We consider an overhead crane system with reduced actuator dynamics. The development of robust control laws that do not result in strong swing angle oscillations is particularly important for the safe movement of payloads. For automatic control of this underactuated system (such systems have fewer controls than regulated variables) under external and parametric disturbances, as well as incomplete measurements, we use smooth and bounded sigmoid functions in the reference action, control law, and dynamic differentiator. The reference trajectory is developed by considering the design constraints on the velocity and acceleration of the trolley. This is the sum of the sigmoid function and the integral of swing angle. Such a trajectory ensures smooth transfer of the payload over a given time while damping its oscillations. To track the trajectory, a discontinuous control (control with sliding mode organization) in the actuator is developed that provides sigmoid local feedbacks in the mechanical subsystem. The sigmoid function is an S-shaped function that is often used as an activation function in neural networks. These local feedbacks allow us to reject unmatched disturbances and satisfy the design constraints on the state variables. In the absence of velocity sensors, we have proposed a first-order dynamic differentiator with a sigmoid-based correction that restores the derivative of the tracking error with any given accuracy based on the measurements of this error. We present the results of numerical simulations that demonstrate the robustness of a closed-loop system. The main contribution and innovation of this work are that, in contrast with the known discontinuous and linear laws, we propose a smooth and bounded law for the control force that can be realized by an actuator. From a practical perspective, the proposed solution allows the use of lower power actuators and increases their lifetime because of the smooth controls. The proposed approaches to trajectory planning and observation problems ensure high-quality performance and are simpler in computational implementation than other efficient solutions.