This dissertation focuses on geometric modeling and control of a quadrotor unmanned aerial vehicle (UAV) in wind fields. Wind deteriorates stability and performance of small aerial vehicles, and even may result in complete failure. To overcome this issue, this dissertation concerns three topics, namely an identification method to understand the effects of wind on the quadrotor UAV, and a mathematical framework to estimate wind in real-time, and a geometric adaptive controller to compensate wind disturbances.

First, a computational approach for system identification for the attitude dynamics of a rigid body is proposed. This is to identify unknown parameters of the system by investigating its dynamic response. System identification or estimation for a rigid body is particularly challenging as they evolve on the compact nonlinear manifold, referred to as the special orthogonal group. Current methods based on local parameterizations or quaternions suffer from inherent singularities or ambiguities associated with them. The proposed method addresses these issues by formulating the system identification problem as an optimization problem on the special orthogonal group. It is solved by a geometric numerical integrator that yields numerical trajectories that are consistent with the geometric structures of Hamiltonian systems on a Lie group. As a result, the proposed method is particularly useful to handle large initial estimation errors that may cause substantial discrepancies between the target attitude trajectories and the initial estimate of them.

Second, the above technique is extended to a computational framework to identify the effects of wind on the dynamics of a quadrotor UAV. Then, using the identified model, the strength and the direction of the wind are estimated without direct measurements from anemometers. The proposed approach is based on the geometric numerical integrator on the special Euclidean group, referred to as a Lie group variational integrator such that singularities or complexities associated with the local coordinates or quaternions are completely avoided. Numerical examples illustrate that the presented methods successfully identify the effects of wind even for the challenging case of large initial estimation errors. Also, numerical results support that the presented method can be implemented for real-time wind estimation during flight.

Finally, a geometric adaptive control scheme for a quadrotor UAV is developed, where the effects of unknown, unstructured disturbances are mitigated by multilayer neural networks that are adjusted online. The stability of the proposed controller is analyzed with Lyapunov stability theory on the special Euclidean group, and it is shown that the tracking errors are uniformly ultimately bounded with an ultimate bound that can be abridged arbitrarily. A mathematical model of wind disturbance on the quadrotor dynamics is presented, and it is shown that the proposed adaptive controller is capable of rejecting the effects of wind disturbances successfully. The efficacy of the proposed approach is illustrated by numerical examples and indoor flight experiments.