This study focuses on the control of a self-balancing unicycle robot equipped with a roll stabilization mechanism and a single drive wheel responsible for both maintaining upright posture and tracking a predefined path. To achieve stable and responsive behavior, Linear Quadratic Regulators (LQRs) are designed and implemented for roll, pitch, and trajectory tracking control loops. While conventional LQR controllers are generally effective in attenuating external disturbances such as road inclines, they may not fully compensate for the dynamic variations induced by significant slope changes. To address this limitation, a gain-scheduled LQR strategy is proposed, in which controller gains are adaptively adjusted based on varying road inclinations. The road slope alters the dynamic behavior of the unicycle system and introduces external forces that can degrade performance if not adequately compensated. The proposed gain scheduling approach enhances the system's adaptability and robustness, ensuring more accurate path tracking and upright stability under non-uniform terrain conditions. A comparative analysis is conducted between the gain-scheduled LQR and a standard fixed-gain LQR design. Simulation results demonstrate the effectiveness of the proposed method in improving the robot’s performance across varying slope conditions, validating its potential for real-world application in unstructured or sloped environments.