A deviation correction control method and system for the entire process of large-scale caisson construction

By installing sensors and computer decomposing motion information on the caisson, combined with PID control technology, the problems of large errors and low efficiency of deviation correction methods in traditional caisson construction are solved, and precise caisson construction control is achieved, which shortens the construction cycle and reduces risks.

CN120193541BActive Publication Date: 2025-09-05SHANGHAI URBAN CONSTRUCTION DESIGN & RESEARCH INSTITUTE (GROUP) CO LTD
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Patent Information

Application Number
CN202510306038.X
Authority / Receiving Office
CN · China
Patent Type
Patents(China)
Current Assignee / Owner
Filing Date
2025-03-14
Publication Date
2025-09-05
Estimated Expiration
2045-03-14

AI Technical Summary

Technical Problem

In traditional caisson construction, the deviation correction method relies on manual measurement and adjustment, which has large errors, slow response speed, low efficiency, and cannot be controlled accurately in real time, resulting in extended construction cycle and increased costs.

Method used

At least three position sensors are used to monitor the caisson point position information, and the caisson movement is decomposed in real time to vertical translation and horizontal rotation through a computer to determine the main force point of correction, and use PID control technology to combine reaction piles and water tanks and other execution modules for precise deviation correction.

Benefits of technology

Accurate deviation correction control during caisson construction has been achieved, significantly shortening the construction cycle, reducing risks, and improving efficiency and accuracy.

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Abstract

The present invention discloses a method and system for controlling deviation correction during the entire process of large-scale caisson construction, which relates to the field of building construction. The method of the present invention comprises: installing at least three position sensors on the caisson to monitor information of at least three points on the caisson that are not located on the same straight line; inputting the position information obtained by the three position sensors into a computer in real time, and the computer decomposes the movement of the caisson within a time step T into vertical translation and rotation along a horizontal axis; based on the orientation information of the horizontal axis, determining the two locations farthest from both sides of the horizontal axis as the main force points for deviation correction, and the deviation correction process at least includes applying downward pressure to the main force points that are tilted upward. The system includes various modules for implementing the above method. Compared with traditional technologies, the present invention has the characteristics of strong real-time performance, high calculation accuracy, and scientific mechanical control, which significantly improves the accuracy of deviation correction in large-scale caisson construction.
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Description

Technical Field

[0001] The present invention relates to the field of building construction, and more particularly to a deviation correction control method and system for the entire construction process of a large-scale caisson. Background Art

[0002] Caissons are a crucial infrastructure construction method, widely used in projects such as subways, bridges, and high-rise buildings. Because their construction often involves deep foundation pits and complex underground environments, caisson positioning and precision are extremely demanding. Caisson construction requires precise adjustments in multiple directions, especially during installation and lowering. These deviations can lead to varying degrees of misalignment and tilt. These deviations not only impact construction quality and safety but can also adversely affect the surrounding environment.

[0003] Traditional methods for correcting caisson deviation typically rely on manual measurement and adjustment, using traditional instruments such as levels and plumb gauges for real-time monitoring, combined with manual operation to correct the caisson's deviation. While this method can achieve a certain degree of correction effectiveness in the short term, it has many drawbacks. For example, manual intervention is prone to errors, and the response speed during the correction process is slow, making it impossible to accurately control and adjust the caisson's movement in real time. Furthermore, manual operation is inefficient, and operators have limited ability to respond to caisson changes, which can easily extend construction periods and increase costs. Summary of the Invention

[0004] The technical problem to be solved by the present invention is to provide a method and system for correcting deviation during the entire process of large-scale caisson construction, so as to solve the problems mentioned in the background technology.

[0005] In order to achieve the above object, the present invention adopts the following technical solutions:

[0006] A method for correcting deviation during the entire process of large-scale caisson construction comprises the following steps:

[0007] S1: Install at least three position sensors on the caisson to monitor the position information of at least three points on the caisson that are not located in the same straight line;

[0008] S2: The position information obtained by the three position sensors is input into the computer in real time. The computer decomposes the movement of the caisson within the time step T into:

[0009] 3) vertical translation;

[0010] 4) Rotation along the horizontal axis;

[0011] S3: Based on the orientation information of the horizontal rotation axis, determine the two locations farthest from both sides of the horizontal rotation axis as the main force application points for correction. The correction process at least includes applying downward pressure to the main force application points that are tilted upward.

[0012] In a preferred embodiment, the computer obtains the vertical translation in the following manner:

[0013] The first spatial position of the caisson at the beginning of the simulation time step T;

[0014] The second spatial position of the caisson at the end of the simulation time step T;

[0015] Calculate the spatial point with the smallest horizontal coordinate change when transforming from the first spatial position to the second spatial position as the vertical translation point, and obtain the coordinates (x1, y1, z1) of the vertical translation point at the beginning of step T and the coordinates (x2, y2, z2) at the end of step T, and use the vector (0, 0, z2-z1) as the vertical translation.

[0016] In a preferred embodiment, a cylindrical area is drawn with the central axis of the caisson as the central axis and a radius of a preset value R, and the vertical translation point is found in the cylindrical area.

[0017] In a preferred embodiment, the step of obtaining the horizontal rotation axis includes:

[0018] 1) Setting the horizontal axis of rotation to pass through the vertical translation point;

[0019] 2) Assume that the direction of the horizontal axis in the horizontal plane is (x0, y0), and assume that the rotation angle in the time step T is θ;

[0020] 3) Solve (x0, y0) and θ so that the change of the caisson within time T predicted by the rotation is closest to the actual change.

[0021] In a preferred embodiment, in step 3), the following steps are specifically included:

[0022] Set the loss function to:

[0023]

[0024] where ΔP p,i is the displacement vector of point i in time T predicted based on (x0, y0) and θ;

[0025] ΔP a,i is the actual displacement vector of point i in time T;

[0026] n is the number of reference points selected;

[0027] By minimizing the loss function, we can solve (x0, y0) and θ.

[0028] In a preferred embodiment, the method further includes setting PID for load control, wherein the input of the PID is the angle α between the caisson axis and the horizontal plane, and the correcting force is output according to the determined main force application point, with α approaching 90 degrees as the control target of the PID.

[0029] In a preferred embodiment, during the deviation correction process, multiple loads are set on the side where the horizontal rotation axis is tilted, and the load forces of the multiple loads increase linearly with increasing distance from the horizontal rotation axis.

[0030] Therefore, the present invention also discloses a large-scale caisson construction whole process correction control system, comprising:

[0031] At least three position sensor modules, used to monitor the position information of at least three points on the caisson that are not located in a straight line;

[0032] A computer processing module is used to receive and process in real time the position information from the position sensor module, and decompose the movement of the caisson within the time step T into a vertical translation and a rotation along the horizontal axis;

[0033] a correction decision module, configured to determine, based on the horizontal axis orientation information output by the computer processing module, two locations farthest from either side of the horizontal axis as main force application points for correction;

[0034] The correction execution module is used to correct the caisson according to the instructions of the correction decision module, at least including applying downward pressure to the main force application point that is tilted upward.

[0035] In a preferred embodiment, the correction execution module also includes any one or two of the following: 1) a reaction pile inserted into the ground, and a pull rope between the reaction pile and the caisson; 2) a water tank arranged at the top of the caisson, which adjusts the load pressure by adjusting the water injection volume.

[0036] In a preferred embodiment, the correction decision module also includes a PID controller, wherein the input of the PID is the angle α between the caisson axis and the horizontal plane, and the correction force is output according to the determined main force application point, with α tending to 90 degrees as the control target of the PID.

[0037] The advantage of the present invention over the prior art is that the present invention can accurately obtain the motion information of the caisson through sensor measurement and computer simulation calculation. By utilizing the velocity decomposition of the rigid body, the quantitative information of the deflection can be decomposed, and force can be applied at the position of the maximum torque of the caisson deflection. By utilizing the lever principle, the precise correction of the caisson can be achieved more effectively. Such a correction process can be accurately calculated once every time step T, so that more precise control can be achieved. Combined with PID control technology, the accuracy of the correction control can be further improved by real-time feedback of the caisson motion error. Compared with traditional manual adjustment, the present invention can greatly improve the accuracy and efficiency through computer intelligent control, significantly shorten the construction period, and reduce construction risks. BRIEF DESCRIPTION OF THE DRAWINGS

[0038] Figure 1 is a flow chart of the method of the present invention;

[0039] Figure 2 is a schematic diagram of the method of the present invention;

[0040] Figure 3 Schematic diagram of the system of the present invention. DETAILED DESCRIPTION

[0041] The specific embodiments of the present invention will be described below with reference to the accompanying drawings.

[0042] like Figure 1 The present invention provides a method for correcting deviation during the entire process of large-scale caisson construction, comprising the following steps:

[0043] S1: Install at least three position sensors on the caisson to monitor the position information of at least three points on the caisson that are not located in the same straight line;

[0044] S2: The position information obtained by the three position sensors is input into the computer in real time. The computer decomposes the movement of the caisson within the time step T into:

[0045] 1) Vertical translation;

[0046] 2) Rotation along a horizontal axis;

[0047] S3: Based on the orientation information of the horizontal rotation axis, determine the two locations farthest from both sides of the horizontal rotation axis as the main force application points for correction. The correction process at least includes applying downward pressure to the main force application points that are tilted upward.

[0048] More specifically, the computer obtains the vertical translation in the following manner:

[0049] The first spatial position of the caisson at the beginning of the simulation time step T;

[0050] The second spatial position of the caisson at the end of the simulation time step T;

[0051] Calculate the spatial point with the smallest horizontal coordinate change when transforming from the first spatial position to the second spatial position as the vertical translation point, and obtain the coordinates (x1, y1, z1) of the vertical translation point at the beginning of step T and the coordinates (x2, y2, z2) at the end of step T, and use the vector (0, 0, z2-z1) as the vertical translation.

[0052] In most cases, the movement of the caisson is both downward and tilted. In a special case, for example, the entire caisson rotates around only one point, such as Figure 2 As shown, the vertical translation point does not move in the horizontal direction nor in the vertical direction, such as the central black dot in the figure. In some cases, for convenience, the center of gravity can be directly used as the vertical translation point.

[0053] The reason why the above process is decomposed into vertical downward translation and horizontal rotation is also due to the special situation of the caisson. The main reason for the caisson's deflection is the unevenness of the bottom area. Therefore, a certain area often sinks downward prematurely, and there is a slight rotation around the vertical direction. Therefore, directly setting the rotation of the horizontal axis can greatly reduce the amount of calculation.

[0054] Typically, the vertical translation point can be found near the central axis. This is because the caisson, due to its positional constraints on both sides, generally rotates around its center, so the vertical translation point is generally located in the central area. To do this, draw a cylindrical area with the central axis of the caisson and a radius of a preset value, R, and search for the vertical translation point within this cylindrical area.

[0055] In a more specific embodiment, the computer may use a region-by-region comparison method, which is relatively fast:

[0056] The space occupied or surrounded by the caisson (in the preferred embodiment described above, a cylindrical area can be drawn with the radius as the preset value R) can be divided into multiple small areas. These small areas can be implemented by simple grid division. Each grid unit contains multiple points. The specific method of grid division can be selected according to the size and accuracy requirements of the caisson monitoring area. For example, if the area is large, it can be divided into a coarser grid; if a more precise search is required, it can be divided into a smaller grid.

[0057] In each grid cell, we select a point as a representative point and calculate its horizontal change. Then, we compare all grid cells and select the one with the smallest change. This is equivalent to finding the area with the smallest horizontal change in the entire space.

[0058] Once we have found the area with the least variation, we can further refine that area:

[0059] The area is divided into smaller grids again and the same comparison is continued.

[0060] In the refined area, the horizontal change of each point is calculated, and the point with the smallest horizontal change is selected.

[0061] This process can be repeated, gradually narrowing the search range until the point of minimum change is found.

[0062] In some cases, multiple similar points may appear, making it difficult to directly identify the point with the smallest change. To avoid this, after selecting the region with the smallest change, we can apply a local optimization algorithm to further identify the optimal point. For example, we can use gradient descent or least squares methods to fine-tune the selected points to further optimize the results.

[0063] Furthermore, the specific steps for solving the rotation part include the following:

[0064] 1) Setting the horizontal axis of rotation to pass through the vertical translation point;

[0065] 2) Assume that the direction of the horizontal axis in the horizontal plane is (x0, y0), and assume that the rotation angle in the time step T is θ;

[0066] 3) Solve (x0, y0) and θ so that the change of the caisson within time T predicted by the rotation is closest to the actual change.

[0067] The solution method can be as follows:

[0068] Set the loss function to:

[0069]

[0070] where ΔP p,i is the displacement vector of point i in time T predicted based on (x0, y0) and θ;

[0071] ΔP a,i is the actual displacement vector of point i in time T;

[0072] n is the number of reference points selected;

[0073] By minimizing the loss function, we can solve (x0, y0) and θ.

[0074] The selection of points can be varied. For example, the positions of three sensors can be directly used as reference points for solving the problem.

[0075] In the above solution steps, the displacement vector ΔP p,i and ΔP a,i In the above equation, the superposition of translation vectors can be included or not, because the translation vector disappears in the loss function through the minus sign, and the result is the same.

[0076] More specifically, the computer can use the least squares method for calculation, and all points can be selected within the horizontal plane, which will be faster.

[0077] During calculation, different calculation strategies can also be adopted. For example, the orientation of (x0, y0) can be divided into multiple regions from 0 to 360°. In each region, a specific orientation is selected to calculate the optimal θ value to minimize the value of the loss function. Then, the values ​​of the loss functions of the orientations of different regions are compared, and the region corresponding to the minimum value is selected. The region is further divided into multiple regions to gradually approach the optimal (x0, y0) and θ values.

[0078] More specific:

[0079] For example, every 30° is an interval. Within each interval, the (x0, y0) direction is fixed, and the corresponding θ value is quickly solved through single-variable optimization to minimize the loss function. After completing the full-circle scan, the interval with the smallest loss value is selected as the fine search range, and the interval is further subdivided into smaller angle steps (such as 5°) to repeat the optimization process. This hierarchical search mechanism can not only avoid falling into local optimality, but also significantly reduce computational complexity. For each determined axis direction, the solution of the θ angle is transformed into a root-finding problem of a one-variable nonlinear equation, and the Newton iteration method can be used to quickly converge.

[0080] After obtaining the preliminary parameters, the system starts the global optimization process. The best (x0, y0, θ) obtained by the hierarchical search is used as the initial value, and the Levenberg-Marquardt algorithm is called for joint optimization. This algorithm balances the advantages of gradient descent and Gauss-Newton method by dynamically adjusting the damping factor, improving the calculation speed while ensuring convergence stability. During the optimization process, the system calculates the Jacobian matrix in real time: partial derivatives of x0, y0, and θ are calculated respectively to quantify the influence of each parameter adjustment on the loss function. The analytical expression of the Jacobian matrix can be obtained by differentiating the rotation matrix. For example, the displacement change caused by a small change in the direction of the axis of rotation can be calculated by vector cross product.

[0081] In order to enhance the robustness of the algorithm, a multiple verification mechanism can be introduced. After the parameter solution is completed, the predicted displacement is cross-validated with the measured data of all sensors (including redundant sensors that are not involved in the optimization). If there is a significant deviation, the iterative reweighted least squares method is triggered to eliminate interference by reducing the weight coefficient of the abnormal data point. At the same time, sometimes the parameter constraints can be set in combination with the mechanical characteristics of the caisson structure: the direction of the axis of rotation must be within the range of ±15° of the central axis, and the absolute value of the angle θ does not exceed 5°. These prior knowledge are integrated into the optimization process through the Lagrange multiplier method to prevent physically unreasonable solutions.

[0082] Of course, there are other solutions. Here we only give an example to illustrate that such calculation is feasible.

[0083] In a more specific embodiment, the time step T can be dynamically adjusted according to the moving speed of the caisson; the moving speed of the caisson can be approximately expressed by the moving speed of the center of gravity. The faster the caisson moves, the smaller the time step T can be selected, so as to grasp the fine deflection of the caisson during the movement process.

[0084]

[0085] Among them, k1 and k2 are adjustment parameters, which can be selected according to specific circumstances.

[0086] After gaining a more accurate understanding of the caisson's movement, by collecting historical caisson motion data (including position, vertical speed, horizontal rotation angular velocity, etc.), a supervised learning algorithm can be used to establish a prediction model to predict the caisson's motion state at the next moment, thereby optimizing the time step and control signal.

[0087] In other embodiments, the data of the movement situation can also be acquired by other methods. The specific implementation is as follows:

[0088] Step 1: Secure at least one camera at the top of the caisson, near the edge, with the lens facing outward and the field of view covering an area containing stable reference points, such as shoreside buildings, trees, or distant landmarks. Use industrial-grade cameras with a resolution of at least 1920×1080 pixels and a frame rate of at least 30 frames per second. Fasten them to the caisson surface with bolts or welded brackets to ensure they remain stable during caisson movement. The cameras are housed in a waterproof and dustproof casing with an IP66 rating or higher. An LED fill light or infrared module is optional for low-light environments. A data transmission cable connects to the processing unit to ensure real-time video streaming.

[0089] Step 2: The camera continuously captures the external environment at fixed intervals (e.g., 0.1 seconds), generating a time-stamped image sequence that is transmitted to the processing unit. The processing unit preprocesses each image frame, identifying at least three feature points of stable reference objects. These feature points are parts of objects with high contrast or geometric significance in the image, such as building corners or tree trunk edges. This recognition process uses a corner detection algorithm or feature extraction algorithm to extract the 2D pixel coordinates of each feature point in the image plane.

[0090] Step 3: For two adjacent image frames, the processing unit calculates the change in coordinates of each feature point, deriving the displacement of each point. Using pre-calibrated camera parameters (including focal length and lens distortion), this pixel-by-pixel change is converted into a displacement of actual physical distance. If the distance to the reference object is unknown, a rough estimate can be made based on the geometric relationship between multiple image frames. The resulting displacement data reflects the relative motion of the reference object with respect to the camera, directly corresponding to the change in the caisson's posture.

[0091] Step 4: Decompose the motion state of the caisson into two parts: vertical translation and horizontal rotation according to the displacement of the reference object:

[0092] a) Vertical translation extraction:

[0093] The processing unit calculates the average displacement of all feature points, which reflects the overall movement of the camera. The size and positive and negative changes in the vertical direction represent the up and down translation distance of the caisson, and the changes in the horizontal direction may correspond to drift.

[0094] b) Horizontal rotation extraction:

[0095] Subtracting the average value from the displacement of each feature point yields a relative displacement, reflecting the caisson's rotational component. Based on the 2D assumption, these relative displacements are used to approximate the direction and center of rotation, and the angle of rotation is estimated by the distance each point moves relative to the center. If 3D analysis is required, the perspective relationship of multiple image frames is combined to infer the direction and angle of rotation about the horizontal axis.

[0096] Step 5: The processing unit outputs the caisson motion parameters at fixed intervals, including the vertical translation distance, the position of the rotation axis, and the rotation angle, for example, "translation -0.05 meters, rotation 1 degree around the X-axis". According to the direction of the rotation axis, the two side points on the caisson surface farthest from the rotation axis are determined as the correction force points. If the reference object displacement shows that one side has moved upward, then that side is the tilted area, and downward pressure is applied by the hydraulic jack, and the magnitude of the pressure is proportional to the rotation angle (for example, 10 tons of force is applied for every 1 degree, and the specific adjustment is based on the quality of the caisson). The camera continuously captures images and updates the motion parameters in real time until the angle is reduced to a preset threshold (such as 0.5 degrees).

[0097] For example, a camera is mounted on the edge of a 10m x 10m caisson, its lens pointed toward the shore structure, with a 0.1-second interval between shots. The first frame identifies the coordinates of three feature points, and the second frame shows these points undergoing slight movement. The calculated average displacement corresponds to a translational shift, resulting in a rotation angle of approximately 0.8 degrees, with the axis of rotation approximately horizontal. During correction, 8 tons of pressure is applied to the tilted side, and after 5 seconds, the angle is reduced to 0.3 degrees.

[0098] The advantage of this embodiment is that a single camera is used to realize caisson rotation detection, the sensor has a simple structure, is easy to install, is low cost, the visual data is intuitive, it is adaptable to various construction environments, and the processing is highly real-time, supporting rapid correction.

[0099] In a more specific embodiment, the method also includes setting PID for load control, wherein the input of the PID is the angle α between the caisson axis and the horizontal plane, and the correcting force is output according to the determined main force application point, with α tending to 90 degrees as the control target of the PID.

[0100] In a more specific embodiment, during the correction process, multiple loads are placed on one side of the tilted horizontal axis, with the loads increasing linearly with distance from the horizontal axis. For example, water tanks can be placed at multiple points on the top of the caisson, with the loads increasing linearly with distance from the horizontal axis. Based on the axis's orientation as measured within a time step T, the water tank loads can be varied in real time. This provides a more flexible approach for measuring and changing loads simultaneously.

[0101] In another embodiment, winches or load-applying devices can be pre-installed at multiple points. Using the rotation axis orientation measured within the time step T, loads can be applied in real time at key correction points. This approach, which allows for real-time and precise control of load points, is more efficient.

[0102] Therefore, if Figure 3 As shown, the present invention actually also discloses a large-scale caisson construction process correction control system, including:

[0103] At least three position sensor modules, used to monitor the position information of at least three points on the caisson that are not located in a straight line;

[0104] A computer processing module is used to receive and process in real time the position information from the position sensor module, and decompose the movement of the caisson within the time step T into a vertical translation and a rotation along the horizontal axis;

[0105] a correction decision module, configured to determine, based on the horizontal axis orientation information output by the computer processing module, two locations farthest from either side of the horizontal axis as main force application points for correction;

[0106] The correction execution module is used to correct the caisson according to the instructions of the correction decision module, at least including applying downward pressure to the main force application point that is tilted upward.

[0107] Among them, the correction execution module also includes any one or two of the following: 1) a reaction pile inserted into the ground, and a pull rope between the reaction pile and the caisson; 2) a water tank arranged at the top of the caisson, which adjusts the load pressure by adjusting the water injection volume.

[0108] Similarly, the correction decision module also includes a PID controller, wherein the input of the PID is the angle α between the caisson axis and the horizontal plane, and the correction force is output according to the determined main force application point, with α approaching 90 degrees as the control target of the PID.

[0109] The above description is only a preferred specific embodiment of the present invention, but the scope of protection of the present invention is not limited thereto. Any technician familiar with the technical field, within the technical scope disclosed by the present invention, who makes equivalent replacements or changes based on the technical solution and inventive concept of the present invention, should be covered by the scope of protection of the present invention.

Claims

1. A method for correcting deviation during the entire process of large-scale caisson construction, characterized in that: The steps include: S1: Install at least three position sensors on the caisson to monitor the position information of at least three points on the caisson that are not located in the same straight line; S2: The position information obtained by the three position sensors is input into the computer in real time. The computer decomposes the movement of the caisson within the time step T into: 1) Vertical translation; 2) Rotation along a horizontal axis; S3: Based on the orientation information of the horizontal rotation axis, determining two locations farthest from both sides of the horizontal rotation axis as main force application points for correction, wherein the correction process at least includes applying downward pressure to the main force application points that are tilted upward; Wherein, a cylindrical area is drawn with the central axis of the caisson as the central axis and a radius of a preset value R, and a vertical translation point is found within the cylindrical area; The computer obtains the vertical translation in the following way: The first spatial position of the caisson at the beginning of the simulation time step T; The second spatial position of the caisson at the end of the simulation time step T; Calculate the spatial point with the smallest horizontal coordinate change when transforming from the first spatial position to the second spatial position as the vertical translation point, obtain the coordinates (x1, y1, z1) of the vertical translation point at the beginning of step length T and the coordinates (x2, y2, z2) at the end of step length T, and use the vector (0, 0, z2-z1) as the vertical translation; The steps to obtain the horizontal axis include: 1) Setting the horizontal axis of rotation to pass through the vertical translation point; 2) Assume that the direction of the horizontal axis in the horizontal plane is (x0, y0), and assume that the rotation angle in the time step T is θ; 3) Solve (x0, y0) and θ so that the change of the caisson within time T predicted by the rotation is closest to the actual change; In step 3), the following steps are specifically included: Set the loss function to: in is the displacement vector of point i in time T predicted based on (x0, y0) and θ; is the actual displacement vector of point i in time T; n is the number of reference points selected; By minimizing the loss function, we can solve (x0, y0) and θ.

2. The method for correcting the deviation of the entire caisson construction process according to claim 1 is characterized in that: The method further includes setting PID for load control, wherein the PID input is the angle α between the caisson axis and the horizontal plane, and the corrective force is output according to the determined main force application point, with α approaching 90 degrees as the PID control target.

3. The method for correcting the deviation of the entire large-scale caisson construction process according to claim 1 is characterized in that: During the deviation correction process, multiple loads are set on the side where the horizontal rotation axis is tilted, and the load forces of the multiple loads increase linearly with the increase of the distance from the horizontal rotation axis.

Citation Information

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