Geometric Control of Underactuated Mechanical Systems with Application Focus on Bipedal Robots

Underactuated mechanical systems (UMS) are mechanical systems with fewer controls than the number of configuration states. The systems have broad applications in robotics, aerospace and marine vehicles and many more areas. The application examples include bipedal robots, quadruped robots, flexible-link robots, underactuated manipulators, snake robots, acrobatic robots, robots on a mobile platform, spacecraft, unmanned aerial vehicles, surface vessels and underwater vehicles. The systems generally have highly nonlinear dynamics, and less control authority due to the underactuation; furthermore, some systems, such as bipedal robots, include a mixture of continuous and discrete dynamics and multiple switching events among different phases. Because of these properties, control of UMS has been an important and challenging problem for years.

The dynamics of almost all mechanical systems can be structured into a form with a distribution of vector fields, and control of the systems can be treated as controlled flows on configuration and velocity manifolds. A major contribution of this dissertation is thus to exploit geometric approaches for control of a class of UMS, which produces general fundamental results. First, in contrast with previous work on controllability of underactuated serial robots, which mostly focused only on a specific number of links, this thesis studies nonlinear controllability for a general N-link serial robot with one unactuated joint. Second, the time reversal symmetry, which is inherent in many mechanical systems, is exploited to develop a general control framework for a class of UMS, and the almost global controllability of the method is proved by following the same line with Lyapunov's method.

This dissertation also addresses robustness issues for underactuated bipedal robots, which can be regarded as some balance between fully actuated and passive walking robots. Thus, the underactuated biped makes a promising solution to balance the competing issues of energy consumption and robustness. For the biped, the coupling between velocities along the actuated and unactuated vector fields has been exploited to define a nonlinear coupling metric, which can be used to quantitatively measure the robustness of gaits. Considering that bipeds will eventually walk in unstructured natural environments, this thesis further examines the problem of bipedal walking on slippery surfaces, and presents some results that illustrate relationships among gait features and the robustness for walking on slippery surfaces. A primary contribution in this aspect is providing a nonlinear mechanical coupling metric and some design insights, such as changing actuation methods, adjusting the center of mass location, speeds and stride lengths, which can be used to improve the robustness of bipedal robots.