Energy-Based Tracking Control with Vertical Vibration Suspending for a 6-DOF Bridge Crane

A bridge crane, or 3D crane, is a classic research subject in the control field and is considered a typical example in the group of underactuated systems. This control problem is relatively fascinating and poses significant challenges when the number of control signals is larger than that of state variables to be controlled. Specifically, the task of the control algorithm is not only to ensure the correct tracking with the desired trajectory but also to reduce and quickly extinguish unwanted oscillations. In this study, we develop a controller for the system with a vertical-oscillation quenching capability, a problem that was rarely mentioned in previous studies. Based on the positive definite and the skewed symmetric characteristics of the matrices of the mathematical model, a controller is designed to guarantee the stability of the system according to Lyapunov criteria. Paralleling the convergence of the tracking errors, the horizontal and vertical oscillations of the cargo are suppressed. Simulation results are given at the end of the paper to investigate the effectiveness and feasibility of the entire study.