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We report on the controllability of a novel mobile robot which comprises two driving wheels and an intermediate body carrying the payload. By virtue of quasiholonomy, a concept introduced elsewhere, the robot is underactuated by design. One challenge here is the control of the motion of the intermediate body, which will tend to rotate about the wheel axis as the wheels are actuated. We prove that it is possible to completely control the robot using only the wheel motors, while tracking a desired trajectory, with apparent advantages in terms of cost, weight and efficiency. To do this, we show that every linearization of the robot dynamics model around an equilibrium point verifies the Kalman rank condition for controllability. Moreover, using modern results from nonlinear control theory, we prove that the robot is locally accessible and small-time locally controllable.