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Rank properties of poincare maps for hybrid systems with applications to bipedal walking

Published: 12 April 2010 Publication History

Abstract

The equivalence of the stability of periodic orbits with the stability of fixed points of a Poincaré map is a well-known fact for smooth dynamical systems. In particular, the eigenvalues of the linearization of a Poincaré map can be used to determine the stability of periodic orbits. The main objective of this paper is to study the properties of Poincaré maps for hybrid systems as they relate to the stability of hybrid periodic orbits. The main result is that the properties of Poincaré maps for hybrid systems are fundamentally different from those for smooth systems, especially with respect to the linearization of the Poincaré map and its eigenvalues. In particular, the linearization of any Poincaré map for a smooth dynamical system will have one trivial eigenvalue equal to 1 that does not affect the stability of the orbit. For hybrid systems, the trivial eigenvalues are equal to 0 and the number of trivial eigenvalues is bounded above by dimensionality differences between the different discrete domains of the hybrid system and the rank of the reset maps. Specifically, if n is the minimum dimension of the domains of the hybrid system, then the Poincaré map on a domain of dimension m ≥ n results in at least m-n+1 trivial 0 eigenvalues, with the remaining eigenvalues determining the stability of the hybrid periodic orbit. These results will be demonstrated on a nontrivial multi-domain hybrid system: a planar bipedal robot with knees.

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cover image ACM Conferences
HSCC '10: Proceedings of the 13th ACM international conference on Hybrid systems: computation and control
April 2010
308 pages
ISBN:9781605589558
DOI:10.1145/1755952
Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]

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Published: 12 April 2010

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Author Tags

  1. hybrid systems
  2. periodic orbits
  3. poincare maps
  4. robotic bipedal walking

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  • (2017)A brief review of dynamics and control of underactuated biped robotsAdvanced Robotics10.1080/01691864.2017.130827031:12(607-623)Online publication date: 7-Apr-2017
  • (2016)Modeling and stability analysis of limit cycles in an integrate-and-fire circuit2016 35th Chinese Control Conference (CCC)10.1109/ChiCC.2016.7553662(2016-2019)Online publication date: Jul-2016
  • (2015)Results on stability and robustness of hybrid limit cycles for a class of hybrid systems2015 54th IEEE Conference on Decision and Control (CDC)10.1109/CDC.2015.7402539(2235-2240)Online publication date: Dec-2015
  • (2014)Human-Inspired Control of Bipedal Walking RobotsIEEE Transactions on Automatic Control10.1109/TAC.2014.229934259:5(1115-1130)Online publication date: May-2014
  • (2012)Dynamically stable bipedal robotic walking with NAO via human-inspired hybrid zero dynamicsProceedings of the 15th ACM international conference on Hybrid Systems: Computation and Control10.1145/2185632.2185655(135-144)Online publication date: 17-Apr-2012
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