A Continuous Control for Stabilizing the Extended Nonholonomic Double Integrator: Theory and Simulation Results

Abstract:

The extended nonholonomic double integrator (ENDI) cannot be asymptotically stabilized by a continuous and time-invariant feedback controller since it violates the Brockett’s condition. In this paper, a new continuous control scheme is proposed to stabilize the ENDI system without drift. The dynamics of an ENDI system is enlarged to a higher dimensional space where a continuous but time-varying control law is designed for its stabilization. Besides the theoretical proofs, simulations conducted on the dynamics of a mobile robot are also presented to demonstrate the validity and performance of the proposed method.

• Periodical: Applied Mechanics and Materials (Volumes 44-47)
• Pages: 3987-3991
• DOI: https://doi.org/10.4028/www.scientific.net/AMM.44-47.3987
• Online since: December 2010
• Authors: Yan Peng, Mei Liu, Shao Rong Xie, Jun Luo
• Keywords: Asymptotically Stabilize, Extended Nonholonomic Double Integrator (ENDI), Higher Dimensional Space

References

[1] R. W. Brockett, Asymptotic stability and feedback stabilization, In: Diff. Geomet. Contr. Theory. Birkhauser, Boston(1983), p.181–191.

[2] P. Morin, and C. Samson, Practical stabilization of driftless systems on lie groups: The transverse function approach, IEEE Transactions on Automatic Control, 48(9) ( 2003), p.1496–1508.

DOI: 10.1109/tac.2003.816963

[3] F. Bullo, A. D. Lewis, Controllability and motion algorithms for underactuated Lagrangian systems on Lie groups, IEEE Trans. Automat. Contr. , vol. 45, No. 8 (2000) , pp.1437-1454.

DOI: 10.1109/9.871753

[4] I. Kolmanovsky, and N. H. McClamroch, Developments in nonholonomic control problems, IEEE Control Systems Magazine 15( 1995), p.20–36.

[5] A. P. Aguiar, and A. M. Pascoal, Regulation of a nonholonomic autonomous underwater vehicle with parametric modeling uncertainty using Lyapunov functions, In Proceedings of the 40th conference on decision and control, Orlando, FL, USA(2001).

DOI: 10.1109/cdc.2001.980841

[6] D. A. Lizárraga, N. P. I. Aneke, and H. Nijmeijer, Robust point stabilization of underactuated mechanical systems via the extended chained form, SIAM Journal of Control Optimization, 42(6) (2004), p.2172–2199.

DOI: 10.1137/s0363012902405571

[7] N. Khaneja and R. Brockett, Dynamic feedback stabilization of nonholonomic systems, Proceedings of the 38th Conference on Decision and Control, Phoenix, Arizona, USA(1999), pp.1640-1645.

DOI: 10.1109/cdc.1999.830259