A　self-balancing　unicycle　robot,　which　has　a　wheel　for　balancing　and　moving　in　the　longitudinal　plane　(pitch　angle),　a　vertical　flywheel　for　balancing　in　the　lateral　plane　(roll　angle),　and　a　flywheel　in　the　horizontal　plane(yaw　angle)　for　yaw　control　is　studied　in　this　paper.　The　non-linear　dynamic　equations　of　the　unicycle　robot　on　level　surface　are　analyzed　using　the　Kane　dynamic　formulation,　then　a　linear　model　of　the　robot　is　derived　at　the　equilibrium　point,　and　linear　quadratic　regulators　(LQR)　are　designed　to　control　the　robot　to　change　its　yaw　angle　from　0　degrees　to　45　degrees.　Simulation　results　validated　that　the　unicycle　robot　can　achieve　good　performance　in　static　standing　and　in　dynamic　moving.