Hostname: page-component-78c5997874-m6dg7 Total loading time: 0 Render date: 2024-11-10T14:25:27.928Z Has data issue: false hasContentIssue false
  • English
  • Français

Is it worth learning differential geometric methods for modeling and control of mechanical systems?

Published online by Cambridge University Press:  01 November 2007

Andrew D. Lewis
Andrew D. Lewis*
Affiliation:
Department of Mathematics and Statistics, Queen's University, Kingston, ON K7L 3N6, Canada.
*
*E-mail: andrew@mast.queensu.ca
Get access Rights & Permissions [Opens in a new window]

Summary

Evidence is presented to indicate that the answer is, “Yes, sometimes.”

Type
Article
Information
Robotica , Volume 25 , Issue 6 , November 2007 , pp. 765 - 777
Copyright
Copyright © Cambridge University Press 2007

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.Arai, H., Tanie, K. and Shiroma, N., “Nonholonomic control of a three-DOF planar underactuated manipulator,” IEEE Trans. Robotics Autom. 14 (5), 681–695 (1998).CrossRefGoogle Scholar
2.Auckly, D. R. and Kapitanski, L. V., “On the λ-equations for matching control laws,” SIAM J. Control Optim. 41 (5), 1372–1388 (2002).CrossRefGoogle Scholar
3.Bhat, S. P. and Bernstein, D. S., “A topological obstruction to continuous global stabilization of rotational motion and the unwinding phenomenon,” Syst. Control Lett. 39, 63–70 (2000).CrossRefGoogle Scholar
4.Bloch, A. M., Nonholonomic Mechanics and Control of Interdisciplinary Applied Mathematics 24 (Springer-Verlag, New York-Heidelberg-Berlin, 2003).CrossRefGoogle Scholar
5.Bloch, A. M., Leonard, N. E. and Marsden, J. E., “Controlled Lagrangians and the stabilization of mechanical systems. I. The first matching theorem,” IEEE Trans. Autom. Control 45 (12), 2253–2270 (2000).CrossRefGoogle Scholar
6.Bloch, A. M., Chang, D. E., Leonard, N. E. and Marsden, J. E., “Controlled Lagrangians and the stabilization of mechanical systems. II. Potential shaping,” IEEE Trans. Autom. Control 46 (10), 1556–1571 (2001).CrossRefGoogle Scholar
7.Bullo, F. and Lewis, A. D., “Kinematic controllability and motion planning for the snakeboard,” IEEE Trans. Robot. Autom. 19 (3), 494–498 (2003).CrossRefGoogle Scholar
8.Bullo, F. and Lewis, A. D., “Low-order controllability and kinematic reductions for affine connection control systems,” SIAM J. Control Optim. 44 (3), 885–908 (2005).CrossRefGoogle Scholar
9.Bullo, F. and Lewis, A. D.. Geometric Control of Mechanical Systems: Modeling, Analysis, and Design for Simple Mechanical Systems. Texts in Applied Mathematics 49. (Springer-Verlag, New York-Heidelberg-Berlin, 2004).Google Scholar
10.Bullo, F. and Lynch, K. M., “Kinematic controllability and decoupled trajectory planning for underactuated mechanical systems,” IEEE Trans. Robot. Autom. 17 (4), 402–412 (2001).CrossRefGoogle Scholar
11.Bullo, F. and Žefran, M., “On mechanical systems with nonholonomic constraints and symmetries,” Syst. Control Lett. 45 (2), 133–143 (2002).CrossRefGoogle Scholar
12.Dalsmo, M. and van der Schaft, A. J., “On representations and integrability of mathematical structures in energy-conserving physical systems,” SIAM J. Control Optim. 37 (1), 54–91 (1998).CrossRefGoogle Scholar
13.Lewis, A. D., “Potential Energy Shaping After Kinetic Energy Shaping,” Proceedings of the 45th IEEE Conference on Decision and Control, San Diego, CA (Dec. 2006) pp. 3339–3344.CrossRefGoogle Scholar
14.Lewis, A. D., “Simple mechanical control systems with constraints,” IEEE Trans. Autom. Control 45 (8), 1420–1436 (2000).CrossRefGoogle Scholar
15.Lewis, A. D. and Murray, R. M., “Controllability of simple mechanical control systems,” SIAM J. Control Optim. 35 (3), 766–790 (1997).CrossRefGoogle Scholar
16.Lewis, A. D. and Tyner, D. R., “Controllability of a Hovercraft Model (and Two General Results),” Proceedings of the 43rd IEEE Conference on Decision and Control, Paradise Island, Bahamas, (Dec. 2004) pp. 1204–1209.Google Scholar
17.Lewis, A. D., Ostrowski, J. P., Murray, R. M. and Burdick, J. W., “Nonholonomic Mechanics and Locomotion: The Snakeboard Example,” Proceedings of the 1994 IEEE International Conference on Robotics and Automation, San Diego, CA, (May 1994) pp. 2391–2400.Google Scholar
18.Lynch, K. M., Shiroma, N., Arai, H. and Tanie, K., “Collision-free trajectory planning for a 3-DOF robot with a passive joint,” Int. J. Robot Res. 19 (12), 1171–1184 (2000).CrossRefGoogle Scholar
19.Murray, R. M., Li, Z. X. and Sastry, S. S., A Mathematical Introduction to Robotic Manipulation (CRC Press, Boca Raton, Florida, 1994).Google Scholar
20.Ortega, R., Spong, M. W., Gómez-Estern, F. and Blankenstein, G., “Stabilization of a class of underactuated mechanical systems via interconnection and damping assignment,” IEEE Trans. Autom. Control 47 (8), 1218–1233 (2002).CrossRefGoogle Scholar
21.Stramigioli, S., Modeling and IPC Control of Interactive Mechanical Systems–-A Coordinate-Free Approach. Lecture Notes in Control and Information Sciences 266 (Springer-Verlag, New York-Heidelberg-Berlin, 2001).Google Scholar
22.Sussmann, H. J., “A general theorem on local controllability,” SIAM J. Control Optim. 25 (1), 158–194 (1987).CrossRefGoogle Scholar
23.Sussmann, H. J. and Jurdjevic, V., “Controllability of nonlinear systems,” J. Differential Equations 12, 95–116 (1972).CrossRefGoogle Scholar
24.Takegaki, M. and Arimoto, S., “A new feedback method for dynamic control of manipulators,” Trans. ASME Ser. G J. Dynam. Syst. Meas. Control 103 (2), 119–125 (1981).CrossRefGoogle Scholar
25.van der Schaft, A. J., “Port-controlled Hamiltonian systems: Towards a theory for control and design of nonlinear physical systems,” J. Soc. Instrum. Control Eng. 39 (2), 91–98 (2000).Google Scholar
26.van der Schaft, A. J., “Stabilization of Hamiltonian systems,” Nonlinear Anal. TMA 10 (10), 1021–1035 (1986).CrossRefGoogle Scholar