Abstract
The purpose of this paper is to apply methods from geometric mechanics to the analysis and control of bipedal robotic walkers. We begin by introducing a generalization of Routhian reduction, functional Routhian Reduction, which allows for the conserved quantities to be functions of the cyclic variables rather than constants. Since bipedal robotic walkers are naturally modeled as hybrid systems, which are inherently nonsmooth, in order to apply this framework to these systems it is necessary to first extend functional Routhian reduction to a hybrid setting. We apply this extension, along with potential shaping and controlled symmetries, to derive a feedback control law that provably results in walking gaits on flat ground for a three-dimensional bipedal walker given walking gaits in two dimensions.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
A. D. Ames. A Categorical Theory of Hybrid Systems. PhD thesis, University of California, Berkeley, 2006.
A. D. Ames and S. Sastry. Hybrid Routhian reduction of hybrid Lagrangians and Lagrangian hybrid systems. In American Control Conference, Minneapolis, MN, 2006.
A. Goswami, B. Thuilot, and B. Espiau. Compass-like biped robot part I: Stability and bifurcation of passive gaits. Rapport de recherche de l’INRIA, 1996.
J. W. Grizzle, G. Abba, and F. Plestan. Asymptotically stable walking for biped robots: Analysis via systems with impulse effects. IEEE Transactions on Automatic Control, 46(1):51–64, 2001.
J. Lygeros, K. H. Johansson, S. Simic, J. Zhang, and S. Sastry. Dynamical properties of hybrid automata. IEEE Transactions on Automatic Control, 48:2–17, 2003.
J. E. Marsden and T. S. Ratiu. Introduction to Mechanics and Symmetry, volume 17 of Texts in Applied Mathematics. Springer, 1999.
T. McGeer. Passive dynamic walking. International Journal of Robotics Research, 9(2):62–82, 1990.
R. M. Murray, Z. Li, and S. Sastry. A Mathematical Introduction to Robotic Manipulation. CRC Press, 1993.
M. W. Spong and F. Bullo. Controlled symmetries and passive walking. In IFAC World Congress, Barcelona, Spain, 2002.
M. W. Spong and F. Bullo. Controlled symmetries and passive walking. IEEE Transactions on Automatic Control, 50(7):1025–1031, 2005.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2007 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Ames, A.D., Gregg, R.D., Wendel, E.D., Sastry, S. (2007). On the Geometric Reduction of Controlled Three-Dimensional Bipedal Robotic Walkers. In: Allgüwer, F., et al. Lagrangian and Hamiltonian Methods for Nonlinear Control 2006. Lecture Notes in Control and Information Sciences, vol 366. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-73890-9_14
Download citation
DOI: https://doi.org/10.1007/978-3-540-73890-9_14
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-73889-3
Online ISBN: 978-3-540-73890-9
eBook Packages: EngineeringEngineering (R0)