The basin of attraction for running robots: Fractals, multistep trajectories, and the choice of control

If the control authority of a running system is insufficient to reach a target state in a single step, i.e. if deadbeat control is not possible, then a stabilizing controller is faced with the decision on how to plan intermediate steps. In this work, we compare the performance of a simple greedy control policy (that computes deadbeat inputs and simply caps them) with the optimal performance found by an exhaustive search through decision space. The performance criterion used in this study is the basin of attraction: the set of all states from where the target state will be reached in a finite number of steps. Using the planar spring-loaded inverted pendulum (SLIP) as a model for a running robot, we compare the two control schemes and a fully passive behavior. To this end, we extended the passive slip model to include a controllable, yet limited variation of the touchdown angle and of the damping in the leg spring. We quantified the number of steps that it would take for the model to fall or converge from arbitrary initial states. The paper highlights how the passive stabilization, that is inherent to the SLIP model, greatly influences the dynamics of the controlled system. Furthermore, it reveals some new insights into the structure of basins of attractions of SLIP-like running models.