The attitude (or orientation) of an object is often crucial in its ability to perform a task, whether the task is driving a car, flying an aircraft, or focusing 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 due to unexpected large motions. More recent works have achieved global stability and avoided these system failures by working directly on the configuration manifold, but these approaches 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 efficient for real time implementation on embedded computers and the tuning parameters have geometric interpretations. These properties make the controllers user friendly and 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 first 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 and to automatically tune the controller gains by combining Riemannian geometry, contraction theory, and offline optimization. The third contribution is the development of a globally, exponentially stable attitude controller. This controller overcomes the topological limitation that prevents continuous, time-invariant controllers from achieving global stability 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.

On the tangent bundle of a Riemannian manifold the most natural choice of metric tensor field is the Sasaki metric. This immediately brings up the question of infinitesimal symmetries associated with the inherent geometry of the tangent bundle arising from the Sasaki metric. The elucidation of the form and the classification of the Killing vector fields have already been effected by the Japanese school of Riemannian geometry in the sixties. In this thesis we shall take up the conformal vector fields of the Sasaki metric with the help of relatively advanced techniques.