KR20170010205A - Diagnostic apparatus for roll - Google Patents

Diagnostic apparatus for roll Download PDF

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Publication number
KR20170010205A
KR20170010205A KR1020150101045A KR20150101045A KR20170010205A KR 20170010205 A KR20170010205 A KR 20170010205A KR 1020150101045 A KR1020150101045 A KR 1020150101045A KR 20150101045 A KR20150101045 A KR 20150101045A KR 20170010205 A KR20170010205 A KR 20170010205A
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South Korea
Prior art keywords
measurement point
roll
point display
display unit
coordinate
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KR1020150101045A
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Korean (ko)
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KR101726056B1 (en
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백선현
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주식회사 포스코
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    • BPERFORMING OPERATIONS; TRANSPORTING
    • B21MECHANICAL METAL-WORKING WITHOUT ESSENTIALLY REMOVING MATERIAL; PUNCHING METAL
    • B21BROLLING OF METAL
    • B21B38/00Methods or devices for measuring, detecting or monitoring specially adapted for metal-rolling mills, e.g. position detection, inspection of the product
    • BPERFORMING OPERATIONS; TRANSPORTING
    • B21MECHANICAL METAL-WORKING WITHOUT ESSENTIALLY REMOVING MATERIAL; PUNCHING METAL
    • B21BROLLING OF METAL
    • B21B38/00Methods or devices for measuring, detecting or monitoring specially adapted for metal-rolling mills, e.g. position detection, inspection of the product
    • B21B38/10Methods or devices for measuring, detecting or monitoring specially adapted for metal-rolling mills, e.g. position detection, inspection of the product for measuring roll-gap, e.g. pass indicators

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  • Engineering & Computer Science (AREA)
  • Mechanical Engineering (AREA)
  • Length Measuring Devices With Unspecified Measuring Means (AREA)

Abstract

The present invention provides a roll diagnosing apparatus. A roll diagnosing apparatus comprises: a measurement point display device; and a coordinate generator. The measurement point display device is attached to a rotation axis of a roll transporting a transportation target goods by rotation, and includes at least one measurement point display portion located each on a plurality of points including two points placed opposite to each other based on the rotation axis of the roll in an operation direction of the transportation target goods, and one point placed vertically based on the rotation axis in an operation direction of the transportation target goods. The coordinate generator acquires each coordinate of the measurement point display portions as a plurality of diagnosis coordinates. Accordingly, the present invention provides a roll diagnosing apparatus capable of easily diagnosing an alignment and level of a roll used in a production line to manufacture steel within a short time.

Description

[0001] DIAGNOSTIC APPARATUS FOR ROLL [0002]

BACKGROUND OF THE INVENTION 1. Field of the Invention The present invention relates to an apparatus for diagnosing the alignment and level of a roll in a cold-rolled annealing process, and more particularly, And a level can be measured at the same time, thereby reducing the measuring time and the setting time, and capable of precise measurement diagnosis.

In general, rolling mills operate line by line for each plant, but alignment of equipment is very important because they have continuity from the inlet to the outlet of the line. Fig. 1 shows the installation state of a roll used in the production line of such a steel product. The rolls 20 of the production line of the cold rolling mill act to turn the strip S, to apply tension, and so on. Therefore, if the alignment and the level of the rolls 20 are not precisely adjusted with respect to the center line 30 in the direction in which the strip S advances, the product will have a center line 30 in the direction in which the strip S advances, This causes a wave, a waveform, and a thickness variation of the strip S, and causes a problem of lowering the productivity, such as stopping the production due to plate breakage during operation . These rolls 20 need maintenance such as repair and exchange as consumables, and are managed according to a strict standard so as to optimize the progress of the strip S due to the characteristics of the continuous line.

Fig. 2 shows a diagnostic device for diagnosing the alignment and level of a conventional roll 20. 2, the alignment and level diagnosis of the conventional roll 20 is performed by simply installing a support bar connected to the weight 41, the piano line 40, the mark 50 and the rotary shaft 21, In order to diagnose the alignment and the level of the roll 20 by detecting the error value by comparing the upper and lower portions, the roll 20 should be dismantled so that the operator manually rotates the roll up, down, left, and right Therefore, a lot of people and time are wasted. In addition, since the position of the reference point is changed according to the various rolls 20, it is necessary to manufacture and diagnose every diagnosis work, resulting in poor work efficiency and time-consuming. In addition, since the accuracy of the error is high due to a high error rate, there is a waste of time such as repeating the diagnosis, and the diagnostic work requires a technique with a high degree of difficulty.

In order to solve such a problem, a magnet having an easy detachable and attachable property is incorporated in the end portion of the roll, and a CCD laser micrometer having a combination of an emitter and a light receiver, A device for measuring the perpendicularity of a roll using a measuring device capable of measuring contact with a piano wire is specifically known in, for example, " Apparatus for measuring the perpendicularity of an assembled roll (Japanese Patent Application Laid-Open No. 10-2004-0043929) ".

[Patent Document 1] Japanese Patent Application Laid-open No. 2004-0043929 (May 27, 2004)

The present invention provides a roll diagnosing apparatus capable of easily diagnosing an alignment and a level diagnosis of a roll used in a steel product production line in a short time.

Further, in a conventional method using a piano line and a weight, there is provided a roll diagnosing apparatus capable of performing precise measurement by diagnosing the alignment and level of a roll by using a three-dimensional laser measuring device for three-dimensionally measuring roll alignment and level .

The present invention also provides a roll diagnosing device capable of diagnosing the alignment and level of rolls without disassembling the rolls by providing a measurement point indicator detachable to the rotation axis of the rolls.

There is also provided a roll diagnosing apparatus capable of diagnosing the alignment and level of a roll without rotating the roll, using a measurement point indicator having three measurement point display portions on the rotation axis of the roll.

An object of the present invention is to provide an apparatus and method for transferring a product to be carried, which is attached to a rotating shaft of a roll that carries a product to be transported by rotation and which has two points opposed to each other about the rotational axis of the roll, A measurement point indicator including at least one measurement point display portion disposed at each of a plurality of points including one point in a direction perpendicular to a traveling direction; And a coordinate generator for obtaining the coordinates of each of the measurement point display units as a plurality of diagnostic coordinates.

According to one embodiment of the present invention, it is possible to easily diagnose the alignment and the level of the roll used in a steel product production line or the like in a short time.

According to an embodiment of the present invention, the alignment and level of rolls are diagnosed by a method using a three-dimensional laser measuring device for three-dimensionally aligning rolls and level in a conventional method using piano lines and weights, Measurement is possible.

Further, according to an embodiment of the present invention, by providing a detachable measurement point indicator on the rotation axis of the roll, it is possible to diagnose the alignment and the level of the roll without disassembling the roll.

According to an embodiment of the present invention, the alignment and level of the roll can be diagnosed without rotating the roll by using a measurement point indicator having three measurement point indicators on the rotation axis of the roll.

Fig. 1 is a view showing the installation state of a roll used in a steel product production line.
2 shows a conventional roll diagnostics device.
Fig. 3 shows a schematic configuration diagram of an embodiment of the roll diagnosing apparatus according to the present invention.
4 shows an example of a reference point indicator of the roll diagnosing apparatus according to the present invention.
5 schematically shows an embodiment of a measurement point indicator according to an embodiment of the present invention.
Figures 6 and 7 schematically illustrate the alignment and level states of the rolls diagnosed using the roll diagnostics device according to the present invention.

Hereinafter, a roll diagnosing apparatus according to an embodiment of the present invention will be described with reference to the drawings. In the drawings, the same or similar reference numerals are given to the same or similar components.

FIG. 3 shows a schematic diagram of a roll diagnostics apparatus 1000 according to an embodiment of the present invention. 3, the roll diagnostics apparatus 1000 according to an embodiment of the present invention includes a roll (not shown) for conveying a conveyed object (for example, the strip S shown in FIG. 1) (20) detachably attached to an axis of rotation of the roll (20) and used to generate a plurality of diagnostic coordinates for diagnosing the alignment and level of the roll (20) Generator (300). The roll diagnostics apparatus 1000 is mounted on a rail 60 extending along the direction of travel of the article to be transported (y direction in the drawing) on the side of the roll 20, And a reference point indicator 100 used to generate reference coordinates for the reference point.

Although various rolls used in the steel manufacturing process and carrying a steel product are described as an example of the roll 20 to be diagnosed in the roll diagnostics apparatus 1000 according to an embodiment of the present invention, But may be applied to any kind of roll that carries the article to be transported by rotation or to any roll that rotates without carrying the article. Such rolls are used to transport steel products, such as strips, for example, and misalignment and level misalignment of the rotation axis of the rolls may occur due to continued use. Such errors may cause phenomena such as stealing of steel products. Therefore, it is desirable to diagnose the alignment and level of such rolls regularly or irregularly.

In order to diagnose a roll, first, a reference point indicator 100 and a coordinate generator 300 are used to generate reference coordinates. 4 shows an example of a reference point indicator of the roll diagnosing apparatus according to the present invention. As shown, the reference point indicator 100 is used to generate reference coordinates for the roll 20, and more specifically, reference coordinates for the measurement coordinates of the measurement point indicator 200, which will be described later, attached to the roll 20 .

FIG. 4 schematically shows an embodiment of a reference point indicator 100 according to the present invention. As shown in FIG. 4, the reference point indicator 100 includes a base 110, an extension 120, and a reference point display 130. The base 110 is disposed on the side of the roll 20 and is attached to the rail 60 arranged side by side with the traveling direction of the article to be transported. The base 110 has an "a" shape and is provided with a coupling member, for example a magnet, to facilitate attachment with the rail 60. The shape of the base 110 and the coupling member may have other configurations.

Next, the extension portion 120 is extended from the base 110 and disposed. The extension 120 may be fixedly attached to the base 110 or may be rotatably attached. The reference point display part 130 is attached to one end of the extension part 120. The coordinate generator 300 shown in FIG. 3 generates coordinates of the reference point display unit 130. For example, the coordinate generator 300 may be a 3D laser light receiver, and the reference point display unit 130 may be a prism. It is possible to generate the light wave from the laser light emitter toward the prism and generate the coordinates of the reference point display section 130 by receiving the light wave reflected by the prism by the laser light emitter.

A method of generating reference coordinates using the reference point indicator 100 and the coordinate generator 300 will be described in detail as follows. First, the reference point indicator 100 is attached to the first point of the rail 60. Next, coordinates of the reference point display unit 130 at the first point are generated using the coordinate generator 300, and the coordinates are set as the first reference coordinates. For example, this reference coordinate may be (0, 0, 0). In this case, the x direction is the rotation axis direction of the roll, the y direction is the direction in which the article to be transported advances, and the height direction of the roll in the z direction.

Then, the reference point indicator 100 is moved to the second point of the rail 60 and attached, and the coordinates of the reference point display unit 130 at the second point are generated using the coordinate generator 300, 2 reference coordinates. The coordinate generator 300 is then calibrated by making the x and z coordinates of the second reference coordinate zero and making the y coordinate the distance between the first point and the second point. Through this process, reference coordinates for generating accurate coordinates of the diagnostic coordinates described later can be generated.

Next, generation of diagnostic coordinates using the measurement point indicator 200 and the coordinate generator 300 will be described. 5 schematically shows a measurement point indicator 200 according to an embodiment of the present invention. 5, the measurement point indicator 200 includes a mounting portion 210, three support portions 220a, 220b and 220c, three measurement point display portions 230a, 230b and 230c (hereinafter, (Also referred to as a second measurement point display section 230b, and a third measurement point display section 230c). The measurement point indicator 200 is mounted on the rotary shaft of the roll 20 through the mounting portion 210. For example, the rotary shaft may have a cylindrical shape, and the mounting portion 210 may include a recessed portion having a shape corresponding to the shape of the rotary shaft by surrounding the rotary shaft. Therefore, the measuring point indicator 200 can be mounted on the rotating shaft of the roll 20 by inserting the rotating shaft into the concave portion of the mounting portion 210. In order to facilitate mounting of the measurement point indicator 200 on the rotation axis, the mounting portion 210 may include a magnet.

Three supports 220a, 220b, and 220c extend radially from the mounting portion 210. [ The three supports 220a, 220b, and 220c are spaced 90 degrees from each other. The two support portions 220a and 220c of the three support portions are arranged to face each other substantially parallel to the traveling direction of the article S to be transported and the other support portion 220b is disposed to face the two support portions 220a and 220c As shown in Fig. Therefore, the three support portions 220a, 220b, and 220c have a "⊥" shape.

Measurement points 230a, 230b and 230c are arranged at the ends of the supports 220a, 220b and 220c, respectively. Accordingly, the measurement point display 200 may include three measurement point display units, i.e., a first measurement point display unit 230a, a second measurement point display unit 230b, and a third measurement point display unit 230c. The three measurement point display portions 230a, 230b, and 230c are arranged so that the distances from the mounting portion 210 are all the same. Accordingly, the measurement point display units 230a, 230b, and 230c are disposed at 90 degrees apart from each other with respect to the mounting unit 210. The measurement point display units 230a, 230b, and 230c include two measurement point display units, The second measurement point display part 230b and the third measurement point display part 230c are disposed opposite to each other substantially parallel to the traveling direction of the article S to be transported and the other measurement point display part, And may be disposed upward between the third measurement point display unit 230a and the third measurement point display unit 230c.

Usually, attachment devices of the whole apparatus are provided under the end of the rotation shaft of the roll 20, so it is difficult to provide the measurement point display section below the end of the rotation axis of the roll. According to the present invention, three measurement point display portions (230a, 230b, and 230c) are provided in the remaining region except for the end portion of the rotation axis to diagnose alignment and level state of the roll (20).

After the measurement unit indicator 200 is attached to the rotation axis of the roll 20, the coordinate generator 300 can generate coordinates of the measurement point display units 230a, 230b, and 230c based on the reference coordinates. In one embodiment, the coordinate generator 300 is a three-dimensional laser light transmitter, and the measurement point indicators 230a, 230b, and 230c may be prisms. It is possible to generate coordinates of the measurement point display portions 230a, 230b and 230c as diagnostic coordinates by generating a light wave toward the prism in the laser light amplifier and receiving the light wave reflected by the prism by the laser light amplifier.

The thus obtained diagnostic coordinates are processed by the controller 400 and used to diagnose the alignment and the coordinates of the roll 20. Hereinafter, the process of diagnosing the roll alignment and the coordinates using the diagnostic coordinates will be described in detail.

The coordinate of the first measurement point display unit 230a is referred to as a first diagnostic coordinate, the coordinate of the second measurement point display unit 230b is referred to as a second diagnostic coordinate, and the coordinate of the third measurement point display unit 230c is referred to as a third diagnostic coordinate . In the steady state, that is, the roll 20 that is correctly aligned, the first and third diagnostic coordinates differ only in the y coordinate value, and the x and z coordinate values are the same. The x coordinate value of the second diagnostic coordinate is the same as the x coordinate value of the first diagnostic coordinate and the y coordinate value of the second diagnostic coordinate is the same as the y coordinate value of the first diagnostic coordinate and the y coordinate value of the third diagnostic coordinate Is the average of the y coordinate values of "

Figures 6 and 7 schematically illustrate the alignment and level states of the rolls diagnosed using the roll diagnostics device according to the present invention. Fig. 6 (a) shows the roll 20 in the steady state, and Fig. 6 (b) shows the roll 20 in the misaligned state. Fig. 7 (a) shows the roll 20 in the steady state, and Fig. 7 (b) shows the roll 20 in the misaligned state. Fig. 6 shows the alignment state of the rolls 20 on the xy plane, and Fig. 7 shows the alignment state of the rolls 20 on the xz plane.

First, the coordinate generator 300 generates a first diagnostic coordinate of the first measurement point indicator 230a, a second diagnostic coordinate of the second measurement point indicator 230b, and a third diagnostic coordinate of the third measurement point indicator 230c.

Then, the difference between the x coordinate value of the first diagnostic coordinate and the x coordinate value of the third diagnostic coordinate, the y coordinate value of the first diagnostic coordinate and the y coordinate value of the third diagnostic coordinate are obtained, The rain is obtained. That is, a value obtained by dividing the difference between the x coordinate value of the first measurement point display section and the x coordinate value of the third measurement point display section by the difference between the y coordinate value of the first measurement point display section and the y coordinate value of the third measurement point display section Value) is obtained, and the alignment of the roll 20 can be determined using this value.

For example, when the roll 20 is in a normal state as shown in FIG. 6A, the x coordinate value of the first measurement point indicator 230a and the x coordinate value of the third measurement point indicator 230c are the same And the difference between the y coordinate value of the first measurement point indicator 230a and the y coordinate value of the third measurement point indicator 230c corresponds to the distance between the first measurement point indicator 230a and the third measurement point indicator 230c, The first diagnostic value becomes zero. In contrast, when the roll 20 is misaligned as shown in FIG. 6 (b), as the misalignment increases, the x coordinate value of the first measurement point indicator 230a and the x coordinate value of the third measurement point indicator 230c And the first diagnostic value obtained by dividing the difference by the difference between the y coordinate value of the first measurement point indicator 230a and the y coordinate value of the third measurement point indicator 230c also becomes large. Therefore, the alignment state of the roll can be diagnosed from the magnitude of the first diagnostic value, and when the value exceeds the predetermined value, repair or the like can be performed.

Next, the first reference value is obtained by averaging the x coordinate value of the first diagnostic coordinate and the x coordinate value of the third diagnostic coordinate. This value corresponds to the x coordinate value of the measurement point indicator placed on the rotation axis of the roll. Similarly, a second reference value is obtained by averaging the z coordinate value of the first diagnostic coordinate and the z coordinate value of the third diagnostic coordinate. This value corresponds to the z coordinate value of the measurement point indicator placed on the rotation axis of the roll.

Next, the value obtained by dividing the difference between the x coordinate value of the second diagnostic coordinate and the first reference value by the difference between the z coordinate value of the second diagnostic coordinate and the second reference value (hereinafter referred to as a second diagnostic value) 20 can be determined.

For example, when the roll 20 is in the steady state as shown in FIG. 7A, the first reference value is equal to the x coordinate value of the second measurement point indicator 230b, and the second measurement point indicator 230b, The difference between the z coordinate value of the measurement point indicator 200 and the second reference value is equal to the distance between the center of the mounting portion 210 of the measurement point indicator 200 and the second measurement point indicator 230b. Therefore, the second diagnostic value becomes zero.

In contrast, when the roll 20 is misaligned as shown in FIG. 7 (b), the larger the misalignment is, the larger the difference between the x coordinate value of the second measurement point indicator 230b and the first reference value becomes, The second diagnostic value obtained by dividing the difference by the difference between the z coordinate value of the second measurement point indicator 230b and the second reference value becomes larger. Therefore, the level state of the roll can be diagnosed from the magnitude of the second diagnostic value, and if the value exceeds the predetermined value, repair or the like can be performed.

Although not shown in the drawings, the roll diagnostics apparatus according to an embodiment of the present invention may further include a microprocessor for performing a process of diagnosing roll alignment and coordinates using diagnostic coordinates. In this case, the microprocessor may be disposed inside the coordinate generator 300 (FIG. 3) and integrated with the coordinate generator 300 (FIG. 3), and may receive diagnostic coordinates from the coordinate generator 300 Or may be disposed in a separate control unit.

As described above, according to the embodiment of the present invention, it is possible to easily diagnose the alignment and the level of rolls used in a steel product production line and the like in a short time, and in the conventional method using piano lines and weights, A three-dimensional laser measuring device for measuring a three-dimensional position of a roll, and a measuring point indicator capable of performing accurate measurement by diagnosing the alignment and level of rolls and detachable to the rotating shaft of the roll, And the level can be diagnosed and the alignment and level of the roll can be diagnosed without rotating the roll by using a measurement point indicator having three measurement point display portions on the rotation axis of the roll.

Although the present invention has been described by way of examples, the technical scope of the present invention is not limited to the range described in the above embodiments. It will be apparent to those skilled in the art that various changes and modifications can be made in the above embodiments. It is apparent from the description of the claims that the form of the addition of such modification or improvement can be included in the technical scope of the present invention.

S strip 20 rolls
21 Rotary shaft 30 Center line
40 piano lines 41 chords
50 Mark 1000 roll diagnostics
100 Base point indicator 110 Base
120 extension part 130 reference point display part
200 Measuring point indicator 210 Mounting part
220a, 220b, 220c Supporting part 230a First measurement point display part
230b Second measurement point display section 230c 3 Measurement point display section
300 coordinate generator

Claims (9)

The article according to any one of claims 1 to 3, wherein the article is attached to a rotary shaft of a roll that carries the article to be transported by rotation, and is located at two points opposed to each other about the rotary shaft of the roll in the traveling direction of the article to be transported A measurement point indicator including at least one measurement point indicator disposed at each of a plurality of points including one point in a vertical direction; And
A coordinate generator for obtaining the coordinates of each of the measurement point display units as a plurality of diagnostic coordinates,
/ RTI >
Roll diagnostics.
The method according to claim 1,
Wherein the measurement point indicator comprises:
A mounting portion for mounting on a rotating shaft of the roll; And
Three support portions extending radially from the mounting portion and spaced 90 degrees apart from each other,
Lt; / RTI >
Wherein the at least one measurement point display unit includes a first measurement point display unit, a second measurement point display unit, and a third measurement point display unit, the first measurement point display unit being attached to an end of the three support units,
Roll diagnostics.
3. The method of claim 2,
Wherein the first measurement point display unit and the third measurement point display unit are arranged so as to face each other in the traveling direction of the article to be transported and the second measurement point display unit is disposed to be vertically upward in the traveling direction of the article to be transported,
Roll diagnostics.
The method of claim 3,
And a microprocessor for determining the alignment of the roll from the coordinate values of the first measurement point display unit and the third measurement point display unit.
Roll diagnostics.
5. The method of claim 4,
When the traveling direction of the rotary shaft of the roll is the x direction, the traveling direction of the article to be transported is the y direction, and the direction perpendicular to the x and y directions is the y direction,
The alignment of the rolls may be performed such that a difference between an x coordinate value of the first measurement point display unit and an x coordinate value of the third measurement point display unit is a difference between a y coordinate value of the first measurement point display unit and a y coordinate value of the third measurement point display unit The value determined using the divided value,
Roll diagnostics.
The method of claim 3,
And determining the level of the roll from the coordinate values of the first measurement point display section, the second measurement point display section and the third measurement point display section,
Roll diagnostics.
The method according to claim 6,
When the traveling direction of the rotary shaft of the roll is the x direction, the traveling direction of the article to be transported is the y direction, and the direction perpendicular to the x and y directions is the y direction,
The level of the roll is determined using a value obtained by dividing the difference between the x coordinate value of the second measurement point display unit and the first reference value by the difference between the z coordinate value of the second measurement point display unit and the second reference value,
Wherein the first reference value is an average of an x coordinate value of the first measurement point display unit and an x coordinate value of the third measurement point display unit,
Wherein the second reference value is an average of a z coordinate value of the first measurement point display unit and a z coordinate value of the third measurement point display unit,
Roll diagnostics.
The method according to claim 1,
Further comprising a reference point indicator mounted on a rail extending in parallel with a traveling direction of the article to be transported on the side of the roll and used to generate reference coordinates for the roll,
Wherein the coordinate generator generates the plurality of diagnostic coordinates based on the reference coordinates,
Roll diagnostics.
9. The method of claim 8,
Wherein the reference point indicator comprises:
A brazed base attached to the rail;
An extension extending from the base; And
A reference point indicator attached to one end of the extension;
Lt; / RTI >
Wherein the coordinate generator generates coordinates of the reference point display unit as the reference coordinates,
Roll diagnostics.
KR1020150101045A 2015-07-16 2015-07-16 Diagnostic apparatus for roll KR101726056B1 (en)

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CN110202011B (en) * 2019-05-24 2021-01-26 首钢京唐钢铁联合有限责任公司 Roller crossing horizontal detection method

Citations (3)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
KR20040043929A (en) 2002-11-20 2004-05-27 주식회사 포스코 Device for measuring right-angled degree of assembling type roll
KR100733396B1 (en) * 2006-03-17 2007-06-29 주식회사 솔루션닉스 Apparatus and method for three dimension measurement data acquisition
KR20080107168A (en) * 2007-06-05 2008-12-10 한국표준과학연구원 Apparatus and method for measuring three-dimensional coordinates

Patent Citations (3)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
KR20040043929A (en) 2002-11-20 2004-05-27 주식회사 포스코 Device for measuring right-angled degree of assembling type roll
KR100733396B1 (en) * 2006-03-17 2007-06-29 주식회사 솔루션닉스 Apparatus and method for three dimension measurement data acquisition
KR20080107168A (en) * 2007-06-05 2008-12-10 한국표준과학연구원 Apparatus and method for measuring three-dimensional coordinates

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