# American Institute of Mathematical Sciences

September  2010, 2(3): 265-302. doi: 10.3934/jgm.2010.2.265

## When is a control system mechanical?

 1 Department of Mathematics, School of Sciences and Technology, University of Trás-os-Montes e Alto Douro, 5001-801 Vila Real, Portugal 2 INSA-Rouen, Laboratoire de Mathématiques, 76801 Saint-Etienne-du-Rouvray, France

Received  May 2010 Published  November 2010

In this work we present a geometric setting for studying mechanical control systems. We distinguish a special class: the class of geodesically accessible mechanical systems, for which the uniqueness of the mechanical structure is guaranteed (up to an extended point transformation). We characterise nonlinear control systems that are state equivalent to a system from this class and we describe the canonical mechanical structure attached to them. Several illustrative examples are given.
Citation: Sandra Ricardo, Witold Respondek. When is a control system mechanical?. Journal of Geometric Mechanics, 2010, 2 (3) : 265-302. doi: 10.3934/jgm.2010.2.265
##### References:
 [1] R. Abraham and J. E. Marsden, "Foundations of Mechanics," Addison-Wesley, 1978.  Google Scholar [2] A. A. Agrachev, Feedback-invariant optimal control theory and differential geometry. II. Jacobi curves for singular extremals, J. Dynam. Control Systems, 4 (1998), 583-604. doi: 10.1023/A:1021871218615.  Google Scholar [3] A. A. Agrachev and R. V. Gamkrelidze, Feedback-invariant optimal control theory and differential geometry. I. Regular extremals, J. Dynam. Control Systems, 3 (1997), 343-389. doi: 10.1007/BF02463256.  Google Scholar [4] A. A. Agrachev and Y. L. Sachkov, "Control Theory from the Geometric Viewpoint," Springer-Verlag Berlin and Heidelberg, 2004.  Google Scholar [5] I. Anderson and G. Thompson, The inverse problem of the calculus of variations for ordinary differential equations, Mem. Amer. Math. Soc., 98 (1992), 108-110.  Google Scholar [6] H. Arai, K. Tanie and N. Shiroma, Nonholonomic control of a three-DOF planar underactuated manipulator, IEEE Trans. Robot. Autom., 14 (1998), 681-695. doi: 10.1109/70.720345.  Google Scholar [7] A. M. Bloch, "Nonholonomics Mechanics and Control," Springer-Verlag, New York, 2003. doi: 10.1007/b97376.  Google Scholar [8] B. Bonnard, Feedback equivalence for nonlinear systems and the time optimal control problem, SIAM J. Control and Optim., 29 (1991), 1300-1321. doi: 10.1137/0329067.  Google Scholar [9] W. Boothby, "An Introduction to Differential Manifolds and Riemannian Geometry," 2nd edition, Academic Press, Inc, 1986. Google Scholar [10] F. Bullo and A. D. Lewis, "Geometric Control of Mechanical Systems," Springer Verlag, New York, 2004.  Google Scholar [11] F. Bullo and K. M. Lynch, Kinematic controllability for decoupled trajectory planning in underactuated mechanical systems, IEEE Trans. Robot. Autom., 17 (2001), 402-412. doi: 10.1109/70.954753.  Google Scholar [12] D. Cheng, A. Astolfi and R. Ortega, On feedback equivalence to port controlled Hamiltonian systems, Systems Control Lett., 54 (2005), 911-917. doi: 10.1016/j.sysconle.2005.02.005.  Google Scholar [13] J. Cortés, A. J. van der Schaft and P. E. Crouch, Characterization of gradient control systems, SIAM J. Control Optim., 44 (2005), 1192-1214. doi: 10.1137/S0363012903425568.  Google Scholar [14] M. Crampin, G. E. Prince and G. Thompson, A geometrical version of the Helmholtz conditions in time-dependent Lagrangian dynamics, J. Phys. A-Math. Gen., 17 (1984), 1437-1447. doi: 10.1088/0305-4470/17/7/011.  Google Scholar [15] P. E. Crouch and A. J. van der Schaft, Hamiltonian and self-adjoint control systems, Systems & Control Letters, 8 (1987), 289-295. doi: 10.1016/0167-6911(87)90093-4.  Google Scholar [16] P. E. Crouch and A. J. van der Schaft, "Variational and Hamiltonian Control Systems," Lectures Notes in Control and Inform. Sci. 101, Springer-Verlag, New York, 1987. Google Scholar [17] J. Douglas, Solution of the inverse problem of the calculus of variations, Trans. Amer. Math. Soc., 50 (1941), 71-128.  Google Scholar [18] R. B. Gardner, "The Method of Equivalence and its Applications," CBMS Regional Conference Series in Applied Mathematics, 58, SIAM, Philadelphia, PA, 1989.  Google Scholar [19] R. B. Gardner and W. F. Shadwick, The GS algorithm for exact linearization to Brunovský normal form, IEEE Trans. Automat. Control, 37 (1992), 224-230. doi: 10.1109/9.121623.  Google Scholar [20] R. B. Gardner, W. F. Shadwick and G. R. Wilkens, Feedback equivalence and symmetries of Brunovský normal forms, Contemp. Math., 97 (1989), 115-130.  Google Scholar [21] J. Hauser, S. Sastry and G. Meyer, Nonlinear control design for slightly non-minimum phase systems: Application to V/STOL aircraft, Automatica J. IFAC, 28 (1992), 665-679. doi: 10.1016/0005-1098(92)90029-F.  Google Scholar [22] A. Isidori, "Nonlinear Control Systems," 3rd edition, Springer Verlag, 1995.  Google Scholar [23] B. Jakubczyk, Equivalence and invariants of nonlinear control systems, in "Nonlinear Controllability and Optimal Control" (eds. H.J. Sussmann), Marcel Dekker, New York-Basel, (1990), 177-218.  Google Scholar [24] B. Jakubczyk, Critical Hamiltonians and feedback invariants, in "Geometry of Feedback and Optimal Control" (eds. B. Jakubczyk and W. Respondek), Marcel Dekker, New York-Basel, (1998), 219-256.  Google Scholar [25] B. Jakubczyk, Feedback invariants and critical trajectories; Hamiltonian formalism for feedback equivalence, in "Nonlinear Control in the Year 2000" 1 (eds. A. Isidori, F. Lamnabhi-Lagarrigue, and W. Respondek), LNCS vol. 258, Springer, London, (2000) 545-568. Google Scholar [26] V. Jurdjevic, "Geometric Control Theory," Cambridge University Press, 1997.  Google Scholar [27] W. Kang and A. J. Krener, Extended quadratic controller normal form and dynamic feedback linearization of nonlinear systems, SIAM J. Control Optim., 30 (1992), 1319-1337. doi: 10.1137/0330070.  Google Scholar [28] J. Koiller, Book review of "Analytical Mechanics: A comprehensive treatise on the dynamics of constrained systems for engineers, physicists and mathematicians," by John G. Papastavridis, Bulletin (New Series) of the American Mathematical Society, 40 (2003), 405-419. Google Scholar [29] P. Kokkonen, "Energy-Shaping Control of Physical Systems (ESC)," Matematiikan Ja Tilastotieteen Laitos, 2007. Google Scholar [30] A. D. Lewis, Affine connections and distributions with applications to nonholonomic mechanics, Rep. Math. Phys., 42 (1998), 135-164. doi: 10.1016/S0034-4877(98)80008-6.  Google Scholar [31] A. D. Lewis, Affine connections control systems, in "Proc. IFAC Workshop on Lagrangian and Hamiltonian Methods for Nonlinear control," (2000), 128-133. Google Scholar [32] A. D. Lewis, The category of affine connection control systems, in "Proc. of the 39th IEEE Conf. on Decision and Control, Sydney, Australia," (2000), 1260-1265. Google Scholar [33] A. D. Lewis and R. M. Murray, Configuration Controllability of Simple Mechanical Control Systems, SIAM J. Control Optim., 35 (1997), 766-790. doi: 10.1137/S0363012995287155.  Google Scholar [34] A. D. Lewis and R. M. Murray, Decompositions for control systems on manifolds with an affine connection, Syst. Contr. Lett., 31 (1997), 199-205. doi: 10.1016/S0167-6911(97)00040-6.  Google Scholar [35] J. E. Marsden and T. Ratiu, "Introduction to Mechanics and Symmetry," Springer-Verlag, 1994.  Google Scholar [36] P. Martin, S. Devasia and B. Paden, A different look at output tracking: control of a VTOL aircraft, in "Proc. of the 33rd IEEE Conf. on Decision and Control," (1994), 2376-2381. Google Scholar [37] E. Martínez, J. F. Cariñena and W. Sarlet, A geometric characterization of separable second-order differential equations, Mathematical Proceedings of the Cambridge Philosophical Society, 113 (1993), 205-224. doi: 10.1017/S0305004100075897.  Google Scholar [38] M. Milam and R. M. Murray, A testbed for nonlinear flight control techniques: The Caltech ducted fan, in "Proc. of the IEEE Int. Conf. on Control Applications," 1 (1999), 345-351. Google Scholar [39] G. Morandi, C. Ferrario, G. Lo Vecchio, G. Marmo and C. Rubano, The inverse problem in the calculus of variations and the geometry of the tangent bundle, Physics Reports, 188 (1990), 147-284. doi: 10.1016/0370-1573(90)90137-Q.  Google Scholar [40] R. M. Murray, Nonlinear control of mechanical systems: A Lagrangian perspective, Annual Reviews in Control, 21 (1997), 31-42. doi: 10.1016/S1367-5788(97)00023-0.  Google Scholar [41] R. M. Murray, Z. Li and S. S. Sastry, "A Mathematical Introduction to Robotic Manipulation," Taylor & Francis Ltd, Boca Raton, 1994.  Google Scholar [42] H. Nijmeijer and A. J. van der Schaft, "Nonlinear Dynamical Control Systems," Springer-Verlag, New York, 1990.  Google Scholar [43] R. Olfati-Saber, Global configuration stabilization for the VTOL aircraft with strong input coupling, IEEE Trans. Automat. Control, 47 (2002), 1949-1952. doi: 10.1109/TAC.2002.804457.  Google Scholar [44] W. M. Oliva, "Geometric Mechanics," Springer-Verlag, Berlin, 2002.  Google Scholar [45] R. Ortega, A. Loria, P. J. Nicklasson and H. Sira-Ramirez, "Passivity-Based Control of Euler-Lagrange Systems: Mechanical, Electrical and Electromechanical Applications," Springer-Verlag, Berlin, 1998. Google Scholar [46] R. H. Rand and D. V. Ramani, Nonlinear normal modes in a system with nonholonomic constraints, Nonlinear Dynamics, 25 (2001), 49-64. doi: 10.1023/A:1012946515772.  Google Scholar [47] W. Respondek, Feedback classification of nonlinear control systems in $\mathbbR^2$ and $\mathbbR^3$, in "Geometry of Feedback and Optimal Control" 207 (eds. B. Jakubczyk and W. Respondek), Marcel Dekker, New York, (1998), 347-382.  Google Scholar [48] W. Respondek, Introduction to geometric nonlinear control; linearization, observability and decoupling, in "Mathematical Control Theory" (ed. A. Agrachev), ICTP Lecture Notes, (2002), 169-222.  Google Scholar [49] W. Respondek and S. Ricardo, Equivariants of mechanical control systems, submitted, (2010). Google Scholar [50] W. Respondek and I. A. Tall, Feedback equivalence of nonlinear control systems: A survey on formal approach, in "Chaos in Automatic Control" (eds. J.-P. Barbot et W. Perruquetti), Taylor and Francis, (2006), 137-262.  Google Scholar [51] W. Respondek and M. Zhitomirskii, Feedback classification of nonlinear control systems on 3-manifolds, Math. Control Signals Systems, 8 (1995), 299-333. doi: 10.1007/BF01209688.  Google Scholar [52] S. Ricardo and W. Respondek, Geometry of second-order nonholonomic chained form systems, submitted, (2010). Google Scholar [53] W. Sarlet, The Helmholtz conditions revisited. A new approach to the inverse problem of Lagrangian dynamics, J. Phys. A-Math. Theor., 15 (1982), 1503-1517. doi: 10.1088/0305-4470/15/5/013.  Google Scholar [54] W. Sarlet, Geometrical structures related to second-order equations, Differential Geometry and Its Applications, (1987), 279-299.  Google Scholar [55] S. Sastry, "Nonlinear Systems: Analysis, Stability, and Control," Springer-Verlag, New York, 1999.  Google Scholar [56] E. D. Sontag, "Mathematical Control Theory: Deterministic Finite Dimensional Systems," Springer-Verlag, New York, 1998.  Google Scholar [57] M. W. Spong, Underactuated mechanical systems, in "Control Problems in Robotics and Automation," 230, Springer Berlin/Heidelberg, (1998), 135-150. Google Scholar [58] P. Tabuada and G. Pappas, From nonlinear to Hamiltonian via feedback, IEEE Trans. Automat. Control, 48 (2003), 1439-1442. doi: 10.1109/TAC.2003.815040.  Google Scholar [59] A. J. van der Schaft, Symmetries, conservation laws and time-reversibility for Hamiltonian systems with external forces, J. Math. Phys., 24 (1983), 2095-2101. doi: 10.1063/1.525962.  Google Scholar [60] J. Vankerschaver, F. Cantrijn, M. de León and D. Martín de Diego, Geometric aspects of nonholonomic field theories, Rep. Math. Phys., 56 (2005), 387-411. doi: 10.1016/S0034-4877(05)80093-X.  Google Scholar [61] M. Zhitomirskii and W. Respondek, Simple germs of corank one affine distributions, Banach Center Publications, 44 (1998), 269-276.  Google Scholar

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##### References:
 [1] R. Abraham and J. E. Marsden, "Foundations of Mechanics," Addison-Wesley, 1978.  Google Scholar [2] A. A. Agrachev, Feedback-invariant optimal control theory and differential geometry. II. Jacobi curves for singular extremals, J. Dynam. Control Systems, 4 (1998), 583-604. doi: 10.1023/A:1021871218615.  Google Scholar [3] A. A. Agrachev and R. V. Gamkrelidze, Feedback-invariant optimal control theory and differential geometry. I. Regular extremals, J. Dynam. Control Systems, 3 (1997), 343-389. doi: 10.1007/BF02463256.  Google Scholar [4] A. A. Agrachev and Y. L. Sachkov, "Control Theory from the Geometric Viewpoint," Springer-Verlag Berlin and Heidelberg, 2004.  Google Scholar [5] I. Anderson and G. Thompson, The inverse problem of the calculus of variations for ordinary differential equations, Mem. Amer. Math. Soc., 98 (1992), 108-110.  Google Scholar [6] H. Arai, K. Tanie and N. Shiroma, Nonholonomic control of a three-DOF planar underactuated manipulator, IEEE Trans. Robot. Autom., 14 (1998), 681-695. doi: 10.1109/70.720345.  Google Scholar [7] A. M. Bloch, "Nonholonomics Mechanics and Control," Springer-Verlag, New York, 2003. doi: 10.1007/b97376.  Google Scholar [8] B. Bonnard, Feedback equivalence for nonlinear systems and the time optimal control problem, SIAM J. Control and Optim., 29 (1991), 1300-1321. doi: 10.1137/0329067.  Google Scholar [9] W. Boothby, "An Introduction to Differential Manifolds and Riemannian Geometry," 2nd edition, Academic Press, Inc, 1986. Google Scholar [10] F. Bullo and A. D. Lewis, "Geometric Control of Mechanical Systems," Springer Verlag, New York, 2004.  Google Scholar [11] F. Bullo and K. M. Lynch, Kinematic controllability for decoupled trajectory planning in underactuated mechanical systems, IEEE Trans. Robot. Autom., 17 (2001), 402-412. doi: 10.1109/70.954753.  Google Scholar [12] D. Cheng, A. Astolfi and R. Ortega, On feedback equivalence to port controlled Hamiltonian systems, Systems Control Lett., 54 (2005), 911-917. doi: 10.1016/j.sysconle.2005.02.005.  Google Scholar [13] J. Cortés, A. J. van der Schaft and P. E. Crouch, Characterization of gradient control systems, SIAM J. Control Optim., 44 (2005), 1192-1214. doi: 10.1137/S0363012903425568.  Google Scholar [14] M. Crampin, G. E. Prince and G. Thompson, A geometrical version of the Helmholtz conditions in time-dependent Lagrangian dynamics, J. Phys. A-Math. Gen., 17 (1984), 1437-1447. doi: 10.1088/0305-4470/17/7/011.  Google Scholar [15] P. E. Crouch and A. J. van der Schaft, Hamiltonian and self-adjoint control systems, Systems & Control Letters, 8 (1987), 289-295. doi: 10.1016/0167-6911(87)90093-4.  Google Scholar [16] P. E. Crouch and A. J. van der Schaft, "Variational and Hamiltonian Control Systems," Lectures Notes in Control and Inform. Sci. 101, Springer-Verlag, New York, 1987. Google Scholar [17] J. Douglas, Solution of the inverse problem of the calculus of variations, Trans. Amer. Math. Soc., 50 (1941), 71-128.  Google Scholar [18] R. B. Gardner, "The Method of Equivalence and its Applications," CBMS Regional Conference Series in Applied Mathematics, 58, SIAM, Philadelphia, PA, 1989.  Google Scholar [19] R. B. Gardner and W. F. Shadwick, The GS algorithm for exact linearization to Brunovský normal form, IEEE Trans. Automat. Control, 37 (1992), 224-230. doi: 10.1109/9.121623.  Google Scholar [20] R. B. Gardner, W. F. Shadwick and G. R. Wilkens, Feedback equivalence and symmetries of Brunovský normal forms, Contemp. Math., 97 (1989), 115-130.  Google Scholar [21] J. Hauser, S. Sastry and G. Meyer, Nonlinear control design for slightly non-minimum phase systems: Application to V/STOL aircraft, Automatica J. IFAC, 28 (1992), 665-679. doi: 10.1016/0005-1098(92)90029-F.  Google Scholar [22] A. Isidori, "Nonlinear Control Systems," 3rd edition, Springer Verlag, 1995.  Google Scholar [23] B. Jakubczyk, Equivalence and invariants of nonlinear control systems, in "Nonlinear Controllability and Optimal Control" (eds. H.J. Sussmann), Marcel Dekker, New York-Basel, (1990), 177-218.  Google Scholar [24] B. Jakubczyk, Critical Hamiltonians and feedback invariants, in "Geometry of Feedback and Optimal Control" (eds. B. Jakubczyk and W. Respondek), Marcel Dekker, New York-Basel, (1998), 219-256.  Google Scholar [25] B. Jakubczyk, Feedback invariants and critical trajectories; Hamiltonian formalism for feedback equivalence, in "Nonlinear Control in the Year 2000" 1 (eds. A. Isidori, F. Lamnabhi-Lagarrigue, and W. Respondek), LNCS vol. 258, Springer, London, (2000) 545-568. Google Scholar [26] V. Jurdjevic, "Geometric Control Theory," Cambridge University Press, 1997.  Google Scholar [27] W. Kang and A. J. Krener, Extended quadratic controller normal form and dynamic feedback linearization of nonlinear systems, SIAM J. Control Optim., 30 (1992), 1319-1337. doi: 10.1137/0330070.  Google Scholar [28] J. Koiller, Book review of "Analytical Mechanics: A comprehensive treatise on the dynamics of constrained systems for engineers, physicists and mathematicians," by John G. Papastavridis, Bulletin (New Series) of the American Mathematical Society, 40 (2003), 405-419. Google Scholar [29] P. Kokkonen, "Energy-Shaping Control of Physical Systems (ESC)," Matematiikan Ja Tilastotieteen Laitos, 2007. Google Scholar [30] A. D. Lewis, Affine connections and distributions with applications to nonholonomic mechanics, Rep. Math. Phys., 42 (1998), 135-164. doi: 10.1016/S0034-4877(98)80008-6.  Google Scholar [31] A. D. Lewis, Affine connections control systems, in "Proc. IFAC Workshop on Lagrangian and Hamiltonian Methods for Nonlinear control," (2000), 128-133. Google Scholar [32] A. D. Lewis, The category of affine connection control systems, in "Proc. of the 39th IEEE Conf. on Decision and Control, Sydney, Australia," (2000), 1260-1265. Google Scholar [33] A. D. Lewis and R. M. Murray, Configuration Controllability of Simple Mechanical Control Systems, SIAM J. Control Optim., 35 (1997), 766-790. doi: 10.1137/S0363012995287155.  Google Scholar [34] A. D. Lewis and R. M. Murray, Decompositions for control systems on manifolds with an affine connection, Syst. Contr. Lett., 31 (1997), 199-205. doi: 10.1016/S0167-6911(97)00040-6.  Google Scholar [35] J. E. Marsden and T. Ratiu, "Introduction to Mechanics and Symmetry," Springer-Verlag, 1994.  Google Scholar [36] P. Martin, S. Devasia and B. Paden, A different look at output tracking: control of a VTOL aircraft, in "Proc. of the 33rd IEEE Conf. on Decision and Control," (1994), 2376-2381. Google Scholar [37] E. Martínez, J. F. Cariñena and W. Sarlet, A geometric characterization of separable second-order differential equations, Mathematical Proceedings of the Cambridge Philosophical Society, 113 (1993), 205-224. doi: 10.1017/S0305004100075897.  Google Scholar [38] M. Milam and R. M. Murray, A testbed for nonlinear flight control techniques: The Caltech ducted fan, in "Proc. of the IEEE Int. Conf. on Control Applications," 1 (1999), 345-351. Google Scholar [39] G. Morandi, C. Ferrario, G. Lo Vecchio, G. Marmo and C. Rubano, The inverse problem in the calculus of variations and the geometry of the tangent bundle, Physics Reports, 188 (1990), 147-284. doi: 10.1016/0370-1573(90)90137-Q.  Google Scholar [40] R. M. Murray, Nonlinear control of mechanical systems: A Lagrangian perspective, Annual Reviews in Control, 21 (1997), 31-42. doi: 10.1016/S1367-5788(97)00023-0.  Google Scholar [41] R. M. Murray, Z. Li and S. S. Sastry, "A Mathematical Introduction to Robotic Manipulation," Taylor & Francis Ltd, Boca Raton, 1994.  Google Scholar [42] H. Nijmeijer and A. J. van der Schaft, "Nonlinear Dynamical Control Systems," Springer-Verlag, New York, 1990.  Google Scholar [43] R. Olfati-Saber, Global configuration stabilization for the VTOL aircraft with strong input coupling, IEEE Trans. Automat. Control, 47 (2002), 1949-1952. doi: 10.1109/TAC.2002.804457.  Google Scholar [44] W. M. Oliva, "Geometric Mechanics," Springer-Verlag, Berlin, 2002.  Google Scholar [45] R. Ortega, A. Loria, P. J. Nicklasson and H. Sira-Ramirez, "Passivity-Based Control of Euler-Lagrange Systems: Mechanical, Electrical and Electromechanical Applications," Springer-Verlag, Berlin, 1998. Google Scholar [46] R. H. Rand and D. V. Ramani, Nonlinear normal modes in a system with nonholonomic constraints, Nonlinear Dynamics, 25 (2001), 49-64. doi: 10.1023/A:1012946515772.  Google Scholar [47] W. Respondek, Feedback classification of nonlinear control systems in $\mathbbR^2$ and $\mathbbR^3$, in "Geometry of Feedback and Optimal Control" 207 (eds. B. Jakubczyk and W. Respondek), Marcel Dekker, New York, (1998), 347-382.  Google Scholar [48] W. Respondek, Introduction to geometric nonlinear control; linearization, observability and decoupling, in "Mathematical Control Theory" (ed. A. Agrachev), ICTP Lecture Notes, (2002), 169-222.  Google Scholar [49] W. Respondek and S. Ricardo, Equivariants of mechanical control systems, submitted, (2010). Google Scholar [50] W. Respondek and I. A. Tall, Feedback equivalence of nonlinear control systems: A survey on formal approach, in "Chaos in Automatic Control" (eds. J.-P. Barbot et W. Perruquetti), Taylor and Francis, (2006), 137-262.  Google Scholar [51] W. Respondek and M. Zhitomirskii, Feedback classification of nonlinear control systems on 3-manifolds, Math. 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