Formal embedding of the Spring Loaded Inverted Pendulum in an Asymmetric hopper

The control of running is discussed in terms of a model called the Asymmetric Spring Loaded Inverted Pendulum (ASLIP), shown in Fig. 1. The ASLIP consists of a Spring Loaded Inverted Pendulum (SLIP) with the addition of pitch dynamics, and can be used to study the sagittal plane motion of bipedal running. A hybrid controller for the ASLIP is developed that acts on two levels. In the first level, continuous in-stride control is used to stabilize the torso at a desired posture, and to create an invariant surface on which the stance dynamics of the closed-loop system is diffeomorphic to the center of mass dynamics of a SLIP. In the second level, event-based control is employed to stabilize the closed-loop hybrid system along a periodic orbit of the SLIP dynamics. These results provide a systematic framework for designing control laws with provable stability properties which take advantage of existing SLIP controllers that are known to induce elegant running motions in legged models.