2(r)$ $x_3=\pm x_2+(c_2\mp c_1)$ if $c_1=c_2$ or $c_1=-c_2$
$(x_2-c_1)^2-(x_3-c_2)^2=c_1^2-c_2^2$ if $c_1\neq \pm c_2$
$H_2^2(r)$ $(x_2-c_1)^2+(x_3-c_2)^2=c_1^2+c_2^2$
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