The equations of motion for underactuated crane systems with 2 up to n parallel payloads are established by the Euler–Lagrange principle.

The equidistant boundary condition in the existing works has been relaxed to the unequal boundary condition from the theoretically viewpoint.

We fabricate a new storage function which consists of the system energy storage function, energy shaping terms, and an asymmetric Barrier function. The Barrier function is inserted with the purpose of preventing the trolley from running out the permitted range.

An adaptive controller is designed to achieve the control aim of simultaneous accurate positioning and swing suppression in the presence of rail length constraint and uncertain parameters.