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This article presents a modified zero vibration (ZV) input shaping technique to address the sensitivity and flexibility limitations of the classic ZV shapers commonly implemented in overhead crane applications. Starting with the classical ZV formulation, new parameters are introduced to optimize the control system performance according to a versatile objective function. The new shaper enhances the design flexibility and operational domain of the shaper, while it inherits the robustness properties and computational efficiency of the ZV scheme. Unlike the original ZV shaper, the proposed shaper allows for the point-to-point maneuver time to be fixed. The sensitivity analysis of the controller confirms that the new shaper effectively reduces the ZV sensitivity to the cable length variations.

Overhead cranes are widely used in manufacturing plants, nuclear facilities, and shipping yards, where heavy payloads need to be transferred [

Researchers have extensively studied the application of various closed-loop control solutions for payload oscillation suppression in crane systems [

Introduced by Singer et al. in 1990 [

In this paper, inspired by the standard ZV formulation, a new shaper is proposed with a flexible objective function that allows the controller to optimize and fix the maneuver time. The existing ZV-based shapers provide limited control over the maneuver time, input acceleration, input jerk, and cable length sensitivity. However, the new shaper can optimize the maneuver time while satisfying the system jerk and sensitivity requirements. This method allows for the input function to be adjusted according to the predefined objective for different operating cases. Simulations are conducted to evaluate the performance of the implemented methodology in mitigating the payload sway and addressing the parameter uncertainties. Moreover, various operating conditions including different cable lengths are investigated. Finally, the effectiveness of the proposed controller is studied in terms of dimensionless metrics, which represent the sensitivity and control jerk.

A standard overhead crane model consists of a massless jib moving in a unidirectional motion along the

Overhead crane model with a massless slider.

The kinetic energy

Assuming a small oscillating angle of the mass

Performing the integral reduces equation (

Differentiating equation (

Substituting the accelerating-phase final-time

Equations (

Rearranging equation (

Equation (

The sensitivity index

Figure

Payload angle versus maneuver time for a specific cable length.

Figure _{4}) for different cable lengths calculated at the standard maneuver time, which is 1 second. The results indicate that the effect of the fourth input constant on the sensitivity index increases as the cable length decreases. For a given range of cable lengths, there are optimum _{4} values for which the sensitivity index is minimized and the system is least sensitive to unexpected cable length variations.

Sensitivity index over different cable lengths and input values (_{4}) with the standard maneuver time.

Figure _{4} and how they change as cable length changes. As the cable length increases, higher input constants are expected due to higher kinetic and potential energies of the system as the pendulum oscillates. It is also apparent that the input constants change in pairs: _{1} with _{4} and _{2} with _{3}. At

Input constant values versus cable lengths for the standard maneuver time.

Figure

Sensitivity index over different cable lengths and a range of maneuver times.

A new metric is introduced to normalize the impact of the maneuver time on the sensitivity index. The normalized sensitivity index

Normalized sensitivity index

Significant differences between input values may be unfavorable due to the existence of a jerk. Therefore, jerk index

Figure _{4}) for different cable length settings. For a given cable length, there is an optimum value of _{4}, for which the jerk index of the system is minimized. Additionally, as expected, increasing the cable length increases the total energy of the system which results in a higher jerk index and optimal _{4} value.

Jerk index versus the fourth input constant (_{4}) for different cable lengths with the standard maneuver time.

Normalized jerk index _{4} for the same cable length as follows:

NJI indicates how the jerk index is affected by adding _{4} to the formulation. Figure _{4} for different cable lengths. Values less than one indicate that the system is improved by adding _{4} compared to the system with three input constants (i.e.,

Normalized jerk index versus the fourth input constant (_{4}) for different cable lengths with the standard maneuver time.

Jerk index variation

Figure _{4} to minimize the jerk index for different cable lengths. Higher values of jerk index variation indicate that the jerks are unfavorably high relative to the input constants.

Input constants and jerk index variation versus cable length for the standard maneuver time.

Figure

Jerk index variation over different maneuver times and cable lengths.

This paper presented a novel input shaping approach to address the limitations of conventional ZV shaping techniques in controlling overhead cranes with demanding design and operational requirements. The presented method results in optimal maneuver time with a minimal jerk and parameter insensitivity. An extensive sensitivity analysis is conducted to quantify the performance of the controller under different feasible operating conditions. The jerk analysis indicates that the optimization process by using the shaper enables it to reduce the input jerk while minimizing maneuver time. The results confirm that the modified ZV shaper provides a powerful yet effective and computationally efficient control solution for overhead cranes with demanding point-to-point maneuver requirements.

No data were used to support this study.

The authors declare that they have no conflicts of interest.