In this paper, we present the balancing and tracking position or known as point-to-point control for an underactuated unicycle robot. It can be a solution for mobile robot applications in certain environments where the robot is required to have a small space and narrow margin of clearance when passing obstacles. This typical robot can also be used as a platform to study self-balancing systems, e.g. personal transportation devices for energy-saving, low-noisy, and pollutant-free vehicles. Here, we consider a one-wheel mobile robot that has a body and a single driving wheel with a flywheel attached vertically to the top of the body. The main objective is to control the robot's motion from one point to other points (along a straight line) while maintaining its balance upright. By decoupling the system for lateral and longitudinal axes, the controllers can be designed separately for each decoupled system. For the lateral and longitudinal controllers, we design state feedback LQR control to balance the robot in the longitudinal axis (roll) and lateral axis (pitch). Moreover, the motion control to track points along a straight line is also considered. To verify the effectiveness of the controllers, we conduct several simulations with 0.1 rad (5.73°) of initial position for roll angle while the reference input is given in the form of step functions. It is shown from the simulation results that the proposed controller effectively stabilizes the robot upright and tracks the predefined points of given commands.