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This paper presents a controller-design methodology for a class of underactuated mechanical systems that are affected by parametric uncertainties and external disturbances. The perturbations due to parametric uncertainties are mismatched, whereas those caused by external disturbances are of the matched type. Their effects are canceled by employing a novel strategy that combines time scaling and Lyapunov redesign. The control methodology is applied to a two-wheeled mobile inverted pendulum and a ball-beam system. Along the way, the nonexistence of a smooth control law for point-to-point stabilization of the mobile inverted pendulum is established. Simulation and experimental studies are used to verify the efficacy of the proposed controller-design method.