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March  2021, 11(1): 143-167. doi: 10.3934/mcrf.2020031

Error-based control systems on Riemannian state manifolds: Properties of the principal pushforward map associated to parallel transport

Simone Fiori

Università Politecnica delle Marche, Via Brecce Bianche, I-60131 Ancona, Italy

Received  August 2019 Revised  March 2020 Published  March 2021 Early access  June 2020

The objective of the paper is to contribute to the theory of error-based control systems on Riemannian manifolds. The present study focuses on system where the control field influences the covariant derivative of a control path. In order to define error terms in such systems, it is necessary to compare tangent vectors at different points using parallel transport and to understand how the covariant derivative of a vector field along a path changes after such field gets parallely transported to a different curve. It turns out that such analysis relies on a specific map, termed principal pushforward map. The present paper aims at contributing to the algebraic theory of the principal pushforward map and of its relationship with the curvature endomorphism of a state manifold.

Keywords: Error-based non-linear dynamical systems, Riemannian state manifolds, Parallel transport, Covariant differentiation, Principal pushforward map, Curvature endomorphism.
Mathematics Subject Classification: Primary:93C10, 93C25, 58C25.
Citation: Simone Fiori. Error-based control systems on Riemannian state manifolds: Properties of the principal pushforward map associated to parallel transport. Mathematical Control & Related Fields, 2021, 11 (1) : 143-167. doi: 10.3934/mcrf.2020031
References:
[1]

W. Ambrose and I. M. Singer, A theorem on holonomy, Transactions of the American Mathematical Society, 75 (1953), 428-443.  doi: 10.1090/S0002-9947-1953-0063739-1.  Google Scholar

[2]

R. Anirudh, V. Venkataraman and P. Turaga, A generalized Lyapunov feature for dynamical systems on Riemannian manifolds,, Proceedings of the 1st International Workshop on Differential Geometry in Computer Vision for Analysis of Shapes, Images and Trajectories, (2015), 4.1–4.10. doi: 10.5244/C.29.DIFFCV.4.  Google Scholar

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F. Bullo, R. M. Murray and A. Sarti, Control on the sphere and reduced attitude stabilization, IFAC Proceedings Volumes, 28 (1995), 495-501.  doi: 10.1016/S1474-6670(17)46878-9.  Google Scholar

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J. Casey, Parallel transport of a vector on a surface, Exploring Curvature, (1996), 250–263. Google Scholar

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Y. Chitour and M. Sigalotti, On the controllability of the Dubins' problem for surfaces, Proceedings of the IFAC System, Structure and Control Conference, (2004), 563–565. Google Scholar

[7]

Y. Chitour and M. Sigalotti, On the controllability of the Dubins problem for surfaces. I Nonnegative curvature, Journal of Geometric Analysis, 15 (2005), 565-587.  doi: 10.1007/BF02922245.  Google Scholar

[8]

Y. Chitour and M. Sigalotti, On the controllability of the Dubins problem for surfaces. Ⅱ Negative curvature, SIAM Journal on Control and Optimization, 45 (2006), 457-482.  doi: 10.1137/040619739.  Google Scholar

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R. Dandoloff, Berry's phase and Fermi-Walker parallel transport, Physics Letters A, 139 (1989), 19-20.  doi: 10.1016/0375-9601(89)90599-9.  Google Scholar

[10]

S. Eftekhar Azam, S. Mariani and N. K. A. Attari, Online damage detection via a synergy of proper orthogonal decomposition and recursive Bayesian filters, Nonlinear Dynamics, 89 (2017), 1489-1511.   Google Scholar

[11]

S. Fiori, On vector averaging over the unit hyphersphere, Digital Signal Processing, 19 (2009), 715-725.   Google Scholar

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S. Fiori, Nonlinear damped oscillators on Riemannian manifolds: Fundamentals, Journal of Systems Science and Complexity, 29 (2016), 22-40.  doi: 10.1007/s11424-015-4063-7.  Google Scholar

[13]

S. Fiori, Non-delayed synchronization of non-autonomous dynamical systems on Riemannian manifolds and its applications, Nonlinear Dynamics, 94 (2018), 3077-3100.  doi: 10.1007/s11071-018-4546-x.  Google Scholar

[14]

S. Fiori, Nonlinear damped oscillators on Riemannian manifolds: Numerical simulation, Communications in Nonlinear Science and Numerical Simulation, 47 (2017), 207-222.  doi: 10.1016/j.cnsns.2016.11.025.  Google Scholar

[15]

R. Fuentes, G. Hicks and J. Osborne, The spring paradigm in tracking control of simple mechanical systems, Automatica, 47 (2011), 993-1000.  doi: 10.1016/j.automatica.2011.01.046.  Google Scholar

[16]

F. A. Goodarzi, D. Lee and T. Lee, Geometric control of a quadrotor UAV transporting a payload connected via flexible cable, International Journal of Control, Automation, and Systems, 13 (2015), 1486-1498.  doi: 10.1007/s12555-014-0304-0.  Google Scholar

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M.-D. Hua, T. Hamel, R. Mahony and J. Trumpf, Gradient-like observer design on the special Euclidean group SE(3) with system outputs on the real projective space, Proceedings of the 54th IEEE Conference on Decision and Control, (2015), 2139–2145. doi: 10.1109/CDC.2015.7402523.  Google Scholar

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M. Kabiri, H. Atrianfar and M. B. Menhaj, Trajectory tracking of a class of under-actuated thrust-propelled vehicle with uncertainties and unknown disturbances, Nonlinear Dynamics, 90 (2017), 1695-1706.  doi: 10.1007/s11071-017-3759-8.  Google Scholar

[19]

L. M. Lopes, S. Fernandes and C. Grácio, Complete synchronization and delayed synchronization in couplings, Nonlinear Dynamics, 79 (2015), 1615-1624.  doi: 10.1007/s11071-014-1764-8.  Google Scholar

[20]

L. Lü, C. Li, S. Liu, Z. Wang, J. Tian and J. Gu, The signal synchronization transmission among uncertain discrete networks with different nodes, Nonlinear Dynamics, 81 (2015), 801-809.   Google Scholar

[21]

C. W. Misner, K. S. Thorne and J. A. Wheeler, Gravitation, W.H. Freeman, 1973.  Google Scholar

[22]

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

[23]

K. Ojo, S. Ogunjo and A. Olagundoye, Projective synchronization via active control of identical chaotic oscillators with parametric and external excitation, International Journal of Nonlinear Science, 24 (2017), 76-83.   Google Scholar

[24]

J. M. Osborne and G. P. Hicks, The geodesic spring on the Euclidean sphere with parallel-transport-based damping, Notices of the AMS, 60 (2013), 544-556.  doi: 10.1090/noti997.  Google Scholar

[25]

P. Piras, L. Teresi, L. Traversetti, V. Varano, S. Gabriele, T. Kotsakis, P. Raia, P. E. Puddu and M. Scalici, The conceptual framework of ontogenetic trajectories: parallel transport allows the recognition and visualization of pure deformation patterns, Evolution & Development, 18 (2016), 182-200.  doi: 10.1111/ede.12186.  Google Scholar

[26] W. H. Press, S. A. Teukolsky, W.T. Vetterling and B.P. Flannery, Numerical Recipes: The Art of Scientific Computing, Third edition. Cambridge University Press, Cambridge, 2007.   Google Scholar
[27]

H. Reckziegel and E. Wilhelmus, How the curvature generates the holonomy of a connection in an arbitrary fibre bundle, Results in Mathematics, 49 (2006), 339-359.  doi: 10.1007/s00025-006-0228-y.  Google Scholar

[28]

T. Soyfıdan, H. Parlatici and M. A. Güngör, On the quaternionic curve according to parallel transport frame, TWMS Journal of Pure and Applied Mathematics, 4 (2013), 194-203.   Google Scholar

[29]

M. Spivak, A Comprehensive Introduction to Differential Geometry, Publish or Perish, Inc., Wilmington, Del., 1979.  Google Scholar

[30]

Z. G. Ying and W. Q. Zhu, Optimal bounded control for nonlinear stochastic smart structure systems based on extended Kalman filter, Nonlinear Dynamics, 90 (2017), 105-114.  doi: 10.1007/s11071-017-3650-7.  Google Scholar

[31]

W. Yu and Z. Pan, Dynamical equations of multibody systems on Lie groups, Advances in Mechanical Engineering, 7 (2015), 1-9.  doi: 10.1177/1687814015575959.  Google Scholar

[32]

Z. Zhang, Z. Ling and A. Sarlette, Modified integral control globally counters symmetry-breaking biases, Symmetry, 11 (2019), 639. doi: 10.3390/sym11050639.  Google Scholar

[33]

Y. Zou, Adaptive trajectory tracking control approach for a model-scaled helicopter, Nonlinear Dynamics, 83 (2016), 2171-2181.  doi: 10.1007/s11071-015-2473-7.  Google Scholar

show all references

References:
[1]

W. Ambrose and I. M. Singer, A theorem on holonomy, Transactions of the American Mathematical Society, 75 (1953), 428-443.  doi: 10.1090/S0002-9947-1953-0063739-1.  Google Scholar

[2]

R. Anirudh, V. Venkataraman and P. Turaga, A generalized Lyapunov feature for dynamical systems on Riemannian manifolds,, Proceedings of the 1st International Workshop on Differential Geometry in Computer Vision for Analysis of Shapes, Images and Trajectories, (2015), 4.1–4.10. doi: 10.5244/C.29.DIFFCV.4.  Google Scholar

[3]

A. Bejenaru and C. Udrişte, Multivariate optimal control with payoffs defined by submanifold integrals, Symmetry, 11 (2019), 893. doi: 10.3390/sym11070893.  Google Scholar

[4]

F. Bullo, R. M. Murray and A. Sarti, Control on the sphere and reduced attitude stabilization, IFAC Proceedings Volumes, 28 (1995), 495-501.  doi: 10.1016/S1474-6670(17)46878-9.  Google Scholar

[5]

J. Casey, Parallel transport of a vector on a surface, Exploring Curvature, (1996), 250–263. Google Scholar

[6]

Y. Chitour and M. Sigalotti, On the controllability of the Dubins' problem for surfaces, Proceedings of the IFAC System, Structure and Control Conference, (2004), 563–565. Google Scholar

[7]

Y. Chitour and M. Sigalotti, On the controllability of the Dubins problem for surfaces. I Nonnegative curvature, Journal of Geometric Analysis, 15 (2005), 565-587.  doi: 10.1007/BF02922245.  Google Scholar

[8]

Y. Chitour and M. Sigalotti, On the controllability of the Dubins problem for surfaces. Ⅱ Negative curvature, SIAM Journal on Control and Optimization, 45 (2006), 457-482.  doi: 10.1137/040619739.  Google Scholar

[9]

R. Dandoloff, Berry's phase and Fermi-Walker parallel transport, Physics Letters A, 139 (1989), 19-20.  doi: 10.1016/0375-9601(89)90599-9.  Google Scholar

[10]

S. Eftekhar Azam, S. Mariani and N. K. A. Attari, Online damage detection via a synergy of proper orthogonal decomposition and recursive Bayesian filters, Nonlinear Dynamics, 89 (2017), 1489-1511.   Google Scholar

[11]

S. Fiori, On vector averaging over the unit hyphersphere, Digital Signal Processing, 19 (2009), 715-725.   Google Scholar

[12]

S. Fiori, Nonlinear damped oscillators on Riemannian manifolds: Fundamentals, Journal of Systems Science and Complexity, 29 (2016), 22-40.  doi: 10.1007/s11424-015-4063-7.  Google Scholar

[13]

S. Fiori, Non-delayed synchronization of non-autonomous dynamical systems on Riemannian manifolds and its applications, Nonlinear Dynamics, 94 (2018), 3077-3100.  doi: 10.1007/s11071-018-4546-x.  Google Scholar

[14]

S. Fiori, Nonlinear damped oscillators on Riemannian manifolds: Numerical simulation, Communications in Nonlinear Science and Numerical Simulation, 47 (2017), 207-222.  doi: 10.1016/j.cnsns.2016.11.025.  Google Scholar

[15]

R. Fuentes, G. Hicks and J. Osborne, The spring paradigm in tracking control of simple mechanical systems, Automatica, 47 (2011), 993-1000.  doi: 10.1016/j.automatica.2011.01.046.  Google Scholar

[16]

F. A. Goodarzi, D. Lee and T. Lee, Geometric control of a quadrotor UAV transporting a payload connected via flexible cable, International Journal of Control, Automation, and Systems, 13 (2015), 1486-1498.  doi: 10.1007/s12555-014-0304-0.  Google Scholar

[17]

M.-D. Hua, T. Hamel, R. Mahony and J. Trumpf, Gradient-like observer design on the special Euclidean group SE(3) with system outputs on the real projective space, Proceedings of the 54th IEEE Conference on Decision and Control, (2015), 2139–2145. doi: 10.1109/CDC.2015.7402523.  Google Scholar

[18]

M. Kabiri, H. Atrianfar and M. B. Menhaj, Trajectory tracking of a class of under-actuated thrust-propelled vehicle with uncertainties and unknown disturbances, Nonlinear Dynamics, 90 (2017), 1695-1706.  doi: 10.1007/s11071-017-3759-8.  Google Scholar

[19]

L. M. Lopes, S. Fernandes and C. Grácio, Complete synchronization and delayed synchronization in couplings, Nonlinear Dynamics, 79 (2015), 1615-1624.  doi: 10.1007/s11071-014-1764-8.  Google Scholar

[20]

L. Lü, C. Li, S. Liu, Z. Wang, J. Tian and J. Gu, The signal synchronization transmission among uncertain discrete networks with different nodes, Nonlinear Dynamics, 81 (2015), 801-809.   Google Scholar

[21]

C. W. Misner, K. S. Thorne and J. A. Wheeler, Gravitation, W.H. Freeman, 1973.  Google Scholar

[22]

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

[23]

K. Ojo, S. Ogunjo and A. Olagundoye, Projective synchronization via active control of identical chaotic oscillators with parametric and external excitation, International Journal of Nonlinear Science, 24 (2017), 76-83.   Google Scholar

[24]

J. M. Osborne and G. P. Hicks, The geodesic spring on the Euclidean sphere with parallel-transport-based damping, Notices of the AMS, 60 (2013), 544-556.  doi: 10.1090/noti997.  Google Scholar

[25]

P. Piras, L. Teresi, L. Traversetti, V. Varano, S. Gabriele, T. Kotsakis, P. Raia, P. E. Puddu and M. Scalici, The conceptual framework of ontogenetic trajectories: parallel transport allows the recognition and visualization of pure deformation patterns, Evolution & Development, 18 (2016), 182-200.  doi: 10.1111/ede.12186.  Google Scholar

[26] W. H. Press, S. A. Teukolsky, W.T. Vetterling and B.P. Flannery, Numerical Recipes: The Art of Scientific Computing, Third edition. Cambridge University Press, Cambridge, 2007.   Google Scholar
[27]

H. Reckziegel and E. Wilhelmus, How the curvature generates the holonomy of a connection in an arbitrary fibre bundle, Results in Mathematics, 49 (2006), 339-359.  doi: 10.1007/s00025-006-0228-y.  Google Scholar

[28]

T. Soyfıdan, H. Parlatici and M. A. Güngör, On the quaternionic curve according to parallel transport frame, TWMS Journal of Pure and Applied Mathematics, 4 (2013), 194-203.   Google Scholar

[29]

M. Spivak, A Comprehensive Introduction to Differential Geometry, Publish or Perish, Inc., Wilmington, Del., 1979.  Google Scholar

[30]

Z. G. Ying and W. Q. Zhu, Optimal bounded control for nonlinear stochastic smart structure systems based on extended Kalman filter, Nonlinear Dynamics, 90 (2017), 105-114.  doi: 10.1007/s11071-017-3650-7.  Google Scholar

[31]

W. Yu and Z. Pan, Dynamical equations of multibody systems on Lie groups, Advances in Mechanical Engineering, 7 (2015), 1-9.  doi: 10.1177/1687814015575959.  Google Scholar

[32]

Z. Zhang, Z. Ling and A. Sarlette, Modified integral control globally counters symmetry-breaking biases, Symmetry, 11 (2019), 639. doi: 10.3390/sym11050639.  Google Scholar

[33]

Y. Zou, Adaptive trajectory tracking control approach for a model-scaled helicopter, Nonlinear Dynamics, 83 (2016), 2171-2181.  doi: 10.1007/s11071-015-2473-7.  Google Scholar

Figure 1.  A mass-spring-damper system
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Figure 2.  Depiction of the parallel transport of a tangent vector field from a smooth curve to another
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Figure 3.  Exemplification of two curves on a curved manifold $ {{{\mathbb{M}}}} $ that meet at a point $ p $
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Figure 4.  Depiction of the homotopic net used in the proof of the Lemma 5.1.
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