Gait Regulation for Bipedal Locomotion

This work explores regulation of forward speed, step length, and slope walking for the passive-dynamic class of bipedal robots. Previously, an energy-shaping control for regulating forward speed has appeared in the literature; here we show that control to be a special case of a more general time-scaling control that allows for speed transitions in arbitrary time. As prior work has focused on potential energy shaping for fully actuated bipeds, we study in detail the shaping of kinetic energy for bipedal robots, giving special treatment to issues of underactuation. Drawing inspiration from features of human walking, an underactuated kinetic-shaping control is presented that provides efficient regulation of walking speed while adjusting step length. Previous results on energetic symmetries of bipedal walking are also extended, resulting in a control that allows regulation of speed and step length while walking on any slope. Finally we formalize the optimal gait regulation problem and propose a dynamic programming solution seeded with passive-dynamic limit cycles. Observations of the optimal solutions generated by this method reveal further similarities between passive dynamic walking and human locomotion and give insight into the structure of minimum-effort controls for walking.

Exact Solution of Bayes and Minimax Change-Detection Problems

The challenge of detecting a change in the distribution of data is a sequential decision problem that is relevant to many engineering solutions, including quality control and machine and process monitoring. This dissertation develops techniques for exact solution of change-detection problems with discrete time and discrete observations. Change-detection problems are classified as Bayes or minimax based on the availability of information on the change-time distribution. A Bayes optimal solution uses prior information about the distribution of the change time to minimize the expected cost, whereas a minimax optimal solution minimizes the cost under the worst-case change-time distribution. Both types of problems are addressed. The most important result of the dissertation is the development of a polynomial-time algorithm for the solution of important classes of Markov Bayes change-detection problems. Existing techniques for epsilon-exact solution of partially observable Markov decision processes have complexity exponential in the number of observation symbols. A new algorithm, called constellation induction, exploits the concavity and Lipschitz continuity of the value function, and has complexity polynomial in the number of observation symbols. It is shown that change-detection problems with a geometric change-time distribution and identically- and independently-distributed observations before and after the change are solvable in polynomial time. Also, change-detection problems on hidden Markov models with a fixed number of recurrent states are solvable in polynomial time. A detailed implementation and analysis of the constellation-induction algorithm are provided. Exact solution methods are also established for several types of minimax change-detection problems. Finite-horizon problems with arbitrary observation distributions are modeled as extensive-form games and solved using linear programs. Infinite-horizon problems with linear penalty for detection delay and identically- and independently-distributed observations can be solved in polynomial time via epsilon-optimal parameterization of a cumulative-sum procedure. Finally, the properties of policies for change-detection problems are described and analyzed. Simple classes of formal languages are shown to be sufficient for epsilon-exact solution of change-detection problems, and methods for finding minimally sized policy representations are described.