Three-axis leveling method based on center adjustment

Abstract

The invention provides a three-axis leveling method based on center adjustment, and relates to the technical field of workpiece table leveling, and the method comprises the following steps: obtaining the initial position of a workpiece table, establishing a space rectangular coordinate system, obtaining space parameters, obtaining the heights of three connection points of a motion axis and the workpiece table, and calculating to obtain a theoretical inclination value of the workpiece table. Calculating the inclination variation according to the theoretical inclination value and the real-time inclination value, calculating the shaft variation according to the space parameters and the inclination variation, calculating the leveling height and the leveling absolute value, and controlling the motion shaft motor to move at a low speed according to the leveling absolute value to drive the workpiece table so that the connection point reaches the leveling absolute position. According to the technical scheme, motion errors caused by mechanical friction and the like are eliminated, the relative level of the workpiece table to a reference system is guaranteed, the method has high accuracy, physical implementation is simple, the algorithm is short, operation is convenient and rapid, and high stability and accuracy of leveling of the workpiece table become possible.

Description

Technical Field

[0001] This invention relates to the field of workpiece stage leveling technology, and in particular, to a three-axis leveling method based on center adjustment. Background Technology

[0002] A workpiece stage is a mechanical plane arranged in multiple axes in space within automated industrial equipment. Such a mechanical plane often needs to maintain a high degree of relative horizontality with respect to an ideal plane of reference (usually the ground). Achieving relative horizontality through leveling ensures the accuracy of workpiece processing and inspection, uniform force distribution, reduces cumulative errors caused by physical factors such as motor movement and equipment wear, and improves operational stability and reliability. Therefore, workpiece stage leveling technology has become an important tool in many industrial fields, including objective lens inspection, coating exposure, and interferometric scanning.

[0003] The leveling process of a workpiece stage involves microscopically controlling the movement data of its axes to achieve leveling relative to a reference physical plane by adjusting the spatial position of these axes. Currently, there is no industry standard for maintaining a fixed point for leveling in space, and the spatial layout of different workpiece stages varies. During leveling, accuracy errors can occur due to motor performance or idling, making it difficult for the workpiece stage to achieve absolute horizontality relative to the ideal plane. This severely restricts high-precision industrial environments such as object inspection and image recognition. Most current leveling equipment uses air flotation devices to reduce friction through pneumatics or utilizes ball joints for leveling. While these methods are simple to operate, the size, weight, and cost of the equipment itself place a significant burden on industrial equipment, hindering automated control. The size and weight of most industrial equipment are already considerable; using leveling equipment linearly related to the size and weight of industrial equipment would make its use, maintenance, and transportation extremely complex. Summary of the Invention

[0004] The purpose of this invention is to provide a three-axis leveling method based on center adjustment, which solves the problems of volume redundancy and weight redundancy caused by the additional leveling system in the prior art.

[0005] To achieve the aforementioned objective, this invention provides a three-axis leveling method based on center adjustment. The method includes: Step S1, obtaining the initial position of the workpiece stage and controlling all motion axes to rise simultaneously and uniformly, so that the entire workpiece stage rises to a safe distance relative to its initial position; Step S2, establishing a spatial rectangular coordinate system based on the motion axes of the workpiece stage; Step S3, obtaining spatial parameters based on the connection points between the motion axes and the workpiece stage and a pre-set center reference point of the workpiece stage; Step S4, measuring the real-time tilt value of the workpiece stage around the x-axis and the real-time tilt value around the y-axis using an inclinometer; Step S5, obtaining the height of the three connection points between the motion axes and the workpiece stage based on motor feedback; Step S6, adjusting the height of the three connection points between the motion axes and the workpiece stage and the spatial parameters... Step S7: Calculate the theoretical tilt values ​​of the workpiece stage around the x-axis and y-axis; Step S8: Calculate the tilt change based on the theoretical tilt values ​​of the workpiece stage around the x-axis and y-axis, as well as the real-time tilt values ​​of the workpiece stage around the x-axis and y-axis; Step S9: Calculate the axis change based on the spatial parameters and the tilt change; Step S10: Calculate the leveling height based on the height of the three connection points between the motion axis and the workpiece stage, the spatial parameters, and the axis change; Step S11: Calculate the absolute leveling value based on the leveling height and the tilt change; and Step S12: Control the motion axis motor to move at low speed according to the absolute leveling value, causing the workpiece stage to reach the absolute leveling position.

[0006] The beneficial effects of this invention are as follows:

[0007] The technical solution provided by this invention allows for the leveling of the workpiece stage by controlling the motor movement, given the spatial positions of the three axes. However, in practice, due to the natural mechanical errors of the motion axes, these errors accumulate during motor movement. Therefore, reconstructing the coordinate system based on leveling reduces the accumulated motion error. Simultaneously, the motor movement causes instability and inaccuracies due to changes in height.

[0008] For a fixed-point leveling workpiece stage, this invention determines the spatial parameters according to the actual situation and requirements, reducing the errors caused by mechanical friction and motor backlash, and ensuring the stability and accuracy of the three-axis motion of the workpiece stage at a fixed point in space, as well as its parallelism to the reference physical plane.

[0009] The technical solution provided by this invention eliminates motion errors caused by mechanical friction and other problems, and ensures the relative horizontality of the workpiece stage with respect to the reference system, with high accuracy. Furthermore, its physical implementation is relatively simple, the algorithm is concise, and the operation is convenient and quick, making it possible to achieve high stability and accuracy in leveling the workpiece stage. Attached Figure Description

[0010] The accompanying drawings are provided to further illustrate embodiments of the present invention and form part of the specification. They are used together with the following detailed description to explain the embodiments of the present invention, but do not constitute a limitation thereof. In the drawings:

[0011] Figure 1 This is a flowchart of the three-axis leveling method based on center adjustment provided by the present invention;

[0012] Figure 2 This is a direction indicator diagram provided by the present invention;

[0013] Figure 3 This is a spatial location and parameter indication diagram provided by the present invention;

[0014] Figure 4 This is a data graph of the inclinometer provided by the present invention. Detailed Implementation

[0015] To make the objectives, technical solutions, and advantages of this invention clearer, the invention will be further described in detail below with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are merely illustrative and not intended to limit the invention. Furthermore, the technical features involved in the various embodiments of this invention described below can be combined with each other as long as they do not conflict with each other. To achieve the above objectives, this invention adopts the following technical solution.

[0016] In this invention, the center refers to an imaginary fixed point on the workpiece stage (e.g., a horizontal workpiece stage) that maintains a constant height during rotation and movement. The workpiece stage may experience significant elevation angle and height fluctuations during axis movement, leading to a series of safety issues and greatly affecting the stability of leveling. A suitable center can be selected based on the actual situation. To reduce elevation angle and height fluctuations during leveling and ensure uniform force distribution across the three axes, the center of gravity is typically used as the leveling center. For three axes with defined positions and homogeneous, regular workpiece stages, the spatial parameters of the uniquely determined center can often be calculated directly using a physical model. For non-homogeneous or irregular workpiece stages, simulation is required to obtain their spatial parameters. During leveling, the spatial parameters of the center are known and confirmed.

[0017] Figure 1 This is a flowchart of the triaxial leveling method based on center adjustment provided by the present invention, as follows: Figure 1 As shown, the method includes:

[0018] Step S1: Obtain the initial position of the workpiece stage, control all motion axes to rise at a constant speed simultaneously, so that the entire workpiece stage rises to a safe distance relative to the initial position.

[0019] Step S2: Establish a spatial rectangular coordinate system based on the motion axes of the workpiece stage;

[0020] Step S3: Obtain spatial parameters based on the connection point between the motion axis and the workpiece stage and the pre-set center reference point of the workpiece stage;

[0021] Step S4: Obtain the real-time tilt value of the workpiece table around the x-axis and the real-time tilt value around the y-axis by measuring the tilt meter.

[0022] Step S5: Obtain the height of the three connection points between the motion axis and the workpiece stage based on motor feedback;

[0023] Step S6: Calculate the theoretical tilt value of the workpiece stage around the x-axis and the theoretical tilt value around the y-axis based on the height and spatial parameters of the three connection points between the motion axis and the workpiece stage.

[0024] Step S7: Calculate the tilt change based on the theoretical tilt values ​​of the workpiece stage around the x-axis and y-axis, as well as the real-time tilt values ​​of the workpiece stage around the x-axis and y-axis.

[0025] Step S8: Calculate the axis change based on the spatial parameters and the tilt change.

[0026] Step S9: Calculate the leveling height based on the height of the three connection points between the motion axis and the workpiece stage, the spatial parameters, and the axis change.

[0027] Step S10: Calculate the absolute value of the leveling based on the leveling height and the change in tilt.

[0028] Step S11: Control the motion axis motor to move at low speed according to the absolute value of the leveling so that the connection point reaches the absolute position of the leveling.

[0029] Before step S1, the ideal plane is determined based on the spatial layout, and the method for obtaining the pitch and roll states of the ideal plane is determined.

[0030] Before step S1, the system status is read. Specifically, the servo system status is read to ensure that the system is running normally. In this embodiment, the horizontal ground is determined to be the ideal plane.

[0031] In step S1, the initial position of the workpiece stage is obtained. This initial position refers to the lowest point of the workpiece stage in space or the desired initial zero position, which is usually determined by electrical limit switches. After the workpiece stage stabilizes, the motion axis of the workpiece stage is controlled to rise at a constant speed, so that the entire workpiece stage rises to a safe distance relative to the initial position. The safe distance here refers to the minimum distance at which the workpiece stage will not touch the initial position under any pitch and roll posture, and is usually set based on engineering experience.

[0032] Figure 2 This is a direction indicator diagram provided by the present invention, such as... Figure 2The plane pitch direction is the P direction, rotating around the y-axis in the spatial coordinate system; the roll state is the R direction, rotating around the x-axis in the spatial coordinate system.

[0033] In step S1, photoelectric limit switches can be installed at the bottom of the three motion axes of the workpiece stage as initial position detection devices, and motion motors can be installed at the three ends of the workpiece stage. The control circuit is designed with a simultaneous triggering mechanism for the three motion axes to ensure the synchronization and continuity of the motion. After the setup is complete, the three motion axes are controlled to move downwards synchronously until any axis triggers its photoelectric limit switch, at which point the movement stops.

[0034] Next, the pre-leveling step is performed: the three motion axes are controlled to rise synchronously, so that the entire workpiece table moves relative to the initial position to an empirical safe distance. In this embodiment, the safe distance is set to 200mm.

[0035] Step S2 includes: the workpiece stage has three non-collinear motion axes, namely the first motion axis, the second motion axis, and the third motion axis. The axial direction of the motion axes is perpendicular to the ideal horizontal plane and is connected to the three endpoints of the workpiece stage respectively; the projection of the line connecting the two intersection points of the second motion axis and the third motion axis with the plane of the workpiece stage on the ideal horizontal plane is the y-axis; the axis perpendicular to the ideal horizontal plane, parallel to the axial direction of the first motion axis, the second motion axis, and the third motion axis, and passing through any point on the y-axis is the z-axis, wherein the z-axis is positive when vertically upward; the axis perpendicular to the yOz plane and orthogonal to both the y-axis and the z-axis, and passing through the intersection point of the yOz plane and the y-axis is the x-axis, satisfying the right-hand rule; wherein the origin of the established spatial rectangular coordinate system is located at the selected point on the y-axis.

[0036] In step S2, when establishing a spatial rectangular coordinate system, a point on the plane where the workpiece stage is located can be selected as the center point. In this embodiment, the selected workpiece stage is an isosceles triangle, with the three motion axes connected to the three vertices of the triangle. The length of the base is 670mm (i.e., the horizontal distance between the two endpoints of the base). The vertical height from the vertex to the base is 780mm. The midpoint of this vertical height (i.e., the midpoint of the line connecting the foot of the perpendicular line drawn from the vertex to the base and the vertex) is taken as the center point of the workpiece stage plane. The projection of the straight line connecting the two intersection points of the two motion axes in the workpiece stage (the second and third motion axes in this embodiment) with the plane where the workpiece stage is located onto the ideal horizontal plane is the y-axis. The axis perpendicular to the ideal horizontal plane, parallel to the direction of the three motion axis axes, and passing through any point on the y-axis is the z-axis, where the z-axis is vertically upward as positive. The axis perpendicular to the yOz plane and orthogonal to both the y-axis and z-axis, and passing through the intersection point of the yOz plane and the y-axis, is the x-axis. A spatial rectangular coordinate system is established using the x-axis, y-axis, and z-axis.

[0037] The three-axis leveling method based on center adjustment provided by the present invention further includes: the motion axis is independently driven by a motor and its height at the connection point with the workpiece stage is fed back in real time; by controlling the lifting and lowering of the three connection points respectively, the pitch, yaw and overall height of the workpiece stage are adjusted.

[0038] Step S3 includes: determining the center of gravity of the workpiece stage as the center reference point, and the coordinates of the center reference point are... The coordinates of the connection points between the first motion axis, the second motion axis, the third motion axis and the workpiece stage are respectively: , , Define spatial parameters in a Cartesian coordinate system. , , , , , for: , , , , , .

[0039] The three motion axes in this invention include a first motion axis, a second motion axis, and a third motion axis.

[0040] In step S4, taking the experimental data from one leveling operation as an example, the inclinometer reading is as follows: Figure 4 As shown, the tilt direction is as follows Figure 2 As shown, the inclinometer reading , . Figure 4 This is a plot of inclinometer data provided by the present invention, such as... Figure 4 The tilt value R of the inclinometer can be read. 实 (Actual tilt of the horizontal workpiece stage on the x-axis) and P 实 (The actual tilt of the horizontal workpiece stage on the y-axis).

[0041] The heights of the three connection points between the motion axis and the workpiece stage obtained in step S5 are respectively , , In this embodiment, the height positions of the three connection points between the current motion axis and the workpiece stage are read as follows: , , ,in, Indicates the first axis of motion. Indicates the second axis of motion. This indicates the third axis of motion.

[0042] Step S6 includes: calculating the theoretical tilt value of the workpiece stage around the x-axis in a spatial rectangular coordinate system. as follows:

[0043] ;

[0044] In a spatial rectangular coordinate system, calculate the theoretical tilt value of the workpiece stage around the y-axis. as follows:

[0045] .

[0046] Step S7 includes: according to the formula The change in tilt of the workpiece stage around the x-axis was calculated. ,in, This is the real-time tilt value of the workpiece stage around the x-axis; according to the formula... The change in tilt of the workpiece stage around the y-axis was calculated. ,in, This is the real-time tilt value of the workpiece stage around the y-axis.

[0047] Step S8 includes:

[0048] According to the formula The axial change corresponding to the first motion axis is calculated. ;

[0049] According to the formula The axial change corresponding to the second motion axis was calculated. ;

[0050] According to the formula The axial change corresponding to the third motion axis was calculated. ;

[0051] Among them, the axial change is the distance between the connection point of the corresponding motion axis and the leveling height when the workpiece table reaches the leveling state. The difference, the shaft change includes the shaft change corresponding to the first motion axis. The axial change corresponding to the second motion axis The axial change corresponding to the third motion axis . (i takes values ​​of 1, 2, and 3) represents the distance from the connection point of each motion axis to the leveling height when the workpiece table reaches the leveling state. The difference.

[0052] Step S9 includes:

[0053] Calculate the center height based on the height of the three connection points between the motion axis and the workpiece stage and the spatial parameters. Center height Calculated using the following formula:

[0054] ;

[0055] Calculate the leveling height using the following formula. :

[0056] .

[0057] Figure 3 This is a spatial location and parameter indication diagram provided by the present invention, such as... Figure 3 As shown, in this embodiment, , , , The center height was calculated. , , , , The leveling height was calculated. and axis change , , .

[0058] According to the formula The absolute value of leveling was calculated. With i taking values ​​of 1, 2, and 3, the absolute positions of the leveling corresponding to the three motion axes are obtained as follows: , , .

[0059] The three-axis leveling method based on center adjustment provided by this invention further includes: controlling the three motion axes of the workpiece stage to descend synchronously and slowly in the negative direction until any one motion axis touches the initial position and then stops; reading the height feedback values ​​of the three motion axes at the time of stopping and recording them as the initial compensation amount. , , The initial compensation amount, which is the motor feedback value at the connection point between the motion axis and the workpiece stage, is used as the actual height feedback of the connection point.

[0060] In actual operation, the controller controls the motor to ensure that the connection point between the motion axis and the workpiece table reaches the absolute leveling position. The inclinometer readings are then read; if the inclinometer readings are all within the specified range... Fluctuation, leveling successful. Specifically, after leveling, the workpiece table is controlled to move synchronously downwards along the three motion axes until any one of the motion axes triggers its photoelectric limit switch and stops. The position readings of the three-axis motors at this time are read and recorded as the zero-return compensation amount. , , In subsequent leveling operations, this compensation amount is superimposed on the target position of each motion axis to eliminate positioning deviations caused by cumulative motion errors and improve the accuracy and efficiency of iterative leveling.

[0061] Step S9 further includes: for the motion axes in space where x and y are known and z is unknown, and the three connection points of the workpiece stage. , , The plane formed (the three connection points of the motion axis and the workpiece stage follow the motor along the motion axis; the projection values ​​of the motion axis on the x and y axes are fixed, therefore x and y are known, and z is unknown), is based on two vectors in the plane. , The cross product yields the components of the normal vector projected onto the x, y, and z axes:

[0062] ;

[0063] ;

[0064] ;

[0065] Calculate the normal vector magnitude ;

[0066] The spatial rectangular coordinate system is based on , If the projection of the connected straight lines onto the ideal plane is the y-axis, then... And there are:

[0067] ;

[0068] ;

[0069] ;

[0070] To reduce the risk of slippage in engineering, a safe upper limit for the tilt angle is set at 3°. Calculations are performed. , ;

[0071] calculate ;

[0072] definition It is the rotation angle of the workpiece stage about the y-axis. It is the rotation angle of the workpiece stage around the x-axis;

[0073] ;

[0074] ;

[0075] Establish the plane equation based on the normal vector and the connection points: ;

[0076] Assumption Let be a point in this plane, then:

[0077] ;

[0078] according to , , , , , Substituting the values, we get:

[0079] .

[0080] The above description is merely a specific embodiment of the present invention, but the scope of protection of the present invention is not limited thereto. Any transformations or substitutions that can be conceived by those skilled in the art within the technical scope disclosed in the present invention should be included within the scope of the present invention. Therefore, the scope of protection of the present invention should be determined by the scope of the claims.

[0081] The optional embodiments of the present invention have been described in detail above with reference to the accompanying drawings. However, the embodiments of the present invention are not limited to the specific details in the above embodiments. Within the scope of the technical concept of the embodiments of the present invention, various simple modifications can be made to the technical solutions of the embodiments of the present invention, and these simple modifications all fall within the protection scope of the embodiments of the present invention.

[0082] It should also be noted that the various specific technical features described in the above embodiments can be combined in any suitable manner without contradiction. To avoid unnecessary repetition, the embodiments of the present invention will not describe the various possible combinations separately.

[0083] Furthermore, various different implementations of the present invention can be combined arbitrarily, as long as they do not violate the spirit of the present invention, they should also be regarded as the content disclosed in the present invention.

Claims

1. A three-axis leveling method based on center adjustment, characterized in that, The method includes: Step S1: Obtain the initial position of the workpiece stage, control all motion axes to rise at a constant speed simultaneously, so that the entire workpiece stage rises to a safe distance relative to the initial position. Step S2: Establish a spatial rectangular coordinate system based on the motion axis of the workpiece stage; Step S3: Obtain spatial parameters based on the connection point between the motion axis and the workpiece stage and the preset center reference point of the workpiece stage; Step S4: Obtain the real-time tilt value of the workpiece table around the x-axis and the real-time tilt value around the y-axis by measuring with an inclinometer; Step S5: Obtain the height of the three connection points between the motion axis and the workpiece stage based on motor feedback; Step S6: Calculate the theoretical tilt value of the workpiece stage around the x-axis and the theoretical tilt value around the y-axis based on the height of the three connection points between the motion axis and the workpiece stage and the spatial parameters. Step S7: Calculate the tilt change based on the theoretical tilt values ​​of the workpiece stage around the x-axis and y-axis, as well as the real-time tilt values ​​of the workpiece stage around the x-axis and y-axis. Step S8: Calculate the axis change based on the spatial parameters and the tilt change. Step S9: Calculate the leveling height based on the height of the three connection points between the motion axis and the workpiece stage, the spatial parameters, and the axis change. Step S10: Calculate the absolute value of the leveling based on the leveling height and the amount of tilt change; and, Step S11: Control the motion axis motor to move at low speed according to the absolute value of the leveling so that the workpiece table reaches the absolute position of the leveling.

2. The three-axis leveling method based on center adjustment according to claim 1, characterized in that, Step S2 includes: The workpiece stage has three non-collinear motion axes, namely the first motion axis, the second motion axis and the third motion axis. The axial direction of the motion axes is perpendicular to the ideal horizontal plane and is respectively connected to the three endpoints of the workpiece stage. The projection of the line connecting the two intersection points of the second motion axis and the third motion axis with the workpiece stage plane on the ideal horizontal plane is the y-axis; The z-axis is a point perpendicular to the ideal horizontal plane, parallel to the axes of the first, second, and third motion axes, and passing through any point on the y-axis. The z-axis is positive when it is vertically upward. The x-axis is perpendicular to the yOz plane and orthogonal to both the y-axis and z-axis, and the x-axis intersects the y-axis in the yOz plane, satisfying the right-hand rule. The origin of the established spatial rectangular coordinate system is located at the selected point on the y-axis.

3. The three-axis leveling method based on center adjustment according to claim 1, characterized in that, The method further includes: the motion axis is independently driven by a motor and its height at the connection point with the workpiece stage is fed back in real time; the pitch, yaw and overall height of the workpiece stage are adjusted by controlling the lifting and lowering of the three connection points respectively.

4. The three-axis leveling method based on center adjustment according to claim 2, characterized in that, Step S3 includes: The center of gravity of the workpiece stage is determined as the central reference point, and the coordinates of the central reference point are: ; The coordinates of the connection points between the first motion axis, the second motion axis, the third motion axis and the workpiece stage are respectively... , , ;as well as, Define spatial parameters in the aforementioned Cartesian coordinate system. , , , , , for: , , , , , .

5. The three-axis leveling method based on center adjustment according to claim 4, characterized in that, The heights of the three connection points between the motion axis and the workpiece stage obtained in step S5 are respectively , , Step S6 includes: In the aforementioned spatial rectangular coordinate system, calculate the theoretical tilt value of the workpiece stage around the x-axis. as follows: ； In the spatial rectangular coordinate system, calculate the theoretical tilt value of the workpiece stage around the y-axis. as follows: 。 6. The three-axis leveling method based on center adjustment according to claim 5, characterized in that, Step S7 includes: According to the formula The tilt change of the workpiece stage around the x-axis was calculated. ,in, The real-time tilt value of the workpiece stage around the x-axis; and, According to the formula The inclination change of the workpiece stage around the y-axis was calculated. ,in, The real-time tilt value of the workpiece stage around the y-axis is given.

7. The three-axis leveling method based on center adjustment according to claim 6, characterized in that, Step S8 includes: According to the formula The axial change corresponding to the first motion axis is calculated. ; According to the formula The axial change corresponding to the second motion axis was calculated. ; According to the formula The axial change corresponding to the third motion axis was calculated. ; Wherein, the axial change is the distance between the connection point of the corresponding motion axis and the leveling height when the workpiece table reaches the leveling state. The difference, wherein the axis change includes the axis change corresponding to the first motion axis. The axial change corresponding to the second motion axis The axial change corresponding to the third motion axis .

8. The three-axis leveling method based on center adjustment according to claim 7, characterized in that, Step S9 includes: The center height is calculated based on the heights of the three connection points between the motion axis and the workpiece stage, and the aforementioned spatial parameters. The center height Calculated using the following formula: ； The leveling height is calculated using the following formula. : 。 9. The three-axis leveling method based on center adjustment according to claim 8, characterized in that, The method also includes: The three motion axes of the control workpiece stage are synchronously and slowly descended in the negative direction until any one of the motion axes touches the initial position and stops. The height feedback values ​​of the three motion axes at the time of stopping are read and recorded as the initialization compensation amount. , , The initial compensation amount, which is the motor feedback value at the connection point between the motion axis and the workpiece stage, is used as the actual height feedback of the connection point.

10. The center-based three-axis leveling method according to claim 8, characterized in that, Step S9 further includes: For the motion axes in space where x and y are known and z is unknown, and the three connection points of the workpiece stage. , , The plane formed, based on two vectors in the plane , The cross product yields the components of the normal vector projected onto the x, y, and z axes: ； ； ； Calculate the normal vector magnitude ; The spatial rectangular coordinate system is based on , If the projection of the connected straight lines onto the ideal plane is the y-axis, then... And there are: ； ； ； To reduce the risk of slippage in engineering, a safe upper limit for the tilt angle is set at 3°. Calculations are performed. , ; calculate ; definition It is the rotation angle of the workpiece stage about the y-axis. It is the rotation angle of the workpiece stage around the x-axis; ； ； Establish the plane equation based on the normal vector and the connection points: ; Assumption Let be a point in this plane, then: ； according to , , , , , Substituting the values, we get: 。

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