doi: 10.3934/jgm.2020016

A unifying approach for rolling symmetric spaces

1. 

Wydziaƚ Matematyczno-Przyrodniczy, Uniwersytet Kardynaƚa Stefana Wy-szyńskie-go w Warszawie, ul. Dewajtis 5, 01-815 Warszawa, Poland

2. 

Department of Mathematics, University of Trás-os-Montes e Alto Douro (UTAD), 5001-801 Vila Real, Portugal, and, Institute of Systems and Robotics, University of Coimbra - Pólo II, 3030-290 Coimbra, Portugal

3. 

Department of Mathematics, University of Coimbra, 3001-501 Coimbra, Portugal, and, Institute of Systems and Robotics, University of Coimbra - Pólo II, 3030-290 Coimbra, Portugal

* Corresponding author: fleite@mat.uc.pt

Received  December 2019 Revised  March 2020 Published  June 2020

Fund Project: The second and third authors acknowledge Fundação para a Ciência e Tecnologia (FCT) and COMPETE 2020 program for the financial support to the project UIDB/00048/2020

The main goal of this paper is to present a unifying theory to describe the pure rolling motions of Riemannian symmetric spaces, which are submanifolds of Euclidean or pseudo-Euclidean spaces. Rolling motions provide interesting examples of nonholonomic systems and symmetric spaces appear associated to important applications. We make a connection between the structure of the kinematic equations of rolling and the natural decomposition of the Lie algebra associated to the symmetric space. This emphasises the relevance of Lie theory in the geometry of rolling manifolds and explains why many particular examples scattered through the existing literature always show a common pattern.

Citation: Krzysztof A. Krakowski, Luís Machado, Fátima Silva Leite. A unifying approach for rolling symmetric spaces. Journal of Geometric Mechanics, doi: 10.3934/jgm.2020016
References:
[1]

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[2]

A. M. Bloch, Nonholonomic Mechanics and Control, Interdisciplinary Applied Mathematics, 24, Springer-Verlag, New York, 2003. doi: 10.1115/1.1641775.  Google Scholar

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A. M. BlochR. W. Brockett and P. E. Crouch, Double bracket equations and geodesic flows on symmetric spaces, Comm. Math. Phys., 187 (1997), 357-373.  doi: 10.1007/s002200050140.  Google Scholar

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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

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F. Bullo and A. D. Lewis, Geometric Control of Mechanical Systems. Modeling, Analysis, and Design for Simple Mechanical Control Systems, Texts in Applied Mathematics, 49, Springer-Verlag, New York, 2005. doi: 10.1007/978-1-4899-7276-7.  Google Scholar

[10]

G. S. Chirikjian, Information theory on Lie groups and mobile robotics applications, IEEE International Conference on Robotics and Automation, Anchorage, AK, 2010, 2751–2757. doi: 10.1109/ROBOT.2010.5509791.  Google Scholar

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Y. Chitour, M. Godoy Molina and P. Kokkonen, The rolling problem: overview and challenges, in Geometric Control Theory and Sub-Riemannian Geometry, Springer INdAM Ser., 5, Springer, Cham, 2014,103–122. doi: 10.1007/978-3-319-02132-4_7.  Google Scholar

[12]

Y. Chitour and P. Kokkonen, Rolling manifolds on space forms, Ann. Inst. H. Poincaré Anal. Non Linéaire, 29 (2012), 927-954.  doi: 10.1016/j.anihpc.2012.05.005.  Google Scholar

[13]

P. E. Crouch and F. Silva Leite, Rolling motions of pseudo-orthogonal groups, IEEE 51st Annual Conference on Decision and Control (CDC), Maui, HI, 2012, 7485–7491. doi: 10.1109/CDC.2012.6426140.  Google Scholar

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S. Helgason, On Riemannian curvature of homogeneous spaces, Proc. Amer. Math. Soc., 9 (1958), 831-838.  doi: 10.1090/S0002-9939-1958-0108811-2.  Google Scholar

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Z. Huang, R. Wang, S. Shan and X. Chen, Projection metric learning on Grassmann manifold with application to video based face recognition, IEEE Conference on Computer Vision and Pattern Recognition (CVPR), Boston, MA, 2015,140–149. doi: 10.1109/CVPR.2015.7298609.  Google Scholar

[23]

K. Hüper, K. A. Krakowski and F. Silva Leite, Rolling maps in a Riemannian framework, in Mathematical Papers in Honour of Fátima Silva Leite, Textos Mat. Sér. B, 43, Univ. Coimbra, Coimbra, 2011, 15–30.  Google Scholar

[24]

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

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V. Jurdjevic and J. Zimmerman, Rolling problems on spaces of constant curvature, in Lagrangian and Hamiltonian Methods for Nonlinear Control, Lect. Notes Control Inf. Sci., 366, Springer, Berlin, 2007,221–231. doi: 10.1007/978-3-540-73890-9_17.  Google Scholar

[26]

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

[27]

S. Kobayashi and K. Nomizu, Foundations of Differential Geometry, Interscience Tracts in Pure and Applied Mathematics, 1, Interscience Publishers John Wiley & Sons, Inc.,, New York-London, 1963.  Google Scholar

[28]

S. Kobayashi and K. Nomizu, Foundations of Differential Geometry, Interscience Tracts in Pure and Applied Mathematics, 2, Interscience Publishers John Wiley & Sons, Inc., New York-London-Sydney, 1969.  Google Scholar

[29]

A. Korolko and F. Silva Leite, Kinematics for rolling a Lorentzian sphere, 50th IEEE Conference on Decision and Control and European Control Conference, Orlando, FL, 2011. doi: 10.1109/CDC.2011.6160592.  Google Scholar

[30]

O. Kowalski, Generalized Symmetric Spaces, Lecture Notes in Mathematics, 805, Springer-Verlag, Berlin-New York, 1980. doi: 10.1007/BFb0103324.  Google Scholar

[31]

K. A. Krakowski, L. Machado and F. Silva Leite, Rolling symmetric spaces, in Geometric Science of Information, Lecture Notes in Comput. Sci., 9389, Springer, Cham, 2015,550–557. doi: 10.1007/978-3-319-25040-3_59.  Google Scholar

[32]

K. A. Krakowski and F. Silva Leite, An algorithm based on rolling to generate smooth interpolating curves on ellipsoids, Kybernetika, 50 (2014), 544-562.  doi: 10.14736/kyb-2014-4-0544.  Google Scholar

[33]

J. M. Lee, Riemannian Manifolds. An Introduction to Curvature, Graduate Texts in Mathematics, 176, Springer-Verlag, New York, 1997. doi: 10.1007/b98852.  Google Scholar

[34] K. Lynch and F. Park, Modern Robotics - Mechanics, Planning, and Control, Cambridge University Press, New York, 2017.   Google Scholar
[35]

L. Machado, F. Pina and F. Silva Leite, Rolling maps for the essential manifold, in Dynamics, Games and Science, CIM Ser. Math. Sci., 1, Springer, Cham, 2015,399–415. doi: 10.1007/978-3-319-16118-1_21.  Google Scholar

[36]

M. A. Magid, Isometric immersions of Lorentz space with parallel second fundamental forms, Tsukuba J. Math, 8 (1984), 31-54.  doi: 10.21099/tkbjm/1496159942.  Google Scholar

[37]

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

[38]

A. Marques and F. Silva Leite, Controllability for the constrained rolling motion of symplectic groups, in Proc. of the 11th Portuguese Conference on Automatic Control, Lecture Notes in Electrical Engineering, 321, Springer, Cham, 2015, 3–12. doi: 10.1007/978-3-319-10380-8_1.  Google Scholar

[39] R. N. MurrayZ. X. Li and S. S. Sastry, A Mathematical Introduction to Robotic Manipulation, CRC Press, Boca Raton, FL, 1994.   Google Scholar
[40]

K. Nomizu, Kinematics and differential geometry of submanifolds. Rolling a ball with a prescribed locus of contact, Tohoku Math. J. (2), 30 (1978), 623-637.  doi: 10.2748/tmj/1178229921.  Google Scholar

[41] B. O'Neill, Elementary Differential Geometry, Academic Press, New York-London, 1966.   Google Scholar
[42] X. PennecS. Sommer and P. T. Fletcher, Riemannian Geometric Statistics in Medical Image Analysis, Academic Press, 2020.  doi: 10.1016/C2017-0-01561-6.  Google Scholar
[43]

A. G. Rojo and A. M. Bloch, The rolling sphere, the quantum spin, and a simple view of the Landau-Zener problem, American J. Physics, 78 (2010), 1014-1022.  doi: 10.1119/1.3456565.  Google Scholar

[44]

R. W. Sharpe, Differential Geometry. Cartan's Generalization of Klein's Erlangen Program, Graduate Texts in Mathematics, 166, Springer-Verlag, New York, 1997.  Google Scholar

[45]

M. Takeuchi and S. Kobayashi, Minimal imbeddings of ${R}$-spaces, J. Differential Geometry, 2 (1968), 203-215.  doi: 10.4310/jdg/1214428257.  Google Scholar

[46]

R. Tron and K. Daniilidis, The space of essential matrices as a Riemannian quotient manifold, SIAM J. Imaging Sci., 10 (2017), 1416-1445.  doi: 10.1137/16M1091332.  Google Scholar

[47]

P. Turaga and R. Chellappa, Locally time-invariant models of human activities using trajectories on the Grassmannian, IEEE Conference on Computer Vision and Pattern Recognition (CVPR), Miami, FL, 2009, 2435–2441. doi: 10.1109/CVPR.2009.5206710.  Google Scholar

[48]

P. TuragaA. VeeraraghavanA. Srivastava and R. Chellappa, Statistical computations on Grassmann and Stiefel manifolds for image and video-based recognition, IEEE Transactions on Pattern Analysis and Machine Intelligence, 33 (2011), 2273-2286.  doi: 10.1109/TPAMI.2011.52.  Google Scholar

[49]

R. Vemulapalli and R. Chellappa, Rolling rotations for recognizing human actions from 3d skeletal data, IEEE Conference on Computer Vision and Pattern Recognition (CVPR), Las Vegas, NV, 2016, 4471–4479. doi: 10.1109/CVPR.2016.484.  Google Scholar

[50]

L. Vrancken, Parallel affine immersions with maximal codimension, Tohoku Math. J. (2), 53, Number 4 (2001), 511–531.  Google Scholar

[51]

J. Zhang, G. Zhu, R. Heath Jr. and K. Huang, Grassmannian learning: Embedding geometry awareness in shallow and deep learning, preprint, arXiv: 1808.02229[cs.LG]. Google Scholar

[52]

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

[2]

A. M. Bloch, Nonholonomic Mechanics and Control, Interdisciplinary Applied Mathematics, 24, Springer-Verlag, New York, 2003. doi: 10.1115/1.1641775.  Google Scholar

[3]

A. M. BlochR. W. Brockett and P. E. Crouch, Double bracket equations and geodesic flows on symmetric spaces, Comm. Math. Phys., 187 (1997), 357-373.  doi: 10.1007/s002200050140.  Google Scholar

[4]

A. M. Bloch, M. Camarinha and L. Colombo, Variational point-obstacle avoidance on Riemannian manifolds, preprint, arXiv: 1909.12321[eess.SY]. Google Scholar

[5]

A. M. Bloch and P. E. Crouch, Optimal control, optimization, and analytical mechanics, in Mathematical Control Theory, Springer, New York, 1999,268–321. doi: 10.1007/978-1-4612-1416-8_8.  Google Scholar

[6]

A. M. Bloch, P. E. Crouch and T. S. Ratiu, Sub-Riemannian optimal control problems, in Hamiltonian and Gradient Flows, Algorithms and Control, Fields Inst. Commun., 3, Amer. Math. Soc., Providence, RI, 1994, 35–48.  Google Scholar

[7]

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

[8]

R. L. Bryant and L. Hsu, Rigidity of integral curves of rank 2 distributions, Invent. Math., 114 (1993), 435-461.  doi: 10.1007/BF01232676.  Google Scholar

[9]

F. Bullo and A. D. Lewis, Geometric Control of Mechanical Systems. Modeling, Analysis, and Design for Simple Mechanical Control Systems, Texts in Applied Mathematics, 49, Springer-Verlag, New York, 2005. doi: 10.1007/978-1-4899-7276-7.  Google Scholar

[10]

G. S. Chirikjian, Information theory on Lie groups and mobile robotics applications, IEEE International Conference on Robotics and Automation, Anchorage, AK, 2010, 2751–2757. doi: 10.1109/ROBOT.2010.5509791.  Google Scholar

[11]

Y. Chitour, M. Godoy Molina and P. Kokkonen, The rolling problem: overview and challenges, in Geometric Control Theory and Sub-Riemannian Geometry, Springer INdAM Ser., 5, Springer, Cham, 2014,103–122. doi: 10.1007/978-3-319-02132-4_7.  Google Scholar

[12]

Y. Chitour and P. Kokkonen, Rolling manifolds on space forms, Ann. Inst. H. Poincaré Anal. Non Linéaire, 29 (2012), 927-954.  doi: 10.1016/j.anihpc.2012.05.005.  Google Scholar

[13]

P. E. Crouch and F. Silva Leite, Rolling motions of pseudo-orthogonal groups, IEEE 51st Annual Conference on Decision and Control (CDC), Maui, HI, 2012, 7485–7491. doi: 10.1109/CDC.2012.6426140.  Google Scholar

[14]

M. P. do Carmo, Riemannian Geometry, Mathematics: Theory & Applications, Birkhäuser Boston, Inc., Boston, MA, 1992.  Google Scholar

[15]

J.-H. Eschenburg and E. Heintze, Extrinsic symmetric spaces and orbits of $s$-representations, Manuscripta Math., 88 (1995), 517-524.  doi: 10.1007/BF02567838.  Google Scholar

[16]

D. Ferus, Immersions with parallel second fundamental form, Math. Z., 140 (1974), 87-93.  doi: 10.1007/BF01218650.  Google Scholar

[17]

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

[18]

M. HarandiR. HartleyC. ShenB. Lovell and C. Sanderson, Extrinsic methods for coding and dictionary learning on Grassmann manifolds, Int. J. Comput. Vis., 114 (2015), 113-136.  doi: 10.1007/s11263-015-0833-x.  Google Scholar

[19] R. Hartley and A. Zisserman, Multiple View Geometry in Computer Vision, Cambridge University Press, Cambridge, 2003.  doi: 10.1017/CBO9780511811685.  Google Scholar
[20] S. Helgason, Differential Geometry, Lie Groups and Symmetric Spaces, Pure and Applied Mathematics, 80, Academic Press, Inc., New York-London, 1978.   Google Scholar
[21]

S. Helgason, On Riemannian curvature of homogeneous spaces, Proc. Amer. Math. Soc., 9 (1958), 831-838.  doi: 10.1090/S0002-9939-1958-0108811-2.  Google Scholar

[22]

Z. Huang, R. Wang, S. Shan and X. Chen, Projection metric learning on Grassmann manifold with application to video based face recognition, IEEE Conference on Computer Vision and Pattern Recognition (CVPR), Boston, MA, 2015,140–149. doi: 10.1109/CVPR.2015.7298609.  Google Scholar

[23]

K. Hüper, K. A. Krakowski and F. Silva Leite, Rolling maps in a Riemannian framework, in Mathematical Papers in Honour of Fátima Silva Leite, Textos Mat. Sér. B, 43, Univ. Coimbra, Coimbra, 2011, 15–30.  Google Scholar

[24]

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

[25]

V. Jurdjevic and J. Zimmerman, Rolling problems on spaces of constant curvature, in Lagrangian and Hamiltonian Methods for Nonlinear Control, Lect. Notes Control Inf. Sci., 366, Springer, Berlin, 2007,221–231. doi: 10.1007/978-3-540-73890-9_17.  Google Scholar

[26]

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

[27]

S. Kobayashi and K. Nomizu, Foundations of Differential Geometry, Interscience Tracts in Pure and Applied Mathematics, 1, Interscience Publishers John Wiley & Sons, Inc.,, New York-London, 1963.  Google Scholar

[28]

S. Kobayashi and K. Nomizu, Foundations of Differential Geometry, Interscience Tracts in Pure and Applied Mathematics, 2, Interscience Publishers John Wiley & Sons, Inc., New York-London-Sydney, 1969.  Google Scholar

[29]

A. Korolko and F. Silva Leite, Kinematics for rolling a Lorentzian sphere, 50th IEEE Conference on Decision and Control and European Control Conference, Orlando, FL, 2011. doi: 10.1109/CDC.2011.6160592.  Google Scholar

[30]

O. Kowalski, Generalized Symmetric Spaces, Lecture Notes in Mathematics, 805, Springer-Verlag, Berlin-New York, 1980. doi: 10.1007/BFb0103324.  Google Scholar

[31]

K. A. Krakowski, L. Machado and F. Silva Leite, Rolling symmetric spaces, in Geometric Science of Information, Lecture Notes in Comput. Sci., 9389, Springer, Cham, 2015,550–557. doi: 10.1007/978-3-319-25040-3_59.  Google Scholar

[32]

K. A. Krakowski and F. Silva Leite, An algorithm based on rolling to generate smooth interpolating curves on ellipsoids, Kybernetika, 50 (2014), 544-562.  doi: 10.14736/kyb-2014-4-0544.  Google Scholar

[33]

J. M. Lee, Riemannian Manifolds. An Introduction to Curvature, Graduate Texts in Mathematics, 176, Springer-Verlag, New York, 1997. doi: 10.1007/b98852.  Google Scholar

[34] K. Lynch and F. Park, Modern Robotics - Mechanics, Planning, and Control, Cambridge University Press, New York, 2017.   Google Scholar
[35]

L. Machado, F. Pina and F. Silva Leite, Rolling maps for the essential manifold, in Dynamics, Games and Science, CIM Ser. Math. Sci., 1, Springer, Cham, 2015,399–415. doi: 10.1007/978-3-319-16118-1_21.  Google Scholar

[36]

M. A. Magid, Isometric immersions of Lorentz space with parallel second fundamental forms, Tsukuba J. Math, 8 (1984), 31-54.  doi: 10.21099/tkbjm/1496159942.  Google Scholar

[37]

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

[38]

A. Marques and F. Silva Leite, Controllability for the constrained rolling motion of symplectic groups, in Proc. of the 11th Portuguese Conference on Automatic Control, Lecture Notes in Electrical Engineering, 321, Springer, Cham, 2015, 3–12. doi: 10.1007/978-3-319-10380-8_1.  Google Scholar

[39] R. N. MurrayZ. X. Li and S. S. Sastry, A Mathematical Introduction to Robotic Manipulation, CRC Press, Boca Raton, FL, 1994.   Google Scholar
[40]

K. Nomizu, Kinematics and differential geometry of submanifolds. Rolling a ball with a prescribed locus of contact, Tohoku Math. J. (2), 30 (1978), 623-637.  doi: 10.2748/tmj/1178229921.  Google Scholar

[41] B. O'Neill, Elementary Differential Geometry, Academic Press, New York-London, 1966.   Google Scholar
[42] X. PennecS. Sommer and P. T. Fletcher, Riemannian Geometric Statistics in Medical Image Analysis, Academic Press, 2020.  doi: 10.1016/C2017-0-01561-6.  Google Scholar
[43]

A. G. Rojo and A. M. Bloch, The rolling sphere, the quantum spin, and a simple view of the Landau-Zener problem, American J. Physics, 78 (2010), 1014-1022.  doi: 10.1119/1.3456565.  Google Scholar

[44]

R. W. Sharpe, Differential Geometry. Cartan's Generalization of Klein's Erlangen Program, Graduate Texts in Mathematics, 166, Springer-Verlag, New York, 1997.  Google Scholar

[45]

M. Takeuchi and S. Kobayashi, Minimal imbeddings of ${R}$-spaces, J. Differential Geometry, 2 (1968), 203-215.  doi: 10.4310/jdg/1214428257.  Google Scholar

[46]

R. Tron and K. Daniilidis, The space of essential matrices as a Riemannian quotient manifold, SIAM J. Imaging Sci., 10 (2017), 1416-1445.  doi: 10.1137/16M1091332.  Google Scholar

[47]

P. Turaga and R. Chellappa, Locally time-invariant models of human activities using trajectories on the Grassmannian, IEEE Conference on Computer Vision and Pattern Recognition (CVPR), Miami, FL, 2009, 2435–2441. doi: 10.1109/CVPR.2009.5206710.  Google Scholar

[48]

P. TuragaA. VeeraraghavanA. Srivastava and R. Chellappa, Statistical computations on Grassmann and Stiefel manifolds for image and video-based recognition, IEEE Transactions on Pattern Analysis and Machine Intelligence, 33 (2011), 2273-2286.  doi: 10.1109/TPAMI.2011.52.  Google Scholar

[49]

R. Vemulapalli and R. Chellappa, Rolling rotations for recognizing human actions from 3d skeletal data, IEEE Conference on Computer Vision and Pattern Recognition (CVPR), Las Vegas, NV, 2016, 4471–4479. doi: 10.1109/CVPR.2016.484.  Google Scholar

[50]

L. Vrancken, Parallel affine immersions with maximal codimension, Tohoku Math. J. (2), 53, Number 4 (2001), 511–531.  Google Scholar

[51]

J. Zhang, G. Zhu, R. Heath Jr. and K. Huang, Grassmannian learning: Embedding geometry awareness in shallow and deep learning, preprint, arXiv: 1808.02229[cs.LG]. Google Scholar

[52]

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.  A sphere $ \mathbf{M} $ is rolling upon a surface $ \mathbf{M}_0 $, along the development curve $ \sigma_0 $, without slipping or twisting
Figure 2.  A vector attached to a rolling manifold generates a vector field along the path $ \chi(t)(p) $ in the ambient space; thus a tangent (normal) vector in $ \mathbf{T}_p \mathbf{M} $ is isometrically carried to a tangent (normal) vector in $ \mathbf{T}_q(\chi(t_0)( \mathbf{M})) = \mathbf{T}_q \mathbf{M}_0 $
Figure 3.  A sphere $ {\bf S}^{2} $ is rolling upon $ \mathbf{M}_0 $ along the development curve $ \sigma_0 $ without slipping or twisting; the infinitesimal action $ ({\dot\chi\,\chi {^{-1}}})_* $ is orthogonal, with respect to an $ {\rm Ad}(H) $-invariant inner product onk__ge $ \mathfrak{so}({3}) $, to the Lie algebra of the isotropy group
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