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The paper considers a specific class of wheeled mobile robots, namely, mobile wheeled pendulums (MWPs). Robots pertaining to this class are composed of two wheels rotating about a central body. The main feature of MWP pertains to the central body, which can rotate about the wheel axis. As such motion is undesirable, the problem of the stabilization of the central body in an MWP is crucial. The novelty of the work reported here resides in the construction of: 1) the system controllability Lie algebra for the purpose of a rigorous controllability analysis and the computation of the largest feedback-linearizable subsystem; 2) a controller by input-output linearization of the system for achieving the desired steering rate of the robot while stabilizing the central body; 3) a controller based on the internal properties of the system to achieve the desired heading velocity of the robot; and 4) a controller based on the sliding-mode approach for controlling both the position and the orientation of the robot. The entire control structure that permits full control of the robot posture comprises three imbricated loops. Simulations showing good performance of the controlled system are provided. Preliminary tests performed on an experimental platform confirm the validity of the controller.