## AbstractThis dissertation addresses the problem of trajectory generation for dynamical robots operating in unstructured environments in the absence of detailed models of the dynamics of the environment or of the robot itself. We factor this problem into the subproblem of task variation, and the subproblem of imprecision in models of dynamics. The problem of task variation is handled by defining task level control strategies in terms of qualitative models that support structurally stable phase space trajectories. Such models define low-dimensional spaces within which it is possible to select trajectories that constitute plans for achieving a desired goal. The second problem, that of model imprecision, arises when embedding the resulting trajectories in the phase space of the more complex higher-dimensional system that actually performs the task of interest. Trajectories in the high-dimensional phase space that are compatible with the low-dimensional plan are restricted to lie on a manifold. In the absence of analytical models of the high-dimensional dynamics, this manifold may be approximated using observed data generated by a randomized exploration of the state space. Approximations driven by such an imperfect set of observations can lead to spurious trajectories, but this problem is solved by regularizing the approximation using the low-dimensional model. This methodology is developed through a sequence of design problems. First, basic notions regarding control with qualitative models are clarified through the design of a global controller for the inverted pendulum and cart-pole systems. This is followed by the more challenging problem of dynamic bipedal walking on irregular terrain, which is the primary motivating problem for this dissertation. Our solution to the dynamic walking problem advances the state of the art by simultaneously achieving several important properties. Our algorithm generates trajectories to walk on irregular terrain, with only a qualitative model of the dynamics of the robot, and with energy usage comparable with actuated walkers utilizing passive dynamic principles. Although the definition of tasks in terms of structurally stable orbits and manifolds is very natural when talking about physical systems, this representation yields benefits in more artificial domains as well. This is demonstrated through the example of spatiotemporal control of polygonal shapes, such as in a robot collective.
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