Fast and save container cranes as bilevel optimal control problems

Figures & data

Table 1. Parameters for a typical crane system

Length of lane	40–80 m
Height of shelf	5–16 m
Maximum payload	5 t
Maximum horizontal velocity	3 m/s
Maximum vertical velocity	1.5 m/s
Maximum horizontal acceleration	1 m/s^2
Maximum vertical acceleration	1 m/s^2

Figure 1. Model of the crane system. (a) Fixed length of cable rope, (b) Variable length of cable rope.

Figure 2. Single, continuous and discrete time dependent coupling between upper level (axis from 0 to 12s) and lower level (axis from 0 to 4s) problem.

Figure 3. Optimal controls, state vectors and adjoints for main trajectory (thick line) and five alternative trajectories (thin lines) for constrained lower level control. (a) Controls, (b) States, in comparison to single-level solution (dotted line), (c) Adjoints.

Figure 4. Optimal state vectors for main trajectory (thick line) and 10 alternative trajectories (thin lines) for different state constraints in the lower level problem. (a) Constraining acceleration of trolley ; (b) Constraining velocity of payload .

Figure 5. A posteriori calculation of 10 old (thick lines) and 20 new (thin lines) alternative trajectories for different main trajectories. (a) Implicitly constraining acceleration of trolley; (b) Implicitly constraining velocity of payload.

Figure 6. Existence of alternative trajectories for a damped main trajectory.