Analysis and control of periodic gaits in legged robots
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The analysis, identi cation and control of legged locomotion have been an interest for various researchers towards building legged robots that move like the animals do in nature. The extensive studies on understanding legged locomotion led to some mathematical models, such as the Spring-Loaded Inverted Pendulum (SLIP) template (and its various derivatives), that can be used to identify, analyze and control legged locomotor systems. Despite their seemingly simple nature, as being a simple point mass attached to a massless spring from dynamics perspective, the SLIP model constitutes a restricted three-body problem formulation, whose non-integrability has been proven long before. Thus, researchers came up with approximate analytical solutions or they used some other different techniques such as partial feedback linearization for the sake of obtaining analytical Poincar e return maps that govern the motion of the desired legged locomotor system. In the first part of this thesis, we consider a SLIP-based legged locomotion model, which we call as Multi-Actuated Dissipative SLIP (MD-SLIP) that extends the simple SLIP model with two additional actuators. The first one is a linear actuator attached serially to the leg spring to ensure direct control on the compression and decompression of the leg spring. The second actuator is a rotatory one that is attached to hip, which provides ability to inject some torque inputs to the system dynamics, which is mainly inspired by biological legged locomotor systems. Following the analysis of MD-SLIP model, we utilize a partial feedback linearization strategy by which we can cancel some nonlinear dynamics of the legged locomotion model and obtain exact analytical solutions without needing any approximation. Having exact analytical solutions is crucial to investigate stability characteristics of the MD-SLIP model during its hopping gait behavior. We illustrate and compare the applicability of our solutions with open-loop and closedloop hopping performances on various rough terrain simulations. Finally, we show how the MD-SLIP model can be anchored to bipedal legged locomotion models, where we assign two independent MD-SLIP models to each leg and investigate the system performance under their simultaneous but independent control. The proposed bipedal legged locomotion model is called as Multi-Actuated Dissipative Bipedal SLIP (MDB-SLIP) model. The key idea here is that we can still utilize the partial feedback linearization concept that we applied for the original MD-SLIP model and ensure exact analytical solutions for the MDB-SLIP model as well. We also provide detailed investigations for open-loop and closed-loop walking gait performance of the MDB-SLIP model on different noisy terrain profiles.
Partial Feedback Linearization
Spring-Loaded Inverted Pendulum (SLIP) Model