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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

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.

Citation: Simone Fiori. Error-based control systems on Riemannian state manifolds: Properties of the principal pushforward map associated to parallel transport. Mathematical Control and 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.

[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.

[3]

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

[4]

F. BulloR. 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.

[5]

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

[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.

[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.

[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.

[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.

[10]

S. Eftekhar AzamS. 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. 

[11]

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

[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.

[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.

[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.

[15]

R. FuentesG. 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.

[16]

F. A. GoodarziD. 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.

[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.

[18]

M. KabiriH. 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.

[19]

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

[20]

L. LüC. LiS. LiuZ. WangJ. Tian and J. Gu, The signal synchronization transmission among uncertain discrete networks with different nodes, Nonlinear Dynamics, 81 (2015), 801-809. 

[21]

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

[22]

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

[23]

K. OjoS. 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. 

[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.

[25]

P. PirasL. TeresiL. TraversettiV. VaranoS. GabrieleT. KotsakisP. RaiaP. 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.

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

[28]

T. SoyfıdanH. 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. 

[29]

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

[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.

[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.

[32]

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

[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.

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.

[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.

[3]

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

[4]

F. BulloR. 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.

[5]

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

[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.

[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.

[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.

[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.

[10]

S. Eftekhar AzamS. 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. 

[11]

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

[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.

[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.

[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.

[15]

R. FuentesG. 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.

[16]

F. A. GoodarziD. 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.

[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.

[18]

M. KabiriH. 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.

[19]

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

[20]

L. LüC. LiS. LiuZ. WangJ. Tian and J. Gu, The signal synchronization transmission among uncertain discrete networks with different nodes, Nonlinear Dynamics, 81 (2015), 801-809. 

[21]

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

[22]

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

[23]

K. OjoS. 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. 

[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.

[25]

P. PirasL. TeresiL. TraversettiV. VaranoS. GabrieleT. KotsakisP. RaiaP. 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.

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

[28]

T. SoyfıdanH. 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. 

[29]

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

[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.

[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.

[32]

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

[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.

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