Accuracy improvement method and optimal design of straightness in five-degree-of-freedom measurement of long guide

长导轨五自由度测量中直线度精度提升方法与优化设计 Accuracy improvement method and optimal design of straightness in five-degree-of-freedom measurement of long guide first-author 张 聪（1994-），男，河北沧州人，博士研究生，2016年于天津大学获得学士学位，主要从事机床误差测量与补偿方面的研究。E-mail： zhangcong2016@tju.edu.cn http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36183785&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36183794&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36183790&type= 22.01332855 32.00400543 张 聪 （1994-），男，河北沧州人，博士研究生，2016年于天津大学获得学士学位，主要从事机床误差测量与补偿方面的研究。E-mail： zhangcong2016@tju.edu.cn zhangcong2016@tju.edu.cn corresp E-mail： fjduan@tju.edu.cn E-mail： fjduan@tju.edu.cn 段发阶（1968-），男，湖南郴州人，教授，博士生导师，1989年、1992年和1994年于天津大学分别获得学士、硕士和博士学位，主要从事激光及光电测试技术、光纤传感技术、机床误差测量与补偿技术、计算机视觉、仪器智能化、海洋声学探测技术等方向研究。E-mail： fjduan@tju.edu.cn http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36183797&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36183804&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36183801&type= 22.01333618 32.00399017 段发阶 （1968-），男，湖南郴州人，教授，博士生导师，1989年、1992年和1994年于天津大学分别获得学士、硕士和博士学位，主要从事激光及光电测试技术、光纤传感技术、机床误差测量与补偿技术、计算机视觉、仪器智能化、海洋声学探测技术等方向研究。E-mail： fjduan@tju.edu.cn fjduan@tju.edu.cn 天津大学 精密测试技术及仪器国家重点实验室，天津 300072 天津大学 精密测试技术及仪器国家重点实验室，天津 300072 State Key Laboratory of Precision Measuring Technology & Instruments， Tianjin University， Tianjin 300072， China State Key Laboratory of Precision Measuring Technology & Instruments， Tianjin University， Tianjin 300072， China 为了提升长导轨五自由度测量中直线度测量精度，对五自由度测量装置中直线度测量单元的测量光路、误差模型和标定方法等进行研究。长距离测量时基准光束角度漂移会造成严重的直线度误差，光路中采用了望远物镜结构测量角度漂移，相比于单透镜，这种可以在有限空间内增加焦距，使得组合焦距大于机械尺寸，从而提高测量精度，并通过补偿提高测量稳定性；分析了角度串扰引入的误差，并结合装置中的角度测量值建立了补偿模型；采用四象限探测器作为直线度测量传感器，为解决传感器测量非线性随量程变大而加剧以及测量灵敏度随长导轨移动非线性变化的问题，提出了一种多项式拟合与三次样条插值拟合相结合的分区间标定方法。试验结果表明：在5 m的测量距离以及±400 μm的量程内，与激光干涉仪测量结果对比，误差标准差小于0.3 μm，3 h内漂移小于±0.5 μm。满足长导轨直线度测量对大量程、高精度、高稳定性的要求。 To improve the straightness accuracy during the five-degree-of-freedom measurement （5DOFM） of long guide， the measurement light path， error model， and calibration method of the straightness measurement unit in the 5DOFM device were examined. During long distance measurement， the angle drift of the reference beam causes severe straightness error. Accordingly， a telescopic objective lens structure was adopted to measure the angle drift in the optical path. Compared with a single lens， the focal length can be increased in a limited space， such that the combined focal length is larger than the mechanical size， thereby improving the measurement accuracy， and compensating for the angle drift to improve the measurement stability. Additionally， the error introduced by angle crosstalk was analyzed， and a compensation model was established based on the angle measurement value in the device. A four-quadrant detector was used as the straightness measurement sensor. To solve the problems that the nonlinearity of the sensor increases as the range increases， and the sensitivity changes nonlinearly with the movement of the long guide， an interval calibration method combining polynomial fitting and cubic spline interpolation fitting was proposed. The experimental results showed that within the 5-m measurement distance and ±400-μm range， the error standard deviation was less than 0.3 μm compared with the measurement results of the laser interferometer， and the drift was less than ±0.5 μm within 3 h. Thus， the straightness measurement unit meets the requirements of large range， high precision， and high stability of long guide straightness measurement. 光束漂移 角度串扰 四象限探测器 长距离 多项式拟合 三次样条插值拟合 beam drift angle crosstalk quadrant detector long distance polynomial fitting cubic spline interpolation fitting 1 引 言 1 多自由度测量是实现机床几何误差快速、高精度检测的重要手段 ［ 1 1 ］ ，为补偿机床空间误差、提高加工精度提供数据支撑。对于大型机床，其最长导轨的行程往往可以达到4米以上 ［ 2 2 ］ ，然而测量距离的增加会影响多自由度的测量精度。多自由度测量装置中的直线度测量单元一般基于激光准直原理研制而成，在长距离测量时，受光束漂移、角度耦合、光斑尺寸变化等影响较大 ［ 3 3 ］ ，将直接影响大型机床误差测量精度。因此，对长导轨多自由度测量中的直线度测量单元进行优化设计具有重要意义。 目前，基于激光准直原理的直线度测量方法被广泛应用于多自由度测量装置中 ［ 4 4 - 6 6 4-6 ］ ，为了提高测量精度，众多学者从不同角度进行了研究。为了抑制或补偿激光光束漂移，冯其波等人 ［ 7 7 ］ 使用单模光纤耦合输出的激光器作为光源，能够有效抑制光源本身的漂移，而且结构小巧易于系统集成；殷纯永等人 ［ 8 8 ］ 在光路结构中使用两束足够靠近的激光，一束用于测量，另一束用于采集噪声，可以在一定程度上去除噪声，但光路结构复杂，难以安装，使用场景受限；由凤玲等人 ［ 9 9 ］ 使测量光路与监测激光漂移光路完全共路，提高补偿效率，对于激光器本身的漂移具有很好的补偿效果，但难以补偿局部空气扰动带来的光束漂移；朱凡等人 ［ 10 10 ］ 使用平动式反射镜补偿光束漂移，杨滨赫等人 ［ 11 11 ］ 使用带压电陶瓷的二维角度调整架补偿光束漂移，有效提高了激光准直精度，但试验均在1米测量距离内进行，不能克服长距离测量时空气扰动的影响。多自由度测量装置工作时，角度测量往往会对直线度测量产生耦合干扰，高帅 ［ 12 12 ］ 、崔存星 ［ 13 13 ］ 分别对五自由度、六自由度测量装置测量中产生的误差串扰进行了分析与补偿，提高了测量精度，但误差串扰模型与光路结构直接相关，当光路结构改变后，需要进行特定的分析与处理。四象限探测器（Quadrant Detector， QD）由于精度高、响应速度快等特点常被用作直线度测量传感器， 但存在非线性随量程变大而加剧以及灵敏度受光斑尺寸影响的问题，当在固定距离下测量时，光束截面尺寸不变，标定时使用多项式拟合 ［ 14 14 - 16 16 14-16 ］ 可以很好地解决QD非线性的问题；当被测物移动时，光束截面尺寸变化，QD测量灵敏度随之变化，蔡引娣等人 ［ 17 17 ］ 和崔存星等人 ［ 18 18 ］ 均按灵敏度随距离线性变化处理，但在长距离测量时，这种处理方法的误差很大，需要一种新的方法来解决灵敏度变化的问题。 本课题组提出了一种基于激光准直与自准直原理的五自由度测量装置 ［ 19 19 ］ ，本文在此基础上对装置中的直线度测量单元进行了优化设计。其中，采用了望远物镜结构，相比于单透镜，可以在有限空间内增加焦距至大于机械尺寸，使光束角度漂移测量更加精确；分析了角度串扰引入的直线度误差，对直线度测量模型进行修正；提出了一种多项式拟合与三次样条插值拟合相结合的分区间标定方法，实现QD长距离大量程下的高精度测量；最后，对优化后的直线度测量单元性能进行了试验验证。 2 直线度测量单元及其测量模型 1 2.1　五自由度测量结构中直线度测量光路 2 五自由度测量结构基于激光准直与自准直原理设计，采用激光器发射与探测器接收分体式结构，简称收发分体式结构。如 图1 图1 所示，发射端安装在导轨基座上固定不动，接收端安装在导轨滑块上同时移动，实现导轨误差测量。为了实现长导轨的五自由度测量，将位置测量传感器全部安装在接收端，这种结构比收发一体式结构 ［ 20 20 ］ 缩短一半的光程，抗干扰能力增强。 该光路结构可以测量俯仰角、偏摆角、滚转角、水平直线度与垂直直线度共五个自由度。激光器（LD）发出的光束经平面镜1（M1）反射后，经分光镜1（BS1）后被分为两束光。其中，透射光经分光镜2（BS2）后又被分为两束光，一束光照射在四象限探测器1（QD1）上，测量水平直线度和垂直直线度，另一束光经凸透镜1（L1）聚焦在位置敏感探测器1（PSD1）上，测量俯仰角与偏摆角；反射光经平面镜2（M2）反射后与经BS1的透射光平行，照射在四象限探测器2（QD2）上，测量垂直直线度，结合QD1测得的垂直直线度解算出滚转角。 构成直线度测量单元的基本元件包括LD、M1、BS1、BS2与QD1，如 图2 图2 所示。当只产生直线度时，光斑在QD1上坐标变化为（ X ， Y ），QD1四个通道输出光电流，并转换成电压用于计算，水平直线度和垂直直线度测量模型表示为： <math display="block" id="M1"><mfenced open="{" close="" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><msub><mrow><mi>δ</mi></mrow><mrow><mi>x</mi></mrow></msub><mo>=</mo><mo>-</mo><mi>X</mi></mtd></mtr><mtr><mtd><msub><mrow><mi>δ</mi></mrow><mrow><mi>y</mi></mrow></msub><mo>=</mo><mo>-</mo><mi>Y</mi></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36183821&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36183817&type= 15.91733265 8.55133343 ， 其中， X 、 Y 是关于电压的函数，需要经过标定后确定，标定方法将在下一节讨论。 2.2　光束漂移测量光路结构及补偿模型 2 长距离测量时，发射端的光束角度漂移对直线度测量产生严重的干扰，需要更精密的光路测量漂移量。在发射端末端增加望远物镜结构，如 图3 图3 所示，光束经分光镜3（BS3）反射后先后经过凸透镜2（L2）、凹透镜3（L3）聚焦在位置敏感探测器2（PSD2）上，从而测量光束角度漂移。 望远物镜的焦距大于机械尺寸，相比于单透镜，可以在相同安装空间内获得更长的焦距，根据自准直测量原理知，当PSD2位置测量精度相同时，透镜焦距越长，角度测量精度越高，因此，望远物镜可以达到更高的测角精度，望远镜光路如 图4 图4 所示。设L2的焦距为 f 2 、L3的焦距为 f 3 、L2与L3之间的距离为 d ，则望远物镜的组合焦距 f 表示为： <math display="block" id="M2"><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>f</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><msub><mrow><mi>f</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mrow></mfrac><mo>+</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><msub><mrow><mi>f</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub></mrow></mfrac><mo>-</mo><mfrac><mrow><mi>d</mi></mrow><mrow><msub><mrow><mi>f</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><msub><mrow><mi>f</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36183856&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36183852&type= 29.46400070 8.80533314 . 当光束角度漂移 <math id="M3"><msub><mrow><mi>θ</mi></mrow><mrow><mi>y</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36183863&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36183859&type= 2.96333337 3.72533321 、 <math id="M4"><msub><mrow><mi>θ</mi></mrow><mrow><mi>z</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36183868&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36183865&type= 2.87866688 3.72533321 时，如 图5 图5 所示，PSD2上聚焦的光点坐标变化 <math id="M5"><mo stretchy="false">(</mo><mi mathvariant="normal">Δ</mi><msub><mrow><mi>z</mi></mrow><mrow><mi mathvariant="normal">P</mi><mi mathvariant="normal">S</mi><mi mathvariant="normal">D</mi><mn 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mathvariant="normal">2</mn></mrow></msub></mrow><mrow><mi>f</mi></mrow></mfrac><mo stretchy="false">(</mo><msub><mrow><mi>l</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>l</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mi mathvariant="normal">Δ</mi><msub><mrow><mi>δ</mi></mrow><mrow><mi>y</mi><mn mathvariant="normal">1</mn></mrow></msub><mo>=</mo><msub><mrow><mi>θ</mi></mrow><mrow><mi>z</mi></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi>l</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>l</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><mi mathvariant="normal">Δ</mi><msub><mrow><mi>y</mi></mrow><mrow><mi mathvariant="normal">P</mi><mi mathvariant="normal">S</mi><mi mathvariant="normal">D</mi><mn mathvariant="normal">2</mn></mrow></msub></mrow><mrow><mi>f</mi></mrow></mfrac><mo stretchy="false">(</mo><msub><mrow><mi>l</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>l</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36183888&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36183886&type= 57.91199875 20.48933220 . 2.3　角度串扰误差分析及补偿模型 2 五自由度测量时各自由度之间产生串扰，需要分析角度测量对直线度测量产生的串扰误差，构建补偿模型并修正直线度测量模型。以水平直线度为例，光束透过BS2照射到QD1上，BS2是平行平板，假设接收端此时无水平直线度误差，当接收端偏摆角变化 <math id="M9"><msub><mrow><mi>ε</mi></mrow><mrow><mi>y</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36184271&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36184270&type= 2.53999996 3.72533321 时，光路以QD1中心旋转 <math id="M10"><msub><mrow><mi>ε</mi></mrow><mrow><mi>y</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36184271&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36184270&type= 2.53999996 3.72533321 ，光线透射后出现偏移，如 图6 图6 （a）所示，设玻璃折射率为 n ，BS2边长为 h ，其他角度对直线度测量的影响可以忽略。同理，当进行垂直直线度测量时，俯仰角 <math id="M11"><msub><mrow><mi>ε</mi></mrow><mrow><mi>x</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36183947&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36183945&type= 2.70933342 3.72533321 产生串扰误差，如 图6 图6 （b）所示。水平直线度、垂直直线度角度串扰误差补偿模型 <math id="M12"><mi mathvariant="normal">Δ</mi><msub><mrow><mi>δ</mi></mrow><mrow><mi>x</mi><mn mathvariant="normal">2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36183915&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36183912&type= 6.60400009 3.72533321 、 <math id="M13"><mi mathvariant="normal">Δ</mi><msub><mrow><mi>δ</mi></mrow><mrow><mi>y</mi><mn mathvariant="normal">2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36183920&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36183918&type= 6.51933336 3.72533321 为： <math display="block" id="M14"><mfenced open="{" close="" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><mi mathvariant="normal">Δ</mi><msub><mrow><mi>δ</mi></mrow><mrow><mi>x</mi><mn mathvariant="normal">2</mn></mrow></msub><mo>=</mo><mo>-</mo><mi>h</mi><msub><mrow><mi>ε</mi></mrow><mrow><mi>y</mi></mrow></msub><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>n</mi></mrow></mfrac></mrow></mfenced></mtd></mtr><mtr><mtd><mi mathvariant="normal">Δ</mi><msub><mrow><mi>δ</mi></mrow><mrow><mi>y</mi><mn mathvariant="normal">2</mn></mrow></msub><mo>=</mo><mi>h</mi><msub><mrow><mi>ε</mi></mrow><mrow><mi>x</mi></mrow></msub><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>n</mi></mrow></mfrac></mrow></mfenced></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36183924&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36183922&type= 34.03599930 21.08200073 ， 其中：接收端偏摆角 <math id="M15"><msub><mrow><mi>ε</mi></mrow><mrow><mi>y</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36184271&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36184270&type= 2.53999996 3.72533321 、俯仰角 <math id="M16"><msub><mrow><mi>ε</mi></mrow><mrow><mi>x</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36183947&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36183945&type= 2.70933342 3.72533321 由PSD1基于自准直原理测得。PSD1测量结果 <math id="M17"><msub><mrow><mi>ε</mi></mrow><mrow><mi>y</mi></mrow></msub><mi>'</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36183935&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36183933&type= 3.55599999 4.23333359 、 <math id="M18"><msub><mrow><mi>ε</mi></mrow><mrow><mi>x</mi></mrow></msub><mi>'</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36183940&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36183937&type= 3.64066648 4.23333359 中包含接收端角度和光束漂移角度，经PSD2补偿后即可得到 <math id="M19"><msub><mrow><mi>ε</mi></mrow><mrow><mi>y</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36184271&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36184270&type= 2.53999996 3.72533321 、 <math id="M20"><msub><mrow><mi>ε</mi></mrow><mrow><mi>x</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36183947&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36183945&type= 2.70933342 3.72533321 ，表示为： 1;2; <math display="block" id="M21"><mfenced open="{" close="" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><msub><mrow><mi>ε</mi></mrow><mrow><mi>y</mi></mrow></msub><mo>=</mo><msub><mrow><mi>ε</mi></mrow><mrow><mi>y</mi></mrow></msub><mi>'</mi><mo>-</mo><msub><mrow><mi>θ</mi></mrow><mrow><mi>y</mi></mrow></msub><mo>=</mo><mfrac><mrow><mi mathvariant="normal">Δ</mi><msub><mrow><mi>z</mi></mrow><mrow><mi mathvariant="normal">P</mi><mi mathvariant="normal">S</mi><mi mathvariant="normal">D</mi><mn mathvariant="normal">1</mn></mrow></msub></mrow><mrow><msub><mrow><mi>f</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></mfrac><mo>-</mo><mfrac><mrow><mi mathvariant="normal">Δ</mi><msub><mrow><mi>z</mi></mrow><mrow><mi mathvariant="normal">P</mi><mi mathvariant="normal">S</mi><mi mathvariant="normal">D</mi><mn mathvariant="normal">2</mn></mrow></msub></mrow><mrow><mi>f</mi></mrow></mfrac></mtd></mtr><mtr><mtd><msub><mrow><mi>ε</mi></mrow><mrow><mi>x</mi></mrow></msub><mo>=</mo><msub><mrow><mi>ε</mi></mrow><mrow><mi>x</mi></mrow></msub><mi>'</mi><mo>+</mo><msub><mrow><mi>θ</mi></mrow><mrow><mi>z</mi></mrow></msub><mo>=</mo><mo>-</mo><mfrac><mrow><mi mathvariant="normal">Δ</mi><msub><mrow><mi>y</mi></mrow><mrow><mi mathvariant="normal">P</mi><mi mathvariant="normal">S</mi><mi mathvariant="normal">D</mi><mn mathvariant="normal">1</mn></mrow></msub></mrow><mrow><msub><mrow><mi>f</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></mfrac><mo>+</mo><mfrac><mrow><mi mathvariant="normal">Δ</mi><msub><mrow><mi>y</mi></mrow><mrow><mi mathvariant="normal">P</mi><mi mathvariant="normal">S</mi><mi mathvariant="normal">D</mi><mn mathvariant="normal">2</mn></mrow></msub></mrow><mrow><mi>f</mi></mrow></mfrac></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36183967&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36183965&type= 54.01733780 20.48933220 ， 其中： f 1 为L1焦距， <math id="M22"><mo stretchy="false">(</mo><mi mathvariant="normal">Δ</mi><msub><mrow><mi>z</mi></mrow><mrow><mi mathvariant="normal">P</mi><mi mathvariant="normal">S</mi><mi mathvariant="normal">D</mi><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><mi mathvariant="normal">Δ</mi><msub><mrow><mi>y</mi></mrow><mrow><mi mathvariant="normal">P</mi><mi mathvariant="normal">S</mi><mi mathvariant="normal">D</mi><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36183971&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36183969&type= 23.45266724 4.91066647 为PSD1上光点坐标变化。经补偿后的直线度测量模型为： <math display="block" id="M23"><mfenced open="{" close="" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><msub><mrow><mi>δ</mi></mrow><mrow><mi>x</mi></mrow></msub><mi>'</mi><mo>=</mo><msub><mrow><mi>δ</mi></mrow><mrow><mi>x</mi></mrow></msub><mo>+</mo><mi mathvariant="normal">Δ</mi><msub><mrow><mi>δ</mi></mrow><mrow><mi>x</mi><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><mi mathvariant="normal">Δ</mi><msub><mrow><mi>δ</mi></mrow><mrow><mi>x</mi><mn mathvariant="normal">2</mn></mrow></msub></mtd></mtr><mtr><mtd><msub><mrow><mi>δ</mi></mrow><mrow><mi>y</mi></mrow></msub><mi>'</mi><mo>=</mo><msub><mrow><mi>δ</mi></mrow><mrow><mi>y</mi></mrow></msub><mo>+</mo><mi mathvariant="normal">Δ</mi><msub><mrow><mi>δ</mi></mrow><mrow><mi>y</mi><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><mi mathvariant="normal">Δ</mi><msub><mrow><mi>δ</mi></mrow><mrow><mi>y</mi><mn mathvariant="normal">2</mn></mrow></msub></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36183974&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36183973&type= 36.99933624 9.65200043 . 3 长距离直线度测量标定方法 1 3.1　QD理论测量模型 2 理想情况下，单模光纤耦合激光器输出的光束可以看做高斯光束，基于高斯光束建立QD测量模型，如 图7 图7 所示。光束传播过程中的截面半径是双曲线，可以表示为 ［ 1 1 ］ ： <math display="block" id="M24"><mi>ω</mi><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mrow><mi>ω</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><msup><mrow><mfenced open="[" close="]" separators="|"><mrow><mn mathvariant="normal">1</mn><mo>+</mo><msup><mrow><mfenced separators="|"><mrow><mfrac><mrow><mi>λ</mi><mo stretchy="false">(</mo><mi>z</mi><mo>-</mo><msub><mrow><mi>z</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">)</mo></mrow><mrow><mi>π</mi><msubsup><mrow><mi>ω</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup></mrow></mfrac></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfenced></mrow><mrow><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36183978&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36183976&type= 47.92133331 13.97000027 ， 其中： <math id="M25"><msub><mrow><mi>ω</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36183981&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36183980&type= 3.64066648 3.72533321 为束腰半径， z 为传播距离， <math id="M26"><msub><mrow><mi>z</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36183986&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36183983&type= 2.96333337 3.72533321 为束腰位置， <math id="M27"><mi>λ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36183990&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36183988&type= 1.77800000 2.79399991 为激光波长。 光斑照射在QD上后，获得四通道电压，分别为 <math id="M28"><msub><mrow><mi>u</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36184001&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36183999&type= 3.04800010 3.72533321 、 <math id="M29"><msub><mrow><mi>u</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36184005&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36184003&type= 3.04800010 3.72533321 、 <math id="M30"><msub><mrow><mi>u</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36184010&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36184007&type= 3.04800010 3.72533321 、 <math id="M31"><msub><mrow><mi>u</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36184014&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36184012&type= 3.04800010 3.72533321 ，可以计算出光斑的相对位置 ［ 21 21 ］ ： <math display="block" id="M32"><mfenced open="{" close="" separators="|"><mrow><mtable><mtr><mtd><msub><mrow><mi>σ</mi></mrow><mrow><mi>x</mi></mrow></msub><mo>=</mo><mfrac><mrow><msub><mrow><mi>u</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>u</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msub><mo>-</mo><msub><mrow><mi>u</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>-</mo><msub><mrow><mi>u</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub></mrow><mrow><msub><mrow><mi>u</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>u</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>+</mo><msub><mrow><mi>u</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mo>+</mo><msub><mrow><mi>u</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msub></mrow></mfrac></mtd></mtr><mtr><mtd><msub><mrow><mi>σ</mi></mrow><mrow><mi>y</mi></mrow></msub><mo>=</mo><mfrac><mrow><msub><mrow><mi>u</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>u</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>-</mo><msub><mrow><mi>u</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mo>-</mo><msub><mrow><mi>u</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msub></mrow><mrow><msub><mrow><mi>u</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>u</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>+</mo><msub><mrow><mi>u</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mo>+</mo><msub><mrow><mi>u</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msub></mrow></mfrac></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36184018&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36184016&type= 37.08399963 19.81200027 . 光斑位置与其相对位置之间具有非线性关系，认为光斑光强是标准的高斯分布，QD感光面积足够大且象限之间缝隙的影响忽略不计，进行推导 ［ 21 21 ］ ，可得： <math display="block" id="M33"><mfenced open="{" close="" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><mi>X</mi><mo>=</mo><mfrac><mrow><mi mathvariant="normal">e</mi><mi mathvariant="normal">r</mi><msup><mrow><mi mathvariant="normal">f</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><mo stretchy="false">(</mo><msub><mrow><mi>σ</mi></mrow><mrow><mi>x</mi></mrow></msub><mo stretchy="false">)</mo></mrow><mrow><mroot><mrow><mn mathvariant="normal">2</mn></mrow><mrow/></mroot></mrow></mfrac><mo>⋅</mo><mi>ω</mi></mtd></mtr><mtr><mtd><mi>Y</mi><mo>=</mo><mfrac><mrow><mi mathvariant="normal">e</mi><mi mathvariant="normal">r</mi><msup><mrow><mi mathvariant="normal">f</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><mo stretchy="false">(</mo><msub><mrow><mi>σ</mi></mrow><mrow><mi>y</mi></mrow></msub><mo stretchy="false">)</mo></mrow><mrow><mroot><mrow><mn mathvariant="normal">2</mn></mrow><mrow/></mroot></mrow></mfrac><mo>⋅</mo><mi>ω</mi></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36184023&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36184021&type= 28.95599937 21.92866707 ， 其中： <math id="M34"><mi mathvariant="normal">e</mi><mi mathvariant="normal">r</mi><msup><mrow><mi mathvariant="normal">f</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36184027&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36184025&type= 7.19666624 3.04800010 （·）为误差函数的反函数。以 x 方向为例，QD上光斑位置与其相对位置关系如 图8 图8 所示，随着测量范围变大，非线性加剧。 3.2　固定距离下标定方法 2 由式（ 7 7 ）、（ 9 9 ）知，固定距离下，光束截面半径不变，测量灵敏度不变，QD的非线性随着量程增加而加剧。实际测量时，光束不是理想的高斯光束，光斑位置与相对位置的关系不严格遵循 式（9） 式（9） ，本文使用多项式描述该函数关系。 水平直线度与垂直直线度的标定方法一致，以水平直线度为例进行说明。搭建固定距离下的直线度标定系统，如 图9 图9 所示，使用激光干涉仪线性测量功能提供标准值，位移台提供水平方向的微小位移，接收端和线性反射镜同时固定在位移台滑块上随之移动，遵循阿贝原则，共同测量水平直线度。 在QD测量范围内，测量 n 个点用于多项式拟合，消除测量非线性，标定精度要求亚微米级别，四阶多项式拟合即可满足要求，多项式阶数增加，一定程度上可以提高标定精度，本文以四阶多项式为例进行说明，在测量距离 L 处的拟合结果表示为： <math display="block" id="M35"><msub><mrow><mi>X</mi></mrow><mrow><mi>L</mi></mrow></msub><mo>=</mo><msub><mrow><mi>A</mi></mrow><mrow><mn mathvariant="normal">4</mn><mo>,</mo><mi>L</mi></mrow></msub><msubsup><mrow><mi>σ</mi></mrow><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msubsup><mo>+</mo><msub><mrow><mi>A</mi></mrow><mrow><mn mathvariant="normal">3</mn><mo>,</mo><mi>L</mi></mrow></msub><msubsup><mrow><mi>σ</mi></mrow><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msubsup><mo>+</mo><msub><mrow><mi>A</mi></mrow><mrow><mn mathvariant="normal">2</mn><mo>,</mo><mi>L</mi></mrow></msub><msubsup><mrow><mi>σ</mi></mrow><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo>+</mo><msub><mrow><mi>A</mi></mrow><mrow><mn mathvariant="normal">1</mn><mo>,</mo><mi>L</mi></mrow></msub><msubsup><mrow><mi>σ</mi></mrow><mrow><mi>x</mi></mrow><mrow/></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36184045&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36184043&type= 62.14533615 4.06400013 ， 其中： <math id="M36"><msub><mrow><mi>A</mi></mrow><mrow><mn mathvariant="normal">4</mn><mo>,</mo><mi>L</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36184050&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36184048&type= 6.43466663 3.72533321 、 <math id="M37"><msub><mrow><mi>A</mi></mrow><mrow><mn mathvariant="normal">3</mn><mo>,</mo><mi>L</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36184054&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36184052&type= 6.43466663 3.72533321 、 <math id="M38"><msub><mrow><mi>A</mi></mrow><mrow><mn mathvariant="normal">2</mn><mo>,</mo><mi>L</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36184059&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36184056&type= 6.43466663 3.72533321 、 <math id="M39"><msub><mrow><mi>A</mi></mrow><mrow><mn mathvariant="normal">1</mn><mo>,</mo><mi>L</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36184062&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36184061&type= 6.43466663 3.72533321 为拟合的多项式系数，常数项暂时归零，即认为相对位置为零处是零点，长距离标定时通过零偏校正统一处理。 3.3　长距离分区间标定方法 2 距离改变后，光束截面尺寸变化，QD灵敏度改变，需要重新标定。长距离测量时，为了应对QD灵敏度变化的问题，提出一种分区间标定方法。 首先，将整个测量距离划分为多个区间，如 图10 图10 所示，区间划分的越多，标定精度越高，但工作量也越大。然后，对区间边界位置的QD分别进行标定，得到对应的多项式： <math display="block" id="M40"><mfenced open="{" close="" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><msub><mrow><mi>X</mi></mrow><mrow><msub><mrow><mi>L</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></msub><mo>=</mo><msub><mrow><mi>A</mi></mrow><mrow><mn mathvariant="normal">4</mn><mo>,</mo><msub><mrow><mi>L</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></msub><msubsup><mrow><mi>σ</mi></mrow><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msubsup><mo>+</mo><msub><mrow><mi>A</mi></mrow><mrow><mn mathvariant="normal">3</mn><mo>,</mo><msub><mrow><mi>L</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></msub><msubsup><mrow><mi>σ</mi></mrow><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msubsup><mo>+</mo><msub><mrow><mi>A</mi></mrow><mrow><mn mathvariant="normal">2</mn><mo>,</mo><msub><mrow><mi>L</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></msub><msubsup><mrow><mi>σ</mi></mrow><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo>+</mo><msub><mrow><mi>A</mi></mrow><mrow><mn mathvariant="normal">1</mn><mo>,</mo><msub><mrow><mi>L</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></msub><msubsup><mrow><mi>σ</mi></mrow><mrow><mi>x</mi></mrow><mrow/></msubsup></mtd></mtr><mtr><mtd><msub><mrow><mi>X</mi></mrow><mrow><msub><mrow><mi>L</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mrow></msub><mo>=</mo><msub><mrow><mi>A</mi></mrow><mrow><mn mathvariant="normal">4</mn><mo>,</mo><msub><mrow><mi>L</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mrow></msub><msubsup><mrow><mi>σ</mi></mrow><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msubsup><mo>+</mo><msub><mrow><mi>A</mi></mrow><mrow><mn mathvariant="normal">3</mn><mo>,</mo><msub><mrow><mi>L</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mrow></msub><msubsup><mrow><mi>σ</mi></mrow><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msubsup><mo>+</mo><msub><mrow><mi>A</mi></mrow><mrow><mn mathvariant="normal">2</mn><mo>,</mo><msub><mrow><mi>L</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mrow></msub><msubsup><mrow><mi>σ</mi></mrow><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo>+</mo><msub><mrow><mi>A</mi></mrow><mrow><mn mathvariant="normal">1</mn><mo>,</mo><msub><mrow><mi>L</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mrow></msub><msubsup><mrow><mi>σ</mi></mrow><mrow><mi>x</mi></mrow><mrow/></msubsup></mtd></mtr><mtr><mtd><msub><mrow><mi>X</mi></mrow><mrow><msub><mrow><mi>L</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub></mrow></msub><mo>=</mo><msub><mrow><mi>A</mi></mrow><mrow><mn mathvariant="normal">4</mn><mo>,</mo><msub><mrow><mi>L</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub></mrow></msub><msubsup><mrow><mi>σ</mi></mrow><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msubsup><mo>+</mo><msub><mrow><mi>A</mi></mrow><mrow><mn mathvariant="normal">3</mn><mo>,</mo><msub><mrow><mi>L</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub></mrow></msub><msubsup><mrow><mi>σ</mi></mrow><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msubsup><mo>+</mo><msub><mrow><mi>A</mi></mrow><mrow><mn mathvariant="normal">2</mn><mo>,</mo><msub><mrow><mi>L</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub></mrow></msub><msubsup><mrow><mi>σ</mi></mrow><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo>+</mo><msub><mrow><mi>A</mi></mrow><mrow><mn mathvariant="normal">1</mn><mo>,</mo><msub><mrow><mi>L</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub></mrow></msub><msubsup><mrow><mi>σ</mi></mrow><mrow><mi>x</mi></mrow><mrow/></msubsup></mtd></mtr><mtr><mtd><mtable><mtr><mtd><mo>⋯</mo></mtd></mtr><mtr><mtd><msub><mrow><mi>X</mi></mrow><mrow><msub><mrow><mi>L</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msub></mrow></msub><mo>=</mo><msub><mrow><mi>A</mi></mrow><mrow><mn mathvariant="normal">4</mn><mo>,</mo><msub><mrow><mi>L</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msub></mrow></msub><msubsup><mrow><mi>σ</mi></mrow><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msubsup><mo>+</mo><msub><mrow><mi>A</mi></mrow><mrow><mn mathvariant="normal">3</mn><mo>,</mo><msub><mrow><mi>L</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msub></mrow></msub><msubsup><mrow><mi>σ</mi></mrow><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msubsup><mo>+</mo><msub><mrow><mi>A</mi></mrow><mrow><mn mathvariant="normal">2</mn><mo>,</mo><msub><mrow><mi>L</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msub></mrow></msub><msubsup><mrow><mi>σ</mi></mrow><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo>+</mo><msub><mrow><mi>A</mi></mrow><mrow><mn mathvariant="normal">1</mn><mo>,</mo><msub><mrow><mi>L</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msub></mrow></msub><msubsup><mrow><mi>σ</mi></mrow><mrow><mi>x</mi></mrow><mrow/></msubsup></mtd></mtr></mtable></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36184067&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36184065&type= 74.84533691 26.83933067 . 最后，对各区间内的多项式系数进行求解，由于多项式系数没有明显的特征，使用三次样条插值法进行拟合。以四次项系数为例，将其看做一个与测量距离相关的函数，定义为： <math id="M41"><msub><mrow><mi>A</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msub><mo>=</mo><mi>R</mi><mo stretchy="false">(</mo><mi>L</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36184078&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36184076&type= 17.69533348 4.48733330 是区间［ <math id="M42"><msub><mrow><mi>L</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>L</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36184081&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36184080&type= 12.69999981 4.14866638 ］上的函数，节点为 <math id="M43"><msub><mrow><mi>L</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36184162&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36184160&type= 3.64066648 3.72533321 < <math id="M44"><msub><mrow><mi>L</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36184165&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36184163&type= 3.64066648 3.72533321 < <math id="M45"><msub><mrow><mi>L</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36184094&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36184091&type= 3.64066648 3.72533321 < <math id="M46"><mo>⋯</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36184172&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36184171&type= 3.97933316 2.79399991 < <math id="M47"><msub><mrow><mi>L</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36184176&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36184175&type= 7.19666624 3.72533321 及相应的函数值为 <math id="M48"><msub><mrow><mi>A</mi></mrow><mrow><mn mathvariant="normal">4</mn><mo>,</mo><msub><mrow><mi>L</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36184105&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36184103&type= 7.11199999 3.55599999 、 <math id="M49"><msub><mrow><mi>A</mi></mrow><mrow><mn mathvariant="normal">4</mn><mo>,</mo><msub><mrow><mi>L</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36184109&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36184107&type= 7.11199999 3.55599999 、 <math id="M50"><msub><mrow><mi>A</mi></mrow><mrow><mn mathvariant="normal">4</mn><mo>,</mo><msub><mrow><mi>L</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36184113&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36184111&type= 7.11199999 3.64066648 、 <math id="M51"><mo>⋯</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36184172&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36184171&type= 3.97933316 2.79399991 、 <math id="M52"><msub><mrow><mi>A</mi></mrow><mrow><mn mathvariant="normal">4</mn><mo>,</mo><msub><mrow><mi>L</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msub></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36184120&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36184118&type= 9.31333351 3.64066648 ，若函数 <math id="M53"><mi>S</mi><mo stretchy="false">(</mo><mi>L</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36184153&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36184151&type= 8.21266651 3.55599999 满足： （1） <math id="M54"><mi>S</mi><mo stretchy="false">(</mo><msub><mrow><mi>L</mi></mrow><mrow><mi>i</mi></mrow></msub><mo stretchy="false">)</mo><mo>=</mo><msub><mrow><mi>A</mi></mrow><mrow><mn mathvariant="normal">4</mn><mo>,</mo><msub><mrow><mi>L</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36184129&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36184127&type= 20.31999969 4.82600021 ，其中 <math id="M55"><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn><mi mathvariant="normal">、</mi><mn mathvariant="normal">2</mn><mi mathvariant="normal">、</mi><mn mathvariant="normal">3</mn><mi mathvariant="normal">、</mi><mo>⋯</mo><mi mathvariant="normal">、</mi><mi>n</mi><mo>+</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36184133&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36184131&type= 31.41133499 3.13266683 ； （2） <math id="M56"><mi>S</mi><mo>/</mo><msup><mrow><mi mathvariant="normal">C</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>∈</mo><mo stretchy="false">[</mo><msub><mrow><mi>L</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>L</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">]</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36184137&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36184135&type= 27.43199921 4.65666676 ； （3）在每个区间［ <math id="M57"><msub><mrow><mi>L</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>,</mo><msub><mrow><mi>L</mi></mrow><mrow><mi>i</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36184142&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36184139&type= 11.93799973 4.14866638 ］上，其中 <math id="M58"><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn><mi mathvariant="normal">、</mi><mn mathvariant="normal">2</mn><mi mathvariant="normal">、</mi><mn mathvariant="normal">3</mn><mi mathvariant="normal">、</mi><mo>⋯</mo><mi mathvariant="normal">、</mi><mi>n</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36184146&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36184144&type= 24.72266769 3.04800010 ， <math id="M59"><mi>S</mi><mo stretchy="false">(</mo><mi>L</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36184153&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36184151&type= 8.21266651 3.55599999 是次数不超过3的多项式。则称 <math id="M60"><mi>S</mi><mo stretchy="false">(</mo><mi>L</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36184153&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36184151&type= 8.21266651 3.55599999 是函数 <math id="M61"><mi>R</mi><mo stretchy="false">(</mo><mi>L</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36184158&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36184156&type= 8.72066593 3.55599999 以 <math id="M62"><msub><mrow><mi>L</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36184162&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36184160&type= 3.64066648 3.72533321 、 <math id="M63"><msub><mrow><mi>L</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36184165&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36184163&type= 3.64066648 3.72533321 、 <math id="M64"><msub><mrow><mi>L</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36184169&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36184167&type= 3.64066648 3.72533321 、 <math id="M65"><mo>⋯</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36184172&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36184171&type= 3.97933316 2.79399991 、 <math id="M66"><msub><mrow><mi>L</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36184176&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36184175&type= 7.19666624 3.72533321 为节点的三次样条插值函数。由于理想高斯光束的截面半径为双曲线，在束腰和远端的截面半径变化率近似常数，因此使用自然边界条件进行求解，即： <math display="block" id="M67"><msup><mrow><mi>S</mi></mrow><mrow><mo>″</mo></mrow></msup><mo stretchy="false">(</mo><msub><mrow><mi>L</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">)</mo><mo>=</mo><msup><mrow><mi>S</mi></mrow><mrow><mo>″</mo></mrow></msup><mo stretchy="false">(</mo><msub><mrow><mi>L</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">)</mo><mo>=</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36184180&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36184178&type= 35.81399918 4.65666676 . 结合节点处函数值、一阶导数连续、二阶导数连续的条件，求解出唯一的三次样条插值函数，进而可以获取区间内任意测量距离下的四次项系数。同理，利用三次样条插值法求解出三次项系数、二次项系数、一次项系数。 激光在长距离传播时，光斑分布均匀性发生变化，不再严格中心对称，因此QD测量的零点不共线，造成零偏，如 图11 图11 所示。 使用激光干涉仪直线度测量功能校正QD测量零偏，如 图12 图12 所示。五自由度测量装置与激光干涉仪同时测量一长导轨的直线度，发射端与激光干涉仪直线度反射镜固定不动，接收端与激光干涉仪沃拉斯顿棱镜同时放置在长导轨的滑块上随之移动，共同测量长导轨的直线度，激光干涉仪测量值与QD测量值的差值是主要由激光干涉仪测量误差、QD测量零偏、QD测量局部非线性误差等造成的。其中，激光干涉仪测量误差主要由本身测量精度、安装误差导致，下文分析了安装时未遵循阿贝原则导致的阿贝误差并提出了补偿方法，而激光干涉仪自身测量精度引入的误差无法消除，在分析QD测量不确定度时应考虑其影响，本文旨在介绍标定方法，不再赘述；QD测量非线性经多项式拟合与三次样条插值拟合后得到改善，远小于QD测量零偏的影响，因此，认为激光干涉仪与QD测量值的差值主要由测量零偏引起的。 受安装空间限制，激光干涉仪的测量不符合阿贝原则，沃拉斯顿棱镜安装在QD1正下方，只存在 Y 方向的阿贝偏位，因此使用电子水平仪测量滑块的滚转角，结合阿贝偏位补偿激光干涉仪的阿贝误差，使QD测量零偏校正得更加准确。在节点处测量得到校正值，将其作为 式（11） 式（11） 的常数项： <math display="block" id="M68"><msub><mrow><mi>A</mi></mrow><mrow><mn mathvariant="normal">0</mn><mo>,</mo><msub><mrow><mi>L</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></msub><mo>=</mo><msub><mrow><mi>X</mi></mrow><mrow><mi>M</mi><mo>,</mo><msub><mrow><mi>L</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></msub><mo>-</mo><msub><mrow><mi>X</mi></mrow><mrow><msub><mrow><mi>L</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36184197&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36184194&type= 28.27866554 3.64066648 ， 其中， <math id="M69"><msub><mrow><mi>X</mi></mrow><mrow><mi>M</mi><mo>,</mo><msub><mrow><mi>L</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36184200&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36184199&type= 7.36600018 3.64066648 为修正阿贝误差后的激光干涉仪测量值。同理，利用三次样条插值法求解出区间内任意测量距离下常数项，完成最终的标定。 4 试验验证与分析 1 搭建了五自由度测量装置，如 图13 图13 所示。其中LD选用单模光纤耦合输出的激光器（PL-FP-633-A-1-SA-14BF，LD-PD），激光波长 <math id="M70"><mi>λ</mi><mo>=</mo><mn mathvariant="normal">633</mn><mtext> </mtext><mi mathvariant="normal">n</mi><mi mathvariant="normal">m</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36184202&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36184201&type= 17.78000069 2.87866688 ，配合光纤准直器（TC18APC-633，Thorlabs）输出准直光束，并选用M1（RAP112-A，Lbtek）转折光束方向。BS1和BS2选用非偏振分束镜（BS1255-A，Lbtek），边长 <math id="M71"><mi>h</mi><mo>=</mo><mn mathvariant="normal">12.7</mn><mtext> </mtext><mi mathvariant="normal">m</mi><mi mathvariant="normal">m</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36184204&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36184203&type= 20.99733353 2.87866688 ，折射率 <math id="M72"><mi>n</mi><mo>=</mo><mn mathvariant="normal">1.52</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36184206&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36184205&type= 14.30866718 2.79399991 。QD1选用低噪声、高精度的四象限探测器（S5981，Hamamatsu）。望远物镜结构中L2选用 f 2 =50 mm的凸透镜（BCX10310-A，Lbtek）、L3选用 f 3 = - 25 mm的凹透镜（CC10305-A，Lbtek），L2与L3之间距离 d =31 mm。PSD2选用低噪声、高精度的位置敏感探测器（DL16-7，First Sensor）。 在中国计量科学研究院的80 m长度标准装置实验室内（温度17±0.5℃，气压98.9~101.8 kPa，湿度20%）进行了标定及其他测试。选取该标准装置的5 m行程作为试验对象（长导轨），测量装置的发射端放置在长导轨平台上固定不动，接收端放置在长导轨滑块上进行移动，发射端与接收端之间用风琴防护罩连接，减弱空气扰动对光路的影响。 4.1　直线度标定试验 2 直线度测量单元的最远测量距离设定为5 m，以1 m处为起点，间隔1 m划分区间，则节点为1 m、2 m、3 m、4 m、5 m，长导轨的测量距离可以通过位置反馈传感器实时获取，分别在节点处进行标定。如 图14 图14 所示，激光干涉仪（SJ6000，Chotest）、接收端、位移台同时固定在长导轨滑块上，滑块移动到对应节点位置固定不动，由位移台提供微小位移，激光干涉仪与接收端同时测量。标定范围为±400 μm，间隔40 μm左右测量一次，共21个点。 固定距离下的标定结果如 图15 图15 所示，最大残差标准差（ S D ）不超过0.3 μm，因此使用四阶多项式拟合可以满足测量精度要求，多项式系数见 表1 表1 。 1;2;3;4;5; 2 节点/m 4 多项式系数 <math id="M73"><msub><mrow><mi>A</mi></mrow><mrow><mn>4</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36184236&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36184235&type= 3.64066648 3.21733332 <math id="M74"><msub><mrow><mi>A</mi></mrow><mrow><mn>3</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36184238&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36184237&type= 3.64066648 3.21733332 <math id="M75"><msub><mrow><mi>A</mi></mrow><mrow><mn>2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36184240&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36184239&type= 3.64066648 3.21733332 <math id="M76"><msub><mrow><mi>A</mi></mrow><mrow><mn>1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36184242&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36184241&type= 3.64066648 3.21733332 1 -90.94 412.40 42.22 1227.18 2 166.29 467.44 13.48 1224.43 3 -49.24 423.02 55.57 1239.39 4 -7.69 375.24 42.29 1260.45 5 -43.58 437.39 34.51 1278.86 利用三次样条插值法求解出区间内任意测量距离下多项式系数： <math display="block" id="M77"><msub><mrow><mi>A</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msub><mo>=</mo><mfenced open="{" close="" separators="|"><mrow><mtable><mtr><mtd><mo>-</mo><mn mathvariant="normal">146.37</mn><msup><mrow><mo stretchy="false">(</mo><mi>L</mi><mo>-</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msup><mo>+</mo><mn mathvariant="normal">403.60</mn><mo stretchy="false">(</mo><mi>L</mi><mo>-</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo><mo>-</mo><mn mathvariant="normal">90.94</mn><mo>,</mo></mtd><mtd><mn mathvariant="normal">1</mn><mo>≤</mo><mi>L</mi><mo><</mo><mn mathvariant="normal">2</mn></mtd></mtr><mtr><mtd><mn mathvariant="normal">259.12</mn><msup><mrow><mo stretchy="false">(</mo><mi>L</mi><mo>-</mo><mn mathvariant="normal">2</mn><mo stretchy="false">)</mo></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msup><mo>-</mo><mn mathvariant="normal">439.12</mn><msup><mrow><mo stretchy="false">(</mo><mi>L</mi><mo>-</mo><mn mathvariant="normal">2</mn><mo stretchy="false">)</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>-</mo><mn mathvariant="normal">35.53</mn><mo stretchy="false">(</mo><mi>L</mi><mo>-</mo><mn mathvariant="normal">2</mn><mo stretchy="false">)</mo><mo>+</mo><mn mathvariant="normal">166.29</mn><mo>,</mo></mtd><mtd><mn mathvariant="normal">2</mn><mo>≤</mo><mi>L</mi><mo><</mo><mn mathvariant="normal">3</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn mathvariant="normal">160.29</mn><msup><mrow><mo stretchy="false">(</mo><mi>L</mi><mo>-</mo><mn mathvariant="normal">3</mn><mo stretchy="false">)</mo></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msup><mo>+</mo><mn mathvariant="normal">338.24</mn><msup><mrow><mo stretchy="false">(</mo><mi>L</mi><mo>-</mo><mn mathvariant="normal">3</mn><mo stretchy="false">)</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>-</mo><mn mathvariant="normal">136.41</mn><mo stretchy="false">(</mo><mi>L</mi><mo>-</mo><mn mathvariant="normal">3</mn><mo stretchy="false">)</mo><mo>-</mo><mn mathvariant="normal">49.24</mn><mo>,</mo></mtd><mtd><mn mathvariant="normal">3</mn><mo>≤</mo><mi>L</mi><mo><</mo><mn mathvariant="normal">4</mn></mtd></mtr><mtr><mtd><mn mathvariant="normal">47.54</mn><msup><mrow><mo stretchy="false">(</mo><mi>L</mi><mo>-</mo><mn mathvariant="normal">4</mn><mo stretchy="false">)</mo></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msup><mo>-</mo><mn mathvariant="normal">142.63</mn><msup><mrow><mo stretchy="false">(</mo><mi>L</mi><mo>-</mo><mn mathvariant="normal">4</mn><mo stretchy="false">)</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><mn mathvariant="normal">59.20</mn><mo stretchy="false">(</mo><mi>L</mi><mo>-</mo><mn mathvariant="normal">4</mn><mo stretchy="false">)</mo><mo>-</mo><mn mathvariant="normal">7.69</mn><mo>,</mo></mtd><mtd><mn mathvariant="normal">4</mn><mo>≤</mo><mi>L</mi><mo>≤</mo><mn mathvariant="normal">5</mn></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36184246&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36184245&type= 133.09599304 22.52133369 ， <math display="block" id="M78"><msub><mrow><mi>A</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mo>=</mo><mfenced open="{" close="" separators="|"><mrow><mtable><mtr><mtd><mo>-</mo><mn mathvariant="normal">24.44</mn><msup><mrow><mo stretchy="false">(</mo><mi>L</mi><mo>-</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msup><mo>+</mo><mn mathvariant="normal">79.48</mn><mo stretchy="false">(</mo><mi>L</mi><mo>-</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo><mo>+</mo><mn mathvariant="normal">412.40</mn><mo>,</mo></mtd><mtd><mn mathvariant="normal">1</mn><mo>≤</mo><mi>L</mi><mo><</mo><mn mathvariant="normal">2</mn></mtd></mtr><mtr><mtd><mn mathvariant="normal">22.73</mn><msup><mrow><mo stretchy="false">(</mo><mi>L</mi><mo>-</mo><mn mathvariant="normal">2</mn><mo stretchy="false">)</mo></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msup><mo>-</mo><mn mathvariant="normal">73.32</mn><msup><mrow><mo stretchy="false">(</mo><mi>L</mi><mo>-</mo><mn mathvariant="normal">2</mn><mo stretchy="false">)</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><mn mathvariant="normal">6.16</mn><mo stretchy="false">(</mo><mi>L</mi><mo>-</mo><mn mathvariant="normal">2</mn><mo stretchy="false">)</mo><mo>+</mo><mn mathvariant="normal">467.44</mn><mo>,</mo></mtd><mtd><mn mathvariant="normal">2</mn><mo>≤</mo><mi>L</mi><mo><</mo><mn mathvariant="normal">3</mn></mtd></mtr><mtr><mtd><mn mathvariant="normal">29.61</mn><msup><mrow><mo stretchy="false">(</mo><mi>L</mi><mo>-</mo><mn mathvariant="normal">3</mn><mo stretchy="false">)</mo></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msup><mo>-</mo><mn mathvariant="normal">5.12</mn><msup><mrow><mo stretchy="false">(</mo><mi>L</mi><mo>-</mo><mn mathvariant="normal">3</mn><mo stretchy="false">)</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>-</mo><mn mathvariant="normal">72.27</mn><mo stretchy="false">(</mo><mi>L</mi><mo>-</mo><mn mathvariant="normal">3</mn><mo stretchy="false">)</mo><mo>+</mo><mn mathvariant="normal">423.02</mn><mo>,</mo></mtd><mtd><mn mathvariant="normal">3</mn><mo>≤</mo><mi>L</mi><mo><</mo><mn mathvariant="normal">4</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn mathvariant="normal">27.91</mn><msup><mrow><mo stretchy="false">(</mo><mi>L</mi><mo>-</mo><mn mathvariant="normal">4</mn><mo stretchy="false">)</mo></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msup><mo>+</mo><mn mathvariant="normal">83.72</mn><msup><mrow><mo stretchy="false">(</mo><mi>L</mi><mo>-</mo><mn mathvariant="normal">4</mn><mo stretchy="false">)</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><mn mathvariant="normal">6.33</mn><mo stretchy="false">(</mo><mi>L</mi><mo>-</mo><mn mathvariant="normal">4</mn><mo stretchy="false">)</mo><mo>+</mo><mn mathvariant="normal">375.24</mn><mo>,</mo></mtd><mtd><mn mathvariant="normal">4</mn><mo>≤</mo><mi>L</mi><mo>≤</mo><mn mathvariant="normal">5</mn></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36184248&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36184247&type= 127.76200104 22.52133369 ， <math display="block" id="M79"><msub><mrow><mi>A</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>=</mo><mfenced open="{" close="" separators="|"><mrow><mtable><mtr><mtd><mn mathvariant="normal">23.02</mn><msup><mrow><mo stretchy="false">(</mo><mi>L</mi><mo>-</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msup><mo>-</mo><mn mathvariant="normal">51.76</mn><mo stretchy="false">(</mo><mi>L</mi><mo>-</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo><mo>+</mo><mn mathvariant="normal">42.22</mn><mo>,</mo></mtd><mtd><mn mathvariant="normal">1</mn><mo>≤</mo><mi>L</mi><mo><</mo><mn mathvariant="normal">2</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn mathvariant="normal">44.29</mn><msup><mrow><mo stretchy="false">(</mo><mi>L</mi><mo>-</mo><mn mathvariant="normal">2</mn><mo stretchy="false">)</mo></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msup><mo>+</mo><mn mathvariant="normal">69.07</mn><msup><mrow><mo stretchy="false">(</mo><mi>L</mi><mo>-</mo><mn mathvariant="normal">2</mn><mo stretchy="false">)</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><mn mathvariant="normal">17.31</mn><mo stretchy="false">(</mo><mi>L</mi><mo>-</mo><mn mathvariant="normal">2</mn><mo stretchy="false">)</mo><mo>+</mo><mn mathvariant="normal">13.48</mn><mo>,</mo></mtd><mtd><mn mathvariant="normal">2</mn><mo>≤</mo><mi>L</mi><mo><</mo><mn mathvariant="normal">3</mn></mtd></mtr><mtr><mtd><mn mathvariant="normal">27.96</mn><msup><mrow><mo stretchy="false">(</mo><mi>L</mi><mo>-</mo><mn mathvariant="normal">3</mn><mo stretchy="false">)</mo></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msup><mo>-</mo><mn mathvariant="normal">63.81</mn><msup><mrow><mo stretchy="false">(</mo><mi>L</mi><mo>-</mo><mn mathvariant="normal">3</mn><mo stretchy="false">)</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><mn mathvariant="normal">22.57</mn><mo stretchy="false">(</mo><mi>L</mi><mo>-</mo><mn mathvariant="normal">3</mn><mo stretchy="false">)</mo><mo>+</mo><mn mathvariant="normal">55.57</mn><mo>,</mo></mtd><mtd><mn mathvariant="normal">3</mn><mo>≤</mo><mi>L</mi><mo><</mo><mn mathvariant="normal">4</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn mathvariant="normal">6.69</mn><msup><mrow><mo stretchy="false">(</mo><mi>L</mi><mo>-</mo><mn mathvariant="normal">4</mn><mo stretchy="false">)</mo></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msup><mo>+</mo><mn mathvariant="normal">20.07</mn><msup><mrow><mo stretchy="false">(</mo><mi>L</mi><mo>-</mo><mn mathvariant="normal">4</mn><mo stretchy="false">)</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>-</mo><mn mathvariant="normal">21.16</mn><mo stretchy="false">(</mo><mi>L</mi><mo>-</mo><mn mathvariant="normal">4</mn><mo stretchy="false">)</mo><mo>+</mo><mn mathvariant="normal">42.29</mn><mo>,</mo></mtd><mtd><mn mathvariant="normal">4</mn><mo>≤</mo><mi>L</mi><mo>≤</mo><mn mathvariant="normal">5</mn></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36184250&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36184249&type= 127.50800323 22.52133369 ， <math display="block" id="M80"><msub><mrow><mi>A</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>=</mo><mfenced open="{" close="" separators="|"><mrow><mtable><mtr><mtd><mn mathvariant="normal">4.26</mn><msup><mrow><mo stretchy="false">(</mo><mi>L</mi><mo>-</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msup><mo>-</mo><mn mathvariant="normal">7.02</mn><mo stretchy="false">(</mo><mi>L</mi><mo>-</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo><mo>+</mo><mn mathvariant="normal">1227.18</mn><mo>,</mo></mtd><mtd><mn mathvariant="normal">1</mn><mo>≤</mo><mi>L</mi><mo><</mo><mn mathvariant="normal">2</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn mathvariant="normal">3.60</mn><msup><mrow><mo stretchy="false">(</mo><mi>L</mi><mo>-</mo><mn mathvariant="normal">2</mn><mo stretchy="false">)</mo></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msup><mo>+</mo><mn mathvariant="normal">12.79</mn><msup><mrow><mo stretchy="false">(</mo><mi>L</mi><mo>-</mo><mn mathvariant="normal">2</mn><mo stretchy="false">)</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><mn mathvariant="normal">5.77</mn><mo stretchy="false">(</mo><mi>L</mi><mo>-</mo><mn mathvariant="normal">2</mn><mo stretchy="false">)</mo><mo>+</mo><mn mathvariant="normal">1224.43</mn><mo>,</mo></mtd><mtd><mn mathvariant="normal">2</mn><mo>≤</mo><mi>L</mi><mo><</mo><mn mathvariant="normal">3</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn mathvariant="normal">1.49</mn><msup><mrow><mo stretchy="false">(</mo><mi>L</mi><mo>-</mo><mn mathvariant="normal">3</mn><mo stretchy="false">)</mo></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msup><mo>+</mo><mn mathvariant="normal">1.99</mn><msup><mrow><mo stretchy="false">(</mo><mi>L</mi><mo>-</mo><mn mathvariant="normal">3</mn><mo stretchy="false">)</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><mn mathvariant="normal">20.55</mn><mo stretchy="false">(</mo><mi>L</mi><mo>-</mo><mn mathvariant="normal">3</mn><mo stretchy="false">)</mo><mo>+</mo><mn mathvariant="normal">1239.39</mn><mo>,</mo></mtd><mtd><mn mathvariant="normal">3</mn><mo>≤</mo><mi>L</mi><mo><</mo><mn mathvariant="normal">4</mn></mtd></mtr><mtr><mtd><mn mathvariant="normal">0.83</mn><msup><mrow><mo stretchy="false">(</mo><mi>L</mi><mo>-</mo><mn mathvariant="normal">4</mn><mo stretchy="false">)</mo></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msup><mo>-</mo><mn mathvariant="normal">2.48</mn><msup><mrow><mo stretchy="false">(</mo><mi>L</mi><mo>-</mo><mn mathvariant="normal">4</mn><mo stretchy="false">)</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><mn mathvariant="normal">20.07</mn><mo stretchy="false">(</mo><mi>L</mi><mo>-</mo><mn mathvariant="normal">4</mn><mo stretchy="false">)</mo><mo>+</mo><mn mathvariant="normal">1260.45</mn><mo>,</mo></mtd><mtd><mn mathvariant="normal">4</mn><mo>≤</mo><mi>L</mi><mo>≤</mo><mn mathvariant="normal">5</mn></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36184252&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36184251&type= 127.67733765 23.70666695 . 为了证明该方法的有效性，在2.5 m和3.5 m处与激光干涉仪进行对比试验。代入式（ 14 14 ）~（ 17 17 ），求得2.5 m和3.5 m处的多项式： <math display="block" id="M81"><mfenced open="{" close="" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><msub><mrow><mi>X</mi></mrow><mrow><mn mathvariant="normal">2.5</mn><mtext> </mtext><mi mathvariant="normal">m</mi></mrow></msub><mo>=</mo><mn mathvariant="normal">71.13</mn><msubsup><mrow><mi>σ</mi></mrow><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msubsup><mo>+</mo><mn mathvariant="normal">455.03</mn><msubsup><mrow><mi>σ</mi></mrow><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msubsup><mo>+</mo><mn mathvariant="normal">33.87</mn><msubsup><mrow><mi>σ</mi></mrow><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo>+</mo><mn mathvariant="normal">1230.06</mn><msub><mrow><mi>σ</mi></mrow><mrow><mi>x</mi></mrow></msub></mtd></mtr><mtr><mtd><msub><mrow><mi>X</mi></mrow><mrow><mn mathvariant="normal">3.5</mn><mtext> </mtext><mi mathvariant="normal">m</mi></mrow></msub><mo>=</mo><mo>-</mo><mn mathvariant="normal">52.92</mn><msubsup><mrow><mi>σ</mi></mrow><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msubsup><mo>+</mo><mn mathvariant="normal">389.31</mn><msubsup><mrow><mi>σ</mi></mrow><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msubsup><mo>+</mo><mn mathvariant="normal">54.40</mn><msubsup><mrow><mi>σ</mi></mrow><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo>+</mo><mn mathvariant="normal">1249.98</mn><msub><mrow><mi>σ</mi></mrow><mrow><mi>x</mi></mrow></msub></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36184254&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36184253&type= 87.96866608 8.80533314 . 为了证明该方法在长导轨直线度测量时的优越性，与直接使用线性插值法 ［ 18 18 ］ 进行对比，测试结果如 图16 图16 所示。试验结果表明，使用分区间三次样条插值法（Spline）的误差明显小于线性插值法（Linear）的误差，误差标准差在0.3 μm以内，因此在固定测量距离下可以实现高精度测量。 1,2; 五自由度测量装置在实际使用中，移动端随被测导轨滑块移动，对于长导轨直线度测量，高斯光束光强分布不再严格的中心对称，使得QD的测量零点存在偏位且不能忽略，因此使用激光干涉仪（Renishaw，XL-80）、电子水平仪（Auleadson，DEG-I）进行了零偏校正试验，如 图17 图17 所示。利用 式（13） 式（13） 求得多项式的常数项，见 表2 表2 。 节点/m 常数项 <math id="M82"><msub><mrow><mi>A</mi></mrow><mrow><mn>0</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36184267&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36184266&type= 3.64066648 3.21733332 1 0.00 2 0.43 3 -1.53 4 -1.99 5 0.00 4.2　角度串扰误差补偿验证试验 2 试验时使用的长导轨精度高，角度误差小于2″，对直线度串扰影响可以忽略。为了测试角度串扰误差补偿的效果，使用旋转台（ZX110-200，Lyseiki）模拟长导轨的偏摆角 <math id="M83"><msub><mrow><mi>ε</mi></mrow><mrow><mi>y</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36184271&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36184270&type= 2.53999996 3.72533321 ，如 图18 图18 所示，旋转台从 - 60″到60″转动，每次转动约12″，共11个测量点，对水平直线度测量造成串扰误差，对比补偿前、补偿后的水平直线度测量结果，从而验证串扰误差补偿效果，如 图19 图19 所示。试验结果表明，偏摆角对水平直线度测量产生串扰误差，补偿前水平直线度测量误差在±1.4 μm以内，存在斜率，补偿后消除了斜率，即偏摆角产生的串扰误差，水平直线度测量误差在±0.2 μm以内，其中还包含其它误差因素，因此，补偿后串扰误差可忽略不计。 4.3　稳定性试验 2 在测量距离5 m处进行了稳定性测试，测试时间为3 h，同时记录未补偿和补偿光束角度漂移的直线度数据，结果如 图20 图20 所示。试验结果表明，光束角度漂移对直线度测量产生误差，补偿后稳定性在±0.5 μm以内，效果显著。同时注意到，测量数据仍然有漂移，分析可能的主要因素有环境温度变化造成光束传播路径上折射率梯度变化，造成光束偏折；测量接收端受温度影响，产生热变形；激光器长时间工作后，激光光强分布变化等。 5 结 论 1 本文针对长导轨五自由度测量装置中直线度的精密测量要求，对直线度测量单元进行了优化设计。首先，在光路中增加了望远物镜结构，建立了光束角度漂移补偿模型；然后，分析了角度串扰对直线度测量引入的误差，建立了角度串扰误差补偿模型；最后，提出了一种多项式拟合与三次样条插值拟合相结合的QD分区间标定方法，解决了长距离下QD非线性及灵敏度变化的问题。试验结果表明：在5 m长导轨上、±400 μm量程内，与激光干涉仪测量结果对比，标定后QD测量误差标准差小于0.3 μm，相比于线性插值法标定具有明显优势，角度串扰补偿与光束漂移补偿均具有良好的效果，其中角度对直线度的影响可忽略不计，3小时内直线度漂移小于±0.5 μm，实现了长导轨直线度的精密测量。 参考文献 焦明星 ， 冯其波 ， 王鸣 . 激光传感与测量 ［M］. 北京 ： 科学出版社 ， 2014 . 激光传感与测量 JIAO M X ， FENG Q B ， WANG M . Sensing and measuring with lasers ［M］. Beijing ： Science Press ， 2014 . （in Chinese） Sensing and measuring with lasers 孙攀攀 . 大型多轴机床几何误差建模及后置补偿求解 ［D］. 武汉 ： 华中科技大学 ， 2014 . 大型多轴机床几何误差建模及后置补偿求解 SUN P P . The geometric error modeling and compensation in post-processing solution for large multi-axis machine tools ［D］. Wuhan ： Huazhong University of Science and Technology ， 2014 . （in Chinese） The geometric error modeling and compensation in post-processing solution for large multi-axis machine tools 匡萃方 ， 冯其波 ， 张斌 ， 等 . 直线导轨四自由度同时测量方法的研究 ［J］. 中国激光 ， 2005 ， 32 （ 9 ）： 1266 - 1270 . doi: 10.3321/j.issn:0258-7025.2005.09.024 http://dx.doi.org/10.3321/j.issn:0258-7025.2005.09.024 直线导轨四自由度同时测量方法的研究 KUANG C F ， FENG Q B ， ZHANG B ， et al . Study of the method for measuring four-degree-of-freedom geometric errors of a linear stage ［J］. Chinese Journal of Lasers ， 2005 ， 32 （ 9 ）： 1266 - 1270 . （in Chinese） . doi: 10.3321/j.issn:0258-7025.2005.09.024 http://dx.doi.org/10.3321/j.issn:0258-7025.2005.09.024 Study of the method for measuring four-degree-of-freedom geometric errors of a linear stage LI J K ， FENG Q B ， BAO C C ， et al . Method for simultaneous measurement of five DOF motion errors of a rotary axis using a single-mode fiber-coupled laser ［J］. Optics Express ， 2018 ， 26 （ 3 ）： 2535 - 2545 . doi: 10.1364/oe.26.002535 http://dx.doi.org/10.1364/oe.26.002535 Method for simultaneous measurement of five DOF motion errors of a rotary axis using a single-mode fiber-coupled laser HUANG P ， LI Y ， WEI H Y ， et al . Five-degrees-of-freedom measurement system based on a monolithic prism and phase-sensitive detection technique ［J］. Applied Optics ， 2013 ， 52 （ 26 ）： 6607 - 6615 . doi: 10.1364/ao.52.006607 http://dx.doi.org/10.1364/ao.52.006607 Five-degrees-of-freedom measurement system based on a monolithic prism and phase-sensitive detection technique YU X Z ， GILLMER S R ， WOODY S C ， et al . Development of a compact， fiber-coupled， six degree-of-freedom measurement system for precision linear stage metrology ［J］. The Review of Scientific Instruments ， 2016 ， 87 （ 6 ）： 065109 . doi: 10.1063/1.4953335 http://dx.doi.org/10.1063/1.4953335 Development of a compact， fiber-coupled， six degree-of-freedom measurement system for precision linear stage metrology 冯其波 ， 刘依真 ， 张斌 ， 等 . 基于半导体激光光纤组件的激光准直仪 ［J］. 仪器仪表学报 ， 2005 ， 26 （ 1 ）： 28 - 31 . doi: 10.3321/j.issn:0254-3087.2005.01.006 http://dx.doi.org/10.3321/j.issn:0254-3087.2005.01.006 基于半导体激光光纤组件的激光准直仪 FENG Q B ， LIU Y Z ， ZHANG B ， et al . A simple laser collimator based on single-mode fiber-coupled laser module ［J］. Chinese Journal of Scientific Instrument ， 2005 ， 26 （ 1 ）： 28 - 31 . （in Chinese） . doi: 10.3321/j.issn:0254-3087.2005.01.006 http://dx.doi.org/10.3321/j.issn:0254-3087.2005.01.006 A simple laser collimator based on single-mode fiber-coupled laser module 殷纯永 ， 曾理江 ， 梁晋文 ， 等 . 自适应直线度测量方法及其装置 ： CN1017090B ［P］. 1992-06-17 . 自适应直线度测量方法及其装置 YIN C Y ， ZENG L J ， LIANG J W . Method and apparatus for adaptive measurement of linearity ： CN1017090B ［P］. 1992-06-17 . （in Chinese） Method and apparatus for adaptive measurement of linearity 由凤玲 ， 冯其波 ， 张斌 . 基于共路光线漂移补偿的直线度测量 ［J］. 光学 精密工程 ， 2011 ， 19 （ 3 ）： 515 - 519 . doi: 10.3788/ope.20111903.0515 http://dx.doi.org/10.3788/ope.20111903.0515 基于共路光线漂移补偿的直线度测量 YOU F L ， FENG Q B ， ZHANG B . Straightness error measurement based on common-path compensation for laser beam drift ［J］. 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ZHANG Cong,LIU Wenzheng,DUAN Fajie,et al.Accuracy improvement method and optimal design of straightness in five-degree-of-freedom measurement of long guide[J].Optics and Precision Engineering,2022,30(20):2467-2478. 国家重点研发计划项目（No.2020YFB2010800）；国家自然科学基金项目（No.61905175，No.61971307）；国家重点实验室探索课题项目（No.Pilt2103）；霍英东教育基金会资助项目（No.171055）；青年人才托举工程项目（No.2021QNRC001）；广东省重点研发计划项目（No.2020B0404030001）；国防科技重点实验室基金项目（No.6142212210304） 2022-02-28 2022-04-12 2022-10-25 2022-10-27 光学精密工程 20 30