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.

[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.

[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.

[4]

A. A. Agrachev and Y. L. Sachkov, "Control Theory from the Geometric Viewpoint," Springer-Verlag Berlin and Heidelberg, 2004.

[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.

[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.

[7]

A. M. Bloch, "Nonholonomics Mechanics and Control," Springer-Verlag, New York, 2003. doi: 10.1007/b97376.

[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.

[9]

W. Boothby, "An Introduction to Differential Manifolds and Riemannian Geometry," 2nd edition, Academic Press, Inc, 1986.

[10]

F. Bullo and A. D. Lewis, "Geometric Control of Mechanical Systems," Springer Verlag, New York, 2004.

[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.

[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.

[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.

[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.

[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.

[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.

[17]

J. Douglas, Solution of the inverse problem of the calculus of variations, Trans. Amer. Math. Soc., 50 (1941), 71-128.

[18]

R. B. Gardner, "The Method of Equivalence and its Applications," CBMS Regional Conference Series in Applied Mathematics, 58, SIAM, Philadelphia, PA, 1989.

[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.

[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.

[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.

[22]

A. Isidori, "Nonlinear Control Systems," 3rd edition, Springer Verlag, 1995.

[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.

[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.

[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.

[26]

V. Jurdjevic, "Geometric Control Theory," Cambridge University Press, 1997.

[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.

[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.

[29]

P. Kokkonen, "Energy-Shaping Control of Physical Systems (ESC)," Matematiikan Ja Tilastotieteen Laitos, 2007.

[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.

[31]

A. D. Lewis, Affine connections control systems, in "Proc. IFAC Workshop on Lagrangian and Hamiltonian Methods for Nonlinear control," (2000), 128-133.

[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.

[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.

[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.

[35]

J. E. Marsden and T. Ratiu, "Introduction to Mechanics and Symmetry," Springer-Verlag, 1994.

[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.

[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.

[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.

[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.

[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.

[41]

R. M. Murray, Z. Li and S. S. Sastry, "A Mathematical Introduction to Robotic Manipulation," Taylor & Francis Ltd, Boca Raton, 1994.

[42]

H. Nijmeijer and A. J. van der Schaft, "Nonlinear Dynamical Control Systems," Springer-Verlag, New York, 1990.

[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.

[44]

W. M. Oliva, "Geometric Mechanics," Springer-Verlag, Berlin, 2002.

[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.

[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.

[47]

W. Respondek, Feedback classification of nonlinear control systems in $\mathbb{R}^2$ and $\mathbb{R}^3$, in "Geometry of Feedback and Optimal Control" 207 (eds. B. Jakubczyk and W. Respondek), Marcel Dekker, New York, (1998), 347-382.

[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.

[49]

W. Respondek and S. Ricardo, Equivariants of mechanical control systems, submitted, (2010).

[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.

[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.

[52]

S. Ricardo and W. Respondek, Geometry of second-order nonholonomic chained form systems, submitted, (2010).

[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.

[54]

W. Sarlet, Geometrical structures related to second-order equations, Differential Geometry and Its Applications, (1987), 279-299.

[55]

S. Sastry, "Nonlinear Systems: Analysis, Stability, and Control," Springer-Verlag, New York, 1999.

[56]

E. D. Sontag, "Mathematical Control Theory: Deterministic Finite Dimensional Systems," Springer-Verlag, New York, 1998.

[57]

M. W. Spong, Underactuated mechanical systems, in "Control Problems in Robotics and Automation," 230, Springer Berlin/Heidelberg, (1998), 135-150.

[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.

[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.

[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.

[61]

M. Zhitomirskii and W. Respondek, Simple germs of corank one affine distributions, Banach Center Publications, 44 (1998), 269-276.

show all references

References:
[1]

R. Abraham and J. E. Marsden, "Foundations of Mechanics," Addison-Wesley, 1978.

[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.

[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.

[4]

A. A. Agrachev and Y. L. Sachkov, "Control Theory from the Geometric Viewpoint," Springer-Verlag Berlin and Heidelberg, 2004.

[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.

[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.

[7]

A. M. Bloch, "Nonholonomics Mechanics and Control," Springer-Verlag, New York, 2003. doi: 10.1007/b97376.

[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.

[9]

W. Boothby, "An Introduction to Differential Manifolds and Riemannian Geometry," 2nd edition, Academic Press, Inc, 1986.

[10]

F. Bullo and A. D. Lewis, "Geometric Control of Mechanical Systems," Springer Verlag, New York, 2004.

[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.

[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.

[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.

[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.

[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.

[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.

[17]

J. Douglas, Solution of the inverse problem of the calculus of variations, Trans. Amer. Math. Soc., 50 (1941), 71-128.

[18]

R. B. Gardner, "The Method of Equivalence and its Applications," CBMS Regional Conference Series in Applied Mathematics, 58, SIAM, Philadelphia, PA, 1989.

[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.

[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.

[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.

[22]

A. Isidori, "Nonlinear Control Systems," 3rd edition, Springer Verlag, 1995.

[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.

[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.

[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.

[26]

V. Jurdjevic, "Geometric Control Theory," Cambridge University Press, 1997.

[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.

[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.

[29]

P. Kokkonen, "Energy-Shaping Control of Physical Systems (ESC)," Matematiikan Ja Tilastotieteen Laitos, 2007.

[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.

[31]

A. D. Lewis, Affine connections control systems, in "Proc. IFAC Workshop on Lagrangian and Hamiltonian Methods for Nonlinear control," (2000), 128-133.

[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.

[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.

[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.

[35]

J. E. Marsden and T. Ratiu, "Introduction to Mechanics and Symmetry," Springer-Verlag, 1994.

[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.

[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.

[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.

[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.

[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.

[41]

R. M. Murray, Z. Li and S. S. Sastry, "A Mathematical Introduction to Robotic Manipulation," Taylor & Francis Ltd, Boca Raton, 1994.

[42]

H. Nijmeijer and A. J. van der Schaft, "Nonlinear Dynamical Control Systems," Springer-Verlag, New York, 1990.

[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.

[44]

W. M. Oliva, "Geometric Mechanics," Springer-Verlag, Berlin, 2002.

[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.

[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.

[47]

W. Respondek, Feedback classification of nonlinear control systems in $\mathbb{R}^2$ and $\mathbb{R}^3$, in "Geometry of Feedback and Optimal Control" 207 (eds. B. Jakubczyk and W. Respondek), Marcel Dekker, New York, (1998), 347-382.

[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.

[49]

W. Respondek and S. Ricardo, Equivariants of mechanical control systems, submitted, (2010).

[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.

[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.

[52]

S. Ricardo and W. Respondek, Geometry of second-order nonholonomic chained form systems, submitted, (2010).

[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.

[54]

W. Sarlet, Geometrical structures related to second-order equations, Differential Geometry and Its Applications, (1987), 279-299.

[55]

S. Sastry, "Nonlinear Systems: Analysis, Stability, and Control," Springer-Verlag, New York, 1999.

[56]

E. D. Sontag, "Mathematical Control Theory: Deterministic Finite Dimensional Systems," Springer-Verlag, New York, 1998.

[57]

M. W. Spong, Underactuated mechanical systems, in "Control Problems in Robotics and Automation," 230, Springer Berlin/Heidelberg, (1998), 135-150.

[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.

[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.

[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.

[61]

M. Zhitomirskii and W. Respondek, Simple germs of corank one affine distributions, Banach Center Publications, 44 (1998), 269-276.

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