Optimal control of an overhead crane with a constraint on load deviation during motion and complete suppression of load oscillations after stopping at a prescribed point

Abstract

Abstract. This work presents the results of a study of the motion of a two mass “trolley–load” model representative of most overhead cranes. During trolley motion, load oscillations naturally arise and must be eliminated after the trolley stops at a prescribed position. The primary objective, however, is to minimize the duration of the handling cycle. Therefore, the optimality criterion for the trolley control law is maximum speed of operation. Such a criterion will enable, when these control laws are implemented on a real crane using microprocessor control, attainment of maximal throughput—an important requirement for handling cranes operating under high cargo flow.

The study shows that to achieve the objective of maximal throughput, maximum acceleration or braking must be applied during transient phases. For long hoist ropes this may produce unacceptable load oscillation amplitudes during motion. Consequently, the problem formulation considered here includes a constraint on the maximum allowable load deviation while preserving all other criteria and constraints. A principal distinction of the proposed approach from many known solutions is the deliberate omission of eliminating load oscillations immediately after the trolley acceleration phase. This omission is intended to produce a consistent phase state prior to the onset of braking, independent of the distance traveled by the trolley. Omitting oscillation suppression during acceleration shortens the handling cycle because time is not spent damping load oscillations during acceleration.

This approach complicates the problem substantially because the optimal control must be found on the terminal segment of the cycle (trolley stop), when the initial phase coordinates vary and depend on the travel distance. Load oscillations are absent only after the trolley comes to rest. Use of a simple two mass crane model enables analytical investigation of trolley and load motion. Employing more complex numerical models forces reliance on numerical methods, which significantly reduces analytical insight and offers no advantage here, since the studied phenomena are low frequency load oscillations that are not strongly affected by other factors (for example, drive dynamics).

It is important to choose the control variable rationally. The control input may be trolley acceleration or the driving force. Selecting the driving force as the control variable is more rational because it allows kinetic energy of the load to be utilized during braking or acceleration on certain cycle segments and prevents drive overload.

The analysis demonstrates that, under the stated constraints, the trolley acceleration process proceeds as follows: with two short control actions (“accelerate–brake”) the load is brought to the maximum allowable deflection behind the trolley and then moves together with the trolley in the deflected state under accelerated motion until the trolley reaches steady speed. During steady motion the load oscillates freely. On the trolley stopping segment the load is brought to the maximum allowable deflection ahead of the trolley; braking of the load and trolley as a single unit then begins in the deflected state down to a prescribed value, followed by the final trolley stop. The acceleration and braking segments are similar in structure, but during acceleration the load is held deflected behind the trolley, whereas during braking it is deflected ahead.