# American Institute of Mathematical Sciences

December 2018, 10(4): 467-502. doi: 10.3934/jgm.2018018

## A coordinate-free theory of virtual holonomic constraints

 1 Dipartimento di Ingegneria dell'Informazione, Università di Parma, Parco Area delle Scienze 181/a, 43124 Parma, Italy 2 Department of Electrical and Computer Engineering, University of Toronto, 10 King's College Road, Toronto, Ontario, M5S 3G4, Canada

* Corresponding author

Some of the ideas of this paper appeared in preliminary form in [11]

Received  September 2017 Revised  September 2018 Published  November 2018

This paper presents a coordinate-free formulation of virtual holonomic constraints for underactuated Lagrangian control systems on Riemannian manifolds. It is shown that when a virtual constraint enjoys a regularity property, the constrained dynamics are described by an affine connection dynamical system. The affine connection of the constrained system has an elegant relationship to the Riemannian connection of the original Lagrangian control system. Necessary and sufficient conditions are given for the constrained dynamics to be Lagrangian. A key condition is that the affine connection of the constrained dynamics be metrizable. Basic results on metrizability of affine connections are first reviewed, then employed in three examples in order of increasing complexity. The last example is a double pendulum on a cart with two different actuator configurations. For this control system, a virtual constraint is employed which confines the second pendulum to within the upper half-plane.

Citation: Luca Consolini, Alessandro Costalunga, Manfredi Maggiore. A coordinate-free theory of virtual holonomic constraints. Journal of Geometric Mechanics, 2018, 10 (4) : 467-502. doi: 10.3934/jgm.2018018
##### References:
 [1] W. Ambrose and I. Singer, A theorem on holonomy, Transactions of the American Mathematical Society, 75 (1953), 428-443. doi: 10.1090/S0002-9947-1953-0063739-1. [2] P. Appell, Exemple de mouvement d'un point assujetti à une liaison exprimèe par une relation non-linéaire entre les composantes de la vitesse, Rend. Circ. Mat. Palermo, 32 (1911), 48-50. [3] V. I. Arnol'd, Mathematical Methods of Classical Mechanics (Graduate Texts in Mathematics, Vol. 60), 2nd edition, Springer, 1989. doi: 10.1007/978-1-4757-2063-1. [4] L. Auslander and L. Markus, Holonomy of flat affinely connected manifolds, Annals of Mathematics, 62 (1955), 139-151. doi: 10.2307/2007104. [5] H. Beghin, Étude Théorique des Compas Gyrostatiques Anschütz et Sperry, PhD thesis, Faculté des sciences de Paris, 1922. [6] W. Boothby, An Introduction to Differentiable Manifolds and Riemannian Geometry, vol. 120, 2nd edition, Academic press, 1986. [7] F. Bullo and A. Lewis, Geometric Control of Mechanical Systems, Texts in Applied Mathematics, Springer, 2005. doi: 10.1007/978-1-4899-7276-7. [8] C. Chevallereau, J. Grizzle and C. Shih, Asymptotically stable walking of a five-link underactuated 3D bipedal robot, IEEE Transactions on Robotics, 25 (2008), 37-50. [9] F. Clarke, Y. S. Ledyaev, R. Stern and P. Wolenski, Nonsmooth Analyisis and Control Theory, Graduate Texts in Mathematics, 178. Springer-Verlag, New York, 1998. [10] L. Consolini, M. Maggiore, C. Nielsen and M. Tosques, Path following for the PVTOL aircraft, Automatica, 46 (2010), 1284-1296. doi: 10.1016/j.automatica.2010.05.014. [11] L. Consolini and A. Costalunga, Induced connections on virtual holonomic constraints, in IEEE Conference on Decision and Control (CDC), IEEE, 2015,139–144. [12] M. do Carmo, Riemannian Geometry, Birkhäuser Boston, 1992. doi: 10.1007/978-1-4757-2201-7. [13] L. Freidovich, A. Robertsson, A. Shiriaev and R. Johansson, Periodic motions of the pendubot via virtual holonomic constraints: Theory and experiments, Automatica, 44 (2008), 785-791. doi: 10.1016/j.automatica.2007.07.011. [14] L. Godinho and J. Natário, An Introduction to Riemannian Geometry: With Applications to Mechanics and Relativity, Springer, 2014. doi: 10.1007/978-3-319-08666-8. [15] A. Isidori, Nonlinear Control Systems, 3rd edition, Springer, New York, 1995. doi: 10.1007/978-1-84628-615-5. [16] D. Jankuloski, M. Maggiore and L. Consolini, Further results on virtual holonomic constraints, in Proceedings of the 4th IFAC Workshop on Lagrangian and Hamiltonian Methods for Nonlinear Control, Bertinoro, Italy, 2012. [17] S. Kobayashi and K. Nomizu, Foundations of Differential Geometry, no. v. 1 in A Wiley Publication in Applied Statistics, Wiley, 1996. [18] O. Kowalski, Metrizability of affine connections on analytic manifolds, Note di Matematica, 8 (1988), 1-11. [19] J. Lee, Riemannian Manifolds. An Introduction to Curvature, Graduate texts in mathematics 0176, Springer, 1997. doi: 10.1007/b98852. [20] J. Lee, Introduction to Smooth Manifolds, Graduate Texts in Mathematics, Springer, 2013. [21] M. Maggiore and L. Consolini, Virtual holonomic constraints for Euler-Lagrange systems, Automatic Control, IEEE Transactions on, 58 (2013), 1001-1008. doi: 10.1109/TAC.2012.2215538. [22] A. Mohammadi, M. Maggiore and L. Consolini, When is a Lagrangian control system with virtual holonomic constraints Lagrangian?, Proceedings of NOLCOS 2013, 9 (2013), 512-517. [23] A. Mohammadi, M. Maggiore and L. Consolini, On the lagrangian structure of reduced dynamics under virtual holonomic constraints, ESAIM: Control, Optimisation and Calculus of Variations, 23 (2017), 913-935. doi: 10.1051/cocv/2016020. [24] J. Nakanishi, T. Fukuda and D. Koditschek, A brachiating robot controller, IEEE Transactions on Robotics and Automation, 16 (2000), 109-123. [25] H. Nijmeijer and A. van der Schaft, Nonlinear Dynamical Control Systems., Springer - Verlag, New York, 1990. doi: 10.1007/978-1-4757-2101-0. [26] K. Nomizu and T. Sasaki, Affine Differential Geometry: Geometry of Affine Immersions, Cambridge University Press, 1994. [27] F. Plestan, J. Grizzle, E. Westervelt and G. Abba, Stable walking of a 7-DOF biped robot, IEEE Transactions on Robotics and Automation, 19 (2003), 653-668. [28] W. Poor, Differential Geometric Structures, McGraw-Hill Book Co., New York, 1981. [29] S. Ricardo and W. Respondek, When is a control system mechanical?, Journal of Geometric Mechanics, 2 (2010), 265-302. doi: 10.3934/jgm.2010.2.265. [30] B. Schmidt, Conditions on a connection to be a metric connection, Communications in Mathematical Physics, 29 (1973), 55-59. doi: 10.1007/BF01661152. [31] A. Shiriaev, J. Perram and C. Canudas-de-Wit, Constructive tool for orbital stabilization of underactuated nonlinear systems: Virtual constraints approach, IEEE Transactions on Automatic Control., 50 (2005), 1164-1176. doi: 10.1109/TAC.2005.852568. [32] A. Shiriaev, A. Robertsson, J. Perram and A. Sandberg, Periodic motion planning for virtually constrained Euler-Lagrange systems, Systems & Control Letters, 55 (2006), 900-907. doi: 10.1016/j.sysconle.2006.06.007. [33] A. Shiriaev, L. Freidovich and S. Gusev, Transverse linearization for controlled mechanical systems with several passive degrees of freedom, IEEE Transactions on Automatic Control, 55 (2010), 893-906. doi: 10.1109/TAC.2010.2042000. [34] G. Thompson, Local and global existence of metrics in two-dimensional affine manifolds, Chinese Journal of Physics, 29 (1991), 529-532. [35] A. Vanžurová, Metrization problem for linear connections and holonomy algebras, Archivum Mathematicum (BRNO), 44 (2008), 511-521. [36] A. Vanžurová and P. Žáčková, Metrizability of connections on two-manifolds, Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica, 48 (2009), 157-170. [37] E. Westervelt, J. Grizzle, C. Chevallereau, J. Choi and B. Morris, Feedback Control of Dynamic Bipedal Robot Locomotion, Taylor & Francis, CRC Press, 2007. [38] E. Westervelt, J. Grizzle and D. Koditschek, Hybrid zero dynamics of planar biped robots, IEEE Transactions on Automatic Control, 48 (2003), 42-56. doi: 10.1109/TAC.2002.806653.

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##### References:
 [1] W. Ambrose and I. Singer, A theorem on holonomy, Transactions of the American Mathematical Society, 75 (1953), 428-443. doi: 10.1090/S0002-9947-1953-0063739-1. [2] P. Appell, Exemple de mouvement d'un point assujetti à une liaison exprimèe par une relation non-linéaire entre les composantes de la vitesse, Rend. Circ. Mat. Palermo, 32 (1911), 48-50. [3] V. I. Arnol'd, Mathematical Methods of Classical Mechanics (Graduate Texts in Mathematics, Vol. 60), 2nd edition, Springer, 1989. doi: 10.1007/978-1-4757-2063-1. [4] L. Auslander and L. Markus, Holonomy of flat affinely connected manifolds, Annals of Mathematics, 62 (1955), 139-151. doi: 10.2307/2007104. [5] H. Beghin, Étude Théorique des Compas Gyrostatiques Anschütz et Sperry, PhD thesis, Faculté des sciences de Paris, 1922. [6] W. Boothby, An Introduction to Differentiable Manifolds and Riemannian Geometry, vol. 120, 2nd edition, Academic press, 1986. [7] F. Bullo and A. Lewis, Geometric Control of Mechanical Systems, Texts in Applied Mathematics, Springer, 2005. doi: 10.1007/978-1-4899-7276-7. [8] C. Chevallereau, J. Grizzle and C. Shih, Asymptotically stable walking of a five-link underactuated 3D bipedal robot, IEEE Transactions on Robotics, 25 (2008), 37-50. [9] F. Clarke, Y. S. Ledyaev, R. Stern and P. Wolenski, Nonsmooth Analyisis and Control Theory, Graduate Texts in Mathematics, 178. Springer-Verlag, New York, 1998. [10] L. Consolini, M. Maggiore, C. Nielsen and M. Tosques, Path following for the PVTOL aircraft, Automatica, 46 (2010), 1284-1296. doi: 10.1016/j.automatica.2010.05.014. [11] L. Consolini and A. Costalunga, Induced connections on virtual holonomic constraints, in IEEE Conference on Decision and Control (CDC), IEEE, 2015,139–144. [12] M. do Carmo, Riemannian Geometry, Birkhäuser Boston, 1992. doi: 10.1007/978-1-4757-2201-7. [13] L. Freidovich, A. Robertsson, A. Shiriaev and R. Johansson, Periodic motions of the pendubot via virtual holonomic constraints: Theory and experiments, Automatica, 44 (2008), 785-791. doi: 10.1016/j.automatica.2007.07.011. [14] L. Godinho and J. Natário, An Introduction to Riemannian Geometry: With Applications to Mechanics and Relativity, Springer, 2014. doi: 10.1007/978-3-319-08666-8. [15] A. Isidori, Nonlinear Control Systems, 3rd edition, Springer, New York, 1995. doi: 10.1007/978-1-84628-615-5. [16] D. Jankuloski, M. Maggiore and L. Consolini, Further results on virtual holonomic constraints, in Proceedings of the 4th IFAC Workshop on Lagrangian and Hamiltonian Methods for Nonlinear Control, Bertinoro, Italy, 2012. [17] S. Kobayashi and K. Nomizu, Foundations of Differential Geometry, no. v. 1 in A Wiley Publication in Applied Statistics, Wiley, 1996. [18] O. Kowalski, Metrizability of affine connections on analytic manifolds, Note di Matematica, 8 (1988), 1-11. [19] J. Lee, Riemannian Manifolds. An Introduction to Curvature, Graduate texts in mathematics 0176, Springer, 1997. doi: 10.1007/b98852. [20] J. Lee, Introduction to Smooth Manifolds, Graduate Texts in Mathematics, Springer, 2013. [21] M. Maggiore and L. Consolini, Virtual holonomic constraints for Euler-Lagrange systems, Automatic Control, IEEE Transactions on, 58 (2013), 1001-1008. doi: 10.1109/TAC.2012.2215538. [22] A. Mohammadi, M. Maggiore and L. Consolini, When is a Lagrangian control system with virtual holonomic constraints Lagrangian?, Proceedings of NOLCOS 2013, 9 (2013), 512-517. [23] A. Mohammadi, M. Maggiore and L. Consolini, On the lagrangian structure of reduced dynamics under virtual holonomic constraints, ESAIM: Control, Optimisation and Calculus of Variations, 23 (2017), 913-935. doi: 10.1051/cocv/2016020. [24] J. Nakanishi, T. Fukuda and D. Koditschek, A brachiating robot controller, IEEE Transactions on Robotics and Automation, 16 (2000), 109-123. [25] H. Nijmeijer and A. van der Schaft, Nonlinear Dynamical Control Systems., Springer - Verlag, New York, 1990. doi: 10.1007/978-1-4757-2101-0. [26] K. Nomizu and T. Sasaki, Affine Differential Geometry: Geometry of Affine Immersions, Cambridge University Press, 1994. [27] F. Plestan, J. Grizzle, E. Westervelt and G. Abba, Stable walking of a 7-DOF biped robot, IEEE Transactions on Robotics and Automation, 19 (2003), 653-668. [28] W. Poor, Differential Geometric Structures, McGraw-Hill Book Co., New York, 1981. [29] S. Ricardo and W. Respondek, When is a control system mechanical?, Journal of Geometric Mechanics, 2 (2010), 265-302. doi: 10.3934/jgm.2010.2.265. [30] B. Schmidt, Conditions on a connection to be a metric connection, Communications in Mathematical Physics, 29 (1973), 55-59. doi: 10.1007/BF01661152. [31] A. Shiriaev, J. Perram and C. Canudas-de-Wit, Constructive tool for orbital stabilization of underactuated nonlinear systems: Virtual constraints approach, IEEE Transactions on Automatic Control., 50 (2005), 1164-1176. doi: 10.1109/TAC.2005.852568. [32] A. Shiriaev, A. Robertsson, J. Perram and A. Sandberg, Periodic motion planning for virtually constrained Euler-Lagrange systems, Systems & Control Letters, 55 (2006), 900-907. doi: 10.1016/j.sysconle.2006.06.007. [33] A. Shiriaev, L. Freidovich and S. Gusev, Transverse linearization for controlled mechanical systems with several passive degrees of freedom, IEEE Transactions on Automatic Control, 55 (2010), 893-906. doi: 10.1109/TAC.2010.2042000. [34] G. Thompson, Local and global existence of metrics in two-dimensional affine manifolds, Chinese Journal of Physics, 29 (1991), 529-532. [35] A. Vanžurová, Metrization problem for linear connections and holonomy algebras, Archivum Mathematicum (BRNO), 44 (2008), 511-521. [36] A. Vanžurová and P. Žáčková, Metrizability of connections on two-manifolds, Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica, 48 (2009), 157-170. [37] E. Westervelt, J. Grizzle, C. Chevallereau, J. Choi and B. Morris, Feedback Control of Dynamic Bipedal Robot Locomotion, Taylor & Francis, CRC Press, 2007. [38] E. Westervelt, J. Grizzle and D. Koditschek, Hybrid zero dynamics of planar biped robots, IEEE Transactions on Automatic Control, 48 (2003), 42-56. doi: 10.1109/TAC.2002.806653.
Transversality condition in the definition of regular VHC
The vector bundle map $\sigma: T {\cal Q}|_{\cal C} \to T {\cal C}$
Coordinate systems used in Section 4.3
The set ${\cal C}$ in Example 1 and its parametrization
The VHC ${\cal C}$ in Example 2 and its parametrization
Illustration of the case when the control accelerations are orthogonal to ${\cal C}$
The parallel transport map at the north pole of the unit sphere in $\mathbb{R}^3$, with Riemannian connection induced by the Euclidean metric in $\mathbb{R}^3$. The loop $\gamma_q$ is a triangle on the sphere
The double pendulum on a cart of Example 3. Case (a): control force on the cart. Case (b): control torque on the last joint. The orthogonal frame in the figure is the inertial reference frame
Configurations of the double pendulum on the VHC ${\cal C}$ of Example 3. The missing configurations on the right-hand side are deduced by symmetry with respect to the vertical axis
Parallel transport on $\mathbb{R} \times \mathbb{S}^1$ from $(0, 0)$ to $(s^1, s^2)$
$(q_2, \dot q_2)$ orbits of a few solutions of the double pendulum on a cart subject to the VHC $q_3 = \rho(q_2)$. On the left, case (a) (force on cart). On the right, case (b) (torque on last joint)
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