Stability of hybrid system limit cycles: application to pass gait biped Robot Ian A. Hiskens' Department of Electrical puter Engineering University of Illinois at Urbana-Champaign Urbana IL 61801 USA Abstract Limit cycles mon in hybrid systems. However the non-smooth dynamics of such systems makes stability analysis difficult. This paper uses recent extensions of trajectory sensitivity analysis to obtain the characteristic multipliers of non-smooth limit cycles. The stability of a limit cycle is determined by its characteristic multipliers. The concepts are illustrated using pass gait biped robot example. 1 Introduction Hybrid system are characterized by interactions between continuous (smooth) dynamics and discrete events. Such systems mon across a diverse range of application areas. Examples include power systems [l], robotics [ 2, 3], manufacturing [4] and air-traffic control [5]. In fact, any system where saturation limits are routinely encountered can be thought of as a hybrid system. The limits introduce discrete events which (often) have a significant influence on overall behaviour. Many hybrid systems exhibit periodic behaviour. Discrete events, such as saturation limits, can act to trap the evolving system state within a constrained region of state space. Therefore even when the underlying continuous dynamics are unstable, discrete events may induce a stable limit set. Limit cycles (periodic behaviour) are often created in this way. Other systems, such as robot motion, are naturally periodic. Limit cycles can be stable (attracting), unstable (repelling) or non-stable (saddle). The stability of periodic behaviour is determined by characteristic (or Floquet) multipliers. A periodic solution corresponds to a fixed point of a Poincare map. Stability of the periodic solution is equivalent to stability of the fixed point. The characteristic multipliers are the eigenvalues of the Poincare map linearized about the fixed point. Section 4 reviews
