A　self-balancing　unicycle　robot,　which　has　a　wheel　for　balancing　and　movement　in　the　longitudinal　plane　(pitch　angle)　as　well　as　a　flywheel　for　balancing　in　the　lateral　plane　(roll　angle),　is　studied　in　this　paper.　The　non-linear　dynamic　equations　of　the　unicycle　robot　on　a　slope　are　analyzed　using　the　Lagrangian　dynamic　formulation,　then　a　linear　model　of　the　robot　is　derived　at　the　equilibrium　point,　and　3　linear　quadratic　regulators　(LQR)　are　designed　to　control　the　robot　on　slopes　with　the　angle　of　inclination　varying　from　-11　degrees　to　11　degrees.　Simulation　results　validated　that　the　unicycle　robot　can　achieve　good　performance　both　on　level　plane　and　on　slope.