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We have developed a single wheel, gyroscopically stabilized robot. This is a novel concept for a mobile robot that provides dynamic stability for rapid locomotion. The robot is a sharp-edged wheel actuated by a spinning flywheel for steering and a drive motor for propulsion. The spinning flywheel acts as a gyroscope to stabilize the robot and it can be steered by tilting. This robot is nonholonomic in nature, underactuated and inherently unstable in the lateral direction. In this paper, we first develop a three-dimensional (3-D) nonlinear dynamic model and investigate the dynamic characteristics of the robot. We conduct simulations and real-time experiments to verify the model. Both simulations and experiments show that the flywheel has a significant stabilizing effect on the robot. Then, we can decouple the longitudinal and lateral motions of the robot by linearization. We propose a linear state feedback to stabilize the robot at different lean angles, so as to control the steering velocity of the robot indirectly, because the robot steers only by leaning itself to a predefined angle. For the task of path following, we design a controller for tracking any desired straight line without falling. In the controller, we first design the linear and steering velocities for driving the robot along the desired straight line by controlling the path curvature. We then apply the linear state feedback to stabilize the robot at the predefined lean angle such that the resulting steering velocity of the robot converges to the given steering velocity. This work is a significant step toward fully autonomous control of such a dynamically stable but statically unstable system.