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TITLE: ATTITUDE CONTROL ON MANIFOLDS VIA OPTIMIZATION AND CONTRACTIONS WITH AUTOMATIC GAIN TUNING.

ABSTRACT: The attitude (or orientation) of an object is often crucial in its ability to perform a task, whether the task is driving a car, ying an aircraft, or orientating a satellite. In traditional control approaches, the attitude is often parameterized by Euler angles or unit quaternions which exhibit problems such as gimbal lock or ambiguity in representation, respectively. These complications prevent the controllers from achieving global stability and worse they may cause real physical harm. More recent works have achieved global stability and avoided these system failures by working directly on the conguration manifold, but are generally complex or lack automatic, user-friendly ways to tune them. The goal of this dissertation is to develop simple geometric attitude controllers that are globally, exponentially stable and can be automatically tuned. By simple, we mean that the controllers are computationally ecient and the tuning parameters have geometric interpretations. These properties make the controllers practical for real hardware implementation even on fast dynamical systems. Furthermore, we aim to obtain an automatic tuning procedure that ensures convergence, and can also quantify and optimize performance guarantees. We achieve our goal through four major contributions. The rst is a substantial generalization on the theory of classical Riemannian metrics for tangent bundles which provides the ability to compare and combine attitude and velocity terms in the stability analysis, allowing us to consider a larger set of feasible controller gains. The second contribution is a framework to study the stability of attitude systems on manifolds by combining Riemannian geometry, contraction theory, and oine optimization. The third contribution is the development of a globally, exponentially stable attitude controller. This controller overcomes the topological limitation by using a time-varying intermediate reference trajectory. The fourth contribution is the improvement of the proposed controllers by way of point-wise-in-time quadratic programming. In this dissertation, we consider the application of our results to attitude control on the space of rotations, SO(3), we believe that they can be readily extended to arbitrary systems with dynamics on Riemannian manifolds such as the manifold of 3-D rigid poses SE(3).

COMMITTEE: ADVISOR Professor Roberto Tron, ME/SE; CHAIR Professor R. Glynn Holt, ME; Professor Calin Belta, ME/SE/ECE; Professor John Baillieul, ME/SE/ECE; Professor Sean Andersson, ME/SE