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CN113048949A - Cylindrical object pose detection device and method based on line structure optical vision - Google Patents

Cylindrical object pose detection device and method based on line structure optical vision Download PDF

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CN113048949A
CN113048949A CN202110247239.9A CN202110247239A CN113048949A CN 113048949 A CN113048949 A CN 113048949A CN 202110247239 A CN202110247239 A CN 202110247239A CN 113048949 A CN113048949 A CN 113048949A
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laser line
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CN113048949B (en
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朱海飞
叶文达
王斌
管贻生
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Guangdong University of Technology
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Abstract

The invention discloses a cylindrical object pose detection device and method based on line structure optical vision, wherein the device comprises: mounting fixture, first laser line structure light generator, second laser line structure light generator, monocular camera, first laser line structure light generator, second laser line generator pass through the parallel horizontal part surface of fixing at mounting fixture of locating rack, the monocular camera is fixed between first laser line structure light generator and the second laser line generator, the monocular camera is used for acquireing the three-dimensional image of projection at the object that awaits measuring, and the output of the power connection end of first laser line structure light generator, second laser line structure light generator, monocular camera all is connected to outside monocular vision measurement system. The invention improves the detection precision, realizes non-contact automatic operation, and has low cost and strong practicability.

Description

Cylindrical object pose detection device and method based on line structure optical vision
Technical Field
The invention relates to the technical field of vision sensing measurement, in particular to a cylindrical object pose detection device and method based on line structure optical vision.
Background
In modern industry, cylindrical objects play a very important role in support structures, such as scaffolding, stadium roofs, glass curtain wall frameworks, etc. The estimation of the pose of the cylindrical object in the space is a necessary key technology for modeling space positioning and realizing an intelligent automation technology.
At present, most of pose estimation work of cylindrical objects is still finished manually or semi-automatically, a large number of tools are required in the estimation process, the intermediate process is multiple and complicated, the measurement error is large, and the uniform precision requirement is difficult to achieve. The estimation is difficult to be carried out manually on the cylindrical rod pieces with special structures and environments. With the development of technology, in the pose estimation of a cylindrical object, more and more researchers have started to improve the detection speed of the object pose estimation by using the line structured light detection technology. Most of the sensors are based on multi-sensor cooperation relative to contact sensors, and therefore pose estimation results are obtained. And because the information comes from multiple sensors, more errors and more complicated solving process are brought. The laser line-based structured light detection technology belongs to one of non-contact measurement, can quickly extract the space characteristic parameters of the cylindrical object, has good robustness and timeliness, surely replaces the traditional visual detection method, and has very important significance on the process planning related to the cylindrical rod piece.
In the prior art, a patent of invention CN109087355A discloses a monocular camera pose measurement device and method based on iterative update in 2018, 12 and 25.s.
Disclosure of Invention
The invention provides a cylindrical object pose detection device and method based on line structure optical vision, aiming at overcoming the defects of complex pose estimation process and low precision of the cylindrical object in the prior art.
The primary objective of the present invention is to solve the above technical problems, and the technical solution of the present invention is as follows:
a cylindrical object pose detection apparatus based on line structured optical vision, comprising: a mounting clamp, a first laser line structured light generator, a second laser line structured light generator and a monocular camera,
first laser line structure light generator, second laser line generator pass through the parallel horizontal part surface of fixing at sectional fixture of locating rack, the monocular camera is fixed between first laser line structure light generator and the second laser line generator, the monocular camera is used for acquireing the three-dimensional image of throwing at the object that awaits measuring, and the power connection end of first laser line structure light generator, second laser line structure light generator, the output of monocular camera all are connected to outside monocular vision measurement system.
Furthermore, the laser lines emitted by the first laser line structure light generator and the second laser line structure light generator are projected on the surface of the object to be measured in parallel.
Further, the monocular camera includes: the front end of the CMOS camera is detachably connected with the industrial lens, and the front end of the industrial lens is detachably connected with the optical filter.
Furthermore, a plurality of mounting holes are formed in the surface of the vertical part of the mounting fixture.
The invention provides a cylindrical object pose detection method based on line structure optical vision, which is applied to a cylindrical object pose detection device based on line structure optical vision and comprises the following steps:
s1: calibrating a monocular camera to obtain internal parameters and external parameters of the monocular camera, and calibrating light planes of a first laser line structured light generator and a second laser line structured light generator to obtain a line structured light plane equation;
s2: projecting the emission light of the first laser line-structured light generator and the second laser line-structured light generator onto a cylindrical object to be detected to generate two three-dimensional arcs, acquiring a three-dimensional arc image through a monocular camera, and performing denoising pretreatment;
s3: extracting two-dimensional points on the circular arc center line in the three-dimensional circular arc image and clustering according to the light source;
s4: mapping the clustered two-dimensional points into spatial three-dimensional points under a monocular camera coordinate system, and carrying out ellipse fitting on the spatial three-dimensional points to obtain pose parameters of the cylinder to be measured;
s5: and acquiring a plurality of groups of three-dimensional arc images, carrying out denoising pretreatment, repeating the steps S3-S4 to obtain the corresponding pose parameters of a plurality of groups of cylinders to be detected, carrying out iterative optimization by using a preset iterative optimization algorithm to obtain an optimal result, and taking the optimal result as the final pose parameter of the cylinder to be detected.
Further, the pose parameters include: axis vector, reference point, and radius.
Further, the intrinsic parameters of the monocular camera include: k is a radical ofx,ky,u0,v0,kx,kyIs the effective focal length in the X, Y directions, (u)0,v0) As principal point coordinates, the extrinsic parameters include: the MCW represents the conversion relation between a camera coordinate system and a world coordinate system;
the line structured light plane equation is expressed by a sensor camera model of line structured light, which is expressed as:
Figure BDA0002964530130000031
wherein, OWXWYWZWAs a world coordinate system, OCXCYCC is a camera coordinate system, and OUV is an image pixel coordinate system;
the equation of the line-structured light plane, i.e. the equation of the line-structured light plane in the world coordinate system, can be expressed as ZW=AXW+BYW+ C, where A, B represents the equation coefficients and C is a constant.
Further, the specific process of extracting the two-dimensional points on the circular arc center line in the three-dimensional circular arc image and clustering according to the light source in step S3 is as follows:
extracting points on two arc center lines in the three-dimensional arc image by using a steger algorithm, marking the points as center points I (u, v), and defining a distance formula from the center points to two reference straight lines, wherein the two reference straight lines are corresponding straight lines of the emission light rays of the first laser line structured light generator and the second laser line structured light generator in the three-dimensional arc image;
and calculating the distances from the central point to the two reference straight lines, and dividing the central point into two types according to the difference of the distances.
Further, the distance formula from the central point to the two reference straight lines is as follows:
Figure BDA0002964530130000032
wherein theta isiIs the installation angle of the line structured light generator with respect to the camera, BiThe intercept of the two arcs under the image coordinate system;
the classification of the center points is represented as follows:
Figure BDA0002964530130000033
wherein l1(u, v) denotes a first type center point, l2(u, v) represents the second type center point, d1(u, v) a reference distance calculated for the center point of the first arc, d2(u, v) is the distance calculated from the center point of the second arc.
Further, the specific process of mapping the clustered two-dimensional points into the spatial three-dimensional points in the monocular camera coordinate system is as follows:
according to the calibration relation between the monocular camera and the line structured light,
Figure BDA0002964530130000034
converting the extracted central point I (u, v) into a three-dimensional space point;
the specific process of carrying out ellipse fitting on the space three-dimensional points to obtain the pose parameters of the cylinder to be measured is as follows:
fitting three-dimensional points to a space ellipse by utilizing Hough circle transformation in an OpenCV library, connecting central points of the ellipses, obtaining the axis of the cylinder to be measured, and expressing an equation of the axis
Figure BDA0002964530130000041
Above formula is converted into
Figure BDA0002964530130000042
Wherein
Figure BDA0002964530130000043
y0=y1-nz1Is provided with
Figure BDA0002964530130000044
Figure BDA0002964530130000045
The direction vector of the cylinder axis to be measured and a point on the cylinder axis are represented as
Figure BDA0002964530130000046
P0=[x0 y0 0]T
Rotating the axis of the cylinder to be measured to the z-axis direction to enable the bottom surface of the cylinder to be parallel to the xoy surface, and calculating a rotation matrix R-Rot by using an axis angle method<d×z,d·z>The point on the cylindrical surface, after rotation, may be expressed as p'i=RPi,PiRepresenting points on the surface of the cylinder before rotation;
fitting the points on the rotated cylindrical curved surface to a circle on the xoy plane by using a least square method, wherein the radius of the obtained circle is the radius of the cylinder to be measured, and the radius of the cylinder to be measured is expressed as
Figure BDA0002964530130000047
And N is the total number of the three-dimensional points of the rotating curved surface.
Compared with the prior art, the technical scheme of the invention has the beneficial effects that:
according to the cylindrical object pose detection device based on the line-structured optical vision, the cylindrical object pose detection device based on the line-structured optical vision is constructed by the mounting clamp, the laser line structured light generator and the monocular camera, the defects of complex structure, complex operation and low detection precision of the traditional detection device are overcome, the non-contact automatic operation is realized, the cost is low, and the practicability is high.
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Fig. 1 is a schematic structural diagram of a cylindrical object pose detection device based on line-structured optical vision.
FIG. 2 is a flow chart of the cylindrical object pose detection method based on line structured optical vision of the present invention.
Detailed Description
In order that the above objects, features and advantages of the present invention can be more clearly understood, a more particular description of the invention will be rendered by reference to the appended drawings. It should be noted that the embodiments and features of the embodiments of the present application may be combined with each other without conflict.
In the following description, numerous specific details are set forth in order to provide a thorough understanding of the present invention, however, the present invention may be practiced in other ways than those specifically described herein, and therefore the scope of the present invention is not limited by the specific embodiments disclosed below.
Example 1
As shown in fig. 1, a cylindrical object pose detection apparatus based on line structured optical vision includes: a mounting clamp 1, a first laser line structured light generator 2, a second laser line structured light generator 3 and a monocular camera 4,
first laser line structure light generator 2, the parallel horizontal part surface of fixing at sectional fixture 1 of second laser line structure light generator through locating rack 5, monocular camera 4 is fixed between first laser line structure light generator 2 and second laser line structure light generator 3, monocular camera 4 is used for acquireing the three-dimensional image of throwing at the object that awaits measuring, and first laser line structure light generator 2, second laser line structure light generator 3's power connection end, monocular camera 4's output all are connected to outside monocular vision measurement system.
In a specific embodiment, the first laser line structured light generator 2 and the second laser line structured light generator 3 are the same type laser line structured light generators, wherein in the specific work, the included angle between the first laser line structured light generator 2 and the second laser line structured light generator 3 and the monocular camera 4 is adjusted to ensure that when the laser line structured light is imaged in the monocular camera 4, a certain radian exists, when the device works, the first laser line structured light generator 2 and the second laser line structured light generator project a line structured light to the surface of the cylinder to be measured and reflect the line structured light on the surface, and the stripes reflecting the characteristics of the cylinder are formed in the CMOS chip of the monocular camera.
Further, the laser lines emitted by the first laser line structure light generator 2 and the second laser line generator 3 are projected on the surface of the object to be measured in parallel.
Further, the monocular camera 4 includes: the device comprises a CMOS camera 401, an industrial lens 402 and an optical filter 403, wherein the front end of the CMOS camera 401 is detachably connected with the industrial lens 402, and the front end of the industrial lens 402 is detachably connected with the optical filter 403.
In a specific embodiment, the CMOS camera 401 and the industrial lens 402 may be screwed or clamped, the front end of the industrial lens 402 and the optical filter 403 may be screwed or clamped, the optical filter may be used to filter out a portion of natural light of environmental interference and light reflected from the surface of the workpiece, and only light with a specific wavelength can be imaged in the camera. The linear structured light can add depth constraint to the image, the obtained image is calibrated through the linear structured light, and the depth information of each point of the stripe can be obtained, so that the space pose information of the cylinder to be measured can be obtained.
Further, a plurality of mounting holes 6 are formed on the surface of the vertical part of the mounting fixture 1.
In a specific embodiment, the installation and arrangement are facilitated by arranging the installation holes 6 in the vertical part of the installation clamp 1, and the device can be installed on a robot for detecting the pose of the robot on a cylindrical object.
As shown in fig. 2, a second aspect of the present invention provides a cylindrical object pose detection method based on line structure optical vision, where the method is applied to the cylindrical object pose detection apparatus based on line structure optical vision, and the method includes the following steps:
s1: calibrating a monocular camera to obtain internal parameters and external parameters of the monocular camera, and calibrating light planes of a first laser line structured light generator and a second laser line structured light generator to obtain a line structured light plane equation;
in a specific embodiment, the monocular camera is calibrated by using a checkerboard as an auxiliary tool and adopting a camera fixing installation mode, the camera captures images of checkerboard calibration plates placed in different postures, 15 images of the calibration plates are obtained, and the processing of experiment images is realized through a Matlab calibration tool box. Obtaining camera internal parameters and external parameters through calibration, wherein the internal parameters of the monocular camera comprise: k is a radical ofx,ky,u0,v0,kx,kyIs the effective focal length in the X, Y directions, (u)0,v0) As principal point coordinates, the extrinsic parameters include: and the camera extrinsic parameters represent the relative pose of the coordinate system established on the checkerboard calibration board under the camera coordinate system.
And calibrating line-structured light, namely calibrating light planes of a first laser line-structured light generator and a second laser line-structured light generator, obtaining a line-structured light plane equation through calibrating the light planes, wherein in a line-structured light sensor line plane measurement model, a parameter determining the spatial position of the light planes under a camera coordinate system is a line-structured light plane parameter, namely the line-structured light plane equation, the calibration of the line-structured light plane parameter is to determine the value of a coefficient A, B, C of the line-structured light plane equation, a checkerboard target used in the camera calibration is still used in the calibration of the line-structured light plane, 3-4 points are taken on laser stripes on each image, and the points are used for calibrating the line-structured light plane.
The line structured light plane equation is expressed by a sensor camera model of line structured light, which is expressed as:
Figure BDA0002964530130000061
wherein, OWXWYWZWAs a world coordinate system, OCXCYCC is a camera coordinate system, and OUV is an image pixel coordinate system; setting the three-dimensional coordinate of any point P on the light plane of the line structure as PW(XW,YW,ZW) In the image pixel plane, the coordinates are p (u, v), where
Figure BDA0002964530130000062
And
Figure BDA0002964530130000063
representing the transformation of the world reference frame into the camera frame, the linear structure equation, i.e. the equation of the linear structure light plane in the world frame, can be expressed as ZW=AXW+BYW+ C, where A, B represents the equation coefficients and C is a constant.
S2: projecting the light emitted by the first laser line-structured light generator and the light emitted by the second laser line-structured light generator onto a cylindrical object to be detected to generate two three-dimensional arcs, acquiring a three-dimensional arc image through a monocular camera, and performing denoising pretreatment; in a specific embodiment, the denoising pre-processing includes gaussian filtering, threshold segmentation.
S3: extracting two-dimensional points on the circular arc center line in the three-dimensional circular arc image and clustering according to the light source;
the specific clustering process comprises the following steps:
extracting points on two arc center lines in the three-dimensional arc image by using a steger algorithm, marking the points as center points I (u, v), and defining a distance formula from the center points to two reference straight lines, wherein the two reference straight lines are corresponding straight lines of the emission light rays of the first laser line structured light generator and the second laser line structured light generator in the three-dimensional arc image;
and calculating the distances from the central point to the two reference straight lines, and dividing the central point into two types according to the difference of the distances.
Further, the distance formula from the central point to the two reference straight lines is as follows:
Figure BDA0002964530130000071
wherein theta isiIs the installation angle of the line structured light generator with respect to the camera, BiThe intercept of the two arcs under the image coordinate system;
the classification of the center points is represented as follows:
Figure BDA0002964530130000072
wherein l1(u, v) denotes a first type center point, l2(u, v) represents the second type center point, d1(u, v) a reference distance calculated for the center point of the first arc, d2(u, v) is the distance calculated from the center point of the second arc.
S4: mapping the clustered two-dimensional points into spatial three-dimensional points under a monocular camera coordinate system, and carrying out ellipse fitting on the spatial three-dimensional points to obtain pose parameters of the cylinder to be measured; the pose parameters include: axis vector, reference point, and radius.
According to the calibration relation between the monocular camera and the line structured light,
Figure BDA0002964530130000073
converting the extracted central point I (u, v) into a three-dimensional space point;
the specific process of carrying out ellipse fitting on the space three-dimensional points to obtain the pose parameters of the cylinder to be measured is as follows:
fitting three-dimensional points to a space ellipse by utilizing Hough circle transformation in an OpenCV library, connecting central points of the ellipses, obtaining the axis of the cylinder to be measured, and expressing an equation of the axis
Figure BDA0002964530130000081
Above formula is converted into
Figure BDA0002964530130000082
Wherein
Figure BDA0002964530130000083
y0=y1-nz1Is provided with
Figure BDA0002964530130000084
Figure BDA0002964530130000085
The direction vector of the cylinder axis to be measured and a point on the cylinder axis are represented as
Figure BDA0002964530130000086
Figure BDA0002964530130000087
P0=[x0 y0 0]T
Rotating the axis of the cylinder to be measured to the z-axis direction to enable the bottom surface of the cylinder to be parallel to the xoy surface, and calculating a rotation matrix R-Rot by using an axis angle method<d×z,d·z>The point on the cylindrical surface, after rotation, may be expressed as p'i=RPi,PiRepresenting points on the surface of the cylinder before rotation;
fitting the points on the rotated cylindrical curved surface to a circle on the xoy plane by using a least square method, wherein the radius of the obtained circle is the radius of the cylinder to be measured, and the radius of the cylinder to be measured is expressed as
Figure BDA0002964530130000088
And N is the total number of the three-dimensional points of the rotating curved surface.
S5: and acquiring a plurality of groups of three-dimensional arc images, performing denoising pretreatment, repeating the steps S3-S4 to obtain the corresponding pose parameters of a plurality of groups of cylinders to be detected, performing iterative optimization by using a preset iterative optimization algorithm to obtain an optimal result, and taking the optimal result as the final pose parameter of the cylinder to be detected.
In a specific embodiment, the iterative optimization algorithm may use a Levenberg-Marquarel algorithm.
The same or similar reference numerals correspond to the same or similar parts;
the terms describing positional relationships in the drawings are for illustrative purposes only and are not to be construed as limiting the patent;
it should be understood that the above-described embodiments of the present invention are merely examples for clearly illustrating the present invention, and are not intended to limit the embodiments of the present invention. Other variations and modifications will be apparent to persons skilled in the art in light of the above description. And are neither required nor exhaustive of all embodiments. Any modification, equivalent replacement, and improvement made within the spirit and principle of the present invention should be included in the protection scope of the claims of the present invention.

Claims (10)

1. The utility model provides a cylindric object position appearance detection device based on line structure optical vision which characterized in that includes: mounting fixture, first laser line structure light generator, second laser line structure light generator, monocular camera, first laser line structure light generator, second laser line generator pass through the parallel horizontal part surface of fixing at mounting fixture of locating rack, the monocular camera is fixed between first laser line structure light generator and the second laser line generator, the monocular camera is used for acquireing the three-dimensional image of projection at the object that awaits measuring, and the output of the power connection end of first laser line structure light generator, second laser line structure light generator, monocular camera all is connected to outside monocular vision measurement system.
2. The linear-structured optical vision-based cylindrical object pose detection device as claimed in claim 1, wherein the laser lines emitted by the first laser line-structured light generator and the second laser line generator are projected on the surface of the object to be detected in parallel.
3. The cylindrical object pose detection apparatus based on line structured optical vision according to claim 1, wherein the monocular camera includes: the front end of the CMOS camera is detachably connected with the industrial lens, and the front end of the industrial lens is detachably connected with the optical filter.
4. The cylindrical object pose detection device based on line structured optical vision of claim 1, wherein the vertical portion surface of the installation fixture is provided with a plurality of installation holes.
5. A cylindrical object pose detection method based on line structure optical vision, which is applied to the cylindrical object pose detection device based on line structure optical vision of any one of claims 1 to 4, and is characterized by comprising the following steps:
s1: calibrating a monocular camera to obtain internal parameters and external parameters of the monocular camera, and calibrating light planes of a first laser line structured light generator and a second laser line structured light generator to obtain a line structured light plane equation;
s2: projecting the emission light of the first laser line-structured light generator and the second laser line-structured light generator onto a cylindrical object to be detected to generate two three-dimensional arcs, acquiring a three-dimensional arc image through a monocular camera, and performing denoising pretreatment;
s3: extracting two-dimensional points on the circular arc center line in the three-dimensional circular arc image and clustering according to the light source;
s4: mapping the clustered two-dimensional points into spatial three-dimensional points under a monocular camera coordinate system, and carrying out ellipse fitting on the spatial three-dimensional points to obtain pose parameters of the cylinder to be measured;
s5: and acquiring a plurality of groups of three-dimensional arc images, carrying out denoising pretreatment, repeating the steps S3-S4 to obtain the corresponding pose parameters of a plurality of groups of cylinders to be detected, carrying out iterative optimization by using a preset iterative optimization algorithm to obtain an optimal result, and taking the optimal result as the final pose parameter of the cylinder to be detected.
6. The cylindrical object pose detection method based on line structure optical vision according to claim 5, wherein the pose parameters comprise: axis vector, reference point, and radius.
7. The cylindrical object pose detection method based on line-structured optical vision according to claim 5, wherein the internal parameters of the monocular camera include: k is a radical ofx,ky,u0,v0,kx,kyIs the effective focal length in the X, Y directions, (u)0,v0) As principal point coordinates, the extrinsic parameters include: the MCW represents the conversion relation between a camera coordinate system and a world coordinate system;
the line structured light plane equation is expressed by a sensor camera model of line structured light, which is expressed as:
Figure FDA0002964530120000021
wherein, OWXWYWZWAs a world coordinate system, OCXCYCC is a camera coordinate system, and OUV is an image pixel coordinate system;
the equation of the line-structured light plane, i.e. the equation of the line-structured light plane in the world coordinate system, can be expressed as ZW=AXW+BYW+ C, where A, B represents the equation coefficients and C is a constant.
8. The cylindrical object pose detection method based on line-structured optical vision according to claim 5, wherein the specific process of extracting two-dimensional points on the circular arc center line in the three-dimensional circular arc image and clustering according to the light source in step S3 is as follows:
extracting points on two arc center lines in the three-dimensional arc image by using a steger algorithm, marking the points as center points I (u, v), and defining a distance formula from the center points to two reference straight lines, wherein the two reference straight lines are corresponding straight lines of the emission light rays of the first laser line structured light generator and the second laser line structured light generator in the three-dimensional arc image;
and calculating the distances from the central point to the two reference straight lines, and dividing the central point into two types according to the difference of the distances.
9. The cylindrical object pose detection method based on line-structured optical vision according to claim 8, wherein the distance formula from the central point to the two reference lines is as follows:
Figure FDA0002964530120000022
wherein theta isiIs the installation angle of the line structured light generator with respect to the camera, biThe intercept of the two arcs under the image coordinate system;
the classification of the center points is represented as follows:
Figure FDA0002964530120000031
wherein l1(u, v) denotes a first type center point, l2(u, v) represents the second type center point, d1(u, v) a reference distance calculated for the center point of the first arc, d2(u, v) is the distance calculated from the center point of the second arc.
10. The cylindrical object pose detection method based on line structured optical vision according to claim 9, wherein the specific process of mapping the clustered two-dimensional points to spatial three-dimensional points in a monocular camera coordinate system comprises:
according to the calibration relation between the monocular camera and the line structured light,
Figure FDA0002964530120000032
converting the extracted central point I (u, v) into a three-dimensional space point;
the specific process of carrying out ellipse fitting on the space three-dimensional points to obtain the pose parameters of the cylinder to be measured is as follows:
fitting three-dimensional points to a space ellipse by utilizing Hough circle transformation in an OpenCV library, connecting central points of the ellipses, and obtaining the axis of the cylinder to be measured, wherein an equation of the axis is expressed as
Figure FDA0002964530120000033
Above formula is converted into
Figure FDA0002964530120000034
Wherein
Figure FDA0002964530120000035
y0=y1-nz1Is provided with
Figure FDA0002964530120000036
Figure FDA0002964530120000037
The direction vector of the cylinder axis to be measured and a point on the cylinder axis are represented as
Figure FDA0002964530120000038
Figure FDA0002964530120000039
P0=[x0 y0 0]T
Rotating the axis of the cylinder to be measured to the z-axis direction to enable the bottom surface of the cylinder to be parallel to the xoy surface, and calculating a rotation matrix R-Rot by using an axis angle method<d×z,d·z>The point on the cylindrical surface, after rotation, may be expressed as p'i=RPi,PiRepresenting points on the surface of the cylinder before rotation;
and fitting the points on the rotated cylindrical curved surface to a circle on the xoy plane by using a least square method, wherein the radius of the obtained circle is the radius of the cylinder to be measured, and the radius of the cylinder to be measured is expressed as:
Figure FDA0002964530120000041
and N is the total number of the three-dimensional points of the rotating curved surface.
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