Identification and stability analysis of periodic motions for a planar legged runner with a rigid body and a compliant leg

Abstract

The Spring-Loaded Inverted Pendulum (SLIP) model is an extensively used and fundamental template for modeling human and animal locomotion. Despite its wide use, the SLIP is a very simple model and considering the effects of body dynamics only as a point mass. Although the assumption of a point mass for the upper body simplifies system dynamics, it prevents us from performing detailed analysis for more realistic robot platforms with upper trunks. Hence, we consider an extension to the classic SLIP model to include the upper body dynamics in order to better understand human and animal locomotion. Due to its coupled rotational dynamics, extending the SLIP model to the Body-Attached Spring-Loaded Inverted Pendulum (BA-SLIP) brings additional difficulties in the analysis process, making it more difficult to obtain analytical solutions. Consequently, simulations have been used to reveal the periodic structure behind locomotion with this model, and to find fixed points of discretized system dynamics. These fixed points correspond to periodic motions of the system and are important in designing controllers since they are used as steady-state control targets for most applications. The main concern of this thesis is to find fixed points of the BA-SLIP model and to investigate the dimension of the fixed point manifold. We performed extensive simulation studies to find fixed points of the system and the properties of the underlying space with a PD controller. Our simulations revealed the existence of periodic gaits, in which the upper body should be downward oriented for stable locomotion. Additionally, a region of stability is found such that the model sustains periodic gaits when it stays inside this region. Finally, we show that fixed points for running with upright body orientation are unstable when system dynamics are regulated with a constant parameter controller. We also present some simulation results which indicate the existence of stable periodic motions when controllers with time varying parameters, that use current state information, are used.