This work presents the design and optimization of an LQR controller for a two-wheeled mobile robot, modeled as an underactuated Acrobot-type system. The model incorporates realistic physical parameters derived from a CAD design considering steel and aluminum components. Two LQR controllers were developed: one tuned for general balance control with minimized control effort, and another focused on reducing the system’s settling time. A genetic algorithm was employed to automatically tune the weighting matrices \( Q \) and \( R \) for both scenarios. Simulation results confirm that both controllers effectively stabilize the robot from large initial disturbances. The second controller achieves faster stabilization—nearly in half the time—but at the cost of significantly higher torque requirements. These results offer a trade-off between speed of response and actuator effort. The proposed controllers are intended for implementation on a real testbed in future work.