Time-Dependent Nonlinear Control of Bipedal Robotic Walking

Yan Gu, Purdue University


Although bipedal walking control has been extensively studied for the past forty years, it remains a challenging task. To achieve high-performance bipedal robotic walking, this dissertation studies and investigates control strategies for both fully actuated and underactuated bipedal robots based on nonlinear control theories and formal stability analysis. Previously, the Hybrid-Zero-Dynamics (HZD) framework, which is a state-based feedback controller design based on the full-order dynamic modeling and the input-output linearization, has successfully realized stable, agile, and efficient bipedal walking for both fully actuated and underactuated bipedal robotic walking. However, the critical issue of achieving high walking versatility has not been fully addressed by the HZD framework. In this dissertation, we propose and develop a time-dependent controller design methodology to achieve not only stable, agile, and efficient but also versatile bipedal walking for fully actuated bipeds. Furthermore, the proposed time-dependent approach can be used to achieve better walking robustness to implementation imperfections for both fully actuated and underactuated bipeds by effectively solving the high-sensitivity issue of the state-based approaches to sensor noises. In our controller design methodology, the full-order hybrid walking dynamics are first modeled, which consist of both continuous-time dynamics and rigid-body impact dynamics. Then, the desired path/motion for a biped to track is planned, and the output function is designed as the tracking error of the desired path/motion. Based on the full-order model of walking dynamics, the input-output linearization is utilized to synthesize a controller that exponentially drives the output function to zero during continuous phases. Finally, sufficient conditions are developed to evaluate the stability of the hybrid, time-varying closed-loop control system. By enforcing these conditions, stable bipedal walking can be automatically realized, and the desired motion can be satisfactorily followed. Both full actuation and underactuation are common in bipedal robotic walking. Full actuation occurs when the number of degrees of freedom equals the number of independent actuators while underactuation occurs when the number of degrees of freedom is greater than the number of independent actuators. Because a fully actuated biped can directly control each of its joints, more objectives may be achieved for a fully actuated biped than an underactuated one. In this dissertation, the exponential tracking of a straight-line contour in Cartesian space is achieved for both planar and three-dimensional (3-D) walking, which greatly improves the versatility of fully actuated bipedal robots. To guarantee the closed-loop stability, the first sufficient stability conditions are developed based on the construction of multiple Lyapunov functions. Underactuated walking is much more difficult to control than fully actuated walking because an underactuated biped cannot directly control each of its joints. In this dissertation, control design of periodic, underactuated walking is investigated, and the first set of sufficient conditions for time-dependent orbitally exponential stabilization is established based on time-dependent nonlinear feedback control. Without modifications, the proposed controller design can be directly applied to both planar and 3-D bipeds that are subject to either underactuation or full actuation. Extensive computer simulation results validated the proposed time-dependent controller design methodology for bipedal robotic walking. Specifically, three bipedal models were simulated: one was a fully actuated, planar bipedal model with three revolute joints, one was a fully actuated, 3-D bipedal model with nine revolute joints, and one was an underactuated, planar bipedal model with five revolute joints.




Lee, Purdue University.

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