Practical numerical algorithms for chaotic systems
Practical numerical algorithms for chaotic systems
Differential equations and dynamical systems
Differential equations and dynamical systems
A Mathematical Introduction to Robotic Manipulation
A Mathematical Introduction to Robotic Manipulation
Characterizing knee-bounce in bipedal robotic walking: a zeno behavior approach
Proceedings of the 14th international conference on Hybrid systems: computation and control
Dynamically stable bipedal robotic walking with NAO via human-inspired hybrid zero dynamics
Proceedings of the 15th ACM international conference on Hybrid Systems: Computation and Control
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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.