Hybrid Modelling of a Legged System

Abstract

This thesis focuses on analysing the hybrid dynamics of legged systems. The work begins with a formal introduction to hybrid dynamical systems, including definitions, theoretical results, and an illustrative example such as the bouncing ball. These concepts are then applied to the hybrid modelling of two simple legged systems: the Inverted Pendulum (IP) and the Linear Inverted Pendulum (LIP). The dynamics of the LIP model are explored in depth, starting with the 0-step dynamics, which describe motion without control-driven steps, and extending to the N-step dynamics, where stepping behaviour is introduced and analysed using the hybrid Poincaré map. Numerical simulations are employed to identify and study periodic orbits, including a specific 2-step periodic orbit. The thesis concludes with the implementation of a GeoGebra simulation tool, designed to visualize these dynamics and validate the theoretical findings. This work demonstrates the suitability of hybrid systems as a mathematical framework for legged locomotion and highlights the utility of simplified models like the LIP for studying stability, control, and trajectory planning in legged systems.