Trajectory tracking control of an underactuated mechanical system is a challenging task when tracking errors of all the system coordinates need to be minimized, while the number of control inputs is limited. This paper addresses the observer-based multivariable control of a class of nonlinear, underactuated Lagrangian systems with application to trajectory tracking and sway control of a 3D overhead gantry crane subject to Coulomb friction. A second-order sliding mode observer is proposed for the estimation of velocities. Based on robust estimates, the sliding function of a second-order sliding mode controller for trajectory tracking and anti-swing control is proposed. The simulation results indicate that the proposed observer-based controller can achieve high performance in following a pre-specified trajectory with minimum tracking errors and swing angle suppression, despite the presence of model uncertainties and disturbances.
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