Self-propelled carriage on wheels comprises: one or two pairs of supporting elements for the wheels of either or both of the axles of the motor vehicle, these elements being movable symmetrically and perpendicularly with respect to the longitudinal axis of the carriage and designed to center, immobilize and lift from beneath the wheels; elements for limiting the mass to be transferred; elements for sensing, continuously during the transfer, the translational position of the carriage; elements for sensing the presence of the motor vehicle on the carriage and measuring the front and rear lengths of the motor vehicle relative to its front axle; and elements for sensing excessive displacement of the longitudinal axis of the motor vehicle relative to the longitudinal axis of the carriage when the motor vehicle is being positioned by the user in the entrance bay.