Crane Bang-bang Sway Reduction Strategy

As shown in fig. 1-1, if a bang-bang command is input through the slewing control lever of a rotary crane, the boom performs a trapezoidal angular velocity variation. At the beginning of the “on” command, the boom undergoes a constant acceleration stage. After the boom reaches the target velocity, the acceleration ends and the boom enters the constant velocity stage. Then, when the “off” command is input, the boom decelerates.

This method aims to utilize the acceleration and deceleration stages to reduce the energy of the pendulum system.

The sway reduction strategy aims to maximize the energy reduced during the acceleration and deceleration stages of the bang-bang control. As shown in fig. 3-2(a), if an acceleration is applied to the upper part of the pendulum system, the pendulum will accelerate in the opposite direction with the same value. This can be observed from the upper part of the pendulum system. According to the work-energy theorem, the work that

the acceleration did to the pendulum equals the variation of the total energy in the pendulum system. As shown in fig. 3-2(b), the acceleration is applied to the pendulum in the opposite direction that the pendulum swings during its travel from point 𝑝₊ to 𝑝_,. In such condition, the acceleration is doing negative work to the pendulum. In other words, the acceleration is reducing the energy of the sway motion. Therefore, the strategy is designed to maximize the negative work done to the pendulum system in the acceleration and deceleration stages.

Fig. 3-2. A pendulum system and the applied acceleration (a) and a swaying pendulum system and the applied acceleration (b).

We now consider the example situation shown in fig. 3-2(b). A pendulum with mass 𝑚 and length 𝑙 is swaying from point 𝑝₊ to 𝑝_,. During the entire travel, an acceleration 𝑎 in the opposite direction is applied on the pendulum. The pendulum traveled from 𝜃₊ to 𝜃_,. The angle denoted here is counterclockwise and starts from the vertical line. Hence, in this case, 𝜃₊ is positive, and 𝜃_, is negative. This suggests that the tangential component of the acceleration 𝑎 can be denoted as 𝑎 cos 𝜃 no matter 𝜃

is positive or negative. Moreover, when 𝜃 is small, cos 𝜃 ≈ 1. This suggests that 𝑎 cos 𝜃 = 𝑎. Therefore, the negative work 𝑊 done to the pendulum by the acceleration can be calculated as follow: negative work is done to the pendulum system. Hence, by maximizing the angle traveled by the pendulum during the acceleration and deceleration stages, the energy reduced can be maximized.

We now consider a pendulum that is swaying from right to left as shown in fig. 3-3.

When the acceleration 𝑎 starts to be applied on the pendulum, it has an initial angular velocity 𝜔_C. The left part of the figure illustrates the applied acceleration 𝑎 and its tangential component 𝑎 cos 𝜃 while the right part illustrates the applied gravity acceleration 𝑔 and its tangential component 𝑔 sin 𝜃. If the acceleration 𝑎 is applied to the pendulum for 𝑡₍ second, the angle traveled during the acceleration 𝜃₍ = (𝜃₊− 𝜃_,) by the pendulum can be yielded as:

𝜃₍ = 9 G𝜔_C −𝑎 cos 𝜃 𝑡

𝜔_C determines the value of traveled angle 𝜃₍ alone. Therefore, a larger 𝜔_C contributes to a larger 𝜃₍.

Fig. 3-3. Acceleration and gravity acceleration applied on the pendulum.

According to the law of conservation of energy, the maximum angular velocity of the pendulum occurs at the equilibrium point while the pendulum is at the lowest height.

This is because the potential energy of the pendulum is completely transformed into kinetic energy. Therefore, if an acceleration starts to be applied to a swaying pendulum in the opposite direction at the equilibrium point, the angle that the pendulum traveled in the original direction can be maximized. In this case, the initial angular velocity 𝜔_C in Eq.

(4) is the velocity of the payload at the equilibrium point. The volume of 𝜔_C can be calculated using the law of conservation of energy. Assuming that the potential energy of the pendulum system at the highest point completely transforms into kinetic energy at the equilibrium point, the relationship between the potential energy 𝑈 and the kinetic energy 𝐸 can be written as:

𝑈 = 𝑚𝑔𝑙(1 − cos 𝜃_O) = 𝐸 = 1

The bang-bang sway reduction strategy is to accelerate the boom when the payload first passes the equilibrium point. Then, it decelerates the boom when the payload sways back and passes the equilibrium point for the second time. The direction that the boom rotates must be the same direction as the payload first passes the equilibrium point. The strategy is illustrated in fig. 3-4. Considering a payload initially sways from right to left, as shown in stage (1) of the figure. When the payload first passes the equilibrium point as shown in stage (2) of the figure, the boom tip starts to accelerate in the swaying direction. At the same moment, the payload has an initial angular velocity 𝜔_C in the original swaying direction and an acceleration 𝑎 applied to it in the opposite direction.

After the acceleration duration 𝑡₍, the boom tip reaches the target velocity 𝑣_)(#R&) as shown in stage (3) of the figure. At this moment, the acceleration of the boom ends.

Consequently, the acceleration applied on the payload ends and a remaining angular velocity 𝜔′ may be observed. Then, when the payload sways back to the equilibrium point in the opposite direction, the boom starts to decelerate from 𝑣_)(#R&). Therefore, an acceleration 𝑎 is applied to the payload in the original direction. After 𝑡₍, the boom stops and the sway reduction ends.

Fig. 3-4. Illustration of the bang-bang sway reduction strategy.

Through the proposed strategy, we can maximize the energy reduced through the bang-bang control. From Eq. (1), (4) and (7), we can express the energy 𝑊 reduced by the control as follow:

𝑊(𝑚, 𝑎, 𝑙, 𝑡₍) = 𝑚𝑎𝑙 U𝑡₍P2𝑔(1 − cos 𝜃_O)

𝑙 −𝑎𝑡₍^,

2𝑙 V. (8)

In Eq. (8), 𝑚, 𝑎, 𝑙 and 𝑡₍ are the variables of the crane system.

To achieve the control of the strategy, the sway reduction method guides the operator to input the “on” command in the corresponding direction when the payload first passes the equilibrium point. Then the method guides the operator to input the “off” command when the payload passes the equilibrium point again.