Traditional quadcopters possess only four controllable inputs and are inherently underactuated, limiting their ability to interact with an environment or manipulate a payload. This work discusses a quadcopter architecture incorporating twisting and tilting joints to fully decouple the position and attitude dynamics, permitting full six-degree-of-freedom maneuvering. A dynamic model is presented which includes both the gyroscopic effects of the propellers and the configuration-dependent inertia of the articulated components. A control framework is developed first for a quarter copter model, then generalized to the full copter on a spherical joint, and finally to the full copter with unconstrained base motion on the special Euclidean group SE(3). Simulations are conducted on a Simscape multibody model, and the results are presented to demonstrate the capabilities of the craft.