Quasivelocities and symmetries in simple hybrid systems

This paper discusses Hamel's formalism for simple hybrid systems and explores the role of reversing symmetries in these system with a continuous-discrete combined dynamics. By extending Hamel's formalism to the class of simple hybrid systems with impulsive effects, we derive, under some conditions, the dynamics of Lagrangian hybrid systems and Hamiltonian hybrid systems. In particular, we derive Euler-Poincare and Lie-Poisson equations for systems with impulsive effects as a simple hybrid system. A reversing symmetry in the phase-space permits one to construct a time reversible hybrid Hamiltonian system. Based on the invariance of a Hamiltonian function by a reversing symmetry, we can find sufficient conditions for the existence of periodic solutions for these simple hybrid systems.

[1]  Jerrold E. Marsden,et al.  Quasivelocities and symmetries in non-holonomic systems , 2009 .

[2]  Philip Holmes,et al.  Mechanical models for insect locomotion: dynamics and stability in the horizontal plane I. Theory , 2000, Biological Cybernetics.

[3]  Aaron D. Ames,et al.  Lyapunov-Like Conditions for the Existence of Zeno Behavior in Hybrid and Lagrangian Hybrid Systems , 2007, 2007 46th IEEE Conference on Decision and Control.

[4]  Stewart D. Johnson.  SIMPLE HYBRID SYSTEMS , 1994 .

[5]  A.D. Ames,et al.  Hybrid cotangent bundle reduction of simple hybrid mechanical systems with symmetry , 2006, 2006 American Control Conference.

[6]  S. Shankar Sastry,et al.  Model Reduction Near Periodic Orbits of Hybrid Dynamical Systems , 2013, IEEE Transactions on Automatic Control.

[7]  Franck Plestan,et al.  Asymptotically stable walking for biped robots: analysis via systems with impulse effects , 2001, IEEE Trans. Autom. Control..

[8]  A. Bloch,et al.  Nonholonomic Mechanics and Control , 2004, IEEE Transactions on Automatic Control.

[9]  Seyed Hamed Razavi,et al.  Symmetry Method for Limit Cycle Walking of Legged Robots. , 2016 .

[10]  M. Coleman,et al.  Motions and stability of a piecewise holonomic system: The discrete Chaplygin Sleigh , 1999 .

[11]  J. Lamb,et al.  Time-reversal symmetry in dynamical systems: a survey , 1998 .

[12]  B. Brogliato.  Nonsmooth Impact Mechanics: Models, Dynamics and Control , 1996 .

[13]  John Guckenheimer,et al.  The Dynamics of Legged Locomotion: Models, Analyses, and Challenges , 2006, SIAM Rev..

[14]  J. Marsden,et al.  Introduction to mechanics and symmetry , 1994 .

[15]  E. Westervelt,et al.  Feedback Control of Dynamic Bipedal Robot Locomotion , 2007 .

[16]  Aaron D. Ames,et al.  A geometric approach to three-dimensional hipped bipedal robotic walking , 2007, 2007 46th IEEE Conference on Decision and Control.

[17]  Philip Holmes,et al.  Mechanical models for insect locomotion: dynamics and stability in the horizontal plane – II. Application , 2000, Biological Cybernetics.

[18]  P. Holmes,et al.  Steering by transient destabilization in piecewise-holonomic models of legged locomotion , 2008 .

[19]  Auke Jan Ijspeert,et al.  Symmetric virtual constraints for periodic walking of legged robots , 2016, 2016 IEEE 55th Conference on Decision and Control (CDC).

[20]  A. Ruina.  Nonholonomic stability aspects of piecewise holonomic systems , 1998 .