Abstract
This paper revises and extends our earlier work in using sinusoids to steer systems with nonholonomic constraints. We show that simple sinusoidal input trajectories are not easily applied to some classes of nonholonomic systems. This leads to the definition of a form of systems which can be steered using our earlier methods. We describe this form in detail and present preliminary efforts towards understanding when systems can be converted into this form.
Research supported in part by the National Science Foundation under grant IRI-90-14490
This research was performed while the author was at the University of California, Berkeley
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
V. I. Arnold. Mathematical Methods of Classical Mechanics. Springer-Verlag, second edition, 1989.
A. M Bloch and N. H. McClamroch. Control of mechanical systems with classical nonholonomic constraints. In IEEE Control and Decision Conference ,pages 201–205, 1989.
A. M Bloch and N. H. McClamroch. Controllability and stabilizability properties of a nonholonomic control system. In IEEE Control and Decision Conference ,1990.
R. W. Brockett. Control theory and singular Riemannian geometry. In New Directions in Applied Mathematics ,pages 11–27. Springer-Verlag, New York, 1981.
R. W. Brockett. On the rectification of vibratory motion. Sensors and Actuators ,20(1–2):91–96, 1989.
M. Grayson and R. Grossman. Models for free nilpotent Lie algebras. Technical Memo PAM-397, Center for Pure and Applied Mathematics, University of California, Berkeley, 1987. (to appear in J. Algebra).
V. Gershkovich and A. Vershik. Nonholonomic manifolds and nilpotent analysis. Journal of Geometry and Physics ,5(3):407–452, 1988.
M. Hall. The Theory of Groups. Macmillan, 1959.
H. Hermes, A. Lundell, and D. Sullivan. Nilpotent bases for distributions and control systems. Journal of Differential Equations ,55:385–400, 1984.
A. Isidori. Nonlinear Control Systems. Springer-Verlag, 2nd edition, 1989.
Z. Li and J. Canny. Motion of two rigid bodies with rolling constraint. IEEE Transactions on Robotics and Automation ,6(1):62–71, 1990.
Z. Li, R. Montgomery, and M. Raibert. Dynamics and optimal control of a legged robot in flight phase. In IEEE International Conference on Robotics and Automation ,pages 1816–1821, 1989.
G. Lafferriere and H. J. Sussmann. Motion planning for controllable systems without drift: a preliminary report. Technical Report SYCON-90-04, Rutgers Center for Systems and Control, 1990.
G. Lafferriere and H. J. Sussmann. Motion planning for controllable systems without drift. In IEEE International Conference on Robotics and Automation ,pages 1148–1153, 1991.
R. M. Murray and S. S. Sastry. Grasping and manipulation using multifingered robot hands. In R. W. Brockett, editor, Robotics: Proceedings of Symposia in Applied Mathematics, Volume 41 ,pages 91–128. American Mathematical Society, 1990.
R. M. Murray and S. S. Sastry. Steering nonholonomic systems using sinusoids. In IEEE Control and Decision Conference ,1990.
R. M. Murray and S. S. Sastry. Nonholonomic motion planning: Steering using sinusoids. Technical Report UCB/ERL M91/45, Electronics Research Laboratory, University of California at Berkeley, 1991.
R. M. Murray. Robotic Control and Nonholonomic Motion Planning. PhD thesis, University of California at Berkeley, 1990.
C. Samson. Velocity and torque feedback control of a nonholonomic cart. In International Workshop in Adaptive and Nonlinear Control: Issues in Robotics ,1990.
J-P. Serre. Lie Algebras and Lie groups. W. A. Benjamin, New York, 1965.
V. S. Varadarajan. Lie Groups, Lie Algebras, and Their Representations. Springer-Verlag, 1984.
A. M. Vershik and V. Ya. Gershkovich. Nonholonomic problems and the theory of distributions. Acta Applicandae Mathematicae ,12:181–209, 1988.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1993 Springer Science+Business Media New York
About this chapter
Cite this chapter
Murray, R.M., Sastry, S.S. (1993). Steering Nonholonomic Control Systems Using Sinusoids. In: Li, Z., Canny, J.F. (eds) Nonholonomic Motion Planning. The Springer International Series in Engineering and Computer Science, vol 192. Springer, Boston, MA. https://doi.org/10.1007/978-1-4615-3176-0_2
Download citation
DOI: https://doi.org/10.1007/978-1-4615-3176-0_2
Publisher Name: Springer, Boston, MA
Print ISBN: 978-1-4613-6392-7
Online ISBN: 978-1-4615-3176-0
eBook Packages: Springer Book Archive