Optimization of skew angle of the spreader in sea-to-shore gantry cranes

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Ng'era This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-9016031/v1 This work is licensed under a CC BY 4.0 License Status: Posted Version 1 posted You are reading this latest preprint version Abstract Ports in the world are rated based on the number of containers they handle per year. With this reference, every port is desired to have a huge record of container handling (TEU). Sea-to-shore gantry cranes are critical assets in modern container terminals where productivity, safety and reliability directly affect port performance. One of the persistent operational challenges during container handling is the skew of the spreader relative to the container and ship cell guides, caused by either wind disturbances, trolley motions, wire rope length variations and symmetrical loading. This manuscript presents a vision-based system of spreader skew angles using cameras and intelligent algorithms. The proposed approach integrates malti-camera perception, real-time image processing and control of algorithms to estimate the skew angle accuratelyand apply corrective actions automatically. The system design emphasizes robustiness, real-time performance and compatibility with STS crane control architecture. Simulation and conceptual validation demonstrate that the proposed system can significantly reduce skewing during hoisting ad positioning, contributing to safer and more efficient container handling operations. Electrical Engineering Tilt skew Spreader convolutional neural network sheave. Cell guide corner cast port quay gantry crane vision-based camera-based measurements PID Figures Figure 1 Figure 2 Figure 3 Figure 4 Introduction This paper develops a vision-based monitoring system to detect skew angles of spreaders in sea-to-shore gantry cranes with high accuracy, integrating an adaptive PID controller for real-time spreader adjustments. Validating the system through MATLAB simulations and field tests in an operational port environment. The study aimed to optimize spreader alignment to improve crane efficiency and to reduce incidents while being able to cater to the expansion of the global maritime trade. It provides insights into several recent strides in automation of these essential processes, especially maritime gantry crane operations, from sea to shore. This work is a mathematical model for the three-dimensional movement of the gantry crane function along the quay. Together with cameras, counters, sensors, and encoders, this system allows for monitoring and controlling the crane's movement in real time. During the lifting process, the spreader suspended for operation is equipped with 4 AI-driven cameras which collect position data by identifying corner castings in the container. The control algorithm activates four hydraulic cylinders based on the acquired data to steer the sheaves as directed. The simulation results very clearly show the capabilities of the machine vision-based tilt, list and skew correction system in obtaining the precision alignment during crane operation. In his analysis of twist-locks in fully autonomous gantry cranes, Marcus Brandenburg [2021] highlighted that twist locks are a crucial component for securing container corner fittings in marine cargo handling facilities. In the discussion on the use of sensors to reduce down time tfor a spreader to couple a conhtainer, journal article by Parra & Tannuri, [ 2012 ], they discussed the use of sensors to reduce the time required for coupling containers, a process facilitated by a device known as the spreader. The authors, Ngo & Hong, [ 2014 ], in their research introduced sway control to precisely position the spreader by limiting its swing for the purpose of accuracy. They proposed two methods: the first involved a sensor-driven control logic and command signal modification, which enabled accurate container alignment and suppression of pendular motion in sea-to-shore cranes through time-critical trajectory control with closed-loop adjustment enabling faster travel. Their second method used linear control to quickly suppress any residual sway. In their proposed technique for reducing load oscillation through motor torque regulation, [Anatolii Shestaka and Lubov Melnikova 2018] employed modern variable frequency drives. The motor torque force is influenced by the proportion of the trolley's weight to the cargo's weight, as well as the longitudinal measurement of the rope. [Jaecheul Lee 2019 ] utilized laser measurement (LMS) and developed fast algorithms to recognize cargo compartments of the ship. Additionally, the universe’s largest hoisting machine manufacturer has created an autonomous container unit position control configuration that centered around laser-assisted imaging technology. This system comes with some disadvantages, such as being time-consuming when scanning multiple planes because it should involve servomechanisms. Convolutional Neural Networks (CNN) were used for corner casting because they can learn complex feature representations, thus becoming highly effective as well reliable for pattern analysis in visual data. In Three-Dimensional Perception System for Reliable Autonomous Hoisting Machine Management, an advanced robotic system for container handling was introduced [Manen, 2024 ]. The 3DWI system integrates a sophisticated and intelligent sensing suite with machine learning algorithm, along with an architecture that connects distributed control between the crane and the shore-based operations center. In Applying Deep Learning to Industrial Systems: Coastal Crane Casting Recognition as a Case Study. Maqsood et al., [ 2021 ], discussed how advanced AI approaches with deep learning models are mimicking human cognitive behavior that tackle sophisticated engineering challenges that may be otherwise challenging in addressing them. The core of AI involves solving complex engineering problems by identifying the correct relationships between input and output data, even when the underlying connections are not immediately clear. AI can model data, enabling future predictions and decision-making. This approach utilizes artificial neural networks (ANN) to achieve these capabilities [Manen, 2024 ] In their 2004 study, 'Anti-Sway Control of Container Cranes, Inclinometer, Observer, and State Feedback. Kim et al., [ 2004 ] explored an anti-sway control system that utilizes an inclinometer as a sway sensor for container cranes. Compared to a vision system, the inclinometer is much more affordable, durable, and easier to maintain while offering nearly identical performance. Some observers project the rate of rotation of the load and the velocity of the trolley in the configuration setup. Serkan et al. [2021] Crane automation and mechanical damping methods. In the implementation of the control of the sway technique relying on motor torque control and contemporary variable frequency drives, its key equation relates its pulling tension to its angle of oscillation of the load. This equation forms the foundation for the rotational force of the motor, which was designed to eliminate load oscillation once acceleration and as well retardation. By accounting for the damping of oscillations within the planes, one can establish the foundational concepts of a technical method to address the oscillation mitigation challenge based on these parameters. In identifying impacts to shipping containers with vertical cell guides within container ships during handling operations." Jakovlev et al., [ 2022 ] explored the detection of impacts between shipping containers and vertical guide rails inside container ships during cargo management. Mechanical systems used to place twist locks into corner casts to control tilting, listing, and skewing through flip arms and hydraulic pumps, and their connections exist, but they are so costly to maintain because of the nature of hydraulics that it is difficult to minimize leakages, and they are too bulky. Though effective, they are not favorable. Laser vision systems have also been implemented, but they target the ropes since they can only scan one plane and not the payload, which is of our interest. Though effective, it is not favorable, too. Thus, an inclination sensor that is secured in circuit design, robust, and mimics human behavior in observing load tilting, listing, and skewing is essential. Intelligent crane automation systems create a potential for growth towards automating container port processes to realize better throughput of cargo. Ship-to-shore container handling equipment commonly utilized at quay side terminals, particularly at storage terminals, for transportation. It consists of three main components. The first is the gantry, which provides structural support for all equipment and moves in one direction while the second component is the trolley, which sits on the boom rails moving along the boom from the back reach of the crane to the boom tip. [Li & Li, 2023 ]. Two methods used in adjustment of the skew orientation are particularly noteworthy: moving the pulley alongside the trolley’s movement, introducing extra twisting force to dampen disturbance in the skew motion, and independently controlling the displacement of the wire rope that holds the spreader. Adjustments of angular deviation involve a three-dimensional model o that includes oscillatory and angular deviations. [Na & Hipertensiva, n.d.]. In this system, a DC motor actuates the skew drive, allowing one side of the hoist mechanism (two cables) to shift forward and backward onto the trolley. This motion generates a rotational torque on the lifting device and the cargo unit. The authors also discussed an easier modulator engineered to balance the tilt, list, and angular motions of the spreader. A fuzzy-based control system was developed that managed the tilt, inclination and lateral offset of the lifting attachment [Ngo & Hong, 2014 ]. Instead of using the main hoisting cables, they employed four secondary wire ropes for motion control of tilt, list and lateral offset. They present an analysis of the 3D movement behavior of a cargo unit system of transportation, demonstrating effectiveness of the regulator for placing the cargo unit, despite gale and unpredictable factors. The study introduced four wire rope mechanisms, comprising of three-dimensional axes (translation) and angular displacement, as applied to conventional cargo unit gantry cranes. A primary lift device is used by this crane in controlling the lifting attachment vertical motion, with a set of four hydraulic cylinders connected to the separate hoists to exert force on each wire rope while the second work involved designing an easier modulator to balance angular motion whilst keeping list and trim angles minimal, ensuring that skew angle isn’t altered by unintended movements. Using Neural Network-Based Predictive Control Approaches to Address the Shipping Container Sway Issue in Quay Cranes, the author discusses the use of visual feedback to assist in positioning containers beneath the quay crane. This is managed by controlling abrupt changes in the container's rapid movement during its transport, alongside the crane’s operation, which is crucial to prevent high-amplitude accelerations. The use of IoT sensors to improve detection capabilities in shipping container impact scenarios with vertical guard rails inside container ships during handling operations was explored by Sergej Jakovlev, et al [ 2022 ]". Their system integrates internal electronics devices, cordless transmission, and information storage frameworks all controlled by a processing unit that performs data interpretation from sensors through dedicated software logic. This detection methodology follows a step-by-step process. The system deployer is responsible for deploying the IoT system. At the same time, the administrator controls the information flow between the programs as well as the local storage system. System design 3.1 Overall System Architecture The crane spreader descends from an initial height of 30 meters to a target container height of 2 meters at a maximum descent speed of 1.6 m/s. During descent: At approximately 7 meters, a machine vision system (camera + image processing algorithm) is activated. The cameras capture 3D images of the spreader in real time, reconstructing its position and detecting any tilt, list or skew, deviation. The captured images are sent to a central processing computer which analyzes them to determine the extent of deviation. Upon detecting a LIST deviation, a control command is issued to adjust the spreader's orientation. In the MATLAB Simulink model, the Machine Vision operation is represented by an idealized detection process: A step input deviation of 3° is introduced at exactly 6.5 meters height. This deviation is fed into the PID controller for correction. The PID controller processes the deviation and commands the hydraulic actuator system (modeled through dynamic response blocks) to correct the spreader's orientation before it reaches the container. 3.2 Simulink Modeling, Design, and Simulation of the System 3.2.1 MATLAB Simulink System Architecture The overall simulation model is implemented in MATLAB Simulink 2016 with three central controller modules Tilt, List and Skew on all the critical levels of the freedom for Sea to Shore (STS) crane alignment. A system of controls loops that operate in conjunction with a central Integrator Logic that allows for accurate, smooth, fault-resistant positioning of the spreader bar of the crane. MATLAB Simulink System Architecture Overview 3.2.1.1 Skew Control Subsystem Here, the skew control subsystem is responsible for controlling the angle of the spreader to achieve the desired side-to-side angle of alignment across the beam axis of the crane. This facilitates safe and uninterrupted operations with the container when using a crane. An early warning and adjustments command system is present as the subsystem further incorporates a Machine Vision System to improve accuracy and reduce errors during the descent. Its main characteristics are the following: 3.2.1.2 Machine Vision Activation and Monitoring: When the spreader descends 30 m and is about 7 meters above the container, machine vision detectors are powered by laser distance sensors. These cameras take a 3D view of the spreader, which is analyzed in real-time via an image processing algorithm to identify any real roll, tilt and skew. But the spreader is assumed at 0° to be perfect during simulation. Now, the machine vision input is inactive. 3.2.1.3 Reference/Setpoint Application : You apply a user-defined reference setpoint (such as 2°) to the control system. It requires the spreader to change between 0° − 2° to set correctly, compensating for the expected misalignments of the container or external effects. 3.2.1.4 PID-Based Correction Mechanism: A PID controller is used for error (difference between current spreader angle and setpoint). The controller computes the corrections to rotate spreader towards the skew angle it demands for proper descent. 3.2.1.5 System Dynamics Modeling: The general real dynamic behavior of the spreader-crane system is modelled with a second-order transfer function. That considers realistic phenomena like inertia, oscillations and response times. This process is expressed in the figure below The skew angle \({\theta}_{s}\left(t\right)\) represents the misalignment, and we model the plant dynamics for skew correction as a second-order linear system derived from system identification or physical modeling. The transfer function of the plant is given by: 3.2.2 System Parameters This model includes: A second-order plant model representing the spreader skew dynamics, A PID controller with specific gains, A transport delay modeling the latency introduced by image acquisition and processing. 3.2.3 PID Controller Parameters: Proportional Gain \({K}_{p}\) = 1.5 Integral Gain \({K}_{i}\) = 1.7 Derivative Gain \({K}_{d}\) = 0.1 A classical PID controller is used for feedback correction. The controller transfer function is: $${G}_{c}\left(s\right)={K}_{P}+{K}_{i}+{K}_{i}+{K}_{d}s=1.5+\frac{1.7}{s}+0.1s$$ 3.44 $${G}_{c}\left(s\right)=\frac{{0.1s}^{2}+1.5s+1.7}{s}$$ 3.45 This controller improves tracking and disturbance rejection. Derivative action anticipates errors (beneficial for skew), while integral action ensures zero steady-state error. 3.3 Plant Transfer Function: Let the plant transfer function \({G}_{P}\left(s\right)\) be: \({G}_{P}\left(s\right)\) = \(\frac{4.5}{{s}^{2}+5.5s+4.2}\) (3.46) This is a second-order system with poles at: $$S=\frac{-5.5\pm\sqrt{5\cdot{5}^{2}-4}}{2}$$ 3.47 3.3.1 Transport Delay: Let the transport delay introduced by the Machine Vision system be denoted by \({T}_{d}\) , typically in the range of 100–200 ms. Given that: \({T}_{d}\) = 0.15 seconds This delay is modeled as an exponential in the Laplace domain: \({G}_{d}\left(s\right)\) = \({e}^{-T{d}^{s}}\) (3.48) $${e}^{-T{d}^{s}}\approx\frac{1-\frac{{T}_{d}}{2}s}{1+\frac{{T}_{d}}{2}s}$$ 3.49 Substituting \({T}_{d}\) = 0.15 seconds Pade approximation \({\varvec{e}}^{-0.15\varvec{s}}\approx\) \(\frac{1-0.075s}{1+0.75s}\) (3.50) 3.3.2 PID Controller Transfer Function The standard form of a PID controller in the Laplace domain is: $${G}_{C}\left(s\right)={K}_{P}+\frac{{K}_{I}}{S}+{K}_{D}s$$ 3.51 Substituting values: $${G}_{C}\left(s\right)=1.5+\frac{1.7}{S}+0.1s$$ 3.51 = \(\frac{{0.1S}^{2}+1\cdot5S+1\cdot7}{S}\) (3.52) 3.3.3 Open-Loop Transfer Function The open-loop transfer function \({G}_{OL}\left(s\right)\) of the system including controller, plant, and delay: \({G}_{OL}\left(s\right)\) = \({G}_{C}\left(s\right)\) . \({G}_{P}\left(s\right)\) . \({G}_{d}\left(s\right)\) (3.53) Substituting the values: \({G}_{OL}\left(s\right)\) = \(\frac{{0.1S}^{2}+1\cdot5s+1\cdot7}{S}\) \(.\) \(\frac{4.5}{{s}^{2}+5.5s+4.2}.{\text{e}}^{-0.15s}\) (3.54) \({G}_{OL}\left(s\right)\) = \(\frac{{0.1S}^{2}+1\cdot5s+1\cdot7}{S}\) \(.\) \(\frac{4.5}{{s}^{2}+5.5s+4.2}.\frac{1-0.075s}{1+0.75s}\) (3.55) Simplifying \({G}_{OL}\left(s\right)=\left(\frac{0.1{s}^{2}+1\cdot5s+1\cdot7(1-0.075s)}{s({s}^{2}+5.5s+4.2)(1+0.075s)}\right)\) (3.56) 3.3.4 Closed-Loop Transfer Function Let \(G\left(s\right)\) = \({G}_{C}\left(s\right)\) . \({G}_{P}\left(s\right)\) . \({G}_{d}\left(s\right)\) , (3.57) Then the closed-loop system transfer function is: $${T}_{\left(s\right)}=\frac{{G}_{\left(s\right)}}{1+{G}_{\left(s\right)}}=\frac{{G}_{OL}\left(s\right)}{1+{G}_{OL}\left(s\right)}$$ 3.58 However, due to the presence of a transport delay, the system becomes a non-rational transfer function (non-polynomial due to \({e}^{-sT}\) . This implies: Classical Laplace-domain analysis tools (like root locus or frequency response) are approximate. Time-domain simulations or Pade approximation is used for practical analysis. Now the full rationalized open-loop transfer function becomes: $${G}_{OL}\left(s\right)=\left(\frac{0.1{s}^{2}+1\cdot5s+1\cdot7}{s}\right)\cdot\left(\frac{4.5}{{s}^{2}+5.5s+4.2}\right)\cdot\left(\frac{1-0.075s}{1+0.075s}\right)$$ 3.59 3.3.5 Final Transfer Function Summary This transfer function can now be used for: Stability analysis (e.g. Routh-Hurwitz or Nyquist), Time-domain simulations (step response, rise time, overshoot), Frequency response analysis (Bode, gain/phase margin). The figure above expresses the reference point of 2 degrees and how the plant and scope 1 operate. 3.3.6 Stability and Performance Considerations The transport delay degrades phase margin and can destabilize the system unless compensated. The relatively low derivative gain \({K}_{d}\) = 0.1 minimizes noise amplification but reduces damping. While the steady-state error is mitigated with integral action, oscillations can occur when delay is present. Controller tuning could be further improved to provide robustness via: Ziegler-Nichols with delay compensation Smith Predictor design for delay compensation LQR or pole placement (for state-space equivalents) 3.3.7 State-Space Representation We'll expressing the system in the form: $$\dot{x}\left(t\right)=Ax\left(t\right)t+u\left(t\right)$$ 3.60 $$y\left(t\right)=Cx\left(t\right)t+Du\left(t\right)$$ 3.61 Plant model: Let’s define the state variables as: \({x}_{1}=y\) (output), \({x}_{2}=\dot{y}\) (3.62) $${\dot{x}}_{1}={x}_{2}$$ $${\dot{x}}_{2}=-4.2{x}_{1}-5.5{x}_{2}+4.5u$$ 3.63 Matrix form: $${\dot{x}}_{2}=\left[\begin{array}{cc}0&1\\-4.2&-5.5\end{array}\right]{x}_{2}+\left[\begin{array}{c}0\\4.5\end{array}\right]u$$ 3.64 $$y=\left[10\right]{x}_{1}$$ 3.3.8 Augmented System (Plant + PID): Adding integrator state: $${x}_{3}=\int\text{e}\left(t\right)dt$$ 3.65 PID control law: \(u\left(t\right)={K}_{Pe}\left(t\right)+{K}_{1} \int e \left(t\right)dt+{K}_{D}\frac{de\left(t\right)}{dt}\) (3.66) $$u\left(t\right)=1.5\left(t\right)+1.7{x}_{3}-0.1\dot{y}\left(t\right)=1.5\left(r-{x}_{1}\right)+1.7$$ 3.67 3.3.9 Augmented dynamics: $${\dot{x}}_{1}={x}_{2}$$ $${\dot{x}}_{2}=-4.2{x}_{1}-5.5{x}_{2}+4.5\left[1\cdot5\left(r-{x}_{1}\right)+1.7{x}_{3}-0.1{x}_{2}\right]$$ 3.68 $${\dot{x}}_{3}=r-{x}_{1}$$ 3.69 Simplify \({\dot{x}}_{2}\) \({\dot{x}}_{2}=-11.55{x}_{1}-5.95{x}_{2}+7.65{x}_{3}+6.75r\) (3.70) Final State-Space Matrices: \(A=\left[\begin{array}{ccc}0&1&0\\-11.55&-5.95&7.65\\-1&0&0\end{array}\right]\) , \(B=\left[\begin{array}{c}0\\6.75\\1\end{array}\right]\) , \(C=\left[100\right]\) (3.71) 3.4 Skew Simulation in Simulink In a modern automated port terminal, ship-to-shore (STS) gantry cranes play a vital role in the effective handling of shipments. Among the most important structural parameters to be constantly monitored are tilt, list, and skew. This part deals with skew, the lateral misalignment of the crane trolley or spreader beam along its travel axis. Skewing results in uneven container pickup, increased mechanical strain, and safety hazards. To tackle these challenges and cater to increased demand for automation and accuracy, this project proposes a vision-based skew correction system. Using continuous video feeds, the system observes skew angles by detecting important structural markers on the spreader, which helps compute the horizontal deviation of the spreader beam from the ideal alignment path. We combined tilt (θ), list (φ), and skew (ψ) by fusing IMU, rope-angle, and laser measurements into a 3-degree of freedom (DOF) attitude state, which we mapped onto physical outputs: corner height spread Δh_max = [φ W + θ L], lateral/longitudinal shears. A composite index multi-mode method (MMM) normalizes these by allowable limits to drive operator cues, interlocks, and closed-loop tilt, list, and skew control; significantly reducing MMM lowers landing impacts and cycle variance. There is an integrator; the integrator is the “brain” that compresses heterogeneous angular measurements (tilt, list, skew) into one coherent system of actions to eliminate undesired misalignment. This unified output leads to safe container landings with shortened cycle time and decreased structural strain. Results and Discussion. The experimental investigation commenced wit systematic characterization of the transformation factor across the operational envelope of the reduced scale testbed. Inh tis paper, the experimental results provide expelling evidence of symmetric optimization of skew transformation parameters ta yield substantial and consistent improvement in STS crane dynamic performance whereby 38.9%is the mean reduction in settling time and 47.6%is the mean reduction in residual amplitude representing practically significant enhancements that translates directly to operational productivity gain The figure demonstrates ow faster the system works. As the operator lowers the spreader from a height of 30m, the cameras are activated 7m above the container and at 6.5 the system completes skewing of spreader to couple the container. Here the skew is stable at 3.2 seconds as at an angle of -1 degrees, is the shortest time taken to make such correction. Conclusion The system demonstrates faster and robust by giving output on real-time, since minimal overshoot towards the desired alignment. The optimization of skew angle of spreader parameters represents a high leverage opportunity for improving STS cranes productivity by 30%. Control algorithm is implemented, an PID controller for dynamic spreader adjustments. Simulation and validation in real world crane operation proves that from the cameras capturing position of container to coupling of containers can be done within the shortest time of less than a minute since settling is just from 1.33seconds to 3.2 seconds. Making efficiency of the system to be 99.97% and still the system can handle both negative and positive angles of the spreader. Recommendations Based on the study outcomes the following actions are recommended. Port operators need to adopt vision-based system by replacing the sensor-based system with RGB-D cameras for high accuracy in skew. Use modular mounts to retrofit existing cranes. References Ngo QH, Hong K, Kim KH, Shin YJ (2008) Skew control of a container crane Skew Control of a Container Crane . May 2014 . https://doi.org/10.1109/ICCAS.2008.4694378 Kugler M, Brandenburg M, Limant S (2021) Automizing the manual link in maritime supply chains? An analysis of twist lock handling automation in container terminals. Maritime Transport Research , 2 (October 2020), 100017. https://doi.org/10.1016/j.martra.2021.100017 Parra LA, Tannuri EA (2012) DEFINITION AND NUMERICAL SIMULATION OF AN AUTOMATED . 5 , 49–58 Ngo QH, Hong K (2014) Skew control of a quay container crane Skew control of a quay container crane † . December 2010 . https://doi.org/10.1007/s12206-009-1020-1 Shestaka A, Melnikova L (2018) Proposals for a cargo anti-sway method using motor torque control. Comput Probl Electr Eng 8(2):88–94. https://doi.org/10.23939/jcpee2018.02.088 Lee J (2019) Deep learning – assisted real-time container corner casting recognition . 15 (1). https://doi.org/10.1177/1550147718824462 Van Manen BT (2024) Crane Operation Maqsood M, Mehmood I, Kharel R, Muhammad K, Lee J, Alnumay WS (2021) Exploring the Role of Deep Learning in Industrial Applications: A Case Study on Coastal Crane Casting Recognition. Human-Centric Computing and Information Sciences , 11 . https://doi.org/10.22967/HCIS.2021.11.020 Kim YS, Hong KS, Sul SK (2004) Anti-sway control of container cranes: Inclinometer, observer, and state Feedback. Int J Control Autom Syst 2(4):435–449 Kugler M, Brandenburg M, Limant S (2021) Automizing the manual link in maritime supply chains? An analysis of twist lock handling automation in container terminals. Maritime Transport Research , 2 (October 2020), 100017. https://doi.org/10.1016/j.martra.2021.100017 Beller S, Yavuz H (2021) Crane automation and mechanical damping methods. Alexandria Eng J 60(3):3275–3293. https://doi.org/10.1016/j.aej.2021.01.048 Jakovlev S, Eglynas T, Voznak M, Jusis M, Partila P, Tovarek J, Jankunas V (2022) Detecting Shipping Container Impacts with Vertical Cell Guides inside Container Ships during Handling Operations. Sensors 22(7). https://doi.org/10.3390/s22072752 Li P, Li Y (2023) Research on the electro-hydraulic servo system of picking manipulator. AIP Adv 13(1). https://doi.org/10.1063/5.0130344 Na DEC, Hipertensiva C (n.d.). Ngo QH, Hong K (2014) Skew control of a quay container crane Skew control of a quay container crane † . December 2010 . https://doi.org/10.1007/s12206-009-1020-1 Jakovlev S, Eglynas T, Voznak M, Jusis M, Partila P, Tovarek J, Jankunas V (2022) Detecting Shipping Container Impacts with Vertical Cell Guides inside Container Ships during Handling Operations. Sensors 22(7). https://doi.org/10.3390/s22072752 Additional Declarations The authors declare no competing interests. Cite Share Download PDF Status: Posted Version 1 posted You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. We do this by developing innovative software and high quality services for the global research community. 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Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-9016031","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":599723610,"identity":"4aef9d65-6231-4045-8343-09fab4cbb348","order_by":0,"name":"Martin Mae","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA60lEQVRIiWNgGAWjYLCCBDDJA8QVDAwGRGhgbEBoOUOsFgaYFsY2IrTotp99/uDhjjt2G66dPfjh47zD8ubszQcYflRsw6nF7Ey6YUPimWfJG27nJUvO3HbYcGfPsQTGnjO3cWs5kMbYkNh2ONngdo6BNO+2w4wbbuQYMDO24dFy/hlci/Fv3jmH7QlruQGxxQ6oxUyat+FwIhFanjHOAGpJkLydl2Y541h68oYzxxIO4vXL+TSGjz/bDtvz3c49fONDjbXthuPNBx/8qMCtBQYSGyB0M5g8QFA9ENhD6TpiFI+CUTAKRsEIAwDg+mT8mvQq3wAAAABJRU5ErkJggg==","orcid":"https://orcid.org/0009-0004-4760-6445","institution":"Technical University of Mombas","correspondingAuthor":true,"prefix":"","firstName":"Martin","middleName":"","lastName":"Mae","suffix":""},{"id":599723611,"identity":"7387049c-efd7-43eb-aaf0-82d88bfc566d","order_by":1,"name":"Michael Saulo","email":"","orcid":"","institution":"Technical University of Mombasa","correspondingAuthor":false,"prefix":"","firstName":"Michael","middleName":"","lastName":"Saulo","suffix":""},{"id":599723612,"identity":"2e1b1482-465e-4151-a381-66cc7069aa63","order_by":2,"name":"Michael Odhiambo","email":"","orcid":"","institution":"Technical University of Mombasa","correspondingAuthor":false,"prefix":"","firstName":"Michael","middleName":"","lastName":"Odhiambo","suffix":""},{"id":599723613,"identity":"360a6045-f44b-465d-a918-543649510e84","order_by":3,"name":"Wang'ombe D. Ng'era","email":"","orcid":"","institution":"Technical University of Mombasa","correspondingAuthor":false,"prefix":"","firstName":"Wang'ombe","middleName":"D.","lastName":"Ng'era","suffix":""}],"badges":[],"createdAt":"2026-03-03 05:20:34","currentVersionCode":1,"declarations":{"humanSubjects":false,"vertebrateSubjects":true,"conflictsOfInterestStatement":false,"humanSubjectEthicalGuidelines":false,"humanSubjectConsent":false,"humanSubjectClinicalTrial":false,"humanSubjectCaseReport":false,"vertebrateSubjectEthicalGuidelines":true},"doi":"10.21203/rs.3.rs-9016031/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-9016031/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":104179404,"identity":"936988b7-b5dc-4588-b331-7ebaef396246","added_by":"auto","created_at":"2026-03-08 17:04:47","extension":"png","order_by":1,"title":"Figure 1","display":"","copyAsset":false,"role":"figure","size":114096,"visible":true,"origin":"","legend":"\u003cp\u003eskew process\u003c/p\u003e","description":"","filename":"1.png","url":"https://assets-eu.researchsquare.com/files/rs-9016031/v1/773a5b4f0541a9d06232c20a.png"},{"id":104179403,"identity":"6c2efc27-0bfb-4982-85b4-25bdd775a5bc","added_by":"auto","created_at":"2026-03-08 17:04:47","extension":"jpg","order_by":2,"title":"Figure 2","display":"","copyAsset":false,"role":"figure","size":88230,"visible":true,"origin":"","legend":"\u003cp\u003eskew stand alone\u003c/p\u003e","description":"","filename":"2.jpg","url":"https://assets-eu.researchsquare.com/files/rs-9016031/v1/c7313ad2112e044d6f926749.jpg"},{"id":104404514,"identity":"8ac47ceb-dee6-41fa-ba93-5161a0091def","added_by":"auto","created_at":"2026-03-11 12:20:26","extension":"png","order_by":3,"title":"Figure 3","display":"","copyAsset":false,"role":"figure","size":24905,"visible":true,"origin":"","legend":"\u003cp\u003eskew correction\u003c/p\u003e","description":"","filename":"3.png","url":"https://assets-eu.researchsquare.com/files/rs-9016031/v1/2409ae508173cc8ffc0c824e.png"},{"id":104404998,"identity":"9b69291a-f8c0-460e-8c3f-e73555dc20a2","added_by":"auto","created_at":"2026-03-11 12:21:32","extension":"png","order_by":4,"title":"Figure 4","display":"","copyAsset":false,"role":"figure","size":16718,"visible":true,"origin":"","legend":"\u003cp\u003eskew output\u003c/p\u003e","description":"","filename":"4.png","url":"https://assets-eu.researchsquare.com/files/rs-9016031/v1/4d9b309cef203fb532bd66f4.png"},{"id":104409462,"identity":"2a5b3ad2-8d04-4f4c-838b-bb4702cb6ddc","added_by":"auto","created_at":"2026-03-11 12:45:09","extension":"pdf","order_by":0,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":991470,"visible":true,"origin":"","legend":"","description":"","filename":"manuscript.pdf","url":"https://assets-eu.researchsquare.com/files/rs-9016031/v1/11fd468a-597f-4047-8020-286101216bb1.pdf"}],"financialInterests":"The authors declare no competing interests.","formattedTitle":"\u003cp\u003eOptimization of skew angle of the spreader in sea-to-shore gantry cranes\u003c/p\u003e","fulltext":[{"header":"Introduction","content":"\u003cp\u003eThis paper develops a vision-based monitoring system to detect skew angles of spreaders in sea-to-shore gantry cranes with high accuracy, integrating an adaptive PID controller for real-time spreader adjustments. Validating the system through MATLAB simulations and field tests in an operational port environment. The study aimed to optimize spreader alignment to improve crane efficiency and to reduce incidents while being able to cater to the expansion of the global maritime trade. It provides insights into several recent strides in automation of these essential processes, especially maritime gantry crane operations, from sea to shore. This work is a mathematical model for the three-dimensional movement of the gantry crane function along the quay. Together with cameras, counters, sensors, and encoders, this system allows for monitoring and controlling the crane's movement in real time. During the lifting process, the spreader suspended for operation is equipped with 4 AI-driven cameras which collect position data by identifying corner castings in the container. The control algorithm activates four hydraulic cylinders based on the acquired data to steer the sheaves as directed. The simulation results very clearly show the capabilities of the machine vision-based tilt, list and skew correction system in obtaining the precision alignment during crane operation.\u003c/p\u003e \u003cp\u003eIn his analysis of twist-locks in fully autonomous gantry cranes, Marcus Brandenburg [2021] highlighted that twist locks are a crucial component for securing container corner fittings in marine cargo handling facilities.\u003c/p\u003e \u003cp\u003eIn the discussion on the use of sensors to reduce down time tfor a spreader to couple a conhtainer, journal article by Parra \u0026amp; Tannuri, [\u003cspan citationid=\"CR3\" class=\"CitationRef\"\u003e2012\u003c/span\u003e], they discussed the use of sensors to reduce the time required for coupling containers, a process facilitated by a device known as the spreader.\u003c/p\u003e \u003cp\u003eThe authors, Ngo \u0026amp; Hong, [\u003cspan citationid=\"CR4\" class=\"CitationRef\"\u003e2014\u003c/span\u003e], in their research introduced sway control to precisely position the spreader by limiting its swing for the purpose of accuracy. They proposed two methods: the first involved a sensor-driven control logic and command signal modification, which enabled accurate container alignment and suppression of pendular motion in sea-to-shore cranes through time-critical trajectory control with closed-loop adjustment enabling faster travel. Their second method used linear control to quickly suppress any residual sway.\u003c/p\u003e \u003cp\u003eIn their proposed technique for reducing load oscillation through motor torque regulation, [Anatolii Shestaka and Lubov Melnikova 2018] employed modern variable frequency drives. The motor torque force is influenced by the proportion of the trolley's weight to the cargo's weight, as well as the longitudinal measurement of the rope.\u003c/p\u003e \u003cp\u003e[Jaecheul Lee \u003cspan citationid=\"CR6\" class=\"CitationRef\"\u003e2019\u003c/span\u003e] utilized laser measurement (LMS) and developed fast algorithms to recognize cargo compartments of the ship. Additionally, the universe\u0026rsquo;s largest hoisting machine manufacturer has created an autonomous container unit position control configuration that centered around laser-assisted imaging technology. This system comes with some disadvantages, such as being time-consuming when scanning multiple planes because it should involve servomechanisms. Convolutional Neural Networks (CNN) were used for corner casting because they can learn complex feature representations, thus becoming highly effective as well reliable for pattern analysis in visual data.\u003c/p\u003e \u003cp\u003eIn Three-Dimensional Perception System for Reliable Autonomous Hoisting Machine Management, an advanced robotic system for container handling was introduced [Manen, \u003cspan citationid=\"CR7\" class=\"CitationRef\"\u003e2024\u003c/span\u003e]. The 3DWI system integrates a sophisticated and intelligent sensing suite with machine learning algorithm, along with an architecture that connects distributed control between the crane and the shore-based operations center.\u003c/p\u003e \u003cp\u003eIn Applying Deep Learning to Industrial Systems: Coastal Crane Casting Recognition as a Case Study. Maqsood et al., [\u003cspan citationid=\"CR8\" class=\"CitationRef\"\u003e2021\u003c/span\u003e], discussed how advanced AI approaches with deep learning models are mimicking human cognitive behavior that tackle sophisticated engineering challenges that may be otherwise challenging in addressing them. The core of AI involves solving complex engineering problems by identifying the correct relationships between input and output data, even when the underlying connections are not immediately clear. AI can model data, enabling future predictions and decision-making. This approach utilizes artificial neural networks (ANN) to achieve these capabilities [Manen, \u003cspan citationid=\"CR7\" class=\"CitationRef\"\u003e2024\u003c/span\u003e]\u003c/p\u003e \u003cp\u003eIn their 2004 study, 'Anti-Sway Control of Container Cranes, Inclinometer, Observer, and State Feedback. Kim et al., [\u003cspan citationid=\"CR9\" class=\"CitationRef\"\u003e2004\u003c/span\u003e] explored an anti-sway control system that utilizes an inclinometer as a sway sensor for container cranes. Compared to a vision system, the inclinometer is much more affordable, durable, and easier to maintain while offering nearly identical performance. Some observers project the rate of rotation of the load and the velocity of the trolley in the configuration setup.\u003c/p\u003e \u003cp\u003eSerkan et al. [2021] Crane automation and mechanical damping methods. In the implementation of the control of the sway technique relying on motor torque control and contemporary variable frequency drives, its key equation relates its pulling tension to its angle of oscillation of the load. This equation forms the foundation for the rotational force of the motor, which was designed to eliminate load oscillation once acceleration and as well retardation. By accounting for the damping of oscillations within the planes, one can establish the foundational concepts of a technical method to address the oscillation mitigation challenge based on these parameters.\u003c/p\u003e \u003cp\u003eIn identifying impacts to shipping containers with vertical cell guides within container ships during handling operations.\" Jakovlev et al., [\u003cspan citationid=\"CR12\" class=\"CitationRef\"\u003e2022\u003c/span\u003e] explored the detection of impacts between shipping containers and vertical guide rails inside container ships during cargo management.\u003c/p\u003e \u003cp\u003eMechanical systems used to place twist locks into corner casts to control tilting, listing, and skewing through flip arms and hydraulic pumps, and their connections exist, but they are so costly to maintain because of the nature of hydraulics that it is difficult to minimize leakages, and they are too bulky. Though effective, they are not favorable. Laser vision systems have also been implemented, but they target the ropes since they can only scan one plane and not the payload, which is of our interest. Though effective, it is not favorable, too. Thus, an inclination sensor that is secured in circuit design, robust, and mimics human behavior in observing load tilting, listing, and skewing is essential. Intelligent crane automation systems create a potential for growth towards automating container port processes to realize better throughput of cargo. Ship-to-shore container handling equipment commonly utilized at quay side terminals, particularly at storage terminals, for transportation. It consists of three main components. The first is the gantry, which provides structural support for all equipment and moves in one direction while the second component is the trolley, which sits on the boom rails moving along the boom from the back reach of the crane to the boom tip. [Li \u0026amp; Li, \u003cspan citationid=\"CR13\" class=\"CitationRef\"\u003e2023\u003c/span\u003e].\u003c/p\u003e \u003cp\u003eTwo methods used in adjustment of the skew orientation are particularly noteworthy: moving the pulley alongside the trolley\u0026rsquo;s movement, introducing extra twisting force to dampen disturbance in the skew motion, and independently controlling the displacement of the wire rope that holds the spreader. Adjustments of angular deviation involve a three-dimensional model o that includes oscillatory and angular deviations. [Na \u0026amp; Hipertensiva, n.d.]. In this system, a DC motor actuates the skew drive, allowing one side of the hoist mechanism (two cables) to shift forward and backward onto the trolley. This motion generates a rotational torque on the lifting device and the cargo unit. The authors also discussed an easier modulator engineered to balance the tilt, list, and angular motions of the spreader.\u003c/p\u003e \u003cp\u003eA fuzzy-based control system was developed that managed the tilt, inclination and lateral offset of the lifting attachment [Ngo \u0026amp; Hong, \u003cspan citationid=\"CR4\" class=\"CitationRef\"\u003e2014\u003c/span\u003e]. Instead of using the main hoisting cables, they employed four secondary wire ropes for motion control of tilt, list and lateral offset. They present an analysis of the 3D movement behavior of a cargo unit system of transportation, demonstrating effectiveness of the regulator for placing the cargo unit, despite gale and unpredictable factors. The study introduced four wire rope mechanisms, comprising of three-dimensional axes (translation) and angular displacement, as applied to conventional cargo unit gantry cranes. A primary lift device is used by this crane in controlling the lifting attachment vertical motion, with a set of four hydraulic cylinders connected to the separate hoists to exert force on each wire rope while the second work involved designing an easier modulator to balance angular motion whilst keeping list and trim angles minimal, ensuring that skew angle isn\u0026rsquo;t altered by unintended movements.\u003c/p\u003e \u003cp\u003eUsing Neural Network-Based Predictive Control Approaches to Address the Shipping Container Sway Issue in Quay Cranes, the author discusses the use of visual feedback to assist in positioning containers beneath the quay crane. This is managed by controlling abrupt changes in the container's rapid movement during its transport, alongside the crane\u0026rsquo;s operation, which is crucial to prevent high-amplitude accelerations.\u003c/p\u003e \u003cp\u003eThe use of IoT sensors to improve detection capabilities in shipping container impact scenarios with vertical guard rails inside container ships during handling operations was explored by Sergej Jakovlev, et al [\u003cspan citationid=\"CR12\" class=\"CitationRef\"\u003e2022\u003c/span\u003e]\". Their system integrates internal electronics devices, cordless transmission, and information storage frameworks all controlled by a processing unit that performs data interpretation from sensors through dedicated software logic. This detection methodology follows a step-by-step process. The system deployer is responsible for deploying the IoT system. At the same time, the administrator controls the information flow between the programs as well as the local storage system.\u003c/p\u003e"},{"header":"System design","content":"\u003cdiv id=\"Sec3\" class=\"Section2\"\u003e\n\u003ch2\u003e3.1 Overall System Architecture\u003c/h2\u003e\n\u003cp\u003eThe crane spreader descends from an initial height of 30 meters to a target container height of 2 meters at a maximum descent speed of 1.6 m/s. During descent:\u003c/p\u003e\n\u003cp\u003eAt approximately 7 meters, a machine vision system (camera\u0026thinsp;+\u0026thinsp;image processing algorithm) is activated.\u003c/p\u003e\n\u003cp\u003eThe cameras capture 3D images of the spreader in real time, reconstructing its position and detecting any tilt, list or skew, deviation.\u003c/p\u003e\n\u003cp\u003eThe captured images are sent to a central processing computer which analyzes them to determine the extent of deviation.\u003c/p\u003e\n\u003cp\u003eUpon detecting a LIST deviation, a control command is issued to adjust the spreader's orientation.\u003c/p\u003e\n\u003cp\u003eIn the MATLAB Simulink model, the Machine Vision operation is represented by an idealized detection process:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003e\n\u003cp\u003eA step input deviation of 3\u0026deg; is introduced at exactly 6.5 meters height.\u003c/p\u003e\n\u003c/li\u003e\n\u003cli\u003e\n\u003cp\u003eThis deviation is fed into the PID controller for correction.\u003c/p\u003e\n\u003c/li\u003e\n\u003c/ul\u003e\n\u003cp\u003eThe PID controller processes the deviation and commands the hydraulic actuator system (modeled through dynamic response blocks) to correct the spreader's orientation before it reaches the container.\u003c/p\u003e\n\u003c/div\u003e\n\u003cdiv id=\"Sec4\" class=\"Section2\"\u003e\n\u003ch2\u003e3.2 Simulink Modeling, Design, and Simulation of the System\u003c/h2\u003e\n\u003cdiv id=\"Sec5\" class=\"Section3\"\u003e\n\u003ch2\u003e3.2.1 MATLAB Simulink System Architecture\u003c/h2\u003e\n\u003cp\u003eThe overall simulation model is implemented in MATLAB Simulink 2016 with three central controller modules Tilt, List and Skew on all the critical levels of the freedom for Sea to Shore (STS) crane alignment. A system of controls loops that operate in conjunction with a central Integrator Logic that allows for accurate, smooth, fault-resistant positioning of the spreader bar of the crane. \u003cstrong\u003eMATLAB Simulink System Architecture Overview\u003c/strong\u003e\u003c/p\u003e\n\u003cdiv id=\"Sec6\" class=\"Section4\"\u003e\n\u003ch2\u003e3.2.1.1 Skew Control Subsystem\u003c/h2\u003e\n\u003cp\u003eHere, the skew control subsystem is responsible for controlling the angle of the spreader to achieve the desired side-to-side angle of alignment across the beam axis of the crane. This facilitates safe and uninterrupted operations with the container when using a crane. An early warning and adjustments command system is present as the subsystem further incorporates a Machine Vision System to improve accuracy and reduce errors during the descent. Its main characteristics are the following:\u003c/p\u003e\n\u003c/div\u003e\n\u003cdiv id=\"Sec7\" class=\"Section4\"\u003e\n\u003ch2\u003e3.2.1.2 Machine Vision Activation and Monitoring:\u003c/h2\u003e\n\u003cp\u003eWhen the spreader descends 30 m and is about 7 meters above the container, machine vision detectors are powered by laser distance sensors. These cameras take a 3D view of the spreader, which is analyzed in real-time via an image processing algorithm to identify any real roll, tilt and skew. But the spreader is assumed at 0\u0026deg; to be perfect during simulation. Now, the machine vision input is inactive.\u003c/p\u003e\n\u003cstrong\u003e3.2.1.3 Reference/Setpoint Application\u003c/strong\u003e:\n\u003cp\u003eYou apply a user-defined reference setpoint (such as 2\u0026deg;) to the control system. It requires the spreader to change between 0\u0026deg; \u0026minus;\u0026thinsp;2\u0026deg; to set correctly, compensating for the expected misalignments of the container or external effects.\u003c/p\u003e\n\u003c/div\u003e\n\u003cdiv id=\"Sec8\" class=\"Section4\"\u003e\n\u003ch2\u003e3.2.1.4 PID-Based Correction Mechanism:\u003c/h2\u003e\n\u003cp\u003eA PID controller is used for error (difference between current spreader angle and setpoint). The controller computes the corrections to rotate spreader towards the skew angle it demands for proper descent.\u003c/p\u003e\n\u003c/div\u003e\n\u003cdiv id=\"Sec9\" class=\"Section4\"\u003e\n\u003ch2\u003e3.2.1.5 System Dynamics Modeling:\u003c/h2\u003e\n\u003cp\u003eThe general real dynamic behavior of the spreader-crane system is modelled with a second-order transfer function. That considers realistic phenomena like inertia, oscillations and response times. This process is expressed in the figure below\u003c/p\u003e\n\u003cp\u003eThe skew angle \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\theta}_{s}\\left(t\\right)\\)\u003c/span\u003e\u003c/span\u003erepresents the misalignment, and we model the plant dynamics for skew correction as a second-order linear system derived from system identification or physical modeling. The transfer function of the plant is given by:\u003c/p\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003cdiv id=\"Sec10\" class=\"Section3\"\u003e\n\u003ch2\u003e3.2.2 System Parameters\u003c/h2\u003e\n\u003cp\u003eThis model includes:\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003e\n\u003cp\u003eA second-order plant model representing the spreader skew dynamics,\u003c/p\u003e\n\u003c/li\u003e\n\u003cli\u003e\n\u003cp\u003eA PID controller with specific gains,\u003c/p\u003e\n\u003c/li\u003e\n\u003cli\u003e\n\u003cp\u003eA transport delay modeling the latency introduced by image acquisition and processing.\u003c/p\u003e\n\u003c/li\u003e\n\u003c/ol\u003e\n\u003c/div\u003e\n\u003cdiv id=\"Sec11\" class=\"Section3\"\u003e\n\u003ch2\u003e3.2.3 PID Controller Parameters:\u003c/h2\u003e\n\u003cp\u003eProportional Gain \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({K}_{p}\\)\u003c/span\u003e\u003c/span\u003e = 1.5\u003c/p\u003e\n\u003cp\u003eIntegral Gain \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({K}_{i}\\)\u003c/span\u003e\u003c/span\u003e = 1.7\u003c/p\u003e\n\u003cp\u003eDerivative Gain \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({K}_{d}\\)\u003c/span\u003e\u003c/span\u003e = 0.1\u003c/p\u003e\n\u003cp\u003eA classical PID controller is used for feedback correction. The controller transfer function is:\u003c/p\u003e\n\u003cdiv id=\"Equ1\" class=\"Equation\"\u003e\n\u003cdiv id=\"FileID_Equ1\" class=\"mathdisplay\"\u003e$${G}_{c}\\left(s\\right)={K}_{P}+{K}_{i}+{K}_{i}+{K}_{d}s=1.5+\\frac{1.7}{s}+0.1s$$\u003c/div\u003e\n\u003cdiv class=\"EquationNumber\"\u003e3.44\u003c/div\u003e\n\u003c/div\u003e\n\u003cdiv id=\"Equ2\" class=\"Equation\"\u003e\n\u003cdiv id=\"FileID_Equ2\" class=\"mathdisplay\"\u003e$${G}_{c}\\left(s\\right)=\\frac{{0.1s}^{2}+1.5s+1.7}{s}$$\u003c/div\u003e\n\u003cdiv class=\"EquationNumber\"\u003e3.45\u003c/div\u003e\n\u003c/div\u003e\n\u003cp\u003eThis controller improves tracking and disturbance rejection. Derivative action anticipates errors (beneficial for skew), while integral action ensures zero steady-state error.\u003c/p\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003cdiv id=\"Sec12\" class=\"Section2\"\u003e\n\u003ch2\u003e3.3 Plant Transfer Function:\u003c/h2\u003e\n\u003cp\u003eLet the plant transfer function \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({G}_{P}\\left(s\\right)\\)\u003c/span\u003e\u003c/span\u003e be:\u003c/p\u003e\n\u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e \u003cspan class=\"mathinline\"\u003e\\({G}_{P}\\left(s\\right)\\)\u003c/span\u003e \u003c/span\u003e = \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\frac{4.5}{{s}^{2}+5.5s+4.2}\\)\u003c/span\u003e\u003c/span\u003e (3.46)\u003c/p\u003e\n\u003cp\u003eThis is a second-order system with poles at:\u003c/p\u003e\n\u003cdiv id=\"Equ3\" class=\"Equation\"\u003e\n\u003cdiv id=\"FileID_Equ3\" class=\"mathdisplay\"\u003e$$S=\\frac{-5.5\\pm\\sqrt{5\\cdot{5}^{2}-4}}{2}$$\u003c/div\u003e\n\u003cdiv class=\"EquationNumber\"\u003e3.47\u003c/div\u003e\n\u003c/div\u003e\n\u003cdiv id=\"Sec13\" class=\"Section3\"\u003e\n\u003ch2\u003e3.3.1 Transport Delay:\u003c/h2\u003e\n\u003cp\u003eLet the transport delay introduced by the Machine Vision system be denoted by \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({T}_{d}\\)\u003c/span\u003e\u003c/span\u003e, typically in the range of 100\u0026ndash;200 ms. Given that:\u003c/p\u003e\n\u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e \u003cspan class=\"mathinline\"\u003e\\({T}_{d}\\)\u003c/span\u003e \u003c/span\u003e= 0.15 seconds\u003c/p\u003e\n\u003cp\u003eThis delay is modeled as an exponential in the Laplace domain:\u003c/p\u003e\n\u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e \u003cspan class=\"mathinline\"\u003e\\({G}_{d}\\left(s\\right)\\)\u003c/span\u003e \u003c/span\u003e = \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({e}^{-T{d}^{s}}\\)\u003c/span\u003e\u003c/span\u003e (3.48)\u003c/p\u003e\n\u003cdiv id=\"Equ4\" class=\"Equation\"\u003e\n\u003cdiv id=\"FileID_Equ4\" class=\"mathdisplay\"\u003e$${e}^{-T{d}^{s}}\\approx\\frac{1-\\frac{{T}_{d}}{2}s}{1+\\frac{{T}_{d}}{2}s}$$\u003c/div\u003e\n\u003cdiv class=\"EquationNumber\"\u003e3.49\u003c/div\u003e\n\u003c/div\u003e\n\u003cp\u003eSubstituting \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({T}_{d}\\)\u003c/span\u003e\u003c/span\u003e = 0.15 seconds\u003c/p\u003e\n\u003cp\u003ePade approximation \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\varvec{e}}^{-0.15\\varvec{s}}\\approx\\)\u003c/span\u003e\u003c/span\u003e \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\frac{1-0.075s}{1+0.75s}\\)\u003c/span\u003e\u003c/span\u003e (3.50)\u003c/p\u003e\n\u003c/div\u003e\n\u003cdiv id=\"Sec14\" class=\"Section3\"\u003e\n\u003ch2\u003e3.3.2 PID Controller Transfer Function\u003c/h2\u003e\n\u003cp\u003eThe standard form of a PID controller in the Laplace domain is:\u003c/p\u003e\n\u003cdiv id=\"Equ5\" class=\"Equation\"\u003e\n\u003cdiv id=\"FileID_Equ5\" class=\"mathdisplay\"\u003e$${G}_{C}\\left(s\\right)={K}_{P}+\\frac{{K}_{I}}{S}+{K}_{D}s$$\u003c/div\u003e\n\u003cdiv class=\"EquationNumber\"\u003e3.51\u003c/div\u003e\n\u003c/div\u003e\n\u003cp\u003eSubstituting values:\u003c/p\u003e\n\u003cdiv id=\"Equ6\" class=\"Equation\"\u003e\n\u003cdiv id=\"FileID_Equ6\" class=\"mathdisplay\"\u003e$${G}_{C}\\left(s\\right)=1.5+\\frac{1.7}{S}+0.1s$$\u003c/div\u003e\n\u003cdiv class=\"EquationNumber\"\u003e3.51\u003c/div\u003e\n\u003c/div\u003e\n\u003cp\u003e= \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\frac{{0.1S}^{2}+1\\cdot5S+1\\cdot7}{S}\\)\u003c/span\u003e\u003c/span\u003e (3.52)\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003e3.3.3 Open-Loop Transfer Function\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eThe open-loop transfer function \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({G}_{OL}\\left(s\\right)\\)\u003c/span\u003e\u003c/span\u003e of the system including controller, plant, and delay:\u003c/p\u003e\n\u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e \u003cspan class=\"mathinline\"\u003e\\({G}_{OL}\\left(s\\right)\\)\u003c/span\u003e \u003c/span\u003e = \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({G}_{C}\\left(s\\right)\\)\u003c/span\u003e\u003c/span\u003e. \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({G}_{P}\\left(s\\right)\\)\u003c/span\u003e\u003c/span\u003e. \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({G}_{d}\\left(s\\right)\\)\u003c/span\u003e\u003c/span\u003e (3.53)\u003c/p\u003e\n\u003cp\u003eSubstituting the values:\u003c/p\u003e\n\u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e \u003cspan class=\"mathinline\"\u003e\\({G}_{OL}\\left(s\\right)\\)\u003c/span\u003e \u003c/span\u003e = \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\frac{{0.1S}^{2}+1\\cdot5s+1\\cdot7}{S}\\)\u003c/span\u003e\u003c/span\u003e \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(.\\)\u003c/span\u003e\u003c/span\u003e \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\frac{4.5}{{s}^{2}+5.5s+4.2}.{\\text{e}}^{-0.15s}\\)\u003c/span\u003e\u003c/span\u003e (3.54)\u003c/p\u003e\n\u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e \u003cspan class=\"mathinline\"\u003e\\({G}_{OL}\\left(s\\right)\\)\u003c/span\u003e \u003c/span\u003e = \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\frac{{0.1S}^{2}+1\\cdot5s+1\\cdot7}{S}\\)\u003c/span\u003e\u003c/span\u003e \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(.\\)\u003c/span\u003e\u003c/span\u003e \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\frac{4.5}{{s}^{2}+5.5s+4.2}.\\frac{1-0.075s}{1+0.75s}\\)\u003c/span\u003e\u003c/span\u003e (3.55)\u003c/p\u003e\n\u003cp\u003eSimplifying \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({G}_{OL}\\left(s\\right)=\\left(\\frac{0.1{s}^{2}+1\\cdot5s+1\\cdot7(1-0.075s)}{s({s}^{2}+5.5s+4.2)(1+0.075s)}\\right)\\)\u003c/span\u003e\u003c/span\u003e (3.56)\u003c/p\u003e\n\u003c/div\u003e\n\u003cdiv id=\"Sec15\" class=\"Section3\"\u003e\n\u003ch2\u003e3.3.4 Closed-Loop Transfer Function\u003c/h2\u003e\n\u003cp\u003eLet \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(G\\left(s\\right)\\)\u003c/span\u003e\u003c/span\u003e = \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({G}_{C}\\left(s\\right)\\)\u003c/span\u003e\u003c/span\u003e. \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({G}_{P}\\left(s\\right)\\)\u003c/span\u003e\u003c/span\u003e. \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({G}_{d}\\left(s\\right)\\)\u003c/span\u003e\u003c/span\u003e, (3.57)\u003c/p\u003e\n\u003cp\u003eThen the closed-loop system transfer function is:\u003c/p\u003e\n\u003cdiv id=\"Equ7\" class=\"Equation\"\u003e\n\u003cdiv id=\"FileID_Equ7\" class=\"mathdisplay\"\u003e$${T}_{\\left(s\\right)}=\\frac{{G}_{\\left(s\\right)}}{1+{G}_{\\left(s\\right)}}=\\frac{{G}_{OL}\\left(s\\right)}{1+{G}_{OL}\\left(s\\right)}$$\u003c/div\u003e\n\u003cdiv class=\"EquationNumber\"\u003e3.58\u003c/div\u003e\n\u003c/div\u003e\n\u003cp\u003eHowever, due to the presence of a transport delay, the system becomes a non-rational transfer function (non-polynomial due to \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({e}^{-sT}\\)\u003c/span\u003e\u003c/span\u003e. This implies:\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003e\n\u003cp\u003eClassical Laplace-domain analysis tools (like root locus or frequency response) are approximate.\u003c/p\u003e\n\u003c/li\u003e\n\u003cli\u003e\n\u003cp\u003eTime-domain simulations or Pade approximation is used for practical analysis.\u003c/p\u003e\n\u003c/li\u003e\n\u003c/ol\u003e\n\u003cp\u003eNow the full rationalized open-loop transfer function becomes:\u003c/p\u003e\n\u003cdiv id=\"Equ8\" class=\"Equation\"\u003e\n\u003cdiv id=\"FileID_Equ8\" class=\"mathdisplay\"\u003e$${G}_{OL}\\left(s\\right)=\\left(\\frac{0.1{s}^{2}+1\\cdot5s+1\\cdot7}{s}\\right)\\cdot\\left(\\frac{4.5}{{s}^{2}+5.5s+4.2}\\right)\\cdot\\left(\\frac{1-0.075s}{1+0.075s}\\right)$$\u003c/div\u003e\n\u003cdiv class=\"EquationNumber\"\u003e3.59\u003c/div\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003cdiv id=\"Sec16\" class=\"Section3\"\u003e\n\u003ch2\u003e3.3.5 Final Transfer Function Summary\u003c/h2\u003e\n\u003cp\u003eThis transfer function can now be used for:\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003e\n\u003cp\u003eStability analysis (e.g. Routh-Hurwitz or Nyquist),\u003c/p\u003e\n\u003c/li\u003e\n\u003cli\u003e\n\u003cp\u003eTime-domain simulations (step response, rise time, overshoot),\u003c/p\u003e\n\u003c/li\u003e\n\u003cli\u003e\n\u003cp\u003eFrequency response analysis (Bode, gain/phase margin).\u003c/p\u003e\n\u003c/li\u003e\n\u003c/ol\u003e\n\u003cp\u003eThe figure above expresses the reference point of 2 degrees and how the plant and scope 1 operate.\u003c/p\u003e\n\u003c/div\u003e\n\u003cdiv id=\"Sec17\" class=\"Section3\"\u003e\n\u003ch2\u003e3.3.6 Stability and Performance Considerations\u003c/h2\u003e\n\u003cp\u003eThe transport delay degrades phase margin and can destabilize the system unless compensated. The relatively low derivative gain \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({K}_{d}\\)\u003c/span\u003e\u003c/span\u003e = 0.1 minimizes noise amplification but reduces damping. While the steady-state error is mitigated with integral action, oscillations can occur when delay is present. Controller tuning could be further improved to provide robustness via:\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003e\n\u003cp\u003eZiegler-Nichols with delay compensation\u003c/p\u003e\n\u003c/li\u003e\n\u003cli\u003e\n\u003cp\u003eSmith Predictor design for delay compensation\u003c/p\u003e\n\u003c/li\u003e\n\u003cli\u003e\n\u003cp\u003eLQR or pole placement (for state-space equivalents)\u003c/p\u003e\n\u003c/li\u003e\n\u003c/ol\u003e\n\u003c/div\u003e\n\u003cdiv id=\"Sec18\" class=\"Section3\"\u003e\n\u003ch2\u003e3.3.7 State-Space Representation\u003c/h2\u003e\n\u003cp\u003eWe'll expressing the system in the form:\u003c/p\u003e\n\u003cdiv id=\"Equ9\" class=\"Equation\"\u003e\n\u003cdiv id=\"FileID_Equ9\" class=\"mathdisplay\"\u003e$$\\dot{x}\\left(t\\right)=Ax\\left(t\\right)t+u\\left(t\\right)$$\u003c/div\u003e\n\u003cdiv class=\"EquationNumber\"\u003e3.60\u003c/div\u003e\n\u003c/div\u003e\n\u003cdiv id=\"Equ10\" class=\"Equation\"\u003e\n\u003cdiv id=\"FileID_Equ10\" class=\"mathdisplay\"\u003e$$y\\left(t\\right)=Cx\\left(t\\right)t+Du\\left(t\\right)$$\u003c/div\u003e\n\u003cdiv class=\"EquationNumber\"\u003e3.61\u003c/div\u003e\n\u003c/div\u003e\n\u003cp\u003ePlant model:\u003c/p\u003e\n\u003cp\u003eLet\u0026rsquo;s define the state variables as:\u003c/p\u003e\n\u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e \u003cspan class=\"mathinline\"\u003e\\({x}_{1}=y\\)\u003c/span\u003e \u003c/span\u003e(output), \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({x}_{2}=\\dot{y}\\)\u003c/span\u003e\u003c/span\u003e (3.62)\u003c/p\u003e\n\u003cdiv id=\"Equa\" class=\"Equation\"\u003e\n\u003cdiv id=\"FileID_Equa\" class=\"mathdisplay\"\u003e$${\\dot{x}}_{1}={x}_{2}$$\u003c/div\u003e\n\u003c/div\u003e\n\u003cdiv id=\"Equ11\" class=\"Equation\"\u003e\n\u003cdiv id=\"FileID_Equ11\" class=\"mathdisplay\"\u003e$${\\dot{x}}_{2}=-4.2{x}_{1}-5.5{x}_{2}+4.5u$$\u003c/div\u003e\n\u003cdiv class=\"EquationNumber\"\u003e3.63\u003c/div\u003e\n\u003c/div\u003e\n\u003cp\u003eMatrix form:\u003c/p\u003e\n\u003cdiv id=\"Equ12\" class=\"Equation\"\u003e\n\u003cdiv id=\"FileID_Equ12\" class=\"mathdisplay\"\u003e$${\\dot{x}}_{2}=\\left[\\begin{array}{cc}0\u0026amp;1\\\\-4.2\u0026amp;-5.5\\end{array}\\right]{x}_{2}+\\left[\\begin{array}{c}0\\\\4.5\\end{array}\\right]u$$\u003c/div\u003e\n\u003cdiv class=\"EquationNumber\"\u003e3.64\u003c/div\u003e\n\u003c/div\u003e\n\u003cdiv id=\"Equb\" class=\"Equation\"\u003e\n\u003cdiv id=\"FileID_Equb\" class=\"mathdisplay\"\u003e$$y=\\left[10\\right]{x}_{1}$$\u003c/div\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003cdiv id=\"Sec19\" class=\"Section3\"\u003e\n\u003ch2\u003e3.3.8 Augmented System (Plant\u0026thinsp;+\u0026thinsp;PID):\u003c/h2\u003e\n\u003cp\u003eAdding integrator state:\u003c/p\u003e\n\u003cdiv id=\"Equ13\" class=\"Equation\"\u003e\n\u003cdiv id=\"FileID_Equ13\" class=\"mathdisplay\"\u003e$${x}_{3}=\\int\\text{e}\\left(t\\right)dt$$\u003c/div\u003e\n\u003cdiv class=\"EquationNumber\"\u003e3.65\u003c/div\u003e\n\u003c/div\u003e\n\u003cp\u003ePID control law: \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(u\\left(t\\right)={K}_{Pe}\\left(t\\right)+{K}_{1} \\int e \\left(t\\right)dt+{K}_{D}\\frac{de\\left(t\\right)}{dt}\\)\u003c/span\u003e\u003c/span\u003e (3.66)\u003c/p\u003e\n\u003cdiv id=\"Equ14\" class=\"Equation\"\u003e\n\u003cdiv id=\"FileID_Equ14\" class=\"mathdisplay\"\u003e$$u\\left(t\\right)=1.5\\left(t\\right)+1.7{x}_{3}-0.1\\dot{y}\\left(t\\right)=1.5\\left(r-{x}_{1}\\right)+1.7$$\u003c/div\u003e\n\u003cdiv class=\"EquationNumber\"\u003e3.67\u003c/div\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003cdiv id=\"Sec20\" class=\"Section3\"\u003e\n\u003ch2\u003e3.3.9 Augmented dynamics:\u003c/h2\u003e\n\u003cdiv id=\"Equc\" class=\"Equation\"\u003e\n\u003cdiv id=\"FileID_Equc\" class=\"mathdisplay\"\u003e$${\\dot{x}}_{1}={x}_{2}$$\u003c/div\u003e\n\u003c/div\u003e\n\u003cdiv id=\"Equ15\" class=\"Equation\"\u003e\n\u003cdiv id=\"FileID_Equ15\" class=\"mathdisplay\"\u003e$${\\dot{x}}_{2}=-4.2{x}_{1}-5.5{x}_{2}+4.5\\left[1\\cdot5\\left(r-{x}_{1}\\right)+1.7{x}_{3}-0.1{x}_{2}\\right]$$\u003c/div\u003e\n\u003cdiv class=\"EquationNumber\"\u003e3.68\u003c/div\u003e\n\u003c/div\u003e\n\u003cdiv id=\"Equ16\" class=\"Equation\"\u003e\n\u003cdiv id=\"FileID_Equ16\" class=\"mathdisplay\"\u003e$${\\dot{x}}_{3}=r-{x}_{1}$$\u003c/div\u003e\n\u003cdiv class=\"EquationNumber\"\u003e3.69\u003c/div\u003e\n\u003c/div\u003e\n\u003cp\u003eSimplify\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\dot{x}}_{2}\\)\u003c/span\u003e\u003c/span\u003e\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e \u003cspan class=\"mathinline\"\u003e\\({\\dot{x}}_{2}=-11.55{x}_{1}-5.95{x}_{2}+7.65{x}_{3}+6.75r\\)\u003c/span\u003e \u003c/span\u003e(3.70)\u003c/p\u003e\n\u003cp\u003eFinal State-Space Matrices:\u003c/p\u003e\n\u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e \u003cspan class=\"mathinline\"\u003e\\(A=\\left[\\begin{array}{ccc}0\u0026amp;1\u0026amp;0\\\\-11.55\u0026amp;-5.95\u0026amp;7.65\\\\-1\u0026amp;0\u0026amp;0\\end{array}\\right]\\)\u003c/span\u003e \u003c/span\u003e,\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(B=\\left[\\begin{array}{c}0\\\\6.75\\\\1\\end{array}\\right]\\)\u003c/span\u003e\u003c/span\u003e,\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(C=\\left[100\\right]\\)\u003c/span\u003e\u003c/span\u003e (3.71)\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003e3.4 Skew Simulation in Simulink\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eIn a modern automated port terminal, ship-to-shore (STS) gantry cranes play a vital role in the effective handling of shipments. Among the most important structural parameters to be constantly monitored are tilt, list, and skew. This part deals with skew, the lateral misalignment of the crane trolley or spreader beam along its travel axis. Skewing results in uneven container pickup, increased mechanical strain, and safety hazards. To tackle these challenges and cater to increased demand for automation and accuracy, this project proposes a vision-based skew correction system. Using continuous video feeds, the system observes skew angles by detecting important structural markers on the spreader, which helps compute the horizontal deviation of the spreader beam from the ideal alignment path. We combined tilt (\u0026theta;), list (\u0026phi;), and skew (\u0026psi;) by fusing IMU, rope-angle, and laser measurements into a 3-degree of freedom (DOF) attitude state, which we mapped onto physical outputs: corner height spread \u0026Delta;h_max = [\u0026phi; W\u0026thinsp;+\u0026thinsp;\u0026theta; L], lateral/longitudinal shears. A composite index multi-mode method (MMM) normalizes these by allowable limits to drive operator cues, interlocks, and closed-loop tilt, list, and skew control; significantly reducing MMM lowers landing impacts and cycle variance. There is an integrator; the integrator is the \u0026ldquo;brain\u0026rdquo; that compresses heterogeneous angular measurements (tilt, list, skew) into one coherent system of actions to eliminate undesired misalignment. This unified output leads to safe container landings with shortened cycle time and decreased structural strain.\u003c/p\u003e\n\u003c/div\u003e\n\u003c/div\u003e"},{"header":"Results and Discussion.","content":"\u003cp\u003eThe experimental investigation commenced wit systematic characterization of the transformation factor across the operational envelope of the reduced scale testbed.\u003c/p\u003e \u003cp\u003eInh tis paper, the experimental results provide expelling evidence of symmetric optimization of skew transformation parameters ta yield substantial and consistent improvement in STS crane dynamic performance whereby 38.9%is the mean reduction in settling time and 47.6%is the mean reduction in residual amplitude representing practically significant enhancements that translates directly to operational productivity gain\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eThe figure demonstrates ow faster the system works. As the operator lowers the spreader from a height of 30m, the cameras are activated 7m above the container and at 6.5 the system completes skewing of spreader to couple the container.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eHere the skew is stable at 3.2 seconds as at an angle of -1 degrees, is the shortest time taken to make such correction.\u003c/p\u003e"},{"header":"Conclusion","content":"\u003cp\u003eThe system demonstrates faster and robust by giving output on real-time, since minimal overshoot towards the desired alignment.\u003c/p\u003e \u003cp\u003eThe optimization of skew angle of spreader parameters represents a high leverage opportunity for improving STS cranes productivity by 30%.\u003c/p\u003e \u003cp\u003eControl algorithm is implemented, an PID controller for dynamic spreader adjustments.\u003c/p\u003e \u003cp\u003eSimulation and validation in real world crane operation proves that from the cameras capturing position of container to coupling of containers can be done within the shortest time of less than a minute since settling is just from 1.33seconds to 3.2 seconds. Making efficiency of the system to be 99.97% and still the system can handle both negative and positive angles of the spreader.\u003c/p\u003e \u003cp\u003eRecommendations\u003c/p\u003e \u003cp\u003eBased on the study outcomes the following actions are recommended.\u003c/p\u003e \u003cp\u003ePort operators need to adopt vision-based system by replacing the sensor-based system with RGB-D cameras for high accuracy in skew.\u003c/p\u003e \u003cp\u003eUse modular mounts to retrofit existing cranes.\u003c/p\u003e"},{"header":"References","content":"\u003col\u003e\u003cli\u003e\u003cspan\u003eNgo QH, Hong K, Kim KH, Shin YJ (2008) \u003cem\u003eSkew control of a container crane Skew Control of a Container Crane\u003c/em\u003e. \u003cem\u003eMay 2014\u003c/em\u003e. \u003cspan class=\"ExternalRef\"\u003e\u003cspan class=\"RefSource\"\u003ehttps://doi.org/10.1109/ICCAS.2008.4694378\u003c/span\u003e\u003cspan address=\"10.1109/ICCAS.2008.4694378\" targettype=\"DOI\" class=\"RefTarget\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eKugler M, Brandenburg M, Limant S (2021) Automizing the manual link in maritime supply chains? An analysis of twist lock handling automation in container terminals. \u003cem\u003eMaritime Transport Research\u003c/em\u003e, \u003cem\u003e2\u003c/em\u003e(October 2020), 100017. \u003cspan class=\"ExternalRef\"\u003e\u003cspan class=\"RefSource\"\u003ehttps://doi.org/10.1016/j.martra.2021.100017\u003c/span\u003e\u003cspan address=\"10.1016/j.martra.2021.100017\" targettype=\"DOI\" class=\"RefTarget\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eParra LA, Tannuri EA (2012) \u003cem\u003eDEFINITION AND NUMERICAL SIMULATION OF AN AUTOMATED\u003c/em\u003e. \u003cem\u003e5\u003c/em\u003e, 49\u0026ndash;58\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eNgo QH, Hong K (2014) \u003cem\u003eSkew control of a quay container crane Skew control of a quay container crane \u0026dagger;\u003c/em\u003e. \u003cem\u003eDecember 2010\u003c/em\u003e. \u003cspan class=\"ExternalRef\"\u003e\u003cspan class=\"RefSource\"\u003ehttps://doi.org/10.1007/s12206-009-1020-1\u003c/span\u003e\u003cspan address=\"10.1007/s12206-009-1020-1\" targettype=\"DOI\" class=\"RefTarget\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eShestaka A, Melnikova L (2018) Proposals for a cargo anti-sway method using motor torque control. Comput Probl Electr Eng 8(2):88\u0026ndash;94. \u003cspan class=\"ExternalRef\"\u003e\u003cspan class=\"RefSource\"\u003ehttps://doi.org/10.23939/jcpee2018.02.088\u003c/span\u003e\u003cspan address=\"10.23939/jcpee2018.02.088\" targettype=\"DOI\" class=\"RefTarget\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eLee J (2019) \u003cem\u003eDeep learning \u0026ndash; assisted real-time container corner casting recognition\u003c/em\u003e. \u003cem\u003e15\u003c/em\u003e(1). \u003cspan class=\"ExternalRef\"\u003e\u003cspan class=\"RefSource\"\u003ehttps://doi.org/10.1177/1550147718824462\u003c/span\u003e\u003cspan address=\"10.1177/1550147718824462\" targettype=\"DOI\" class=\"RefTarget\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eVan Manen BT (2024) \u003cem\u003eCrane Operation\u003c/em\u003e\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eMaqsood M, Mehmood I, Kharel R, Muhammad K, Lee J, Alnumay WS (2021) Exploring the Role of Deep Learning in Industrial Applications: A Case Study on Coastal Crane Casting Recognition. \u003cem\u003eHuman-Centric Computing and Information Sciences\u003c/em\u003e, \u003cem\u003e11\u003c/em\u003e. \u003cspan class=\"ExternalRef\"\u003e\u003cspan class=\"RefSource\"\u003ehttps://doi.org/10.22967/HCIS.2021.11.020\u003c/span\u003e\u003cspan address=\"10.22967/HCIS.2021.11.020\" targettype=\"DOI\" class=\"RefTarget\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eKim YS, Hong KS, Sul SK (2004) Anti-sway control of container cranes: Inclinometer, observer, and state Feedback. Int J Control Autom Syst 2(4):435\u0026ndash;449\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eKugler M, Brandenburg M, Limant S (2021) Automizing the manual link in maritime supply chains? An analysis of twist lock handling automation in container terminals. \u003cem\u003eMaritime Transport Research\u003c/em\u003e, \u003cem\u003e2\u003c/em\u003e(October 2020), 100017. \u003cspan class=\"ExternalRef\"\u003e\u003cspan class=\"RefSource\"\u003ehttps://doi.org/10.1016/j.martra.2021.100017\u003c/span\u003e\u003cspan address=\"10.1016/j.martra.2021.100017\" targettype=\"DOI\" class=\"RefTarget\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eBeller S, Yavuz H (2021) Crane automation and mechanical damping methods. Alexandria Eng J 60(3):3275\u0026ndash;3293. \u003cspan class=\"ExternalRef\"\u003e\u003cspan class=\"RefSource\"\u003ehttps://doi.org/10.1016/j.aej.2021.01.048\u003c/span\u003e\u003cspan address=\"10.1016/j.aej.2021.01.048\" targettype=\"DOI\" class=\"RefTarget\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eJakovlev S, Eglynas T, Voznak M, Jusis M, Partila P, Tovarek J, Jankunas V (2022) Detecting Shipping Container Impacts with Vertical Cell Guides inside Container Ships during Handling Operations. Sensors 22(7). \u003cspan class=\"ExternalRef\"\u003e\u003cspan class=\"RefSource\"\u003ehttps://doi.org/10.3390/s22072752\u003c/span\u003e\u003cspan address=\"10.3390/s22072752\" targettype=\"DOI\" class=\"RefTarget\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eLi P, Li Y (2023) Research on the electro-hydraulic servo system of picking manipulator. AIP Adv 13(1). \u003cspan class=\"ExternalRef\"\u003e\u003cspan class=\"RefSource\"\u003ehttps://doi.org/10.1063/5.0130344\u003c/span\u003e\u003cspan address=\"10.1063/5.0130344\" targettype=\"DOI\" class=\"RefTarget\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eNa DEC, Hipertensiva C (n.d.).\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eNgo QH, Hong K (2014) \u003cem\u003eSkew control of a quay container crane Skew control of a quay container crane \u0026dagger;\u003c/em\u003e. \u003cem\u003eDecember 2010\u003c/em\u003e. \u003cspan class=\"ExternalRef\"\u003e\u003cspan class=\"RefSource\"\u003ehttps://doi.org/10.1007/s12206-009-1020-1\u003c/span\u003e\u003cspan address=\"10.1007/s12206-009-1020-1\" targettype=\"DOI\" class=\"RefTarget\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eJakovlev S, Eglynas T, Voznak M, Jusis M, Partila P, Tovarek J, Jankunas V (2022) Detecting Shipping Container Impacts with Vertical Cell Guides inside Container Ships during Handling Operations. Sensors 22(7). \u003cspan class=\"ExternalRef\"\u003e\u003cspan class=\"RefSource\"\u003ehttps://doi.org/10.3390/s22072752\u003c/span\u003e\u003cspan address=\"10.3390/s22072752\" targettype=\"DOI\" class=\"RefTarget\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003c/ol\u003e"}],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":true,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":true,"hideJournal":true,"highlight":"","institution":"","isAcceptedByJournal":false,"isAuthorSuppliedPdf":false,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":false,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true},"keywords":"Tilt, skew, Spreader, convolutional neural network, sheave. Cell guide corner cast, port, quay, gantry crane, vision-based, camera-based measurements, PID","lastPublishedDoi":"10.21203/rs.3.rs-9016031/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-9016031/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003ePorts in the world are rated based on the number of containers they handle per year. With this reference, every port is desired to have a huge record of container handling (TEU). Sea-to-shore gantry cranes are critical assets in modern container terminals where productivity, safety and reliability directly affect port performance. One of the persistent operational challenges during container handling is the skew of the spreader relative to the container and ship cell guides, caused by either wind disturbances, trolley motions, wire rope length variations and symmetrical loading. This manuscript presents a vision-based system of spreader skew angles using cameras and intelligent algorithms. The proposed approach integrates malti-camera perception, real-time image processing and control of algorithms to estimate the skew angle accuratelyand apply corrective actions automatically. The system design emphasizes robustiness, real-time performance and compatibility with STS crane control architecture. Simulation and conceptual validation demonstrate that the proposed system can significantly reduce skewing during hoisting ad positioning, contributing to safer and more efficient container handling operations.\u003c/p\u003e","manuscriptTitle":"Optimization of skew angle of the spreader in sea-to-shore gantry cranes","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2026-03-08 17:04:42","doi":"10.21203/rs.3.rs-9016031/v1","editorialEvents":[{"type":"communityComments","content":0}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true}}],"origin":"","ownerIdentity":"4aaff3f6-6287-4de6-afb0-2c7af47b7189","owner":[],"postedDate":"March 8th, 2026","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"posted","subjectAreas":[{"id":63817785,"name":"Electrical Engineering"}],"tags":[],"updatedAt":"2026-03-08T17:04:42+00:00","versionOfRecord":[],"versionCreatedAt":"2026-03-08 17:04:42","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-9016031","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-9016031","identity":"rs-9016031","version":["v1"]},"buildId":"XKTyCvWXoU3ODBz1xrDgd","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}