CN106247944A - Code targets and vision coordinate measurement method based on Code targets - Google Patents

Abstract

Translated fromChinese

本发明公开的编码靶标，包括有长方体状的壳体，壳体内设置有驱动电路；壳体的外侧壁上分别设置有状态灯、触发开关及把手；壳体的外侧壁上还设有工作面，且工作面上设置有编码单元，编码单元包括有12个大小相同的定位光点和5个大小相同的小光点，均为红外LED，12个定位光点在工作面上沿中线等距离设置，5个小光点分布于12个定位光点附近不同位置，形成编码；壳体的底部设置有靶标测头，且靶标测头与12个定位光点位于同一条直线上。本发明还公开了基于编码靶标的视觉坐标测量方法，测量时只需获得任意一段三个编码以上的靶标图像，即可获得靶标测头的坐标。将本发明编码靶标应用于视觉坐标测量系统后能同时实现测量的大范围和高精度。

The coding target disclosed by the present invention includes a cuboid-shaped casing, and a driving circuit is arranged inside the casing; a status light, a trigger switch and a handle are respectively arranged on the outer wall of the casing; a working surface is also arranged on the outer wall of the casing , and a coding unit is set on the working surface, the coding unit includes 12 positioning light spots of the same size and 5 small light spots of the same size, all of which are infrared LEDs, and the 12 positioning light spots are equidistant along the midline on the working surface Setting, 5 small light spots are distributed in different positions near the 12 positioning light spots to form a code; the bottom of the housing is provided with a target probe, and the target probe and the 12 positioning light spots are located on the same straight line. The invention also discloses a visual coordinate measuring method based on coded targets. During the measurement, only any segment of target images with more than three codes can be obtained to obtain the coordinates of the target measuring head. After the coded target of the present invention is applied to a visual coordinate measurement system, a large range and high precision of measurement can be realized simultaneously.

Description

Translated fromChinese

编码靶标及基于编码靶标的视觉坐标测量方法Coded target and visual coordinate measurement method based on coded target

技术领域technical field

本发明属于测试计量设备及方法技术领域，具体涉及一种编码靶标，本发明还涉及基于编码靶标的视觉坐标测量方法。The invention belongs to the technical field of test and measurement equipment and methods, in particular to a coded target, and also relates to a visual coordinate measurement method based on the coded target.

背景技术Background technique

三维坐标测量技术被广泛应用于对汽车、飞机、船舶以及航天器的外形测量。如今，对大型工件表面轮廓、几何尺寸的现场测量工作越来越多，精度要求越来越高，以三坐标式测量机(CMM)为代表的传统测量设备具有体积大、不便于携带及难以满足现场的测量要求。Three-dimensional coordinate measurement technology is widely used in the shape measurement of automobiles, aircrafts, ships and spacecraft. Nowadays, there are more and more on-site measurement work on the surface profile and geometric dimensions of large workpieces, and the accuracy requirements are getting higher and higher. The traditional measuring equipment represented by three-coordinate measuring Meet on-site measurement requirements.

随着计算机、激光、CCD传感器技术的飞速发展，近年来国内外出现了靶标式坐标测量系统，该系统因具有体积小、重量轻及便于携带的优点，得到越来越多的研究与应用。With the rapid development of computer, laser, and CCD sensor technology, target-type coordinate measurement systems have emerged at home and abroad in recent years. Because of the advantages of small size, light weight and portability, this system has been researched and applied more and more.

使用靶标作为探测工具，测量时需要靶标的全部或大部进入视场，测量范围与靶标大小成比例，而分辨率取决于覆盖靶标范围的视觉传感器像素的数量，同时获得大的测量范围和高的测量精度是困难的。另外，靶标上的靶点的质量影响系统的测量精度，常见印制图像靶点成本低、制作简单，但是测量距离有限；直接采用红外LED点光源作为靶点是提高系统测量范围的一个重要途径，但是点光源靶点的成像光斑质量较难保证。Using the target as a detection tool, all or most of the target needs to enter the field of view during measurement, the measurement range is proportional to the size of the target, and the resolution depends on the number of visual sensor pixels covering the target range, while obtaining a large measurement range and high The measurement accuracy is difficult. In addition, the quality of the target point on the target affects the measurement accuracy of the system. The common printed image target point is low in cost and easy to manufacture, but the measurement distance is limited; directly using infrared LED point light source as the target point is an important way to improve the measurement range of the system , but the quality of the imaging spot of the point light source target is difficult to guarantee.

由此可见，如何突破以往视觉坐标测量的局限性，同时实现测量的大范围和高精度是迫切要解决的问题。It can be seen that how to break through the limitations of the previous visual coordinate measurement and realize the large-scale and high-precision measurement is an urgent problem to be solved.

发明内容Contents of the invention

本发明的第一目的在于提供一种编码靶标，将其应用于视觉坐标测量系统后能突破以往视觉坐标测量的局限性，同时实现测量的大范围和高精度。The first object of the present invention is to provide a coded target, which can break through the limitations of the previous visual coordinate measurement after it is applied to the visual coordinate measurement system, and at the same time realize a wide range and high precision of measurement.

本发明的第二目的在于提供基于编码靶标的视觉坐标测量方法。The second object of the present invention is to provide a method for measuring visual coordinates based on coded targets.

本发明所采用的第一种技术方案是，编码靶标，包括有长方体状的壳体，壳体内设置有驱动电路；壳体的外侧壁上分别设置有状态灯、触发开关及把手；壳体的外侧壁上还设有工作面，且工作面上设置有编码单元，编码单元包括有12个大小相同的定位光点和5个大小相同的小光点，且定位光点2和小光点均为红外LED，定位光点的尺寸大于小光点的尺寸，12个定位光点在工作面上沿中线等距离设置，5个小光点分布于12个定位光点之间；壳体的底部设置有靶标测头，且靶标测头与12个定位光点位于同一条直线上。The first technical solution adopted by the present invention is that the coding target includes a cuboid housing, and a drive circuit is arranged in the housing; a status light, a trigger switch and a handle are respectively arranged on the outer wall of the housing; There is also a working surface on the outer wall, and a coding unit is arranged on the working surface. The coding unit includes 12 positioning light spots of the same size and 5 small light spots of the same size, and the positioning light spot 2 and the small light spot are both It is an infrared LED, the size of the positioning light spot is larger than the size of the small light spot, 12 positioning light spots are arranged equidistantly along the midline on the working surface, and 5 small light spots are distributed among the 12 positioning light spots; the bottom of the shell A target probe is provided, and the target probe and the 12 positioning light spots are located on the same straight line.

本发明第一种技术方案的特点还在于：The feature of the first technical solution of the present invention is also:

5个小光点与12个定位光点的位置关系具体如下：在一个定位光点的左边设置一个小光点或在其右边设置一个小光点或者不设置小光点；若小光点设置于定位光点的左边则编码为C；若小光点设置于定位光点的右边则编码为B；若定位光点附近不设置小光点则编码为A；在工作面上构成的编码，任意三个连续位置编码不同。The positional relationship between the 5 small light spots and the 12 positioning light spots is as follows: set a small light spot on the left of a positioning light spot or set a small light spot on the right side or do not set a small light spot; if the small light spot is set If the small light spot is set on the left side of the positioning light spot, it is coded as C; if the small light spot is set on the right side of the positioning light spot, it is coded as B; Any three consecutive positions are coded differently.

定位光点和小光点均为红外LED，且定位光点的尺寸大于小光点。Both the positioning light spot and the small light spot are infrared LEDs, and the size of the positioning light spot is larger than the small light spot.

本发明所采用的第二种技术方案是，基于编码靶标的视觉坐标测量系统的测量方法，具体按照以下步骤实施：The second technical solution adopted in the present invention is a measurement method based on a coded target visual coordinate measurement system, which is specifically implemented according to the following steps:

步骤1、利用编码靶标、计算机、CCD摄像机构建一个基于编码靶标的视觉坐标测量系统；Step 1, utilize coded target, computer, CCD camera to construct a visual coordinate measurement system based on coded target;

步骤2、先启动步骤1中构建的基于编码靶标的视觉坐标测量系统，待靶标测头接触到被测物上的某一点，将其作为被测点；然后按动触发开关，点亮所有定位光点和小光点，同时将触发开关的信号经无线收发装置无线传输给计算机；之后再由计算机控制CCD摄像机采集一副图像，此时状态灯处于点亮状态，此时表示该点测量启动；Step 2. Start the visual coordinate measurement system based on the coded target built in step 1. When the target probe touches a certain point on the measured object, use it as the measured point; then press the trigger switch to light up all positioning points. At the same time, the signal of the trigger switch is wirelessly transmitted to the computer through the wireless transceiver device; after that, the computer controls the CCD camera to collect an image. At this time, the status light is on, indicating that the measurement of this point is started. ;

步骤3、待步骤2完成后，对成像位置进行解码；Step 3, after step 2 is completed, decode the imaging position;

步骤4、待步骤3完成后，对视场内定位光点进行亮度计算与调节；Step 4. After step 3 is completed, calculate and adjust the brightness of the positioned light spot in the field of view;

步骤5、待步骤4完成后，由亮度调整后的高质量图像精确确定定位光点中心图像坐标，由定位光点中心的图像坐标和编码靶标上已知点坐标即能求出靶标测头的坐标；Step 5. After step 4 is completed, the image coordinates of the center of the positioning light spot are accurately determined from the high-quality image after brightness adjustment, and the position of the target probe can be obtained from the image coordinates of the center of the positioning light spot and the coordinates of the known points on the coded target. coordinate;

步骤6、待步骤5完成后，由计算机发送结束指令经无线收发装置传输至编码靶标，编码靶标上的所有定位光点和小光点熄灭，同时编码靶标上状态灯熄灭，表示该点测量结束；Step 6. After step 5 is completed, the end command sent by the computer is transmitted to the coded target through the wireless transceiver device. All positioning light spots and small light spots on the coded target are extinguished, and the status light on the coded target is extinguished at the same time, indicating that the measurement of this point is over. ;

若以上任意一步出现错误，则无线收发装置发送错误指令至编码靶标，编码靶标上的状态灯闪烁，表示要重新进行测量；If an error occurs in any of the above steps, the wireless transceiver device will send an error command to the coded target, and the status light on the coded target will flash, indicating that the measurement needs to be re-measured;

在测量过程中，步骤2至步骤6均由计算机自动完成，只要按下编码靶标上的触发开关即能进行靶标测头的坐标测量；一点测量完成后，将编码靶标移动到下一测量点，重复步骤2至步骤6，以此类推，直到所有测点测量完成。During the measurement process, steps 2 to 6 are automatically completed by the computer, as long as the trigger switch on the coded target is pressed, the coordinate measurement of the target probe can be carried out; after one-point measurement is completed, the coded target is moved to the next measurement point, Repeat steps 2 to 6, and so on, until all measuring points are measured.

本发明第二种技术方案的特点还在于：The second technical solution of the present invention is characterized in that:

步骤1中基于编码靶标的视觉坐标测量系统，包括有编码靶标和计算机，且计算机内集成有测量算法模块；计算机与无线收发装置通过信号线连接，计算机还通过CCD传感器数据线与CCD摄像机连接，CCD摄像机支撑于摄像机支架上，编码靶标与CCD摄像机呈相对设置，CCD摄像机13成像视场要大于编码靶标的三个编码间距。The visual coordinate measurement system based on the coded target in step 1 includes a coded target and a computer, and a measurement algorithm module is integrated in the computer; the computer is connected to the wireless transceiver device through a signal line, and the computer is also connected to the CCD camera through a CCD sensor data line. The CCD camera is supported on the camera bracket, and the encoding target is set opposite to the CCD camera. The imaging field of view of the CCD camera 13 is greater than the three encoding distances of the encoding target.

步骤3具体按照以下方法实施：Step 3 is specifically implemented as follows:

由计算机内的图像处理单元对CCD摄像机采集到的图像进行处理，得到光点中心的图像坐标和光点大小；再由光点大小区分定位光点与小光点；The image processing unit in the computer processes the image collected by the CCD camera to obtain the image coordinates of the center of the light spot and the size of the light spot;

由定位光点附近小光点的分布得到该段图像对应的编码组合；小光点不出现在定位光点附近编码为A，小光点位于定位光点右边编码为B，小光点位于定位光点左边编码为C；The code combination corresponding to this segment of image is obtained from the distribution of small light spots near the positioning light spot; if the small light spot does not appear near the positioning light spot, it is coded as A, if the small light spot is located on the right side of the positioning light spot, it is coded as B, and if the small light spot is located The code on the left side of the dot is C;

查询计算机事先存储的编码表，即能解码获得该段图像在编码靶标上的位置。By querying the code table stored in advance by the computer, the position of the segment of the image on the coded target can be obtained by decoding.

步骤4具体按照以下方法实施：Step 4 is specifically implemented as follows:

图像处理判断光点亮度，与期望亮度进行比较，之后给出调节参数，具体按照以下步骤实施：Image processing judges the brightness of the light spot, compares it with the expected brightness, and then gives the adjustment parameters. The specific implementation is as follows:

步骤a、计算机根据步骤3解码得到的光点位置，给出视场内不同光点的调节参数，形成调节指令，并将调节指令经无线收发装置无线传输给编码靶标；Step a, the computer gives the adjustment parameters of different light spots in the field of view according to the position of the light spot obtained by decoding in step 3, forms an adjustment instruction, and wirelessly transmits the adjustment instruction to the coded target through the wireless transceiver device;

步骤b、经步骤a后，编码靶标接收到调节指令，通过驱动电路对相应的定位光点的亮度进行调节；Step b. After step a, the encoded target receives the adjustment instruction, and adjusts the brightness of the corresponding positioning light spot through the drive circuit;

步骤c、经步骤b亮度调节后，驱动电路发送控制信号并经无线收发装置传输给计算机，控制CCD摄像机重新采集一副图像。Step c. After the brightness is adjusted in step b, the drive circuit sends a control signal and transmits it to the computer through the wireless transceiver device to control the CCD camera to reacquire an image.

步骤5具体按照以下方法实施：Step 5 is specifically implemented as follows:

以CCD摄像机透视中心作为原点，光轴方向为z轴，平行于CCD像素的横纵方向分别作为x轴和y轴，建立摄像机三维坐标系o-xyz，以靶标测头点P为原点，建立测头空间坐标系P-x′y′z′；Taking the perspective center of the CCD camera as the origin, the optical axis as the z-axis, and the horizontal and vertical directions parallel to the CCD pixels as the x-axis and y-axis respectively, establish the three-dimensional coordinate system o-xyz of the camera, and take the target probe point P as the origin, establish Probe space coordinate system P-x'y'z';

若CCD摄像机的内参数：焦距、图像中心已知，给予足够多控制点(x_i′，y_i′，z_i′)及对应像点坐标(U_i，V_i)，由式(1)能求解得到其中的旋转、平移矩阵R、T；If the internal parameters of the CCD camera: focal length and image center are known, given enough control points (_xi ′, y_i ′,_zi ′) and corresponding image point coordinates (U_i , V_i ), the formula (1) It can be solved to obtain the rotation and translation matrices R and T;

矩阵T为两坐标系的平移矩阵，即将控制点靶标坐标系与摄像机o-xyz坐标系之间的平移矩阵，其物理意义是靶标坐标系原点与摄像机坐标系原点之间的平移位置关系；Matrix T is the translation matrix of the two coordinate systems, that is, the translation matrix between the control point target coordinate system and the camera o-xyz coordinate system, and its physical meaning is the translation position relationship between the origin of the target coordinate system and the origin of the camera coordinate system;

将靶标测头的中心作为测头坐标系的原点，则能确定靶标测头在CCD摄像机坐标系中的坐标，也就是平移矩阵T；在靶标定位光点共线条件下，设定空间点y′、z′坐标为0，则式(1)能转化为式(2)，变量代换得到未知量a_i的线性方程(3)，则有三对以上物像对应点，即能解出未知量a_i的最小二乘解；再由正交约束，解出T矩阵见式(4)，即靶标测头坐标系的原点在CCD摄像机坐标系中的位置能唯一确定；Taking the center of the target probe as the origin of the probe coordinate system, the coordinates of the target probe in the CCD camera coordinate system can be determined, that is, the translation matrix T; under the condition that the target positioning light points are collinear, set the space point y ', z' coordinates are 0, then formula (1) can be transformed into formula (2), and the linear equation (3) of unknown quantity a_i is obtained by variable substitution, then there are more than three pairs of corresponding points of the object image, which can solve the unknown The least squares solution of the quantity a_i ; then by the orthogonal constraint, solve the T matrix and see formula (4), that is, the position of the origin of the target probe coordinate system in the CCD camera coordinate system can be uniquely determined;

其中涉及的具体算法分别如下：The specific algorithms involved are as follows:

$\mathrm{<span><span>\&rho;</span>\&rho;</span>}<span><span>\&CenterDot;</span>\&CenterDot;</span>\left[\begin{array}{c}{\mathrm{<span><span>U</span>u</span>}}_{\mathrm{<span><span>i</span>i</span>}}\\ {\mathrm{<span><span>V</span>V</span>}}_{\mathrm{<span><span>i</span>i</span>}}\\ \mathrm{<span><span>1</span>1</span>}\end{array}\right]<span><span>=</span>=</span>\left[\begin{array}{ccc}\mathrm{<span><span>f</span>f</span>}& \mathrm{<span><span>0</span>0</span>}& \mathrm{<span><span>0</span>0</span>}\\ \mathrm{<span><span>0</span>0</span>}& \mathrm{<span><span>f</span>f</span>}& \mathrm{<span><span>0</span>0</span>}\\ \mathrm{<span><span>0</span>0</span>}& \mathrm{<span><span>0</span>0</span>}& \mathrm{<span><span>1</span>1</span>}\end{array}\right]<span><span>\&CenterDot;</span>\&CenterDot;</span><span><span>\&lsqb;</span>\&lsqb;</span>\begin{array}{cc}\mathrm{<span><span>R</span>R</span>}& \mathrm{<span><span>T</span>T</span>}\end{array}<span><span>\&rsqb;</span>\&rsqb;</span><span><span>\&CenterDot;</span>\&CenterDot;</span>\left[\begin{array}{c}{\mathrm{<span><span>x</span>x</span>}}_{\mathrm{<span><span>i</span>i</span>}}^{<span><span>\&prime;</span>\&prime;</span>}\\ {\mathrm{<span><span>y</span>the\; y</span>}}_{\mathrm{<span><span>i</span>i</span>}}^{<span><span>\&prime;</span>\&prime;</span>}\\ {\mathrm{<span><span>z</span>z</span>}}_{\mathrm{<span><span>i</span>i</span>}}^{<span><span>\&prime;</span>\&prime;</span>}\\ \mathrm{<span><span>1</span>1</span>}\end{array}\right]<span><span>-</span>-</span><span><span>-</span>-</span><span><span>-</span>-</span><span><span>(</span>(</span>\mathrm{<span><span>1</span>1</span>}<span><span>)</span>)</span><span><span>;</span>;</span>$

$\mathrm{<span><span>\&rho;</span>\&rho;</span>}<span><span>\&CenterDot;</span>\&CenterDot;</span>\left[\begin{array}{c}{\mathrm{<span><span>U</span>u</span>}}_{\mathrm{<span><span>i</span>i</span>}}\\ {\mathrm{<span><span>V</span>V</span>}}_{\mathrm{<span><span>i</span>i</span>}}\\ \mathrm{<span><span>1</span>1</span>}\end{array}\right]<span><span>=</span>=</span>\left[\begin{array}{cc}{\mathrm{<span><span>fr</span>fr</span>}}_{\mathrm{<span><span>1</span>1</span>}}& {\mathrm{<span><span>fT</span>f</span>}}_{\mathrm{<span><span>x</span>x</span>}}\\ {\mathrm{<span><span>fr</span>fr</span>}}_{\mathrm{<span><span>4</span>4</span>}}& {\mathrm{<span><span>fT</span>f</span>}}_{\mathrm{<span><span>y</span>the\; y</span>}}\\ {\mathrm{<span><span>r</span>r</span>}}_{\mathrm{<span><span>7</span>7</span>}}& {\mathrm{<span><span>fT</span>f</span>}}_{\mathrm{<span><span>z</span>z</span>}}\end{array}\right]<span><span>.</span>.</span>\left[\begin{array}{c}{\mathrm{<span><span>x</span>x</span>}}_{\mathrm{<span><span>i</span>i</span>}}^{<span><span>\&prime;</span>\&prime;</span>}\\ \mathrm{<span><span>1</span>1</span>}\end{array}\right]<span><span>-</span>-</span><span><span>-</span>-</span><span><span>-</span>-</span><span><span>(</span>(</span>\mathrm{<span><span>2</span>2</span>}<span><span>)</span>)</span><span><span>;</span>;</span>$

$\frac{\mathrm{<span><span>\&rho;</span>\&rho;</span>}}{{\mathrm{<span><span>T</span>T</span>}}_{\mathrm{<span><span>z</span>z</span>}}}<span><span>\&CenterDot;</span>\&CenterDot;</span>\left[\begin{array}{c}{\mathrm{<span><span>U</span>u</span>}}_{\mathrm{<span><span>i</span>i</span>}}\\ {\mathrm{<span><span>V</span>V</span>}}_{\mathrm{<span><span>i</span>i</span>}}\\ \mathrm{<span><span>1</span>1</span>}\end{array}\right]<span><span>=</span>=</span>\left[\begin{array}{cc}{\mathrm{<span><span>a</span>a</span>}}_{\mathrm{<span><span>1</span>1</span>}}& {\mathrm{<span><span>a</span>a</span>}}_{\mathrm{<span><span>2</span>2</span>}}\\ {\mathrm{<span><span>a</span>a</span>}}_{\mathrm{<span><span>3</span>3</span>}}& {\mathrm{<span><span>a</span>a</span>}}_{\mathrm{<span><span>4</span>4</span>}}\\ {\mathrm{<span><span>a</span>a</span>}}_{\mathrm{<span><span>5</span>5</span>}}& \mathrm{<span><span>1</span>1</span>}\end{array}\right]<span><span>\&CenterDot;</span>\&CenterDot;</span>\left[\begin{array}{c}{\mathrm{<span><span>x</span>x</span>}}_{\mathrm{<span><span>i</span>i</span>}}^{<span><span>\&prime;</span>\&prime;</span>}\\ \mathrm{<span><span>1</span>1</span>}\end{array}\right]<span><span>-</span>-</span><span><span>-</span>-</span><span><span>-</span>-</span><span><span>(</span>(</span>\mathrm{<span><span>3</span>3</span>}<span><span>)</span>)</span><span><span>;</span>;</span>$

$\left\{\begin{array}{c}{\mathrm{<span><span>T</span>T</span>}}_{\mathrm{<span><span>Z</span>Z</span>}}<span><span>=</span>=</span>\mathrm{<span><span>f</span>f</span>}<span><span>/</span>/</span>\sqrt{{\mathrm{<span><span>a</span>a</span>}}_{\mathrm{<span><span>1</span>1</span>}}^{\mathrm{<span><span>2</span>2</span>}}<span><span>+</span>+</span>{\mathrm{<span><span>a</span>a</span>}}_{\mathrm{<span><span>3</span>3</span>}}^{\mathrm{<span><span>2</span>2</span>}}<span><span>+</span>+</span>{\mathrm{<span><span>fa</span>fa</span>}}_{\mathrm{<span><span>5</span>5</span>}}^{\mathrm{<span><span>2</span>2</span>}}}\\ {\mathrm{<span><span>T</span>T</span>}}_{\mathrm{<span><span>X</span>x</span>}}<span><span>=</span>=</span>{\mathrm{<span><span>a</span>a</span>}}_{\mathrm{<span><span>2</span>2</span>}}{\mathrm{<span><span>T</span>T</span>}}_{\mathrm{<span><span>Z</span>Z</span>}}<span><span>/</span>/</span>\mathrm{<span><span>f</span>f</span>}\\ {\mathrm{<span><span>T</span>T</span>}}_{\mathrm{<span><span>Y</span>Y</span>}}<span><span>=</span>=</span>{\mathrm{<span><span>a</span>a</span>}}_{\mathrm{<span><span>4</span>4</span>}}{\mathrm{<span><span>T</span>T</span>}}_{\mathrm{<span><span>z</span>z</span>}}<span><span>/</span>/</span>\mathrm{<span><span>f</span>f</span>}\end{array}\right.<span><span>-</span>-</span><span><span>-</span>-</span><span><span>-</span>-</span><span><span>(</span>(</span>\mathrm{<span><span>4</span>4</span>}<span><span>)</span>)</span><span><span>;</span>;</span>$

在式(1)～式(4)中：(x_i′，y_i′，z_i′)为标志点在靶标坐标系中的坐标，(U_i，V_i)为对应像点坐标，R、T分别为靶标坐标系和摄像机坐标系之间的旋转、平移矩阵，f为已知的成像焦距，ρ为设定的系数；(T_x，T_y，T_z)为平移矩阵的展开形式，当定位光点共线，(r₁，r₄，r₇)为旋转矩阵的展开形式，a_i为变量代换的中间量。In formulas (1) to (4): (_xi ′, y_i ′,_zi ′) are the coordinates of the marker points in the target coordinate system, (U_i , V_i ) are the coordinates of the corresponding image points, R , T are the rotation and translation matrices between the target coordinate system and the camera coordinate system respectively, f is the known imaging focal length, ρ is the set coefficient; (T_x ,_Ty , T_z ) is the expanded form of the translation matrix , when the positioning light points are collinear, (r₁ , r₄ , r₇ ) is the expanded form of the rotation matrix, and a_i is the intermediate quantity of variable substitution.

本发明的有益效果在于：The beneficial effects of the present invention are:

(1)本发明编码靶标，其靶标设计简单紧凑，编码解码简单，只要成像视场大于三个编码间距即可正确解码和测量。(1) The coded target of the present invention has a simple and compact design, simple coding and decoding, and can be correctly decoded and measured as long as the imaging field of view is larger than three coding distances.

(2)利用本发明基于编码靶标的视觉坐标测量方法，通过获取编码靶标的局部信息，即可在相对大的范围内获得测点空间坐标，同时实现测量的大范围和高分辨率。(2) By using the coding target-based visual coordinate measurement method of the present invention, by obtaining the local information of the coding target, the spatial coordinates of the measuring point can be obtained in a relatively large range, and at the same time, a large range and high resolution of the measurement can be realized.

(3)本发明编码靶标，通过LED位置解码识别与LED亮度调节解决了靶点成像质量问题，不同测量条件下的成像质量较好，有效提高测量精度。(3) The coded target of the present invention solves the problem of target imaging quality through LED position decoding identification and LED brightness adjustment, the imaging quality under different measurement conditions is better, and the measurement accuracy is effectively improved.

(4)在本发明基于编码靶标的视觉坐标测量方法中涉及基于编码靶标的视觉坐标测量系统，其内部的测量系统采用无线收发，靶标移动自由同时和系统有效通讯，靶标设计兼顾使用简便性与功能完善性。(4) In the visual coordinate measurement method based on coded targets in the present invention, it involves a visual coordinates measurement system based on coded targets. The internal measurement system adopts wireless transmission and reception, and the target moves freely and communicates effectively with the system at the same time. Functional perfection.

附图说明Description of drawings

图1是本发明编码靶标的结构示意图；Fig. 1 is a schematic diagram of the structure of the coding target of the present invention;

图2是本发明基于编码靶标的视觉坐标测量方法中构建的基于编码靶标的视觉坐标测量系统的结构示意图；Fig. 2 is a schematic structural diagram of a coded target-based visual coordinate measurement system constructed in the coded target-based visual coordinate measurement method of the present invention;

图3是本发明基于编码靶标的视觉坐标测量方法中涉及的成像模型示意图。FIG. 3 is a schematic diagram of an imaging model involved in the coded target-based visual coordinate measurement method of the present invention.

图中，1.触发开关，2.定位光点，3.状态灯，4.把手，5.壳体，6.靶标测头，7.被测物，8.驱动电路，9.无线收发装置，10.计算机，11.CCD传感器数据线，12.摄像机支架，13.CCD摄像机，14.摄像机像平面，15.小光点。In the figure, 1. Trigger switch, 2. Positioning light spot, 3. Status light, 4. Handle, 5. Shell, 6. Target probe, 7. Object under test, 8. Driving circuit, 9. Wireless transceiver , 10. Computer, 11. CCD sensor data line, 12. Camera bracket, 13. CCD camera, 14. Camera image plane, 15. Small light spot.

具体实施方式detailed description

下面结合附图和具体实施方式对本发明进行详细说明。The present invention will be described in detail below in conjunction with the accompanying drawings and specific embodiments.

本发明编码靶标，其结构如图1所示，包括有长方体状的壳体5，壳体5内设置有驱动电路8；壳体5的外侧壁上分别设置有状态灯3、触发开关1及把手4；壳体5的外侧壁上还设有工作面，且工作面上设置有编码单元，编码单元包括有12个大小相同的定位光点2和5个大小相同的小光点15，且定位光点2和小光点15均为红外LED，定位光点2的尺寸大于小光点15的尺寸，12个定位光点2在工作面上沿中线等距离设置，5个小光点15分布于12个定位光点2之间；壳体5的底部设置有靶标测头6，且靶标测头6与12个定位光点2位于同一条直线上；其中，5个小光点15与12个定位光点2之间的分布方式具体如下：The coded target of the present invention, its structure as shown in Figure 1, comprises the housing 5 of cuboid shape, and the drive circuit 8 is arranged in the housing 5; The handle 4; the outer wall of the housing 5 is also provided with a working surface, and a coding unit is arranged on the working surface, and the coding unit includes 12 positioning light spots 2 of the same size and 5 small light spots 15 of the same size, and Both the positioning light spot 2 and the small light spot 15 are infrared LEDs, the size of the positioning light spot 2 is larger than the size of the small light spot 15, 12 positioning light spots 2 are arranged equidistantly along the midline on the working surface, and 5 small light spots 15 Distributed between the 12 positioning light spots 2; the bottom of the housing 5 is provided with a target probe 6, and the target probe 6 and the 12 positioning light spots 2 are located on the same straight line; among them, 5 small light spots 15 and The distribution among the 12 positioning light spots 2 is as follows:

在一个定位光点2的左边设置一个小光点15或在其右边设置一个小光点15或不设置小光点15；若小光点15设置于定位光点2的左边则编码为C；若小光点15设置于定位光点2的右边则编码为B；若定位光点2附近不设置小光点15则编码为A。Set a small light spot 15 on the left side of a positioning light spot 2 or set a small light spot 15 on its right side or not set a small light spot 15; if the small light spot 15 is arranged on the left side of the positioning light spot 2, then the code is C; If the small light spot 15 is arranged on the right side of the positioning light spot 2, the code is B; if the small light spot 15 is not set near the positioning light spot 2, the code is A.

如图1所示，仅用5个小光点15就实现了12个位置的编码，在工作面上，自下而上的12个编码为(AAABAACABBAB)，且任意三个连续位置编码不同。As shown in FIG. 1 , only 5 small light spots 15 are used to realize the coding of 12 positions. On the working surface, the 12 codings from bottom to top are (AAABAACABBAB), and any three consecutive positions are coded differently.

本发明基于编码靶标的视觉坐标测量系统的测量方法，具体按照以下步骤实施：The measurement method of the present invention based on the visual coordinate measurement system of the coded target is specifically implemented according to the following steps:

步骤1、利用编码靶标、计算机10及CCD摄像机13构建一个基于编码靶标的视觉坐标测量系统；Step 1, utilize coded target, computer 10 and CCD camera 13 to construct a visual coordinate measurement system based on coded target;

基于编码靶标的视觉坐标测量系统，其结构如图2所示，包括有编码靶标和计算机10，且计算机10内集成有测量算法模块；计算机10与无线收发装置9通过信号线连接，计算机10还通过CCD传感器数据线11与CCD摄像机13连接，CCD摄像机13支撑于摄像机支架12上，编码靶标与CCD摄像机13呈相对设置，CCD摄像机13成像视场要大于编码靶标的三个编码间距。Based on the visual coordinate measurement system of coded target, its structure is as shown in Figure 2, comprises coded target and computer 10, and is integrated with measurement algorithm module in computer 10; Computer 10 is connected with wireless transceiver 9 by signal line, and computer 10 also Connect with CCD camera 13 by CCD sensor data line 11, CCD camera 13 is supported on the camera bracket 12, and coding target and CCD camera 13 are oppositely set, and CCD camera 13 imaging field of view will be larger than three coding intervals of coding target.

在上述结构中，编码靶标用于提供测量参数，CCD摄像机13用于提供测量基准，无线收发装置9用于实现通讯，计算机10用于提供测量算法；各部分通过有机结合，能实现坐标测量。In the above structure, the coding target is used to provide measurement parameters, the CCD camera 13 is used to provide measurement benchmarks, the wireless transceiver 9 is used to realize communication, and the computer 10 is used to provide measurement algorithms; each part can realize coordinate measurement through organic combination.

采用基于编码靶标的视觉坐标测量系统测量之前，已知的参数包括有：(1)CCD摄像机13内参数；(2)编码靶标上各光点在靶标坐标系下的坐标；(3)亮度调节参数与光点亮度偏差的之间的调节函数。Before adopting the visual coordinate measurement system based on the coded target to measure, the known parameters include: (1) internal parameters of the CCD camera 13; (2) coordinates of each light spot on the coded target under the target coordinate system; (3) brightness adjustment The adjustment function between the parameter and the deviation of the brightness of the light spot.

另外，CCD摄像机13作为测量坐标系，在测量过程中保持不变；将靶标光点一面朝向CCD摄像机13的摄像头，由靶标测头6依次接触被测点进行坐标测量，每一点测量只需按下编码靶标上的触发开关1即可进行，其余过程全部由计算机10自动完成。In addition, the CCD camera 13 is used as the measurement coordinate system, which remains unchanged during the measurement process; the target light point faces the camera head of the CCD camera 13, and the target measuring head 6 contacts the measured points in turn for coordinate measurement, and each point measurement only needs to be pressed The trigger switch 1 on the coding target can be carried out, and the remaining processes are all automatically completed by the computer 10 .

步骤2、先启动步骤1中构建的基于编码靶标的视觉坐标测量系统，待靶标测头6接触到被测物7上的某一点，将其作为被测点；然后按动触发开关1，点亮所有定位光点2和小光点15，同时将触发开关1的信号经无线收发装置9无线传输给计算机10；之后再由计算机10控制CCD摄像机13采集一副图像，此时状态灯3处于点亮状态，此时表示该点测量启动。Step 2. First start the visual coordinate measurement system based on the coded target constructed in step 1. When the target probe 6 touches a certain point on the measured object 7, use it as the measured point; then press the trigger switch 1, point All positioning light spots 2 and small light spots 15 are bright, and the signal of the trigger switch 1 is wirelessly transmitted to the computer 10 through the wireless transceiver 9; then the computer 10 controls the CCD camera 13 to collect an image, and the status light 3 is in the When it is on, it means that the measurement of this point is started.

步骤3、待步骤2完成后，对成像位置进行解码，具体按照以下方法实施：Step 3. After step 2 is completed, the imaging position is decoded, and the specific implementation is as follows:

由计算机10内的图像处理单元对CCD摄像机13采集到的图像进行处理，得到光点中心的图像坐标和光点大小；再由光点大小区分定位光点2与小光点15；Process the image collected by the CCD camera 13 by the image processing unit in the computer 10 to obtain the image coordinates and the size of the light spot center; then distinguish and position the light spot 2 and the small light spot 15 by the size of the light spot;

由定位光点2附近小光点15的分布得到该段图像对应的编码组合；其中，小光点15不出现在定位光点2附近编码为A，小光点15位于定位光点2右边编码为B，小光点15位于定位光点2左边编码为C；The code combination corresponding to this segment of image is obtained from the distribution of the small light spot 15 near the positioning light spot 2; where the small light spot 15 does not appear near the positioning light spot 2 is coded as A, and the small light spot 15 is coded on the right side of the positioning light spot 2 It is B, and the small light spot 15 is located on the left side of the positioning light spot 2, and the code is C;

查询计算机10事先存储的编码表，即能解码获得该段图像在编码靶标上的位置。By querying the encoding table stored in advance by the computer 10, the position of the segment of the image on the encoding target can be obtained through decoding.

步骤4、待步骤3完成后，对视场内定位光点2进行亮度计算与调节，具体按照以下方法实施：Step 4. After step 3 is completed, calculate and adjust the brightness of the positioning light spot 2 in the field of view, and implement it according to the following method:

图像处理判断光点亮度，与期望亮度进行比较，之后给出调节参数，具体按照以下步骤实施：Image processing judges the brightness of the light spot, compares it with the expected brightness, and then gives the adjustment parameters, which are implemented according to the following steps:

步骤a、计算机10根据步骤3解码得到的光点位置，给出视场内不同光点的调节参数，形成调节指令，并将调节指令经无线收发装置9无线传输给编码靶标；Step a, the computer 10 decodes the position of the light spot obtained according to step 3, provides the adjustment parameters of different light spots in the field of view, forms an adjustment instruction, and wirelessly transmits the adjustment instruction to the coded target through the wireless transceiver device 9;

步骤b、经步骤a后，编码靶标接收到调节指令，通过驱动电路8对相应的定位光点2的亮度进行调节；Step b, after step a, the coded target receives an adjustment instruction, and adjusts the brightness of the corresponding positioning light spot 2 through the drive circuit 8;

步骤c、经步骤b亮度调节后，驱动电路8发送控制信号并经无线收发装置9传输给计算机10，控制CCD摄像机13重新采集一副图像。Step c: After the brightness is adjusted in step b, the drive circuit 8 sends a control signal and transmits it to the computer 10 through the wireless transceiver 9 to control the CCD camera 13 to reacquire an image.

步骤5、待步骤4完成后，由亮度调整后的高质量图像精确确定定位光点2中心图像坐标，由定位光点2中心的图像坐标和编码靶标上已知点坐标即能求出靶标测头6的坐标，计算方法具体如下：Step 5. After step 4 is completed, the image coordinates of the center of the positioning light spot 2 are accurately determined from the high-quality image after brightness adjustment, and the target measurement can be obtained from the image coordinates of the center of the positioning light spot 2 and the coordinates of the known points on the coding target. The coordinates of head 6 are calculated as follows:

以CCD摄像机13透视中心作为原点，光轴方向为z轴，平行于CCD像素的横纵方向分别作为x轴和y轴，建立摄像机三维坐标系o-xyz，以靶标测头6点P为原点，建立测头空间坐标系P-x′y′z′，具体如图3所示；Take the perspective center of the CCD camera 13 as the origin, the optical axis direction as the z-axis, and the horizontal and vertical directions parallel to the CCD pixels as the x-axis and y-axis respectively, establish the three-dimensional coordinate system o-xyz of the camera, and take the 6 point P of the target probe as the origin , establish the probe space coordinate system P-x′y′z′, as shown in Figure 3;

若CCD摄像机13的内参数，如：焦距、图像中心已知，给予足够多控制点(x_i′，y_i′，z_i′)及对应像点坐标(U_i，V_i)，由式(1)能求解得到其中的旋转、平移矩阵R、T；If the internal parameters of the CCD camera 13, such as: focal length and image center are known, given enough control points (_xi ′, y_i ′,_zi ′) and corresponding image point coordinates (U_i , V_i ), the formula (1) It can be solved to obtain the rotation and translation matrices R and T;

矩阵T为两坐标系的平移矩阵，即将控制点靶标坐标系与摄像机o-xyz坐标系之间的平移矩阵，其物理意义是靶标坐标系原点与摄像机坐标系原点之间的平移位置关系；Matrix T is the translation matrix of the two coordinate systems, that is, the translation matrix between the control point target coordinate system and the camera o-xyz coordinate system, and its physical meaning is the translation position relationship between the origin of the target coordinate system and the origin of the camera coordinate system;

将靶标测头6的中心作为测头坐标系的原点，则能确定靶标测头6在CCD摄像机13坐标系中的坐标，也就是平移矩阵T；在靶标定位光点共线条件下，设定空间点y′、z′坐标为0，则式(1)能转化为式(2)，变量代换得到未知量a_i的线性方程(3)，则有三对以上物像对应点，即能解出未知量a_i的最小二乘解；再由正交约束，解出T矩阵见式(4)，即靶标测头6坐标系的原点在CCD摄像机13坐标系中的位置能唯一确定，其中涉及的具体算法分别如下：Taking the center of the target probe 6 as the origin of the probe coordinate system, the coordinates of the target probe 6 in the CCD camera 13 coordinate system can be determined, that is, the translation matrix T; Space point y′, z′ coordinates are 0, then formula (1) can be transformed into formula (2), variable substitution can get the linear equation (3) of unknown quantity a_i , then there are more than three pairs of corresponding points of the object image, that is, Solve the least squares solution of the unknown quantity a_i ; then by the orthogonal constraint, solve the T matrix and see formula (4), that is, the position of the origin of the coordinate system of the target probe 6 in the coordinate system of the CCD camera 13 can be uniquely determined, The specific algorithms involved are as follows:

$\mathrm{<span><span>\&rho;</span>\&rho;</span>}<span><span>\&CenterDot;</span>\&CenterDot;</span>\left[\begin{array}{c}{\mathrm{<span><span>U</span>u</span>}}_{\mathrm{<span><span>i</span>i</span>}}\\ {\mathrm{<span><span>V</span>V</span>}}_{\mathrm{<span><span>i</span>i</span>}}\\ \mathrm{<span><span>1</span>1</span>}\end{array}\right]<span><span>=</span>=</span>\left[\begin{array}{ccc}\mathrm{<span><span>f</span>f</span>}& \mathrm{<span><span>0</span>0</span>}& \mathrm{<span><span>0</span>0</span>}\\ \mathrm{<span><span>0</span>0</span>}& \mathrm{<span><span>f</span>f</span>}& \mathrm{<span><span>0</span>0</span>}\\ \mathrm{<span><span>0</span>0</span>}& \mathrm{<span><span>0</span>0</span>}& \mathrm{<span><span>1</span>1</span>}\end{array}\right]<span><span>\&CenterDot;</span>\&CenterDot;</span><span><span>\&lsqb;</span>\&lsqb;</span>\begin{array}{cc}\mathrm{<span><span>R</span>R</span>}& \mathrm{<span><span>T</span>T</span>}\end{array}<span><span>\&rsqb;</span>\&rsqb;</span><span><span>\&CenterDot;</span>\&CenterDot;</span>\left[\begin{array}{c}{\mathrm{<span><span>x</span>x</span>}}_{\mathrm{<span><span>i</span>i</span>}}^{<span><span>\&prime;</span>\&prime;</span>}\\ {\mathrm{<span><span>y</span>the\; y</span>}}_{\mathrm{<span><span>i</span>i</span>}}^{<span><span>\&prime;</span>\&prime;</span>}\\ {\mathrm{<span><span>z</span>z</span>}}_{\mathrm{<span><span>i</span>i</span>}}^{<span><span>\&prime;</span>\&prime;</span>}\\ \mathrm{<span><span>1</span>1</span>}\end{array}\right]<span><span>-</span>-</span><span><span>-</span>-</span><span><span>-</span>-</span><span><span>(</span>(</span>\mathrm{<span><span>1</span>1</span>}<span><span>)</span>)</span><span><span>;</span>;</span>$

$\mathrm{<span><span>\&rho;</span>\&rho;</span>}<span><span>\&CenterDot;</span>\&Center\; Dot;</span>\left[\begin{array}{c}{\mathrm{<span><span>U</span>u</span>}}_{\mathrm{<span><span>i</span>i</span>}}\\ {\mathrm{<span><span>V</span>V</span>}}_{\mathrm{<span><span>i</span>i</span>}}\\ \mathrm{<span><span>1</span>1</span>}\end{array}\right]<span><span>=</span>=</span>\left[\begin{array}{cc}{\mathrm{<span><span>fr</span>fr</span>}}_{\mathrm{<span><span>1</span>1</span>}}& {\mathrm{<span><span>fT</span>f</span>}}_{\mathrm{<span><span>x</span>x</span>}}\\ {\mathrm{<span><span>fr</span>fr</span>}}_{\mathrm{<span><span>4</span>4</span>}}& {\mathrm{<span><span>fT</span>f</span>}}_{\mathrm{<span><span>y</span>the\; y</span>}}\\ {\mathrm{<span><span>r</span>r</span>}}_{\mathrm{<span><span>7</span>7</span>}}& {\mathrm{<span><span>fT</span>f</span>}}_{\mathrm{<span><span>z</span>z</span>}}\end{array}\right]<span><span>.</span>.</span>\left[\begin{array}{c}{\mathrm{<span><span>x</span>x</span>}}_{\mathrm{<span><span>i</span>i</span>}}^{<span><span>\&prime;</span>\&prime;</span>}\\ \mathrm{<span><span>1</span>1</span>}\end{array}\right]<span><span>-</span>-</span><span><span>-</span>-</span><span><span>-</span>-</span><span><span>(</span>(</span>\mathrm{<span><span>2</span>2</span>}<span><span>)</span>)</span><span><span>;</span>;</span>$

$\frac{\mathrm{<span><span>\&rho;</span>\&rho;</span>}}{{\mathrm{<span><span>T</span>T</span>}}_{\mathrm{<span><span>z</span>z</span>}}}<span><span>\&CenterDot;</span>\&CenterDot;</span>\left[\begin{array}{c}{\mathrm{<span><span>U</span>u</span>}}_{\mathrm{<span><span>i</span>i</span>}}\\ {\mathrm{<span><span>V</span>V</span>}}_{\mathrm{<span><span>i</span>i</span>}}\\ \mathrm{<span><span>1</span>1</span>}\end{array}\right]<span><span>=</span>=</span>\left[\begin{array}{cc}{\mathrm{<span><span>a</span>a</span>}}_{\mathrm{<span><span>1</span>1</span>}}& {\mathrm{<span><span>a</span>a</span>}}_{\mathrm{<span><span>2</span>2</span>}}\\ {\mathrm{<span><span>a</span>a</span>}}_{\mathrm{<span><span>3</span>3</span>}}& {\mathrm{<span><span>a</span>a</span>}}_{\mathrm{<span><span>4</span>4</span>}}\\ {\mathrm{<span><span>a</span>a</span>}}_{\mathrm{<span><span>5</span>5</span>}}& \mathrm{<span><span>1</span>1</span>}\end{array}\right]<span><span>\&CenterDot;</span>\&CenterDot;</span>\left[\begin{array}{c}{\mathrm{<span><span>x</span>x</span>}}_{\mathrm{<span><span>i</span>i</span>}}^{<span><span>\&prime;</span>\&prime;</span>}\\ \mathrm{<span><span>1</span>1</span>}\end{array}\right]<span><span>-</span>-</span><span><span>-</span>-</span><span><span>-</span>-</span><span><span>(</span>(</span>\mathrm{<span><span>3</span>3</span>}<span><span>)</span>)</span><span><span>;</span>;</span>$

$\left\{\begin{array}{c}{\mathrm{<span><span>T</span>T</span>}}_{\mathrm{<span><span>Z</span>Z</span>}}<span><span>=</span>=</span>\mathrm{<span><span>f</span>f</span>}<span><span>/</span>/</span>\sqrt{{\mathrm{<span><span>a</span>a</span>}}_{\mathrm{<span><span>1</span>1</span>}}^{\mathrm{<span><span>2</span>2</span>}}<span><span>+</span>+</span>{\mathrm{<span><span>a</span>a</span>}}_{\mathrm{<span><span>3</span>3</span>}}^{\mathrm{<span><span>2</span>2</span>}}<span><span>+</span>+</span>{\mathrm{<span><span>fa</span>fa</span>}}_{\mathrm{<span><span>5</span>5</span>}}^{\mathrm{<span><span>2</span>2</span>}}}\\ {\mathrm{<span><span>T</span>T</span>}}_{\mathrm{<span><span>X</span>x</span>}}<span><span>=</span>=</span>{\mathrm{<span><span>a</span>a</span>}}_{\mathrm{<span><span>2</span>2</span>}}{\mathrm{<span><span>T</span>T</span>}}_{\mathrm{<span><span>Z</span>Z</span>}}<span><span>/</span>/</span>\mathrm{<span><span>f</span>f</span>}\\ {\mathrm{<span><span>T</span>T</span>}}_{\mathrm{<span><span>Y</span>Y</span>}}<span><span>=</span>=</span>{\mathrm{<span><span>a</span>a</span>}}_{\mathrm{<span><span>4</span>4</span>}}{\mathrm{<span><span>T</span>T</span>}}_{\mathrm{<span><span>z</span>z</span>}}<span><span>/</span>/</span>\mathrm{<span><span>f</span>f</span>}\end{array}\right.<span><span>-</span>-</span><span><span>-</span>-</span><span><span>-</span>-</span><span><span>(</span>(</span>\mathrm{<span><span>4</span>4</span>}<span><span>)</span>)</span><span><span>;</span>;</span>$

在式(1)～式(4)中：(x_i′，y_i′，z_i′)为标志点在靶标坐标系中的坐标，(U_i，V_i)为对应像点坐标，R、T分别为靶标坐标系和摄像机坐标系之间的旋转、平移矩阵，f为已知的成像焦距，ρ为设定的系数；(T_x，T_y，T_z)为平移矩阵的展开形式，当定位光点共线，(r₁，r₄，r₇)为旋转矩阵的展开形式，a_i为变量代换的中间量。In formulas (1) to (4): (_xi ′, y_i ′,_zi ′) are the coordinates of the marker points in the target coordinate system, (U_i , V_i ) are the coordinates of the corresponding image points, R , T are the rotation and translation matrices between the target coordinate system and the camera coordinate system respectively, f is the known imaging focal length, ρ is the set coefficient; (T_x ,_Ty , T_z ) is the expanded form of the translation matrix , when the positioning light points are collinear, (r₁ , r₄ , r₇ ) is the expanded form of the rotation matrix, and a_i is the intermediate quantity of variable substitution.

步骤6、待步骤5完成后，由计算机10发送结束指令经无线收发装置9传输至编码靶标，编码靶标上的所有定位光点2和小光点15熄灭，同时编码靶标上状态灯3熄灭，表示该点测量结束；Step 6. After step 5 is completed, the computer 10 sends an end command and transmits it to the coded target through the wireless transceiver 9. All positioning light spots 2 and small light spots 15 on the coded target are extinguished, and the status light 3 on the coded target is extinguished at the same time. Indicates the end of the point measurement;

若以上任意一步出现错误，则无线收发装置9发送错误指令至编码靶标，编码靶标上的状态灯3闪烁，表示要重新进行测量；If an error occurs in any of the above steps, the wireless transceiver 9 sends an error command to the coded target, and the status light 3 on the coded target flickers, indicating that the measurement will be performed again;

在以上测量过程中，步骤2至步骤6均由计算机10自动完成，只要按下编码靶标上的触发开关1即能进行靶标测头6的坐标测量；一点测量完成后，将编码靶标移动到下一测量点，重复步骤2至步骤6，以此类推，直到所有测点测量完成。In the above measurement process, steps 2 to 6 are all automatically completed by the computer 10, as long as the trigger switch 1 on the coded target is pressed, the coordinate measurement of the target probe 6 can be carried out; after one-point measurement is completed, the coded target is moved to the next One measurement point, repeat step 2 to step 6, and so on until all measurement points are measured.

本发明编码靶标及其基于编码靶标的视觉坐标测量方法，能突破以往视觉坐标测量的局限性，同时实现测量的大范围和高精度。The coded target and the visual coordinate measurement method based on the coded target of the present invention can break through the limitation of the previous visual coordinate measurement, and simultaneously realize the large-scale and high-precision measurement.

Claims (8)

Translated fromChinese

1.编码靶标，其特征在于，包括有长方体状的壳体(5)，所述壳体(5)内设置有驱动电路(8)；1. The coding target is characterized in that, comprising a cuboid housing (5), the housing (5) is provided with a drive circuit (8);所述壳体(5)的外侧壁上分别设置有状态灯(3)、触发开关(1)及把手(4)；所述壳体(5)的外侧壁上还设有工作面，且工作面上设置有编码单元，所述编码单元包括有12个大小相同的定位光点(2)和5个大小相同的小光点(15)，且所述定位光点2和小光点(15)均为红外LED，所述定位光点(2)的尺寸大于小光点(15)的尺寸，所述12个定位光点(2)在工作面上沿中线等距离设置，所述5个小光点(15)分布于12个定位光点(2)之间；A status light (3), a trigger switch (1) and a handle (4) are respectively provided on the outer wall of the housing (5); a working surface is also provided on the outer wall of the housing (5), and the working A coding unit is arranged on the surface, and the coding unit includes 12 positioning light spots (2) of the same size and 5 small light spots (15) of the same size, and the positioning light spots 2 and small light spots (15) ) are infrared LEDs, the size of the positioning light spot (2) is greater than the size of the small light spot (15), the 12 positioning light spots (2) are arranged equidistantly along the midline on the working surface, and the 5 positioning light spots The small light spots (15) are distributed between the 12 positioning light spots (2);所述壳体(5)的底部设置有靶标测头(6)，且所述靶标测头(6)与12个定位光点(2)位于同一条直线上。The bottom of the casing (5) is provided with a target measuring head (6), and the target measuring head (6) and the 12 positioning light spots (2) are located on the same straight line.2.根据权利要求1所述的编码靶标，其特征在于，所述5个小光点(15)与12个定位光点(2)的位置关系具体如下：2. The coded target according to claim 1, wherein the positional relationship between the 5 small light spots (15) and the 12 positioning light spots (2) is as follows:在一个定位光点(2)的左边设置一个小光点(15)或在其右边设置一个小光点(15)或者不设置小光点(15)；A small light spot (15) is arranged on the left side of a positioning light spot (2) or a small light spot (15) is arranged on its right side or no small light spot (15) is set;若小光点(15)设置于定位光点(2)的左边则编码为C；若小光点(15)设置于定位光点(2)的右边则编码为B；若定位光点(2)附近不设置小光点(15)则编码为A；在工作面上构成的编码，任意三个连续位置编码不同。If the small light spot (15) is arranged on the left side of the positioning light spot (2), then it is coded as C; if the small light spot (15) is arranged on the right side of the positioning light spot (2), then it is coded as B; if the positioning light spot (2) ) is not provided with a small light spot (15) in the vicinity, then the encoding is A; the encoding formed on the working surface has different encodings at any three consecutive positions.3.根据权利要求1或2所述的编码靶标，其特征在于，所述定位光点(2)和所述小光点(15)均为红外LED，且所述定位光点(2)的尺寸大于小光点(15)。3. The coded target according to claim 1 or 2, characterized in that, the positioning light spot (2) and the small light spot (15) are infrared LEDs, and the positioning light spot (2) The size is larger than the small light spot (15).4.基于编码靶标的视觉坐标测量系统的测量方法，其特征在于，该方法依赖于本权利要求1中的编码靶标，具体按照以下步骤实施：4. The measurement method based on the visual coordinate measurement system of the coding target, it is characterized in that, the method relies on the coding target in the claim 1, specifically implements according to the following steps:步骤1、利用编码靶标、计算机(10)及CCD摄像机(13)构建一个基于编码靶标的视觉坐标测量系统；Step 1, utilize coded target, computer (10) and CCD camera (13) to construct a visual coordinate measurement system based on coded target;步骤2、先启动步骤1中构建的基于编码靶标的视觉坐标测量系统，待靶标测头(6)接触到被测物(7)上的某一点，将其作为被测点；然后按动触发开关(1)，点亮所有定位光点(2)和小光点(15)，同时将触发开关(1)的信号经无线收发装置(9)无线传输给计算机(10)；之后再由计算机(10)控制CCD摄像机(13)采集一副图像，此时状态灯3处于点亮状态，此时表示该点测量启动；Step 2. First start the visual coordinate measurement system based on the coded target constructed in step 1. When the target probe (6) touches a certain point on the measured object (7), use it as the measured point; then press the trigger Switch (1), lights all positioning light spots (2) and small light spots (15), simultaneously the signal of trigger switch (1) is wirelessly transmitted to computer (10) through wireless transceiver (9); then by computer (10) control the CCD camera (13) to gather a pair of image, this moment status light 3 is in the state of lighting, this moment represents that this point measurement starts;步骤3、待步骤2完成后，对成像位置进行解码；Step 3, after step 2 is completed, decode the imaging position;步骤4、待步骤3完成后，视场内定位光点(2)进行亮度计算与调节；Step 4. After step 3 is completed, locate the light spot (2) in the field of view to perform brightness calculation and adjustment;步骤5、待步骤4完成后，由亮度调整后的高质量图像精确确定定位光点(2)中心图像坐标，由定位光点(2)中心的图像坐标和编码靶标上已知点坐标即能求出靶标测头(6)的坐标；Step 5, after step 4 is completed, accurately determine the image coordinates of the center of the positioning light spot (2) from the high-quality image after brightness adjustment, and the image coordinates of the center of the positioning light spot (2) and the known point coordinates on the coding target can be Find the coordinates of the target probe (6);步骤6、待步骤5完成后，由计算机(10)发送结束指令经无线收发装置(9)传输至编码靶标，编码靶标上的所有定位光点(2)和小光点(15)熄灭，同时编码靶标上状态灯(3)熄灭，表示该点测量结束；Step 6, after step 5 is completed, the computer (10) sends an end command and transmits it to the coded target through the wireless transceiver (9), and all positioning light spots (2) and small light spots (15) on the coded target are extinguished, and at the same time The status light (3) on the coded target goes out, indicating that the measurement of this point is over;若以上任意一步出现错误，则无线收发装置(9)发送错误指令至编码靶标，编码靶标上的状态灯(3)闪烁，表示要重新进行测量；If an error occurs in any of the above steps, the wireless transceiver (9) sends an error command to the coded target, and the status light (3) on the coded target flickers, indicating that the measurement will be performed again;在以上测量过程中，步骤2至步骤6均由计算机(10)自动完成，只要按下编码靶标上的触发开关(1)即能进行靶标测头(6)的坐标测量；一点测量完成后，将编码靶标移动到下一测量点，重复步骤2至步骤6，以此类推，直到所有测点测量完成。In the above measurement process, steps 2 to 6 are all automatically completed by the computer (10), as long as the trigger switch (1) on the coded target is pressed, the coordinate measurement of the target probe (6) can be carried out; after one point measurement is completed, Move the coded target to the next measurement point, repeat steps 2 to 6, and so on until all measurement points are measured.5.根据权利要求4所述的基于编码靶标的视觉坐标测量系统的测量方法，其特征在于，所述步骤1中基于编码靶标的视觉坐标测量系统，包括有编码靶标和计算机(10)，且所述计算机(10)内集成有测量算法模块；所述计算机(10)与无线收发装置(9)通过信号线连接，所述计算机(10)还通过CCD传感器数据线(11)与CCD摄像机(13)连接，所述CCD摄像机(13)支撑于摄像机支架(12)上，所述编码靶标与CCD摄像机(13)呈相对设置，所述CCD摄像机(13)成像视场要大于编码靶标的三个编码间距。5. the measuring method based on the visual coordinate measuring system of coded target according to claim 4, is characterized in that, in the described step 1, based on the visual coordinate measuring system of coded target, comprises coded target and computer (10), and The measurement algorithm module is integrated in the computer (10); the computer (10) is connected with the wireless transceiver (9) by a signal line, and the computer (10) is also connected with the CCD camera ( 13) connect, described CCD camera (13) is supported on the camera support (12), and described encoding target and CCD camera (13) are relative setting, and described CCD camera (13) imaging field of view will be greater than three-dimensional of encoding target. coding distance.6.根据权利要求4所述的基于编码靶标的视觉坐标测量系统的测量方法，其特征在于，所述步骤3具体按照以下方法实施：6. the measuring method based on the visual coordinate measuring system of coding target according to claim 4, is characterized in that, described step 3 is specifically implemented according to the following method:由计算机(10)内的图像处理单元对CCD摄像机(13)采集到的图像进行处理，得到光点中心的图像坐标和光点大小；再由光点大小区分定位光点(2)与小光点(15)；The image processing unit in the computer (10) processes the image collected by the CCD camera (13), to obtain the image coordinates and the size of the light spot center; (15);由定位光点(2)附近小光点(15)的分布得到该段图像对应的编码组合；其中，小光点(15)不出现在定位光点(2)附近编码为A，小光点(15)位于定位光点(2)右边编码为B，所述小光点(15)位于定位光点(2)左边编码为C；Obtain the coding combination corresponding to this section image by the distribution of the small light spot (15) near the positioning light spot (2); wherein, the small light spot (15) does not appear in the vicinity of the positioning light spot (2) and is coded as A, and the small light spot (15) is coded as B on the right side of the positioning light spot (2), and is coded as C on the left side of the positioning light spot (2) of the small light spot (15);查询计算机(10)事先存储的编码表，即能解码获得该段图像在编码靶标上的位置。The position of the segment of the image on the coding target can be decoded by querying the coding table stored in advance by the computer (10).7.根据权利要求4所述的基于编码靶标的视觉坐标测量系统的测量方法，其特征在于，所述步骤4具体按照以下方法实施：7. the measuring method based on the visual coordinate measuring system of coded target according to claim 4, is characterized in that, described step 4 is specifically implemented according to the following method:图像处理判断光点亮度，与期望亮度进行比较，之后给出调节参数，具体按照以下步骤实施：Image processing judges the brightness of the light spot, compares it with the expected brightness, and then gives the adjustment parameters, which are implemented according to the following steps:步骤a、计算机(10)根据步骤3解码得到的光点位置，给出视场内不同光点的调节参数，形成调节指令，并将调节指令经无线收发装置(9)无线传输给编码靶标；Step a, the computer (10) provides the adjustment parameters of different light spots in the field of view according to the position of the light spot obtained by decoding in step 3, forms an adjustment instruction, and wirelessly transmits the adjustment instruction to the coded target through the wireless transceiver device (9);步骤b、经步骤a后，编码靶标接收到调节指令，通过驱动电路(8)对相应的定位光点(2)的亮度进行调节；Step b, after step a, the coded target receives an adjustment instruction, and adjusts the brightness of the corresponding positioning light spot (2) through the drive circuit (8);步骤c、经步骤b亮度调节后，驱动电路(8)发送控制信号并经无线收发装置(9)传输给计算机(10)，控制CCD摄像机(13)重新采集一副图像。Step c, after the brightness is adjusted in step b, the drive circuit (8) sends a control signal and transmits it to the computer (10) via the wireless transceiver (9) to control the CCD camera (13) to reacquire an image.8.根据权利要求4所述的基于编码靶标的视觉坐标测量系统的测量方法，其特征在于，所述步骤5具体按照以下方法实施：8. the measuring method based on the visual coordinate measuring system of coded target according to claim 4, is characterized in that, described step 5 is specifically implemented according to the following method:以CCD摄像机(13)透视中心作为原点，光轴方向为z轴，平行于CCD像素的横纵方向分别作为x轴和y轴，建立摄像机三维坐标系o-xyz，以靶标测头(6)点P为原点，建立测头空间坐标系P-x′y′z′；Taking the perspective center of the CCD camera (13) as the origin, the optical axis direction as the z-axis, and the horizontal and vertical directions parallel to the CCD pixels as the x-axis and y-axis respectively, establish the three-dimensional coordinate system o-xyz of the camera, and measure the head with the target (6) Point P is the origin, and the probe space coordinate system P-x′y′z′ is established;若CCD摄像机(13)的内参数：焦距、图像中心已知，给予足够多控制点(x_i′，y_i′，z_i′)及对应像点坐标(U_i，V_i)，由式(1)能求解得到其中的旋转、平移矩阵R、T；If the internal parameters of the CCD camera (13): focal length and image center are known, given enough control points (_xi ′, y_i ′,_zi ′) and corresponding image point coordinates (U_i , V_i ), the formula (1) It can be solved to obtain the rotation and translation matrices R and T;矩阵T为两坐标系的平移矩阵，即将控制点靶标坐标系与摄像机o-xyz坐标系之间的平移矩阵，其物理意义是靶标坐标系原点与摄像机坐标系原点之间的平移位置关系；Matrix T is the translation matrix of the two coordinate systems, that is, the translation matrix between the control point target coordinate system and the camera o-xyz coordinate system, and its physical meaning is the translation position relationship between the origin of the target coordinate system and the origin of the camera coordinate system;将靶标测头(6)的中心作为测头坐标系的原点，则能确定靶标测头(6)在CCD摄像机(13)坐标系中的坐标，也就是平移矩阵T；在靶标定位光点共线条件下，设定空间点y′、z′坐标为0，则式(1)能转化为式(2)，变量代换得到未知量a_i的线性方程(3)，则有三对以上物像对应点，即能解出未知量a_i的最小二乘解；再由正交约束，解出T矩阵见式(4)，即靶标测头(6)坐标系的原点在CCD摄像机(13)坐标系中的位置能唯一确定；With the center of the target probe (6) as the origin of the probe coordinate system, the coordinates of the target probe (6) in the CCD camera (13) coordinate system can be determined, that is, the translation matrix T; Under the linear condition, if the coordinates of the space point y′ and z′ are set to 0, the formula (1) can be transformed into the formula (2), and the variable substitution can obtain the linear equation (3) of the unknown quantity a_i , then there are more than three pairs of objects Like the corresponding point, the least squares solution of the unknown quantity a_i can be solved; then by the orthogonal constraint, the T matrix can be solved as shown in formula (4), that is, the origin of the coordinate system of the target probe (6) is at the CCD camera (13 ) The position in the coordinate system can be uniquely determined;其中涉及的具体算法分别如下：The specific algorithms involved are as follows:$\mathrm{<span><span>\&rho;</span>\&rho;</span>}<span><span>\&CenterDot;</span>\&Center\; Dot;</span>\left[\begin{array}{c}{\mathrm{<span><span>U</span>u</span>}}_{\mathrm{<span><span>i</span>i</span>}}\\ {\mathrm{<span><span>V</span>V</span>}}_{\mathrm{<span><span>i</span>i</span>}}\\ \mathrm{<span><span>1</span>1</span>}\end{array}\right]<span><span>=</span>=</span>\left[\begin{array}{ccc}\mathrm{<span><span>f</span>f</span>}& \mathrm{<span><span>0</span>0</span>}& \mathrm{<span><span>0</span>0</span>}\\ \mathrm{<span><span>0</span>0</span>}& \mathrm{<span><span>f</span>f</span>}& \mathrm{<span><span>0</span>0</span>}\\ \mathrm{<span><span>0</span>0</span>}& \mathrm{<span><span>0</span>0</span>}& \mathrm{<span><span>1</span>1</span>}\end{array}\right]<span><span>\&CenterDot;</span>\&Center\; Dot;</span><span><span>\&lsqb;</span>\&lsqb;</span>\begin{array}{cc}\mathrm{<span><span>R</span>R</span>}& \mathrm{<span><span>T</span>T</span>}\end{array}<span><span>\&rsqb;</span>\&rsqb;</span><span><span>\&CenterDot;</span>\&CenterDot;</span>\left[\begin{array}{c}{\mathrm{<span><span>x</span>x</span>}}_{\mathrm{<span><span>i</span>i</span>}}^{<span><span>\&prime;</span>\&prime;</span>}\\ {\mathrm{<span><span>y</span>the\; y</span>}}_{\mathrm{<span><span>i</span>i</span>}}^{<span><span>\&prime;</span>\&prime;</span>}\\ {\mathrm{<span><span>z</span>z</span>}}_{\mathrm{<span><span>i</span>i</span>}}^{<span><span>\&prime;</span>\&prime;</span>}\\ \mathrm{<span><span>1</span>1</span>}\end{array}\right]<span><span>-</span>-</span><span><span>-</span>-</span><span><span>-</span>-</span><span><span>(</span>(</span>\mathrm{<span><span>1</span>1</span>}<span><span>)</span>)</span><span><span>;</span>;</span>$$\mathrm{<span><span>\&rho;</span>\&rho;</span>}<span><span>\&CenterDot;</span>\&CenterDot;</span>\left[\begin{array}{c}{\mathrm{<span><span>U</span>u</span>}}_{\mathrm{<span><span>i</span>i</span>}}\\ {\mathrm{<span><span>V</span>V</span>}}_{\mathrm{<span><span>i</span>i</span>}}\\ \mathrm{<span><span>1</span>1</span>}\end{array}\right]<span><span>=</span>=</span>\left[\begin{array}{cc}{\mathrm{<span><span>fr</span>fr</span>}}_{\mathrm{<span><span>1</span>1</span>}}& {\mathrm{<span><span>fT</span>f</span>}}_{\mathrm{<span><span>x</span>x</span>}}\\ {\mathrm{<span><span>fr</span>fr</span>}}_{\mathrm{<span><span>4</span>4</span>}}& {\mathrm{<span><span>fT</span>f</span>}}_{\mathrm{<span><span>y</span>the\; y</span>}}\\ {\mathrm{<span><span>r</span>r</span>}}_{\mathrm{<span><span>7</span>7</span>}}& {\mathrm{<span><span>fT</span>f</span>}}_{\mathrm{<span><span>z</span>z</span>}}\end{array}\right]<span><span>\&CenterDot;</span>\&Center\; Dot;</span>\left[\begin{array}{c}{\mathrm{<span><span>x</span>x</span>}}_{\mathrm{<span><span>i</span>i</span>}}^{<span><span>\&prime;</span>\&prime;</span>}\\ \mathrm{<span><span>1</span>1</span>}\end{array}\right]<span><span>-</span>-</span><span><span>-</span>-</span><span><span>-</span>-</span><span><span>(</span>(</span>\mathrm{<span><span>2</span>2</span>}<span><span>)</span>)</span><span><span>;</span>;</span>$$\frac{\mathrm{<span><span>\&rho;</span>\&rho;</span>}}{{\mathrm{<span><span>T</span>T</span>}}_{\mathrm{<span><span>z</span>z</span>}}}<span><span>\&CenterDot;</span>\&CenterDot;</span>\left[\begin{array}{c}{\mathrm{<span><span>U</span>u</span>}}_{\mathrm{<span><span>i</span>i</span>}}\\ {\mathrm{<span><span>V</span>V</span>}}_{\mathrm{<span><span>i</span>i</span>}}\\ \mathrm{<span><span>1</span>1</span>}\end{array}\right]<span><span>=</span>=</span>\left[\begin{array}{cc}{\mathrm{<span><span>a</span>a</span>}}_{\mathrm{<span><span>1</span>1</span>}}& {\mathrm{<span><span>a</span>a</span>}}_{\mathrm{<span><span>2</span>2</span>}}\\ {\mathrm{<span><span>a</span>a</span>}}_{\mathrm{<span><span>3</span>3</span>}}& {\mathrm{<span><span>a</span>a</span>}}_{\mathrm{<span><span>4</span>4</span>}}\\ {\mathrm{<span><span>a</span>a</span>}}_{\mathrm{<span><span>5</span>5</span>}}& \mathrm{<span><span>1</span>1</span>}\end{array}\right]<span><span>\&CenterDot;</span>\&CenterDot;</span>\left[\begin{array}{c}{\mathrm{<span><span>x</span>x</span>}}_{\mathrm{<span><span>i</span>i</span>}}^{<span><span>\&prime;</span>\&prime;</span>}\\ \mathrm{<span><span>1</span>1</span>}\end{array}\right]<span><span>-</span>-</span><span><span>-</span>-</span><span><span>-</span>-</span><span><span>(</span>(</span>\mathrm{<span><span>3</span>3</span>}<span><span>)</span>)</span><span><span>;</span>;</span>$$\left\{\begin{array}{c}{\mathrm{<span><span>T</span>T</span>}}_{\mathrm{<span><span>Z</span>Z</span>}}<span><span>=</span>=</span>\mathrm{<span><span>f</span>f</span>}<span><span>/</span>/</span>\sqrt{{\mathrm{<span><span>a</span>a</span>}}_{\mathrm{<span><span>1</span>1</span>}}^{\mathrm{<span><span>2</span>2</span>}}<span><span>+</span>+</span>{\mathrm{<span><span>a</span>a</span>}}_{\mathrm{<span><span>3</span>3</span>}}^{\mathrm{<span><span>2</span>2</span>}}<span><span>+</span>+</span>{\mathrm{<span><span>fa</span>fa</span>}}_{\mathrm{<span><span>5</span>5</span>}}^{\mathrm{<span><span>2</span>2</span>}}}\\ {\mathrm{<span><span>T</span>T</span>}}_{\mathrm{<span><span>X</span>x</span>}}<span><span>=</span>=</span>{\mathrm{<span><span>a</span>a</span>}}_{\mathrm{<span><span>2</span>2</span>}}{\mathrm{<span><span>T</span>T</span>}}_{\mathrm{<span><span>Z</span>Z</span>}}<span><span>/</span>/</span>\mathrm{<span><span>f</span>f</span>}\\ {\mathrm{<span><span>T</span>T</span>}}_{\mathrm{<span><span>Y</span>Y</span>}}<span><span>=</span>=</span>{\mathrm{<span><span>a</span>a</span>}}_{\mathrm{<span><span>4</span>4</span>}}{\mathrm{<span><span>T</span>T</span>}}_{\mathrm{<span><span>z</span>z</span>}}<span><span>/</span>/</span>\mathrm{<span><span>f</span>f</span>}\end{array}\right.<span><span>-</span>-</span><span><span>-</span>-</span><span><span>-</span>-</span><span><span>(</span>(</span>\mathrm{<span><span>4</span>4</span>}<span><span>)</span>)</span><span><span>;</span>;</span>$在式(1)～式(4)中：(x_i′，y_i′，z_i′)为标志点在靶标坐标系中的坐标，(U_i，V_i)为对应像点坐标，R、T分别为靶标坐标系和摄像机坐标系之间的旋转、平移矩阵，f为已知的成像焦距，ρ为设定的系数；(T_x，T_y，T_z)为平移矩阵的展开形式，当定位光点共线，(r₁，r₄，r₇)为旋转矩阵的展开形式，a_i为变量代换的中间量。In formulas (1) to (4): (_xi ′, y_i ′,_zi ′) are the coordinates of the marker points in the target coordinate system, (U_i , V_i ) are the coordinates of the corresponding image points, R , T are the rotation and translation matrices between the target coordinate system and the camera coordinate system respectively, f is the known imaging focal length, ρ is the set coefficient; (T_x ,_Ty , T_z ) is the expanded form of the translation matrix , when the positioning light points are collinear, (r₁ , r₄ , r₇ ) is the expanded form of the rotation matrix, and a_i is the intermediate quantity of variable substitution.