This paper presents a control design technique which enables approximate reference trajectory tracking for a class of underactuated mechanical systems. The control law comprises two terms. The first involves feedback of the trajectory tracking error in the actuated coordinates. Building on the concept of vibrational control, the second term imposes high-frequency periodic inputs that are modulated by the tracking error in the unactuated coordinates. Under appropriate conditions on the system structure and the commanded trajectory, and with sufficient separation between the time scales of the vibrational forcing and the commanded trajectory, the approach provides convergence in both the actuated and unactuated coordinates. The procedure is first described for a two degree-of-freedom (DOF) system with one input. Generalizing to higher-dimensional, underactuated systems, the approach is then applied to a 4DOF system with two inputs. A final example involves control of a rigid plate that is flapping in a uniform flow, a 3DOF system with one input. More general applications include biomimetic locomotion systems, such as underwater vehicles with articulating fins and flapping wing micro-air vehicles.

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