Reachability and Real-Time Actuation Strategies for the Active SLIP Model

Running and hopping follow similar patterns for different animals, independent of the number of legs employed. An aerial phase alternates with a ground contact phase, during which the center of mass moves as if a spring were compressed and then extended to recover stored elastic energy. Hence, consisting of a point mass mounted on a massless spring leg, the Spring Loaded Inverted Pendulum (SLIP) is a prevalent model for analyzing running and hopping. In this work we consider an actuated version of the SLIP model, with a series elastic actuator added to the leg, serving the purposes of adding/removing energy to/from the system and of modifying dynamics during stance, toward achieving non-steady locomotion on varying terrain. While the SLIP model has been a topic of research in legged locomotion for several decades, studies on the effect of actuation on the system's behavior are still not complete.The goal of this thesis is to explore how a series elastic actuator applied to the SLIP model's leg can change the system's dynamics. This, in turn, enables a variety of long-term planning strategies for using limited footholds and design non-steady gaits while simultaneously recovering from unexpected perturbations, both sensorial and due to a limited knowledge of the terrain profile.We principally investigate how, through actuation, we can solve partially or completely the system's equations of motion, to enforce a desired trajectory and reach a desired state. We also determine the reachable state space of the model using several different actuation strategies, investigating the variation of the reachable set with respect to particular actuator motions and providing relationships between local actuator displacements throughout stance and location of the reached apex state. We then propose a control strategy based on graphical and numerical studies of the reachability space to drive the system to a desired state, with the ability to reduce the effects of sensing errors and disturbances happening at landing as well as during ground contact