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The Dual Smart Drive is a specially designed nonlinear actuator intended for use in climbing and walking legged robots. It features a continuously changing transmission ratio and dual properties and is very suitable for situations where the same drive is required to perform two different types of start-stop motions of a mobile link. Then, the associated control problem to this nonlinear actuator is established and a backstepping design strategy adopted to develop Lyapunov-based nonlinear controllers that ensure asymptotic tracking of the desired laws of motion, which have been properly selected using time-optimal control. The approach is extended for bounded control inputs. Both simulation and experimental results are presented to show the effectiveness and feasibility of the proposed nonlinear control methods for the Dual Smart Drive.