There is a great demand for vision-based robotics solutions that can operate using Global Positioning Systems (GPS), but are also robust against GPS signal loss and gyroscope failure. This paper investigates the estimation and tracking control in application to a Vertical Take-Off and Landing (VTOL) Unmanned Aerial Vehicle (UAV) in six degrees of freedom (6 DoF). A full state observer for the estimation of VTOL-UAV motion parameters (attitude, angular velocity, position, and linear velocity) is proposed on the Lie Group of $\mathbb{SE}_{2}\left(3\right)\times\mathbb{R}^{3}$ $=\mathbb{SO}\left(3\right)\times\mathbb{R}^{9}$ with almost globally exponentially stable closed loop error signals. Thereafter, a full state observer-based controller for the VTOL-UAV motion parameters is proposed on the Lie Group with a guaranteed almost global exponential stability. The proposed approach produces good results without the need for angular and linear velocity measurements (without a gyroscope and GPS signals) utilizing only a set of known landmarks obtained by a vision-aided unit (monocular or stereo camera). The equivalent quaternion representation on $\mathbb{S}^{3}\times\mathbb{R}^{9}$ is provided in the Appendix. The observer-based controller is presented in a continuous form while its discrete version is tested using a VTOL-UAV simulation that incorporates large initial error and uncertain measurements. The proposed observer is additionally tested experimentally on a real-world UAV flight dataset. Keywords: Unmanned aerial vehicle, nonlinear filter algorithm, autonomous navigation, tracking control, feature measurement, observer-based controller, localization, exponential stability, asymptotic stability, inertial measurement unit (IMU), Global Positioning Systems (GPS), vision aided inertial navigation system.