多自由度激光干涉空间惯性质量块位姿测量 Multi-degree-of-freedom measurement for space inertial test mass via laser interference 1 3 first-author 马 瑞（1998-），女，河南开封人，研究助理，2020年于北京科技大学获得学士学位，主要研究方向为精密测量。E-mail： marui20@csu.ac.cn http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45410427&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45410453&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45410434&type= 22.01333237 32.00401306 马 瑞 （1998-），女，河南开封人，研究助理，2020年于北京科技大学获得学士学位，主要研究方向为精密测量。E-mail： marui20@csu.ac.cn marui20@csu.ac.cn 1 corresp E-mail： linan@csu.ac.cn E-mail： linan@csu.ac.cn 李 楠（1980-），男，北京人，博士，高级工程师，2003年于北京航空航天大学获得学士学位，2006年于中科院国家空间科学中心获得硕士学位，2019年于中科院空间应用工程与技术中心获得博士学位，主要从事基于激光干涉的微重力及绝对重力测量技术等方面的研究。E-mail： linan@csu.ac.cn http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45410473&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45410471&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45410466&type= 22.01332855 32.00399780 李 楠 （1980-），男，北京人，博士，高级工程师，2003年于北京航空航天大学获得学士学位，2006年于中科院国家空间科学中心获得硕士学位，2019年于中科院空间应用工程与技术中心获得博士学位，主要从事基于激光干涉的微重力及绝对重力测量技术等方面的研究。E-mail： linan@csu.ac.cn linan@csu.ac.cn 2 1 2 3 中国科学院 空间应用工程与技术中心 太空应用重点实验室，北京 100094 中国科学院 空间应用工程与技术中心 太空应用重点实验室，北京 100094 Key Laboratory of Space Utilization， Technology and Engineering Center for Space Utilization， Chinese Academy of Sciences， Beijing 100094， China Key Laboratory of Space Utilization， Technology and Engineering Center for Space Utilization， Chinese Academy of Sciences， Beijing 100094， China 中国科学院 空天信息创新研究院 计算光学成像技术实验室，北京 100094 中国科学院 空天信息创新研究院 计算光学成像技术实验室，北京 100094 Optical Imaging Technology Lab， Aerospace Information Research Institute， Chinese Academy of Sciences， Beijing 100094， China Optical Imaging Technology Lab， Aerospace Information Research Institute， Chinese Academy of Sciences， Beijing 100094， China 中国科学院大学，北京 100094 中国科学院大学，北京 100094 University of Chinese Academy of Sciences， Beijing 100094， China University of Chinese Academy of Sciences， Beijing 100094， China 鉴于通过测量高精度的位移数据可以获得高精度的微重力加速度数据，进而服务于多种空间科学载荷的研究任务，提出了一种基于三组正交对称角锥棱镜的双频光路，利用外差干涉测量技术实现空间惯性质量块的六自由度位移和角度测量的方法。通过光路矢量分析建立了实际角锥棱镜的光路模型，考虑质量块在运动过程中带来的附加光程差，推导了各测量光路的光程变化与质量块六自由度位姿的函数关系。为了克服小角度近似法精度不高的缺陷，提出了利用数值计算法解耦姿态角进而获得相对位移的位姿解算算法。利用空间在轨位姿数据和随机位姿数据进行系统仿真。仿真结果表明：数值计算的位移误差小于0.02 fm，且该方法的计算误差不会随着飞行器振动的增大而变大，算法具有更高的精度和更好的适应性。最后，分析了系统的误差来源，在保证角度安装误差小于5 mrad、距离安装误差小于10 μm且平行度小于2 mrad时，系统的姿态角测量误差小于0.017°，位移测量误差小于10 nm。本文提出的六自由度测量及解算方法也可以服务于其他精密加工与检测领域。 High-precision displacement measurement can help realize high-precision microgravity， which can further sever a variety of space science payloads for research missions. In this study， we devised a dual-frequency interferometer based on three sets of orthogonal symmetric cube corner retroreflectors （CCRs） for measuring the six-degree-of-freedom （6DOF） pose of the spatial inertial test mass （TM） through heterodyne detection. We first established an optical model of the actual CCR， which considers the optical path difference caused by the motion of the TM， and derived the analytical relationship between the pose of the TM and the measured optical path change. Then， the attitude angles and displacements of the TM were calculated using the method of numerical analysis， which affords remarkably higher accuracy than does the traditional small-angle approximation method. Furthermore， the performance of our system was estimated using space in-orbit data and random data. The simulation results show that the displacement error was less than 0.02 fm even when the TM vibrated significantly， indicating that our method affords high accuracy and good adaptability. In addition， the error sources of the system were analyzed. The measurement errors for the attitude angle and displacement are less than 0.017° and 10 nm， respectively， when the angular installation error is below 5 mrad， the distance installation error is less than 10 μm， and the parallelism is less than 2 mrad. The proposed 6DOF measurement and calculation method can also be used in various precision machining and detection applications. 精密测量 外差干涉测量 双频激光干涉 微重力 惯性传感 六自由度 precision measurement heterodyne interference measurement dual frequency laser interference microgravity inertial sensing six-degree-of-freedom 1 引 言 1 航天器在轨运行时会受到地球引力以外各种干扰力的作用，从而达到不完全失重的状态，此时航天器处于“微重力”环境 ［ 1 1 ］ 。提高微重力下惯性加速度的测量精度，在空间科学与基础物理研究、精密导航与定轨以及基于重力卫星的地球重力场测量等方面具有重要的科学意义 ［ 2 2 ］ 。由于加速度是位移对于时间的二阶导数，一种测量加速度的方式是测量空间惯性质量块（Test Mass， TM）的位姿，然后结合精确的原子时钟计算加速度 ［ 3 3 - 4 4 3-4 ］ 。利用该方法得到的加速度精度依赖位姿测量的精度。 激光干涉测量系统具有高分辨率、宽动态范围以及非接触等优点，广泛应用于精密测量领域中 ［ 5 5 ］ 。国内外许多研究机构对激光传感多自由度位姿测量的相关技术进行了深入研究。在迈克尔逊干涉仪和自准直仪的基础上，学者们研制了多种同时测量角度和位移的系统 ［ 6 6 - 7 7 6-7 ］ 。一种新型的位移角度干涉仪采用一个角锥棱镜（Cube Corner Retroreflector， CCR）作为参考镜，使用两个角锥棱镜作为测量镜，解决了双频激光干涉仪中相对位移引起的角度测量误差，实现了角度和位移的同时测量 ［ 8 8 ］ 。然而，该系统仅能测量一个平面内的二自由度运动信息，无法读取另外两个垂直平面的位姿数据，并且垂直平面方向上的运动所带来的附加光程差也会影响系统的测量精度。华中科技大学将质量块的表面镀上反射金膜，通过构建光学传感的六自由度响应反演了质量块的质心运动，实现了五自由度光学传感系统的设计 ［ 9 9 ］ 。但该方法是通过保留角度的二阶分量获得的，并且平面反射还可能造成参考光和测量光不能发生干涉的问题。一种结合磁悬浮控制技术的测量系统利用含有角锥棱镜的干涉仪和平面镜干涉仪的优点，通过在扁平的六棱柱上安装6个角锥棱镜，能够同时测量绕三正交轴的大横向运动和绕三正交轴的大旋转运动 ［ 10 10 - 11 11 10-11 ］ 。该系统通过巧妙的机械设计和合理的欧拉角定义，对运动目标的旋转方向进行分解，从6个干涉仪读数中获得3个方位角的解析解，从而解算出位移。而其他多自由度测量系统的姿态角和位移往往是高度耦合的。为了避免数据间的耦合问题，一些系统通常采用分光技术将各自由度的信息分解到不同的位置传感器或图像传感器中，通过获取并分析传感器对应的数值从而得到各自由度信息 ［ 12 12 - 16 16 12-16 ］ ，但此类系统的精度受限于传感器的精度。对于运动范围较小的测量系统，可采用小角度转动下的近似，降低求解位姿耦合方程组的难度 ［ 17 17 ］ ，但对于高精密的测量系统，近似求解会引入较大的系统误差。 本文针对3组正交对称角锥棱镜的双频激光光路展开质量块六自由度位姿解算的研究。考虑到实际角锥棱镜的光线传播模型，以及质量块在运动过程中带来的附加光程差，通过光路矢量分析了各测量光路的光程变化与质量块六自由度的位姿关系，利用数值计算的方法解决了多自由度位姿间的耦合问题，减小了因小角度假设带来的计算误差，从而提高系统的测量精度。 2 原 理 1 2.1　外差干涉测量技术 2 一组双频激光干涉光路可以同时测量角度和位移两种信息，对三组双频干涉光路进行合理布局能求出六自由度位姿信息。本文通过设置3组互相垂直的双频激光立体干涉光路实现六自由度的同时测量。外差干涉质量块位姿测量原理如 图1 图1 所示。测量系统包含测量单元和质量块两部分，图中只给出了一组双频测量光路部分，另外两组光路的布局类似，每组双频干涉光路共用一个双频激光源。质量块的3个正交平面上部署了6个角锥棱镜，每对角锥棱镜中心对称并且所在轴线方向正交。该设计采用对偏转不敏感的角锥棱镜作为测量镜，克服了平面反射镜可能带来的参考光和测量光无法干涉的问题。正交布局不仅可以更好地平衡质量块的重心，而且使每一组双频干涉光路具有相似的模型，降低了系统设计的复杂度。 图1 图1 中，双频激光源发射频率分别为 <math id="M1"><msub><mrow><mi>f</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45410738&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45410733&type= 2.96333337 4.57200003 （水平偏振光）和 <math id="M2"><msub><mrow><mi>f</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45410767&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45410763&type= 2.96333337 4.57200003 （垂直偏振光）的正交偏振光，双频激光经过反射率和透射率比值为9∶1的分光棱镜BS，透射光经45°通光方向的检偏器进入光电探测器PD3，用于产生频率为（ <math id="M3"><msub><mrow><mi>f</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>-</mo><msub><mrow><mi>f</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45410578&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45410565&type= 10.07533360 4.57200003 ）的参考信号；反射光由反射镜M1反射后，到达消偏振分光棱镜NPBS，随后激光被分为参考光束B1和目标光束B2。频率为 <math id="M4"><msub><mrow><mi>f</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>f</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45410658&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45410634&type= 7.11199999 4.57200003 的参考光束B1经过 <math id="M5"><mrow><mrow><mi>λ</mi></mrow><mo>/</mo><mrow><mn mathvariant="normal">4</mn></mrow></mrow></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45410629&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45410618&type= 4.74133301 3.30200005 波片到达固定的角锥棱镜CCRa并返回，再次经过 <math id="M6"><mrow><mrow><mi>λ</mi></mrow><mo>/</mo><mrow><mn mathvariant="normal">4</mn></mrow></mrow></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45410629&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45410618&type= 4.74133301 3.30200005 波片后，光束B3的偏振方向旋转了90°。同时，频率为 <math id="M7"><msub><mrow><mi>f</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>f</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45410658&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45410634&type= 7.11199999 4.57200003 的目标光束B2到偏振分光棱镜PBS1后被分光，透过PBS1的水平偏振光B6（频率为 <math id="M8"><msub><mrow><mi>f</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45410738&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45410733&type= 2.96333337 4.57200003 ）到达CCR1并返回，由于多普勒效应，回射光束B8的频率为 <math id="M9"><msub><mrow><mi>f</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>±</mo><mi mathvariant="normal">Δ</mi><msub><mrow><mi>f</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45410803&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45410789&type= 12.53066730 4.65666676 ；垂直偏振光B5（频率为 <math id="M10"><msub><mrow><mi>f</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45410767&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45410763&type= 2.96333337 4.57200003 ）经M2反射后到达CCR2并返回，回射光束B7的频率为 <math id="M11"><msub><mrow><mi>f</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>±</mo><mi mathvariant="normal">Δ</mi><msub><mrow><mi>f</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45410774&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45410755&type= 12.53066730 4.65666676 。两束光B7和B8在PBS1相遇并到达NPBS形成目标光束B4。经NPBS反射后，目标光束与参考光束重叠。经PBS2分光后，分别经过透镜组（Lens1，Lens2）聚焦、光强衰减器（Attenuator1，Attenuator2）进行信号的衰减，最后被两个光电探测器（PD1，PD2）检测到。PD1检测垂直偏振光B9（频率为 <math id="M12"><msub><mrow><mi>f</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45410738&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45410733&type= 2.96333337 4.57200003 和 <math id="M13"><msub><mrow><mi>f</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>±</mo><mi mathvariant="normal">Δ</mi><msub><mrow><mi>f</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45410774&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45410755&type= 12.53066730 4.65666676 ）所形成的干涉拍频信号，PD2检测水平偏振光B10（频率为 <math id="M14"><msub><mrow><mi>f</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45410767&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45410763&type= 2.96333337 4.57200003 和 <math id="M15"><msub><mrow><mi>f</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>±</mo><mi mathvariant="normal">Δ</mi><msub><mrow><mi>f</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45410803&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45410789&type= 12.53066730 4.65666676 ）所形成的干涉拍频信号。通过相位计检测PD1和PD2输出的拍频信号和PD3输出的参考信号，可以得到携带CCR1和CCR2运动信息的相位。其中，透镜组对目标光束和参考光束进行聚焦，可以解决因目标光束和参考光束距离较远而造成光束重合范围较小、拍频信号功率较小的问题；光强衰减器可以解决因透镜组聚焦造成的光信号功率过大的问题，保证经光电探测器的输出值在相位计电压输入值的额定范围之内。 若双频激光源发射波长均为 <math id="M16"><mi>λ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45410813&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45410798&type= 1.77800000 2.79399991 的正交偏振光，由多普勒效应可知，当质量块上的每个角锥棱镜沿运动光轴有相对速率 <math id="M17"><mi>v</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45410825&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45410833&type= 6.68866634 3.55599999 时，从角锥棱镜返回的测量光产生的多普勒频移 <math id="M18"><mi mathvariant="normal">Δ</mi><mi>f</mi><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45410851&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45410830&type= 9.31333351 4.06400013 为 ［ 18 18 ］ ： <math display="block" id="M19"><mi mathvariant="normal">Δ</mi><mi>f</mi><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced><mo>=</mo><mn mathvariant="normal">2</mn><mi>v</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>/</mo><mi>λ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45410859&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45410856&type= 26.75466537 4.06400013 . 在信号处理时间 <math id="M20"><mi mathvariant="normal">Δ</mi><mi>t</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45410968&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45410954&type= 3.72533321 2.79399991 内，经过积分可以获得周期数 <math id="M21"><mi>N</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45410914&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45410908&type= 3.30200005 2.87866688 ，从而获得测量光和参考光之间光程差变化所对应的相位变化 <math id="M22"><mi mathvariant="normal">Δ</mi><mi>φ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45410947&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45410935&type= 4.82600021 3.72533321 ，即： <math display="block" id="M23"><mi mathvariant="normal">Δ</mi><mi>φ</mi><mo>=</mo><mn mathvariant="normal">2</mn><mi mathvariant="normal">π</mi><mi>N</mi><mo>=</mo><mn mathvariant="normal">2</mn><mi mathvariant="normal">π</mi><msubsup><mo>∫</mo><mrow><mi>t</mi></mrow><mrow><mi>t</mi><mo>+</mo><mi mathvariant="normal">Δ</mi><mi>t</mi></mrow></msubsup><mrow><mi mathvariant="normal">Δ</mi><mi>f</mi><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced><mi mathvariant="normal">d</mi><mi>t</mi></mrow></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45410933&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45410930&type= 46.05867004 7.87400007 . 由于参考光的光程为定值，因此根据 <math id="M24"><mi mathvariant="normal">Δ</mi><mi>φ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45410947&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45410935&type= 4.82600021 3.72533321 可以得到 <math id="M25"><mi mathvariant="normal">Δ</mi><mi>t</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45410968&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45410954&type= 3.72533321 2.79399991 时间内测量光的光程变化，即： <math display="block" id="M26"><mi mathvariant="normal">Δ</mi><mi>L</mi><mo>=</mo><mfrac><mrow><mi mathvariant="normal">Δ</mi><mi>φ</mi><mi>λ</mi></mrow><mrow><mn mathvariant="normal">2</mn><mi mathvariant="normal">π</mi></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45410978&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45410973&type= 16.93333435 7.95866632 . 采集测量光和参考光所形成的干涉拍频信号，通过相位计获得相位差，可以得到采样时间内测量光路的光程变化。对于平动系统而言，角锥棱镜的位移是光程变化的一半。 整个系统的位姿测量流程如 图2 图2 所示。质量块在运动时会带来多普勒频差，利用相位计检测并计算多普勒频差和相位差，计算测量光路的光程变化，通过建立合理的位姿解算模型，得到质量块的六自由度位姿信息。 2.2　实际角锥棱镜的光路模型 2 理想角锥棱镜是由3个互相垂直且全等的反射面和一个折射面（底面）组成，具有后反射特性 ［ 19 19 ］ 。对于从底面正入射的光，在角锥棱镜内部的传播路程等于棱镜高度的两倍。当光线斜入射角锥棱镜底面时，光线在棱镜内的光程大于垂直入射时的光程，存在附加光程差，从而造成测距误差 ［ 20 20 ］ 。如 图1 图1 所示，质量块在平面运动时可以通过CCR1和CCR2对应的位移差求解旋转角度。但对于立体干涉光路而言，质量块在多自由度运动时相位计的输出结果是相互耦合的，并且角锥棱镜产生的附加光程差会影响测量系统的精度。实际工程中采用的角锥棱镜并非正三棱锥，而是由三个互相垂直的直角面、一个弧面以及一个圆形底面组成。本文针对光线在实际角锥棱镜的传播路径进行建模，分析光线沿任意角度入射实际角锥棱镜时所产生的附加光程差。 角锥棱镜及其光线传播路径如 图3 图3 所示，并分别建立 <math id="M27"><mi mathvariant="normal">Σ</mi><mi>C</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45411040&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45411021&type= 4.99533367 2.87866688 坐标系和 <math id="M28"><mi mathvariant="normal">Σ</mi><mi>R</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45411701&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45411707&type= 5.07999992 2.87866688 坐标系。 <math id="M29"><mi mathvariant="normal">Σ</mi><mi>C</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45411040&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45411021&type= 4.99533367 2.87866688 坐标系的原点 <math id="M30"><msub><mrow><mi>O</mi></mrow><mrow><mi>C</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45411024&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45411046&type= 4.48733330 3.72533321 位于角锥棱镜顶点，坐标轴分别沿3条直角边方向； <math id="M31"><mi mathvariant="normal">Σ</mi><mi>R</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45411701&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45411707&type= 5.07999992 2.87866688 坐标系原点 <math id="M32"><msub><mrow><mi>O</mi></mrow><mrow><mi>R</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45411077&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45411065&type= 4.48733330 3.72533321 位于角锥棱镜底面中心， <math id="M33"><msub><mrow><mi>x</mi></mrow><mrow><mi>R</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45411082&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45411068&type= 3.80999994 3.72533321 轴指向角锥棱镜的顶点， <math id="M34"><msub><mrow><mi>y</mi></mrow><mrow><mi>R</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45411091&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45411086&type= 3.64066648 4.57200003 轴指向 <math id="M35"><mi>E</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45411113&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45411109&type= 2.70933342 2.79399991 点在底面的投影， <math id="M36"><msub><mrow><mi>z</mi></mrow><mrow><mi>R</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45411121&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45411096&type= 3.38666677 3.72533321 轴平行于 <math id="M37"><mi>F</mi><mi>G</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45411129&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45411125&type= 5.50333309 2.79399991 连线。记空气的折射率为 <math id="M38"><mi>n</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45411138&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45411134&type= 1.86266661 2.79399991 ，角锥棱镜的折射率为 <math id="M39"><mi>n</mi><mi mathvariant="normal">'</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45411154&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45411143&type= 2.62466669 3.30200005 ，角锥棱镜的底面直径为 <math id="M40"><mi>D</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45411168&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45411163&type= 2.96333337 2.79399991 ，高度为 <math id="M41"><mi>h</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45411178&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45411172&type= 1.94733346 2.79399991 。假设入射光线 <math id="M42"><msub><mrow><mi mathvariant="bold-italic">L</mi></mrow><mrow><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45411200&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45411196&type= 4.14866638 3.72533321 以 <math id="M43"><mi>θ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45411477&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45411483&type= 1.86266661 2.79399991 角从角锥棱镜底面入射，折射后依次到达平面 <math id="M44"><mi>E</mi><mi>F</mi><msub><mrow><mi>O</mi></mrow><mrow><mi>C</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45411224&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45411191&type= 9.56733322 3.72533321 ， <math id="M45"><mi>F</mi><mi>G</mi><msub><mrow><mi>O</mi></mrow><mrow><mi>C</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45411220&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45411217&type= 9.82133293 3.72533321 ， <math id="M46"><mi>E</mi><mi>G</mi><msub><mrow><mi>O</mi></mrow><mrow><mi>C</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45411235&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45411232&type= 9.82133293 3.72533321 ，最后再次到达底面后并折射出来。 记入射光线 <math id="M47"><msub><mrow><mi mathvariant="bold-italic">L</mi></mrow><mrow><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45411277&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45411273&type= 4.14866638 3.72533321 在 <math id="M48"><mi mathvariant="normal">Σ</mi><mi>R</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45411701&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45411707&type= 5.07999992 2.87866688 坐标系下的矢量表达式为： <math display="block" id="M49"><mmultiscripts><mrow/><mprescripts/><mrow/><mrow><mi>R</mi></mrow></mmultiscripts><msub><mrow><mi mathvariant="bold-italic">L</mi></mrow><mrow><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi></mrow></msub><mo>=</mo><mfenced open="[" close="]" separators="|"><mrow><mmultiscripts><mrow/><mprescripts/><mrow/><mrow><mi>R</mi></mrow></mmultiscripts><msub><mrow><mi mathvariant="bold-italic">P</mi></mrow><mrow><msub><mrow><mi mathvariant="bold-italic">L</mi></mrow><mrow><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi></mrow></msub></mrow></msub><mo>,</mo><mmultiscripts><mrow><mi mathvariant="bold-italic">N</mi></mrow><mprescripts/><mrow/><mrow><mi>R</mi></mrow></mmultiscripts><msub><mrow><mi mathvariant="bold-italic">N</mi></mrow><mrow><msub><mrow><mi mathvariant="bold-italic">L</mi></mrow><mrow><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi></mrow></msub></mrow></msub></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45411303&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45411285&type= 31.07266808 4.65666676 ， 式中： <math id="M50"><mmultiscripts><mrow/><mprescripts/><mrow/><mrow><mi>R</mi></mrow></mmultiscripts><msub><mrow><mi mathvariant="bold-italic">P</mi></mrow><mrow><msub><mrow><mi mathvariant="bold-italic">L</mi></mrow><mrow><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi></mrow></msub></mrow></msub><mo>=</mo><msup><mrow><mfenced separators="|"><mrow><mmultiscripts><mrow/><mprescripts/><mrow/><mrow><mi>R</mi></mrow></mmultiscripts><msub><mrow><mi mathvariant="bold-italic">P</mi></mrow><mrow><msub><mrow><mi mathvariant="bold-italic">L</mi></mrow><mrow><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi></mrow></msub><mi>X</mi></mrow></msub><mo>,</mo><mmultiscripts><mrow><mi mathvariant="bold-italic">P</mi></mrow><mprescripts/><mrow/><mrow><mi>R</mi></mrow></mmultiscripts><msub><mrow><mi mathvariant="bold-italic">P</mi></mrow><mrow><msub><mrow><mi mathvariant="bold-italic">L</mi></mrow><mrow><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi></mrow></msub><mi>Y</mi></mrow></msub><mo>,</mo><mmultiscripts><mrow><mi mathvariant="bold-italic">P</mi></mrow><mprescripts/><mrow/><mrow><mi>R</mi></mrow></mmultiscripts><msub><mrow><mi mathvariant="bold-italic">P</mi></mrow><mrow><msub><mrow><mi mathvariant="bold-italic">L</mi></mrow><mrow><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi></mrow></msub><mi>Z</mi></mrow></msub></mrow></mfenced></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45411308&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45411290&type= 47.58266449 5.33400011 ， <math id="M51"><mmultiscripts><mrow/><mprescripts/><mrow/><mrow><mi>R</mi></mrow></mmultiscripts><msub><mrow><mi mathvariant="bold-italic">N</mi></mrow><mrow><msub><mrow><mi mathvariant="bold-italic">L</mi></mrow><mrow><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi></mrow></msub></mrow></msub><mo>=</mo><msup><mrow><mfenced separators="|"><mrow><mmultiscripts><mrow/><mprescripts/><mrow/><mrow><mi>R</mi></mrow></mmultiscripts><msub><mrow><mi mathvariant="bold-italic">N</mi></mrow><mrow><msub><mrow><mi mathvariant="bold-italic">L</mi></mrow><mrow><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi></mrow></msub><mi>X</mi></mrow></msub><mo>,</mo><mmultiscripts><mrow><mi mathvariant="bold-italic">N</mi></mrow><mprescripts/><mrow/><mrow><mi>R</mi></mrow></mmultiscripts><msub><mrow><mi mathvariant="bold-italic">N</mi></mrow><mrow><msub><mrow><mi mathvariant="bold-italic">L</mi></mrow><mrow><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi></mrow></msub><mi>Y</mi></mrow></msub><mo>,</mo><mmultiscripts><mrow><mi mathvariant="bold-italic">N</mi></mrow><mprescripts/><mrow/><mrow><mi>R</mi></mrow></mmultiscripts><msub><mrow><mi mathvariant="bold-italic">N</mi></mrow><mrow><msub><mrow><mi mathvariant="bold-italic">L</mi></mrow><mrow><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi></mrow></msub><mi>Z</mi></mrow></msub></mrow></mfenced></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45411328&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45411313&type= 49.86866760 5.33400011 ， <math id="M52"><mfenced open="|" close="|" separators="|"><mrow><mmultiscripts><mrow/><mprescripts/><mrow/><mrow><mi>R</mi></mrow></mmultiscripts><msub><mrow><mi mathvariant="bold-italic">N</mi></mrow><mrow><msub><mrow><mi mathvariant="bold-italic">L</mi></mrow><mrow><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi></mrow></msub></mrow></msub></mrow></mfenced><mo>=</mo><mi>n</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45411323&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45411332&type= 16.42533302 4.65666676 。 由坐标系间的相互转换关系和折射定律可以得到折射光线的矢量表达，即有： <math display="block" id="M53"><mmultiscripts><mrow/><mprescripts/><mrow/><mrow><mi>C</mi></mrow></mmultiscripts><msub><mrow><mi mathvariant="bold-italic">L</mi></mrow><mrow><mi mathvariant="normal">R</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">f</mi></mrow></msub><mo>=</mo><mfenced open="[" close="]" separators="|"><mrow><mmultiscripts><mrow/><mprescripts/><mrow/><mrow><mi>C</mi></mrow></mmultiscripts><msub><mrow><mi mathvariant="bold-italic">P</mi></mrow><mrow><msub><mrow><mi mathvariant="bold-italic">L</mi></mrow><mrow><mi mathvariant="normal">R</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">f</mi></mrow></msub></mrow></msub><mo>,</mo><mmultiscripts><mrow><mi mathvariant="bold-italic">N</mi></mrow><mprescripts/><mrow/><mrow><mi>C</mi></mrow></mmultiscripts><msub><mrow><mi mathvariant="bold-italic">N</mi></mrow><mrow><msub><mrow><mi mathvariant="bold-italic">L</mi></mrow><mrow><mi mathvariant="normal">R</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">f</mi></mrow></msub></mrow></msub></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45411338&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45411336&type= 33.95133591 4.65666676 ， 式中： <math display="block" id="M54"><mmultiscripts><mrow/><mprescripts/><mrow/><mrow><mi>C</mi></mrow></mmultiscripts><msub><mrow><mi mathvariant="bold-italic">P</mi></mrow><mrow><msub><mrow><mi mathvariant="bold-italic">L</mi></mrow><mrow><mi mathvariant="normal">R</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">f</mi></mrow></msub></mrow></msub><mo>=</mo><mfenced separators="|"><mrow><mtable><mtr><mtd><mo>-</mo><mfrac><mrow><mroot><mrow><mn mathvariant="normal">3</mn></mrow><mrow/></mroot></mrow><mrow><mn mathvariant="normal">3</mn></mrow></mfrac><mmultiscripts><mrow><mi mathvariant="bold-italic">P</mi></mrow><mprescripts/><mrow/><mrow><mi>R</mi></mrow></mmultiscripts><msub><mrow><mi mathvariant="bold-italic">P</mi></mrow><mrow><msub><mrow><mi mathvariant="bold-italic">L</mi></mrow><mrow><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi></mrow></msub><mi>X</mi></mrow></msub><mo>+</mo><mfrac><mrow><mroot><mrow><mn mathvariant="normal">6</mn></mrow><mrow/></mroot></mrow><mrow><mn mathvariant="normal">3</mn></mrow></mfrac><mmultiscripts><mrow><mi mathvariant="bold-italic">P</mi></mrow><mprescripts/><mrow/><mrow><mi>R</mi></mrow></mmultiscripts><msub><mrow><mi mathvariant="bold-italic">P</mi></mrow><mrow><msub><mrow><mi mathvariant="bold-italic">L</mi></mrow><mrow><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi></mrow></msub><mi>Y</mi></mrow></msub><mo>+</mo><mfrac><mrow><mroot><mrow><mn mathvariant="normal">3</mn></mrow><mrow/></mroot><mi>h</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></mfrac></mtd></mtr><mtr><mtd><mo>-</mo><mfrac><mrow><mroot><mrow><mn mathvariant="normal">3</mn></mrow><mrow/></mroot></mrow><mrow><mn mathvariant="normal">3</mn></mrow></mfrac><mmultiscripts><mrow><mi mathvariant="bold-italic">P</mi></mrow><mprescripts/><mrow/><mrow><mi>R</mi></mrow></mmultiscripts><msub><mrow><mi mathvariant="bold-italic">P</mi></mrow><mrow><msub><mrow><mi mathvariant="bold-italic">L</mi></mrow><mrow><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi></mrow></msub><mi>X</mi></mrow></msub><mo>-</mo><mfrac><mrow><mroot><mrow><mn mathvariant="normal">6</mn></mrow><mrow/></mroot></mrow><mrow><mn mathvariant="normal">6</mn></mrow></mfrac><mmultiscripts><mrow><mi mathvariant="bold-italic">P</mi></mrow><mprescripts/><mrow/><mrow><mi>R</mi></mrow></mmultiscripts><msub><mrow><mi mathvariant="bold-italic">P</mi></mrow><mrow><msub><mrow><mi mathvariant="bold-italic">L</mi></mrow><mrow><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi></mrow></msub><mi>Y</mi></mrow></msub><mo>-</mo><mfrac><mrow><mroot><mrow><mn mathvariant="normal">2</mn></mrow><mrow/></mroot></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><mmultiscripts><mrow><mi mathvariant="bold-italic">P</mi></mrow><mprescripts/><mrow/><mrow><mi>R</mi></mrow></mmultiscripts><msub><mrow><mi mathvariant="bold-italic">P</mi></mrow><mrow><msub><mrow><mi mathvariant="bold-italic">L</mi></mrow><mrow><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi></mrow></msub><mi>Z</mi></mrow></msub><mo>+</mo><mfrac><mrow><mroot><mrow><mn mathvariant="normal">3</mn></mrow><mrow/></mroot><mi>h</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></mfrac></mtd></mtr><mtr><mtd><mo>-</mo><mfrac><mrow><mroot><mrow><mn mathvariant="normal">3</mn></mrow><mrow/></mroot></mrow><mrow><mn mathvariant="normal">3</mn></mrow></mfrac><mmultiscripts><mrow><mi mathvariant="bold-italic">P</mi></mrow><mprescripts/><mrow/><mrow><mi>R</mi></mrow></mmultiscripts><msub><mrow><mi mathvariant="bold-italic">P</mi></mrow><mrow><msub><mrow><mi mathvariant="bold-italic">L</mi></mrow><mrow><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi></mrow></msub><mi>X</mi></mrow></msub><mo>-</mo><mfrac><mrow><mroot><mrow><mn mathvariant="normal">6</mn></mrow><mrow/></mroot></mrow><mrow><mn mathvariant="normal">6</mn></mrow></mfrac><mmultiscripts><mrow><mi mathvariant="bold-italic">P</mi></mrow><mprescripts/><mrow/><mrow><mi>R</mi></mrow></mmultiscripts><msub><mrow><mi mathvariant="bold-italic">P</mi></mrow><mrow><msub><mrow><mi mathvariant="bold-italic">L</mi></mrow><mrow><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi></mrow></msub><mi>Y</mi></mrow></msub><mo>+</mo><mfrac><mrow><mroot><mrow><mn mathvariant="normal">2</mn></mrow><mrow/></mroot></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><mmultiscripts><mrow><mi mathvariant="bold-italic">P</mi></mrow><mprescripts/><mrow/><mrow><mi>R</mi></mrow></mmultiscripts><msub><mrow><mi mathvariant="bold-italic">P</mi></mrow><mrow><msub><mrow><mi mathvariant="bold-italic">L</mi></mrow><mrow><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi></mrow></msub><mi>Z</mi></mrow></msub><mo>+</mo><mfrac><mrow><mroot><mrow><mn mathvariant="normal">3</mn></mrow><mrow/></mroot><mi>h</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></mfrac></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45411352&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45411341&type= 88.13800049 32.25799942 ， <math display="block" id="M55"><mmultiscripts><mrow/><mprescripts/><mrow/><mrow><mi>C</mi></mrow></mmultiscripts><msub><mrow><mi mathvariant="bold-italic">N</mi></mrow><mrow><msub><mrow><mi mathvariant="bold-italic">L</mi></mrow><mrow><mi mathvariant="normal">R</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">f</mi></mrow></msub></mrow></msub><mo>=</mo><mfenced separators="|"><mrow><mtable><mtr><mtd><mo>-</mo><mfrac><mrow><mroot><mrow><mn mathvariant="normal">3</mn></mrow><mrow/></mroot></mrow><mrow><mn mathvariant="normal">3</mn></mrow></mfrac><mmultiscripts><mrow><mi mathvariant="bold-italic">N</mi></mrow><mprescripts/><mrow/><mrow><mi>R</mi></mrow></mmultiscripts><msub><mrow><mi mathvariant="bold-italic">N</mi></mrow><mrow><msub><mrow><mi mathvariant="bold-italic">L</mi></mrow><mrow><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi></mrow></msub><mi>X</mi></mrow></msub><mo>+</mo><mfrac><mrow><mroot><mrow><mn mathvariant="normal">6</mn></mrow><mrow/></mroot></mrow><mrow><mn mathvariant="normal">3</mn></mrow></mfrac><mmultiscripts><mrow><mi mathvariant="bold-italic">N</mi></mrow><mprescripts/><mrow/><mrow><mi>R</mi></mrow></mmultiscripts><msub><mrow><mi mathvariant="bold-italic">N</mi></mrow><mrow><msub><mrow><mi mathvariant="bold-italic">L</mi></mrow><mrow><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi></mrow></msub><mi>Y</mi></mrow></msub><mo>+</mo><mfrac><mrow><mroot><mrow><mn mathvariant="normal">3</mn></mrow><mrow/></mroot></mrow><mrow><mn mathvariant="normal">3</mn></mrow></mfrac><mi>p</mi></mtd></mtr><mtr><mtd><mo>-</mo><mfrac><mrow><mroot><mrow><mn mathvariant="normal">3</mn></mrow><mrow/></mroot></mrow><mrow><mn mathvariant="normal">3</mn></mrow></mfrac><mmultiscripts><mrow><mi mathvariant="bold-italic">N</mi></mrow><mprescripts/><mrow/><mrow><mi>R</mi></mrow></mmultiscripts><msub><mrow><mi mathvariant="bold-italic">N</mi></mrow><mrow><msub><mrow><mi mathvariant="bold-italic">L</mi></mrow><mrow><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi></mrow></msub><mi>X</mi></mrow></msub><mo>-</mo><mfrac><mrow><mroot><mrow><mn mathvariant="normal">6</mn></mrow><mrow/></mroot></mrow><mrow><mn mathvariant="normal">6</mn></mrow></mfrac><mmultiscripts><mrow><mi mathvariant="bold-italic">N</mi></mrow><mprescripts/><mrow/><mrow><mi>R</mi></mrow></mmultiscripts><msub><mrow><mi mathvariant="bold-italic">N</mi></mrow><mrow><msub><mrow><mi mathvariant="bold-italic">L</mi></mrow><mrow><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi></mrow></msub><mi>Y</mi></mrow></msub><mo>-</mo><mfrac><mrow><mroot><mrow><mn mathvariant="normal">2</mn></mrow><mrow/></mroot></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><mmultiscripts><mrow><mi mathvariant="bold-italic">N</mi></mrow><mprescripts/><mrow/><mrow><mi>R</mi></mrow></mmultiscripts><msub><mrow><mi mathvariant="bold-italic">N</mi></mrow><mrow><msub><mrow><mi mathvariant="bold-italic">L</mi></mrow><mrow><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi></mrow></msub><mi>Z</mi></mrow></msub><mo>+</mo><mfrac><mrow><mroot><mrow><mn mathvariant="normal">3</mn></mrow><mrow/></mroot></mrow><mrow><mn mathvariant="normal">3</mn></mrow></mfrac><mi>p</mi></mtd></mtr><mtr><mtd><mo>-</mo><mfrac><mrow><mroot><mrow><mn mathvariant="normal">3</mn></mrow><mrow/></mroot></mrow><mrow><mn mathvariant="normal">3</mn></mrow></mfrac><mmultiscripts><mrow><mi mathvariant="bold-italic">N</mi></mrow><mprescripts/><mrow/><mrow><mi>R</mi></mrow></mmultiscripts><msub><mrow><mi mathvariant="bold-italic">N</mi></mrow><mrow><msub><mrow><mi mathvariant="bold-italic">L</mi></mrow><mrow><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi></mrow></msub><mi>X</mi></mrow></msub><mo>-</mo><mfrac><mrow><mroot><mrow><mn mathvariant="normal">6</mn></mrow><mrow/></mroot></mrow><mrow><mn mathvariant="normal">6</mn></mrow></mfrac><mmultiscripts><mrow><mi mathvariant="bold-italic">N</mi></mrow><mprescripts/><mrow/><mrow><mi>R</mi></mrow></mmultiscripts><msub><mrow><mi mathvariant="bold-italic">N</mi></mrow><mrow><msub><mrow><mi mathvariant="bold-italic">L</mi></mrow><mrow><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi></mrow></msub><mi>Y</mi></mrow></msub><mo>+</mo><mfrac><mrow><mroot><mrow><mn mathvariant="normal">2</mn></mrow><mrow/></mroot></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><mmultiscripts><mrow><mi mathvariant="bold-italic">N</mi></mrow><mprescripts/><mrow/><mrow><mi>R</mi></mrow></mmultiscripts><msub><mrow><mi mathvariant="bold-italic">N</mi></mrow><mrow><msub><mrow><mi mathvariant="bold-italic">L</mi></mrow><mrow><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi></mrow></msub><mi>Z</mi></mrow></msub><mo>+</mo><mfrac><mrow><mroot><mrow><mn mathvariant="normal">3</mn></mrow><mrow/></mroot></mrow><mrow><mn mathvariant="normal">3</mn></mrow></mfrac><mi>p</mi></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45411357&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45411363&type= 90.93199921 30.14133072 ， <math display="block" id="M56"><mtable columnalign="left"><mtr><mtd><mi>p</mi><mo>=</mo><mi>n</mi><mi mathvariant="normal">c</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">s</mi><mi>θ</mi><mo>-</mo><mi>n</mi><mi mathvariant="normal">'</mi><mi mathvariant="normal">c</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">s</mi><mi>γ</mi><mo>=</mo><mi>n</mi><mo>-</mo><mi>n</mi><mi mathvariant="normal">'</mi><mo>,</mo></mtd></mtr><mtr><mtd><mi mathvariant="normal">c</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">s</mi><mi>γ</mi><mo>=</mo><mroot><mrow><mn mathvariant="normal">1</mn><mo>-</mo><msup><mrow><mfenced separators="|"><mrow><mfrac><mrow><mi>n</mi></mrow><mrow><mi>n</mi><mi mathvariant="normal">'</mi></mrow></mfrac></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><msup><mrow><mi mathvariant="normal">n</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mi>θ</mi></mrow><mrow/></mroot><mo>.</mo></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45411380&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45411371&type= 48.42933273 17.10266685 根据线面交点公式可以依次得到折射光线与角锥棱镜各反射面的交点；由光线的反射定律可以得到反射后各光线的方向矢量，从而得到出射点坐标和出射光线的方向矢量。 <math display="block" id="M57"><mmultiscripts><mrow/><mprescripts/><mrow/><mrow><mi>R</mi></mrow></mmultiscripts><msub><mrow><mi mathvariant="bold-italic">L</mi></mrow><mrow><mi mathvariant="normal">o</mi><mi mathvariant="normal">u</mi><mi mathvariant="normal">t</mi></mrow></msub><mo>=</mo><mfenced open="[" close="]" separators="|"><mrow><mmultiscripts><mrow/><mprescripts/><mrow/><mrow><mi>R</mi></mrow></mmultiscripts><msub><mrow><mi mathvariant="bold-italic">P</mi></mrow><mrow><msub><mrow><mi mathvariant="bold-italic">L</mi></mrow><mrow><mi mathvariant="normal">o</mi><mi mathvariant="normal">u</mi><mi mathvariant="normal">t</mi></mrow></msub></mrow></msub><mo>,</mo><mmultiscripts><mrow><mi mathvariant="bold-italic">N</mi></mrow><mprescripts/><mrow/><mrow><mi>R</mi></mrow></mmultiscripts><msub><mrow><mi mathvariant="bold-italic">N</mi></mrow><mrow><msub><mrow><mi mathvariant="bold-italic">L</mi></mrow><mrow><mi mathvariant="normal">o</mi><mi mathvariant="normal">u</mi><mi mathvariant="normal">t</mi></mrow></msub></mrow></msub></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45411398&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45411385&type= 33.52799988 4.65666676 ， 式中： <math display="block" id="M58"><mmultiscripts><mrow/><mprescripts/><mrow/><mrow><mi>R</mi></mrow></mmultiscripts><msub><mrow><mi mathvariant="bold-italic">P</mi></mrow><mrow><msub><mrow><mi mathvariant="bold-italic">L</mi></mrow><mrow><mi mathvariant="normal">o</mi><mi mathvariant="normal">u</mi><mi mathvariant="normal">t</mi></mrow></msub></mrow></msub><mo>=</mo><mfenced separators="|"><mrow><mtable><mtr><mtd><mn mathvariant="normal">0</mn></mtd></mtr><mtr><mtd><mfrac><mrow><mn mathvariant="normal">2</mn><mfenced separators="|"><mrow><mi>a</mi><mo>+</mo><mroot><mrow><mn mathvariant="normal">3</mn></mrow><mrow/></mroot><mi>d</mi></mrow></mfenced><mmultiscripts><mrow><mi mathvariant="bold-italic">N</mi></mrow><mprescripts/><mrow/><mrow><mi>R</mi></mrow></mmultiscripts><msub><mrow><mi mathvariant="bold-italic">N</mi></mrow><mrow><msub><mrow><mi mathvariant="bold-italic">L</mi></mrow><mrow><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi></mrow></msub><mi>Y</mi></mrow></msub></mrow><mrow><mroot><mrow><mn mathvariant="normal">3</mn></mrow><mrow/></mroot><mfenced separators="|"><mrow><mi>p</mi><mo>-</mo><mmultiscripts><mrow><mi mathvariant="bold-italic">N</mi></mrow><mprescripts/><mrow/><mrow><mi>R</mi></mrow></mmultiscripts><msub><mrow><mi mathvariant="bold-italic">N</mi></mrow><mrow><msub><mrow><mi mathvariant="bold-italic">L</mi></mrow><mrow><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi></mrow></msub><mi>X</mi></mrow></msub></mrow></mfenced></mrow></mfrac><mo>+</mo><mfrac><mrow><mroot><mrow><mn mathvariant="normal">2</mn></mrow><mrow/></mroot></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><mmultiscripts><mrow><mi mathvariant="bold-italic">P</mi></mrow><mprescripts/><mrow/><mrow><mi>R</mi></mrow></mmultiscripts><msub><mrow><mi mathvariant="bold-italic">P</mi></mrow><mrow><msub><mrow><mi mathvariant="bold-italic">L</mi></mrow><mrow><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi></mrow></msub><mi>X</mi></mrow></msub><mo>-</mo><mmultiscripts><mrow><mi mathvariant="bold-italic">P</mi></mrow><mprescripts/><mrow/><mrow><mi>R</mi></mrow></mmultiscripts><msub><mrow><mi mathvariant="bold-italic">P</mi></mrow><mrow><msub><mrow><mi mathvariant="bold-italic">L</mi></mrow><mrow><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi></mrow></msub><mi>Y</mi></mrow></msub></mtd></mtr><mtr><mtd><mfrac><mrow><mn mathvariant="normal">2</mn><mfenced separators="|"><mrow><mi>a</mi><mo>+</mo><mroot><mrow><mn mathvariant="normal">3</mn></mrow><mrow/></mroot><mi>d</mi></mrow></mfenced><mmultiscripts><mrow><mi mathvariant="bold-italic">N</mi></mrow><mprescripts/><mrow/><mrow><mi>R</mi></mrow></mmultiscripts><msub><mrow><mi mathvariant="bold-italic">N</mi></mrow><mrow><msub><mrow><mi mathvariant="bold-italic">L</mi></mrow><mrow><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi></mrow></msub><mi>Z</mi></mrow></msub></mrow><mrow><mroot><mrow><mn mathvariant="normal">3</mn></mrow><mrow/></mroot><mfenced separators="|"><mrow><mi>p</mi><mo>-</mo><mmultiscripts><mrow><mi mathvariant="bold-italic">N</mi></mrow><mprescripts/><mrow/><mrow><mi>R</mi></mrow></mmultiscripts><msub><mrow><mi mathvariant="bold-italic">N</mi></mrow><mrow><msub><mrow><mi mathvariant="bold-italic">L</mi></mrow><mrow><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi></mrow></msub><mi>X</mi></mrow></msub></mrow></mfenced></mrow></mfrac><mo>-</mo><mfrac><mrow><mroot><mrow><mn mathvariant="normal">6</mn></mrow><mrow/></mroot></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><mmultiscripts><mrow><mi mathvariant="bold-italic">P</mi></mrow><mprescripts/><mrow/><mrow><mi>R</mi></mrow></mmultiscripts><msub><mrow><mi mathvariant="bold-italic">P</mi></mrow><mrow><msub><mrow><mi mathvariant="bold-italic">L</mi></mrow><mrow><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi></mrow></msub><mi>X</mi></mrow></msub><mo>-</mo><mmultiscripts><mrow><mi mathvariant="bold-italic">P</mi></mrow><mprescripts/><mrow/><mrow><mi>R</mi></mrow></mmultiscripts><msub><mrow><mi mathvariant="bold-italic">P</mi></mrow><mrow><msub><mrow><mi mathvariant="bold-italic">L</mi></mrow><mrow><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi></mrow></msub><mi>Z</mi></mrow></msub></mtd></mtr></mtable></mrow></mfenced><mo>,</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45411419&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45411400&type= 82.80400085 33.52799988 <math display="block" id="M59"><mmultiscripts><mrow/><mprescripts/><mrow/><mrow><mi>R</mi></mrow></mmultiscripts><msub><mrow><mi mathvariant="bold-italic">N</mi></mrow><mrow><msub><mrow><mi mathvariant="bold-italic">L</mi></mrow><mrow><mi mathvariant="normal">o</mi><mi mathvariant="normal">u</mi><mi mathvariant="normal">t</mi></mrow></msub></mrow></msub><mo>=</mo><mfenced separators="|"><mrow><mtable><mtr><mtd><mo>-</mo><mmultiscripts><mrow><mi mathvariant="bold-italic">N</mi></mrow><mprescripts/><mrow/><mrow><mi>R</mi></mrow></mmultiscripts><msub><mrow><mi mathvariant="bold-italic">N</mi></mrow><mrow><msub><mrow><mi mathvariant="bold-italic">L</mi></mrow><mrow><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi></mrow></msub><mi>X</mi></mrow></msub></mtd></mtr><mtr><mtd><mo>-</mo><mmultiscripts><mrow><mi mathvariant="bold-italic">N</mi></mrow><mprescripts/><mrow/><mrow><mi>R</mi></mrow></mmultiscripts><msub><mrow><mi mathvariant="bold-italic">N</mi></mrow><mrow><msub><mrow><mi mathvariant="bold-italic">L</mi></mrow><mrow><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi></mrow></msub><mi>Y</mi></mrow></msub></mtd></mtr><mtr><mtd><mo>-</mo><mmultiscripts><mrow><mi mathvariant="bold-italic">N</mi></mrow><mprescripts/><mrow/><mrow><mi>R</mi></mrow></mmultiscripts><msub><mrow><mi mathvariant="bold-italic">N</mi></mrow><mrow><msub><mrow><mi mathvariant="bold-italic">L</mi></mrow><mrow><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi></mrow></msub><mi>Z</mi></mrow></msub></mtd></mtr></mtable></mrow></mfenced><mo>.</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45411434&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45411415&type= 31.91933441 15.83266735 根据各个点的坐标可以求出光线在角锥棱镜内部传播的总光程，即有： <math display="block" id="M60"><mi>P</mi><mfenced separators="|"><mrow><mi>θ</mi></mrow></mfenced><mo>=</mo><mfrac><mrow><mn mathvariant="normal">2</mn><mi>n</mi><msup><mrow><mi mathvariant="normal">'</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mi>h</mi></mrow><mrow><mroot><mrow><mi>n</mi><msup><mrow><mi mathvariant="normal">'</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>-</mo><msup><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><msup><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mi mathvariant="normal">c</mi><mi mathvariant="normal">o</mi><msup><mrow><mi mathvariant="normal">s</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mi>θ</mi></mrow><mrow/></mroot></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45411449&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45411437&type= 45.12733078 10.24466610 . 光线相较垂直入射 <math id="M61"><mfenced separators="|"><mrow><mi>θ</mi><mo>=</mo><mn mathvariant="normal">0</mn></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45411447&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45411454&type= 11.43000031 3.97933316 时产生的附加光程差为： <math display="block" id="M62"><mi mathvariant="normal">Δ</mi><mi>P</mi><mfenced separators="|"><mrow><mi>θ</mi></mrow></mfenced><mo>=</mo><mfrac><mrow><mn mathvariant="normal">2</mn><mi>n</mi><msup><mrow><mi mathvariant="normal">'</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mi>h</mi></mrow><mrow><mroot><mrow><mi>n</mi><msup><mrow><mi mathvariant="normal">'</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>-</mo><msup><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><msup><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mi mathvariant="normal">c</mi><mi mathvariant="normal">o</mi><msup><mrow><mi mathvariant="normal">s</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mi>θ</mi></mrow><mrow/></mroot></mrow></mfrac><mo>-</mo><mn mathvariant="normal">2</mn><mi>n</mi><mi mathvariant="normal">'</mi><mi>h</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45411479&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45411466&type= 58.67399979 10.24466610 . 由此可知，角锥棱镜内部的附加光程差与入射角度 <math id="M63"><mi>θ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45411477&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45411483&type= 1.86266661 2.79399991 有关。 2.3　系统光程变化模型 2 干涉系统中，测量光路的长度包括角锥棱镜的外部光路长度和内部光路长度 ［ 10 10 ］ 。因此，在质量块自由运动的过程中，测量光路相对于初始状态的光程变化（Optical Path Change， OPC）也包含角锥棱镜内外两部分。其中，角锥棱镜内部的光程变化 <math id="M64"><mi mathvariant="normal">O</mi><mi mathvariant="normal">P</mi><msub><mrow><mi mathvariant="normal">C</mi></mrow><mrow><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45411498&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45411496&type= 9.39799976 3.72533321 只与光线的入射角有关，角锥棱镜外部的光程变化 <math id="M65"><mi mathvariant="normal">O</mi><mi mathvariant="normal">P</mi><msub><mrow><mi mathvariant="normal">C</mi></mrow><mrow><mi mathvariant="normal">o</mi><mi mathvariant="normal">u</mi><mi mathvariant="normal">t</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45411521&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45411523&type= 10.49866676 3.72533321 受质量块的位置和姿态影响。 图1 图1 中，光线在NPBS，PBS1，PBS2中的光程之和不随质量块的运动而改变，透镜组使光线经过PBS2和两个光电探测器之间的光程也不随质量块的运动而改变。因此， <math id="M66"><mi mathvariant="normal">O</mi><mi mathvariant="normal">P</mi><msub><mrow><mi mathvariant="normal">C</mi></mrow><mrow><mi mathvariant="normal">o</mi><mi mathvariant="normal">u</mi><mi mathvariant="normal">t</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45411521&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45411523&type= 10.49866676 3.72533321 仅与光线在真空中的光程变化有关。 为表征测量光路的光程变化，定义质量块参考坐标系 <math id="M67"><mi mathvariant="normal">Σ</mi><mi>f</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413127&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413125&type= 4.14866638 3.64066648 、固定参考坐标系 <math id="M68"><mi mathvariant="normal">Σ</mi><mi>o</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412170&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412166&type= 4.23333359 2.87866688 及各个角锥棱镜坐标系 <math id="M69"><mi mathvariant="normal">Σ</mi><msub><mrow><mi>r</mi></mrow><mrow><mi>i</mi></mrow></msub><mfenced separators="|"><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1,2</mn><mo>,</mo><mo>…</mo><mo>,</mo><mn mathvariant="normal">6</mn></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45411576&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45411571&type= 28.87133408 3.97933316 ，各坐标系的定义及位置关系如 图1 图1 所示。 <math id="M70"><mi mathvariant="normal">Σ</mi><mi>f</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413127&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413125&type= 4.14866638 3.64066648 坐标系原点 <math id="M71"><msub><mrow><mi>O</mi></mrow><mrow><mi>f</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45411590&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45411599&type= 3.64066648 3.72533321 位于该质量块的几何中心，坐标轴平行于立方体的各边，即 <math id="M72"><msub><mrow><mi>x</mi></mrow><mrow><mi>f</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413080&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413090&type= 2.96333337 3.72533321 轴沿 <math id="M73"><msub><mrow><mi>D</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><msub><mrow><mi>A</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45411631&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45411628&type= 8.63599968 3.72533321 方向， <math id="M74"><msub><mrow><mi>y</mi></mrow><mrow><mi>f</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45411626&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45411635&type= 2.79399991 4.57200003 轴沿 <math id="M75"><msub><mrow><mi>B</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><msub><mrow><mi>A</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413170&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413165&type= 8.29733276 3.72533321 方向， <math id="M76"><msub><mrow><mi>z</mi></mrow><mrow><mi>f</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45411673&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45411662&type= 2.53999996 3.72533321 轴沿 <math id="M77"><msub><mrow><mi>A</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><msub><mrow><mi>A</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45411669&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45411679&type= 8.72066593 3.72533321 方向； <math id="M78"><mi mathvariant="normal">Σ</mi><msub><mrow><mi>r</mi></mrow><mrow><mi>i</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45411850&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45411846&type= 4.65666676 3.80999994 坐标系定义同 图3 图3 中的 <math id="M79"><mi mathvariant="normal">Σ</mi><mi>R</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45411701&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45411707&type= 5.07999992 2.87866688 坐标系，原点 <math id="M80"><msub><mrow><mi>O</mi></mrow><mrow><msub><mrow><mi>r</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45411733&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45411719&type= 4.31799984 3.64066648 位于角锥棱镜的底面中心， <math id="M81"><msub><mrow><mi>x</mi></mrow><mrow><msub><mrow><mi>r</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45411740&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45411725&type= 3.64066648 3.64066648 轴指向角锥棱镜顶点， <math id="M82"><msub><mrow><mi>y</mi></mrow><mrow><msub><mrow><mi>r</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45411764&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45411746&type= 3.47133350 3.72533321 轴指向 <math id="M83"><msub><mrow><mi>E</mi></mrow><mrow><mi>i</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45411755&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45411753&type= 3.38666677 3.72533321 点在底面的投影， <math id="M84"><msub><mrow><mi>z</mi></mrow><mrow><msub><mrow><mi>r</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45411777&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45411774&type= 3.21733332 3.64066648 轴平行于 <math id="M85"><msub><mrow><mi>F</mi></mrow><mrow><mi>i</mi></mrow></msub><msub><mrow><mi>G</mi></mrow><mrow><mi>i</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45411799&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45411782&type= 7.11199999 3.72533321 。 <math id="M86"><mi mathvariant="normal">Σ</mi><mi>f</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413127&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413125&type= 4.14866638 3.64066648 坐标系相对于 <math id="M87"><mi mathvariant="normal">Σ</mi><mi>o</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412170&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412166&type= 4.23333359 2.87866688 坐标系的位置和姿态可以用平移向量 <math id="M88"><mmultiscripts><mrow/><mprescripts/><mrow><mi>f</mi></mrow><mrow><mi>o</mi></mrow></mmultiscripts><mi mathvariant="bold-italic">t</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45411960&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45411956&type= 2.79399991 4.48733330 和旋转矩阵 <math id="M89"><mmultiscripts><mrow/><mprescripts/><mrow><mi>f</mi></mrow><mrow><mi>o</mi></mrow></mmultiscripts><mi mathvariant="bold-italic">R</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412225&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412233&type= 4.57200003 4.48733330 表示； <math id="M90"><mi mathvariant="normal">Σ</mi><msub><mrow><mi>r</mi></mrow><mrow><mi>i</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45411850&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45411846&type= 4.65666676 3.80999994 坐标系相对于 <math id="M91"><mi mathvariant="normal">Σ</mi><mi>f</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413127&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413125&type= 4.14866638 3.64066648 坐标系的位置和姿态可以用平移向量 <math id="M92"><mmultiscripts><mrow/><mprescripts/><mrow><msub><mrow><mi>r</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow><mrow><mi>f</mi></mrow></mmultiscripts><mi mathvariant="bold-italic">t</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412221&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412206&type= 3.04800010 3.80999994 和旋转矩阵 <math id="M93"><mmultiscripts><mrow/><mprescripts/><mrow><msub><mrow><mi>r</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow><mrow><mi>f</mi></mrow></mmultiscripts><mi mathvariant="bold-italic">R</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45411887&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45411884&type= 4.82600021 3.80999994 表示。 初始时刻，各个入射光都垂直入射角锥棱镜，且指向 <math id="M94"><mi mathvariant="normal">Σ</mi><mi>f</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413127&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413125&type= 4.14866638 3.64066648 坐标轴正方向，将初始位置定义为固定参考坐标系 <math id="M95"><mi mathvariant="normal">Σ</mi><mi>o</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412170&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412166&type= 4.23333359 2.87866688 和质量块坐标系 <math id="M96"><mi mathvariant="normal">Σ</mi><mi>f</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413127&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413125&type= 4.14866638 3.64066648 重合。初始状态下，各角锥棱镜的内部光程为 <math id="M97"><mn mathvariant="normal">2</mn><mi>n</mi><mi mathvariant="normal">'</mi><mi>h</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45411945&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45411926&type= 6.34999990 3.30200005 ，外部光程为 <math id="M98"><mn mathvariant="normal">2</mn><mmultiscripts><mrow><mover accent="true"><mi mathvariant="bold-italic">ρ</mi><mo>^</mo></mover></mrow><mprescripts/><mrow/><mrow><mi>o</mi></mrow></mmultiscripts><msup><mrow><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">ρ</mi><mo>^</mo></mover></mrow><mrow><mi>i</mi></mrow></msub></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup><msub><mrow><mi mathvariant="bold-italic">I</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mmultiscripts><mrow><mi mathvariant="bold-italic">t</mi></mrow><mprescripts/><mrow><msub><mrow><mi>r</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow><mrow><mi>f</mi></mrow></mmultiscripts></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45411936&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45411933&type= 15.32466698 5.58799982 。本文所涉及的位姿解算及测量都是相对位姿。质量块在运动过程中，相对于初始状态的位姿可由 <math id="M99"><mmultiscripts><mrow/><mprescripts/><mrow><mi>f</mi></mrow><mrow><mi>o</mi></mrow></mmultiscripts><mi mathvariant="bold-italic">t</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45411960&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45411956&type= 2.79399991 4.48733330 和 <math id="M100"><mmultiscripts><mrow/><mprescripts/><mrow><mi>f</mi></mrow><mrow><mi>o</mi></mrow></mmultiscripts><mi mathvariant="bold-italic">R</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412225&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412233&type= 4.57200003 4.48733330 表示。角锥棱镜内外部的光程变化分别为： <math display="block" id="M101"><mi mathvariant="normal">O</mi><mi mathvariant="normal">P</mi><msub><mrow><mi mathvariant="normal">C</mi></mrow><mrow><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi></mrow></msub><mo>=</mo><mfrac><mrow><mn mathvariant="normal">2</mn><mi>n</mi><msup><mrow><mi mathvariant="normal">'</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mi>h</mi></mrow><mrow><mroot><mrow><mi>n</mi><msup><mrow><mi mathvariant="normal">'</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>-</mo><msup><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><msup><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mi mathvariant="normal">c</mi><mi mathvariant="normal">o</mi><msup><mrow><mi mathvariant="normal">s</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mi>θ</mi></mrow><mrow/></mroot></mrow></mfrac><mo>-</mo><mn mathvariant="normal">2</mn><mi>n</mi><mi mathvariant="normal">'</mi><mi>h</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45411990&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45411976&type= 57.57332993 10.24466610 ， <math display="block" id="M102"><mi mathvariant="normal">O</mi><mi mathvariant="normal">P</mi><msub><mrow><mi mathvariant="normal">C</mi></mrow><mrow><mi mathvariant="normal">o</mi><mi mathvariant="normal">u</mi><mi mathvariant="normal">t</mi></mrow></msub><mo>=</mo><mn mathvariant="normal">2</mn><mi>n</mi><mmultiscripts><mrow><mover accent="true"><mi mathvariant="bold-italic">ρ</mi><mo>^</mo></mover></mrow><mprescripts/><mrow/><mrow><mi>o</mi></mrow></mmultiscripts><msup><mrow><mover accent="true"><mi mathvariant="bold-italic">ρ</mi><mo>^</mo></mover></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup><mfenced open="[" close="]" separators="|"><mrow><mmultiscripts><mrow/><mprescripts/><mrow><mi>f</mi></mrow><mrow><mi>o</mi></mrow></mmultiscripts><mi mathvariant="bold-italic">t</mi><mo>+</mo><mfenced separators="|"><mrow><mmultiscripts><mrow/><mprescripts/><mrow><mi>f</mi></mrow><mrow><mi>o</mi></mrow></mmultiscripts><mi mathvariant="bold-italic">R</mi><mo>-</mo><msub><mrow><mi mathvariant="bold-italic">I</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub></mrow></mfenced><mmultiscripts><mrow><mi mathvariant="bold-italic">t</mi></mrow><mprescripts/><mrow><mi>r</mi></mrow><mrow><mi>f</mi></mrow></mmultiscripts></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45411995&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45411992&type= 54.01733780 5.58799982 . 在真空状态下，取 <math id="M103"><mi>n</mi><mo>=</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412002&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45411999&type= 8.63599968 2.79399991 ，第 i 个角锥棱镜的测量光路的光程变化可表示为： <math display="block" id="M104"><mi mathvariant="normal">O</mi><mi mathvariant="normal">P</mi><msub><mrow><mi mathvariant="normal">C</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>=</mo><mn mathvariant="normal">2</mn><mmultiscripts><mrow><mover accent="true"><mi mathvariant="bold-italic">ρ</mi><mo>^</mo></mover></mrow><mprescripts/><mrow/><mrow><mi>o</mi></mrow></mmultiscripts><msup><mrow><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">ρ</mi><mo>^</mo></mover></mrow><mrow><mi>i</mi></mrow></msub></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup><mfenced open="[" close="]" separators="|"><mrow><mmultiscripts><mrow/><mprescripts/><mrow><mi>f</mi></mrow><mrow><mi>o</mi></mrow></mmultiscripts><mi mathvariant="bold-italic">t</mi><mo>+</mo><mfenced separators="|"><mrow><mmultiscripts><mrow/><mprescripts/><mrow><mi>f</mi></mrow><mrow><mi>o</mi></mrow></mmultiscripts><mi mathvariant="bold-italic">R</mi><mo>-</mo><msub><mrow><mi mathvariant="bold-italic">I</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub></mrow></mfenced><mmultiscripts><mrow><mi mathvariant="bold-italic">t</mi></mrow><mprescripts/><mrow><msub><mrow><mi>r</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow><mrow><mi>f</mi></mrow></mmultiscripts></mrow></mfenced><mo>+</mo><mfrac><mrow><mn mathvariant="normal">2</mn><mi>n</mi><msup><mrow><mi mathvariant="normal">'</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mi>h</mi></mrow><mrow><mroot><mrow><mi>n</mi><msup><mrow><mi mathvariant="normal">'</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>-</mo><mn mathvariant="normal">1</mn><mo>+</mo><mi mathvariant="normal">c</mi><mi mathvariant="normal">o</mi><msup><mrow><mi mathvariant="normal">s</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><msub><mrow><mi>θ</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow><mrow/></mroot></mrow></mfrac><mo>-</mo><mn mathvariant="normal">2</mn><mi>n</mi><mi mathvariant="normal">'</mi><mi>h</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412025&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412005&type= 95.84266663 10.58333302 ， 式中： <math id="M105"><mi mathvariant="normal">c</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">s</mi><msub><mrow><mi>θ</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>=</mo><msup><mrow><mover accent="true"><mi mathvariant="bold-italic">x</mi><mo>^</mo></mover></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup><mmultiscripts><mrow><mi mathvariant="bold-italic">R</mi></mrow><mprescripts/><mrow><mi>f</mi></mrow><mrow><msub><mrow><mi>r</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></mmultiscripts><mmultiscripts><mrow><mi mathvariant="bold-italic">R</mi></mrow><mprescripts/><mrow><mi>o</mi></mrow><mrow><mi>f</mi></mrow></mmultiscripts><msup><mrow><mi mathvariant="bold-italic">R</mi></mrow><mrow><mi>o</mi></mrow></msup><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">ρ</mi><mo>^</mo></mover></mrow><mrow><mi>i</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412013&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412010&type= 31.32666588 5.07999992 ， <math id="M106"><mover accent="true"><mi mathvariant="bold-italic">x</mi><mo>^</mo></mover><mo>=</mo><msup><mrow><mfenced open="[" close="]" separators="|"><mrow><mn mathvariant="normal">1,0</mn><mo>,</mo><mn mathvariant="normal">0</mn></mrow></mfenced></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412051&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412035&type= 21.08200073 4.74133301 ， <math id="M107"><mmultiscripts><mrow/><mprescripts/><mrow/><mrow><mi>o</mi></mrow></mmultiscripts><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">ρ</mi><mo>^</mo></mover></mrow><mrow><mi>i</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412044&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412055&type= 4.40266657 5.41866684 为第 i 个角锥棱镜对应测量光轴的方向矢量， <math id="M108"><msub><mrow><mi mathvariant="bold-italic">I</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412068&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412063&type= 2.79399991 3.72533321 为三阶单位矩阵。记 H 为质量块的边长， L 为共面的一组角锥棱镜的中心距，则 <math id="M109"><mmultiscripts><mrow/><mprescripts/><mrow/><mrow><mi>o</mi></mrow></mmultiscripts><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">ρ</mi><mo>^</mo></mover></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>=</mo><mmultiscripts><mrow><mover accent="true"><mi mathvariant="bold-italic">ρ</mi><mo>^</mo></mover></mrow><mprescripts/><mrow/><mrow><mi>o</mi></mrow></mmultiscripts><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">ρ</mi><mo>^</mo></mover></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>=</mo><msup><mrow><mfenced separators="|"><mrow><mn mathvariant="normal">0,0</mn><mo>,</mo><mn mathvariant="normal">1</mn></mrow></mfenced></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412084&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412082&type= 33.27399826 4.74133301 ， <math id="M110"><mmultiscripts><mrow/><mprescripts/><mrow/><mrow><mi>o</mi></mrow></mmultiscripts><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">ρ</mi><mo>^</mo></mover></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mo>=</mo><mmultiscripts><mrow><mover accent="true"><mi mathvariant="bold-italic">ρ</mi><mo>^</mo></mover></mrow><mprescripts/><mrow/><mrow><mi>o</mi></mrow></mmultiscripts><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">ρ</mi><mo>^</mo></mover></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msub><mo>=</mo><msup><mrow><mfenced separators="|"><mrow><mn mathvariant="normal">1,0</mn><mo>,</mo><mn mathvariant="normal">0</mn></mrow></mfenced></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412088&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412074&type= 33.27399826 4.74133301 ， <math id="M111"><mmultiscripts><mrow/><mprescripts/><mrow/><mrow><mi>o</mi></mrow></mmultiscripts><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">ρ</mi><mo>^</mo></mover></mrow><mrow><mn mathvariant="normal">5</mn></mrow></msub><mo>=</mo><mmultiscripts><mrow><mover accent="true"><mi mathvariant="bold-italic">ρ</mi><mo>^</mo></mover></mrow><mprescripts/><mrow/><mrow><mi>o</mi></mrow></mmultiscripts><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">ρ</mi><mo>^</mo></mover></mrow><mrow><mn mathvariant="normal">6</mn></mrow></msub><mo>=</mo><msup><mrow><mfenced separators="|"><mrow><mn mathvariant="normal">0,1</mn><mo>,</mo><mn mathvariant="normal">0</mn></mrow></mfenced></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412095&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412090&type= 29.63333130 4.74133301 ， <math id="M112"><mmultiscripts><mrow/><mprescripts/><mrow><msub><mrow><mi>r</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow><mrow><mi>f</mi></mrow></mmultiscripts><mi mathvariant="bold-italic">t</mi><mo>=</mo><msup><mrow><mfenced separators="|"><mrow><mfrac><mrow><mi>L</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><mo>,</mo><mn mathvariant="normal">0</mn><mo>,</mo><mo>-</mo><mfrac><mrow><mi>H</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac></mrow></mfenced></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup><mo>,</mo><mmultiscripts><mrow><mi mathvariant="bold-italic">t</mi></mrow><mprescripts/><mrow><msub><mrow><mi>r</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mrow><mrow><mi>f</mi></mrow></mmultiscripts><mo>=</mo><mfenced close="" separators="|"><mrow><mo>-</mo><mfrac><mrow><mi>L</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac></mrow></mfenced><mo>,</mo><mn mathvariant="normal">0</mn><mo>,</mo><mo>-</mo><msup><mrow><mfenced open="" separators="|"><mrow><mfrac><mrow><mi>H</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac></mrow></mfenced></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412114&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412098&type= 59.94400024 9.90600014 ， <math id="M113"><mmultiscripts><mrow/><mprescripts/><mrow><msub><mrow><mi>r</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub></mrow><mrow><mi>f</mi></mrow></mmultiscripts><mi mathvariant="bold-italic">t</mi><mo>=</mo><msup><mrow><mfenced separators="|"><mrow><mo>-</mo><mfrac><mrow><mi>H</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><mo>,</mo><mfrac><mrow><mi>L</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><mo>,</mo><mn mathvariant="normal">0</mn></mrow></mfenced></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup><mo>,</mo><mmultiscripts><mrow><mi mathvariant="bold-italic">t</mi></mrow><mprescripts/><mrow><msub><mrow><mi>r</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msub></mrow><mrow><mi>f</mi></mrow></mmultiscripts><mo>=</mo><mfenced close="" separators="|"><mrow><mo>-</mo><mfrac><mrow><mi>H</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><mo>,</mo></mrow></mfenced><mo>-</mo><mfrac><mrow><mi>L</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><mo>,</mo><msup><mrow><mfenced open="" separators="|"><mrow><mn mathvariant="normal">0</mn></mrow></mfenced></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup><mo>,</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412122&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412101&type= 70.35800171 9.90600014 <math id="M114"><mmultiscripts><mrow/><mprescripts/><mrow><msub><mrow><mi>r</mi></mrow><mrow><mn mathvariant="normal">5</mn></mrow></msub></mrow><mrow><mi>f</mi></mrow></mmultiscripts><mi mathvariant="bold-italic">t</mi><mo>=</mo><msup><mrow><mfenced separators="|"><mrow><mn mathvariant="normal">0</mn><mo>,</mo><mo>-</mo><mfrac><mrow><mi>H</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><mo>,</mo><mfrac><mrow><mi>L</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac></mrow></mfenced></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412131&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412127&type= 28.95599937 9.90600014 ， <math id="M115"><mmultiscripts><mrow/><mprescripts/><mrow><msub><mrow><mi>r</mi></mrow><mrow><mn mathvariant="normal">6</mn></mrow></msub></mrow><mrow><mi>f</mi></mrow></mmultiscripts><mi mathvariant="bold-italic">t</mi><mo>=</mo><msup><mrow><mfenced separators="|"><mrow><mn mathvariant="normal">0</mn><mo>,</mo><mo>-</mo><mfrac><mrow><mi>H</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><mo>,</mo><mo>-</mo><mfrac><mrow><mi>L</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac></mrow></mfenced></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412138&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412106&type= 35.64466476 9.90600014 。 3 六自由度干涉系统的位姿解算 1 在2.3节定义的坐标系下，采用 z-y-x 内旋的方式进行变换，质量块的姿态角可以分解为绕 <math id="M116"><msub><mrow><mi>x</mi></mrow><mrow><mi>o</mi></mrow></msub><mo>,</mo><msub><mrow><mi>y</mi></mrow><mrow><mi>o</mi></mrow></msub><mo>,</mo><msub><mrow><mi>z</mi></mrow><mrow><mi>o</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412142&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412154&type= 13.03866673 4.57200003 轴分别旋转 <math id="M117"><msub><mrow><mi>θ</mi></mrow><mrow><mi>x</mi></mrow></msub><mo>,</mo><msub><mrow><mi>θ</mi></mrow><mrow><mi>y</mi></mrow></msub><mo>,</mo><msub><mrow><mi>θ</mi></mrow><mrow><mi>z</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412160&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412144&type= 12.61533356 4.14866638 ，即 <math id="M118"><mi mathvariant="normal">Σ</mi><mi>f</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413127&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413125&type= 4.14866638 3.64066648 坐标系相对于 <math id="M119"><mi mathvariant="normal">Σ</mi><mi>o</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412170&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412166&type= 4.23333359 2.87866688 坐标系的旋转矩阵为： <math display="block" id="M120"><mmultiscripts><mrow/><mprescripts/><mrow><mi>f</mi></mrow><mrow><mi>o</mi></mrow></mmultiscripts><mi mathvariant="bold-italic">R</mi><mo>=</mo><msub><mrow><mi mathvariant="bold-italic">R</mi></mrow><mrow><mi>z</mi></mrow></msub><mfenced separators="|"><mrow><msub><mrow><mi>θ</mi></mrow><mrow><mi>z</mi></mrow></msub></mrow></mfenced><msub><mrow><mi mathvariant="bold-italic">R</mi></mrow><mrow><mi>y</mi></mrow></msub><mfenced separators="|"><mrow><msub><mrow><mi>θ</mi></mrow><mrow><mi>y</mi></mrow></msub></mrow></mfenced><msub><mrow><mi mathvariant="bold-italic">R</mi></mrow><mrow><mi>x</mi></mrow></msub><mfenced separators="|"><mrow><msub><mrow><mi>θ</mi></mrow><mrow><mi>x</mi></mrow></msub></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412188&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412184&type= 41.40200043 4.91066647 ， 式中： <math id="M121"><msub><mrow><mi mathvariant="bold-italic">R</mi></mrow><mrow><mi>z</mi></mrow></msub><mo>=</mo><mfenced separators="|"><mrow><mtable><mtr><mtd><mi mathvariant="normal">c</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">s</mi><msub><mrow><mi>θ</mi></mrow><mrow><mi>z</mi></mrow></msub></mtd><mtd><mo>-</mo><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><msub><mrow><mi>θ</mi></mrow><mrow><mi>z</mi></mrow></msub></mtd><mtd><mn mathvariant="normal">0</mn></mtd></mtr><mtr><mtd><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><msub><mrow><mi>θ</mi></mrow><mrow><mi>z</mi></mrow></msub></mtd><mtd><mi mathvariant="normal">c</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">s</mi><msub><mrow><mi>θ</mi></mrow><mrow><mi>z</mi></mrow></msub></mtd><mtd><mn mathvariant="normal">0</mn></mtd></mtr><mtr><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">1</mn></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412176&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412190&type= 40.63999939 13.54666710 ， <math id="M122"><msub><mrow><mi mathvariant="bold-italic">R</mi></mrow><mrow><mi>y</mi></mrow></msub><mo>=</mo><mfenced separators="|"><mrow><mtable><mtr><mtd><mi mathvariant="normal">c</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">s</mi><msub><mrow><mi>θ</mi></mrow><mrow><mi>y</mi></mrow></msub></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><msub><mrow><mi>θ</mi></mrow><mrow><mi>y</mi></mrow></msub></mtd></mtr><mtr><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">1</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd></mtr><mtr><mtd><mo>-</mo><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><msub><mrow><mi>θ</mi></mrow><mrow><mi>y</mi></mrow></msub></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mi mathvariant="normal">c</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">s</mi><msub><mrow><mi>θ</mi></mrow><mrow><mi>y</mi></mrow></msub></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412181&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412178&type= 40.89400101 13.54666710 ， <math id="M123"><msub><mrow><mi mathvariant="bold-italic">R</mi></mrow><mrow><mi>x</mi></mrow></msub><mo>=</mo><mfenced separators="|"><mrow><mtable><mtr><mtd><mn mathvariant="normal">1</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd></mtr><mtr><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mi mathvariant="normal">c</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">s</mi><msub><mrow><mi>θ</mi></mrow><mrow><mi>x</mi></mrow></msub></mtd><mtd><mo>-</mo><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><msub><mrow><mi>θ</mi></mrow><mrow><mi>x</mi></mrow></msub></mtd></mtr><mtr><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><msub><mrow><mi>θ</mi></mrow><mrow><mi>x</mi></mrow></msub></mtd><mtd><mi mathvariant="normal">c</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">s</mi><msub><mrow><mi>θ</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=45412215&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412198&type= 41.23266602 13.54666710 。 将 <math id="M124"><mmultiscripts><mrow/><mprescripts/><mrow><msub><mrow><mi>r</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow><mrow><mi>f</mi></mrow></mmultiscripts><mi mathvariant="bold-italic">R</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412204&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412201&type= 4.82600021 3.80999994 ， <math id="M125"><mmultiscripts><mrow/><mprescripts/><mrow><msub><mrow><mi>r</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow><mrow><mi>f</mi></mrow></mmultiscripts><mi mathvariant="bold-italic">t</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412221&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412206&type= 3.04800010 3.80999994 ， <math id="M126"><mmultiscripts><mrow/><mprescripts/><mrow/><mrow><mi>o</mi></mrow></mmultiscripts><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">ρ</mi><mo>^</mo></mover></mrow><mrow><mi>i</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412230&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412212&type= 4.40266657 5.41866684 和 <math id="M127"><mmultiscripts><mrow/><mprescripts/><mrow><mi>f</mi></mrow><mrow><mi>o</mi></mrow></mmultiscripts><mi mathvariant="bold-italic">R</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412225&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412233&type= 4.57200003 4.48733330 代入 式（11） 式（11） ，CCR1和CCR2对应的光程变化相减，CCR3和CCR4对应的光程变化相减，CCR5和CCR6对应的光程变化相减，分别得到： <math display="block" id="M128"><mi mathvariant="normal">O</mi><mi mathvariant="normal">P</mi><msub><mrow><mi mathvariant="normal">C</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>-</mo><mi mathvariant="normal">O</mi><mi mathvariant="normal">P</mi><msub><mrow><mi mathvariant="normal">C</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>=</mo><mo>-</mo><mn mathvariant="normal">2</mn><mi>L</mi><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><msub><mrow><mi>θ</mi></mrow><mrow><mi>y</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412243&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412237&type= 43.68799973 3.72533321 ， <math display="block" id="M129"><mtable><mtr><mtd><mi mathvariant="normal">O</mi><mi mathvariant="normal">P</mi><msub><mrow><mi mathvariant="normal">C</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mo>-</mo><mi mathvariant="normal">O</mi><mi mathvariant="normal">P</mi><msub><mrow><mi mathvariant="normal">C</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msub><mo>=</mo></mtd></mtr><mtr><mtd><mn mathvariant="normal">2</mn><mi>L</mi><mfenced separators="|"><mrow><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><msub><mrow><mi>θ</mi></mrow><mrow><mi>x</mi></mrow></msub><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><msub><mrow><mi>θ</mi></mrow><mrow><mi>y</mi></mrow></msub><mi mathvariant="normal">c</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">s</mi><msub><mrow><mi>θ</mi></mrow><mrow><mi>z</mi></mrow></msub><mo>-</mo><mi mathvariant="normal">c</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">s</mi><msub><mrow><mi>θ</mi></mrow><mrow><mi>x</mi></mrow></msub><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><msub><mrow><mi>θ</mi></mrow><mrow><mi>z</mi></mrow></msub></mrow></mfenced></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412242&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412246&type= 55.71067047 9.48266602 ， <math display="block" id="M130"><mtable><mtr><mtd><mi mathvariant="normal">O</mi><mi mathvariant="normal">P</mi><msub><mrow><mi mathvariant="normal">C</mi></mrow><mrow><mn mathvariant="normal">5</mn></mrow></msub><mo>-</mo><mi mathvariant="normal">O</mi><mi mathvariant="normal">P</mi><msub><mrow><mi mathvariant="normal">C</mi></mrow><mrow><mn mathvariant="normal">6</mn></mrow></msub><mo>=</mo></mtd></mtr><mtr><mtd><mn mathvariant="normal">2</mn><mi>L</mi><mfenced separators="|"><mrow><mi mathvariant="normal">c</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">s</mi><msub><mrow><mi>θ</mi></mrow><mrow><mi>x</mi></mrow></msub><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><msub><mrow><mi>θ</mi></mrow><mrow><mi>y</mi></mrow></msub><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><msub><mrow><mi>θ</mi></mrow><mrow><mi>z</mi></mrow></msub><mo>-</mo><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><msub><mrow><mi>θ</mi></mrow><mrow><mi>x</mi></mrow></msub><mi mathvariant="normal">c</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">s</mi><msub><mrow><mi>θ</mi></mrow><mrow><mi>z</mi></mrow></msub></mrow></mfenced></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412252&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412259&type= 55.71067047 9.48266602 . 3.1 数值解算方法 2 由于 式（13） 式（13） 中OPC 1 ，OPC 2 可通过测量得到，因此只含有 <math id="M131"><msub><mrow><mi>θ</mi></mrow><mrow><mi>y</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412433&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412439&type= 2.96333337 3.72533321 一个未知量，可以直接求出姿态角 <math id="M132"><msub><mrow><mi>θ</mi></mrow><mrow><mi>y</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412433&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412439&type= 2.96333337 3.72533321 。将 式（13） 式（13） 代入 式（14） 式（14） 和 式（15） 式（15） 中进行化简可得： <math display="block" id="M133"><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mtext> </mtext><mi>q</mi><mo>=</mo><mfrac><mrow><mfenced separators="|"><mrow><mi mathvariant="normal">O</mi><mi mathvariant="normal">P</mi><msub><mrow><mi mathvariant="normal">C</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>-</mo><mi mathvariant="normal">O</mi><mi mathvariant="normal">P</mi><msub><mrow><mi mathvariant="normal">C</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><mn mathvariant="normal">2</mn><mi>L</mi></mrow></mfenced><mfenced separators="|"><mrow><mi mathvariant="normal">O</mi><mi mathvariant="normal">P</mi><msub><mrow><mi mathvariant="normal">C</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mo>-</mo><mi mathvariant="normal">O</mi><mi mathvariant="normal">P</mi><msub><mrow><mi mathvariant="normal">C</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msub><mo>+</mo><mi mathvariant="normal">O</mi><mi mathvariant="normal">P</mi><msub><mrow><mi mathvariant="normal">C</mi></mrow><mrow><mn mathvariant="normal">5</mn></mrow></msub><mo>-</mo><mi mathvariant="normal">O</mi><mi mathvariant="normal">P</mi><msub><mrow><mi mathvariant="normal">C</mi></mrow><mrow><mn mathvariant="normal">6</mn></mrow></msub></mrow></mfenced></mrow><mrow><msup><mrow><mfenced separators="|"><mrow><mi mathvariant="normal">O</mi><mi mathvariant="normal">P</mi><msub><mrow><mi mathvariant="normal">C</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>-</mo><mi mathvariant="normal">O</mi><mi mathvariant="normal">P</mi><msub><mrow><mi mathvariant="normal">C</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>-</mo><mn mathvariant="normal">4</mn><msup><mrow><mi>L</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412279&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412278&type= 100.92266846 10.24466610 . 根据 式（16） 式（16） 求出 <math id="M134"><mi>q</mi><mfenced separators="|"><mrow><mi>q</mi><mo>=</mo><msub><mrow><mi>θ</mi></mrow><mrow><mi>x</mi></mrow></msub><mo>+</mo><msub><mrow><mi>θ</mi></mrow><mrow><mi>z</mi></mrow></msub></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412292&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412280&type= 22.94466591 4.82600021 ，并代入 式（14） 式（14） 和 式（15） 式（15） 进行化简，可以得到： <math display="block" id="M135"><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><msub><mrow><mi>θ</mi></mrow><mrow><mi>x</mi></mrow></msub><mi mathvariant="normal">c</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">s</mi><mfenced separators="|"><mrow><mi>q</mi><mo>-</mo><msub><mrow><mi>θ</mi></mrow><mrow><mi>x</mi></mrow></msub></mrow></mfenced><mo>=</mo><mfrac><mrow><mfenced separators="|"><mrow><mi mathvariant="normal">O</mi><mi mathvariant="normal">P</mi><msub><mrow><mi mathvariant="normal">C</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>-</mo><mi mathvariant="normal">O</mi><mi mathvariant="normal">P</mi><msub><mrow><mi mathvariant="normal">C</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></mfenced><mfenced separators="|"><mrow><mi mathvariant="normal">O</mi><mi mathvariant="normal">P</mi><msub><mrow><mi mathvariant="normal">C</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mo>-</mo><mi mathvariant="normal">O</mi><mi mathvariant="normal">P</mi><msub><mrow><mi mathvariant="normal">C</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msub></mrow></mfenced><mo>+</mo><mn mathvariant="normal">2</mn><mi>L</mi><mfenced separators="|"><mrow><mi mathvariant="normal">O</mi><mi mathvariant="normal">P</mi><msub><mrow><mi mathvariant="normal">C</mi></mrow><mrow><mn mathvariant="normal">5</mn></mrow></msub><mo>-</mo><mi mathvariant="normal">O</mi><mi mathvariant="normal">P</mi><msub><mrow><mi mathvariant="normal">C</mi></mrow><mrow><mn mathvariant="normal">6</mn></mrow></msub></mrow></mfenced></mrow><mrow><msup><mrow><mfenced separators="|"><mrow><mi mathvariant="normal">O</mi><mi mathvariant="normal">P</mi><msub><mrow><mi mathvariant="normal">C</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>-</mo><mi mathvariant="normal">O</mi><mi mathvariant="normal">P</mi><msub><mrow><mi mathvariant="normal">C</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>-</mo><mn mathvariant="normal">4</mn><msup><mrow><mi>L</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412304&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412285&type= 120.14199829 10.15999985 . 利用牛顿迭代法对 式（17） 式（17） 进行求解，可以得到姿态角 <math id="M136"><msub><mrow><mi>θ</mi></mrow><mrow><mi>x</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412437&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412441&type= 3.04800010 3.72533321 ，再代入 式（16） 式（16） 得到姿态角 <math id="M137"><msub><mrow><mi>θ</mi></mrow><mrow><mi>z</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412446&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412444&type= 2.87866688 3.72533321 。 记姿态角 <math id="M138"><msub><mrow><mi>θ</mi></mrow><mrow><mi>x</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412437&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412441&type= 3.04800010 3.72533321 的牛顿迭代输出结果为 <math id="M139"><mi>x</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412346&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412344&type= 2.20133328 2.79399991 ，求解 式（17） 式（17） 可化简为求解方程： <math display="block" id="M140"><mi>F</mi><mfenced separators="|"><mrow><mi>x</mi></mrow></mfenced><mo>=</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412331&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412338&type= 15.40933418 3.97933316 ， <math display="block" id="M141"><mi>F</mi><mfenced separators="|"><mrow><mi>x</mi></mrow></mfenced><mo>=</mo><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mi>x</mi><mi mathvariant="normal">c</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">s</mi><mfenced separators="|"><mrow><mi>q</mi><mo>-</mo><mi>x</mi></mrow></mfenced><mo>-</mo><mfrac><mrow><mfenced separators="|"><mrow><mi mathvariant="normal">O</mi><mi mathvariant="normal">P</mi><msub><mrow><mi mathvariant="normal">C</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>-</mo><mi mathvariant="normal">O</mi><mi mathvariant="normal">P</mi><msub><mrow><mi mathvariant="normal">C</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></mfenced><mfenced separators="|"><mrow><mi mathvariant="normal">O</mi><mi mathvariant="normal">P</mi><msub><mrow><mi mathvariant="normal">C</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mo>-</mo><mi mathvariant="normal">O</mi><mi mathvariant="normal">P</mi><msub><mrow><mi mathvariant="normal">C</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msub></mrow></mfenced><mo>+</mo><mn mathvariant="normal">2</mn><mi>L</mi><mfenced separators="|"><mrow><mi mathvariant="normal">O</mi><mi mathvariant="normal">P</mi><msub><mrow><mi mathvariant="normal">C</mi></mrow><mrow><mn mathvariant="normal">5</mn></mrow></msub><mo>-</mo><mi mathvariant="normal">O</mi><mi mathvariant="normal">P</mi><msub><mrow><mi mathvariant="normal">C</mi></mrow><mrow><mn mathvariant="normal">6</mn></mrow></msub></mrow></mfenced></mrow><mrow><msup><mrow><mfenced separators="|"><mrow><mi mathvariant="normal">O</mi><mi mathvariant="normal">P</mi><msub><mrow><mi mathvariant="normal">C</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>-</mo><mi mathvariant="normal">O</mi><mi mathvariant="normal">P</mi><msub><mrow><mi mathvariant="normal">C</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>-</mo><mn mathvariant="normal">4</mn><msup><mrow><mi>L</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412349&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412340&type= 119.04132843 8.89000034 。 记 <math id="M142"><mi>x</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412346&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412344&type= 2.20133328 2.79399991 的第 <math id="M143"><mi>k</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412364&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412356&type= 1.69333339 2.79399991 次迭代结果为 <math id="M144"><msub><mrow><mi>x</mi></mrow><mrow><mi>k</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412362&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412361&type= 3.21733332 3.72533321 。由于该系统在空间自由漂移的过程中角位移较小，设定 <math id="M145"><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>=</mo><mn mathvariant="normal">0</mn><mo>,</mo><mi>k</mi><mo>=</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412371&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412369&type= 20.74333191 4.14866638 为算法的初始值，然后进行迭代，如果误差大于设定值 <math id="M146"><mi>ε</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412375&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412374&type= 1.43933344 2.79399991 则继续迭代；如果误差小于设定值 <math id="M147"><mi>ε</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412395&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412393&type= 1.43933344 2.79399991 则返回结果并退出循环。具体步骤如下： 第一步：假设初始值为 <math id="M148"><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>=</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412398&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412385&type= 10.07533360 3.72533321 ， <math id="M149"><mi>k</mi><mo>=</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412402&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412400&type= 8.38199997 2.79399991 ； 第二步：计算 <math id="M150"><msub><mrow><mi>x</mi></mrow><mrow><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msub><mo>=</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>-</mo><mrow><mrow><mi>F</mi><mfenced separators="|"><mrow><msub><mrow><mi>x</mi></mrow><mrow><mi>k</mi></mrow></msub></mrow></mfenced></mrow><mo>/</mo><mrow><mi>F</mi><mi mathvariant="normal">'</mi><mfenced separators="|"><mrow><msub><mrow><mi>x</mi></mrow><mrow><mi>k</mi></mrow></msub></mrow></mfenced></mrow></mrow></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412409&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412405&type= 41.14799881 4.57200003 。 3.2 小角度近似方法 2 在某些精度要求不是很高的应用场景，可以采用小角度近似求解进而降低系统姿态角的解耦难度，即 <math id="M151"><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><msub><mrow><mi>θ</mi></mrow><mrow><mi>x</mi></mrow></msub><mo>~</mo><msub><mrow><mi>θ</mi></mrow><mrow><mi>x</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412414&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412413&type= 14.47799969 3.72533321 ， <math id="M152"><mi mathvariant="normal">c</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">s</mi><msub><mrow><mi>θ</mi></mrow><mrow><mi>x</mi></mrow></msub><mo>~</mo><mn mathvariant="normal">1</mn><mo>,</mo><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><msub><mrow><mi>θ</mi></mrow><mrow><mi>y</mi></mrow></msub><mo>~</mo><msub><mrow><mi>θ</mi></mrow><mrow><mi>y</mi></mrow></msub><mo>,</mo><mi mathvariant="normal">c</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">s</mi><msub><mrow><mi>θ</mi></mrow><mrow><mi>y</mi></mrow></msub><mo>~</mo><mn mathvariant="normal">1</mn><mo>,</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412428&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412416&type= 47.75199890 4.14866638 <math id="M153"><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><msub><mrow><mi>θ</mi></mrow><mrow><mi>z</mi></mrow></msub><mo>~</mo><msub><mrow><mi>θ</mi></mrow><mrow><mi>z</mi></mrow></msub><mo>,</mo><mi mathvariant="normal">c</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">s</mi><msub><mrow><mi>θ</mi></mrow><mrow><mi>z</mi></mrow></msub><mo>~</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412420&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412418&type= 29.80266571 4.14866638 。根据 式（13） 式（13） 可直接求出姿态角 <math id="M154"><msub><mrow><mi>θ</mi></mrow><mrow><mi>y</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412433&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412439&type= 2.96333337 3.72533321 ，将 式（13） 式（13） 代入 式（14） 式（14） 和 式（15） 式（15） 中进行近似化简可求得 <math id="M155"><msub><mrow><mi>θ</mi></mrow><mrow><mi>x</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412437&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412441&type= 3.04800010 3.72533321 和 <math id="M156"><msub><mrow><mi>θ</mi></mrow><mrow><mi>z</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412446&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412444&type= 2.87866688 3.72533321 ，即： <math display="block" id="M157"><mfenced separators="|"><mrow><mtable><mtr><mtd><msub><mrow><mi>θ</mi></mrow><mrow><mi>x</mi></mrow></msub></mtd></mtr><mtr><mtd><msub><mrow><mi>θ</mi></mrow><mrow><mi>y</mi></mrow></msub></mtd></mtr><mtr><mtd><msub><mrow><mi>θ</mi></mrow><mrow><mi>z</mi></mrow></msub></mtd></mtr></mtable></mrow></mfenced><mo>=</mo><mfenced separators="|"><mrow><mtable><mtr><mtd><mfrac><mrow><mfenced separators="|"><mrow><mi mathvariant="normal">O</mi><mi mathvariant="normal">P</mi><msub><mrow><mi mathvariant="normal">C</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>-</mo><mi mathvariant="normal">O</mi><mi mathvariant="normal">P</mi><msub><mrow><mi mathvariant="normal">C</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></mfenced><mfenced separators="|"><mrow><mi mathvariant="normal">O</mi><mi mathvariant="normal">P</mi><msub><mrow><mi mathvariant="normal">C</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mo>-</mo><mi mathvariant="normal">O</mi><mi mathvariant="normal">P</mi><msub><mrow><mi mathvariant="normal">C</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msub></mrow></mfenced><mo>+</mo><mn mathvariant="normal">2</mn><mi>L</mi><mfenced separators="|"><mrow><mi mathvariant="normal">O</mi><mi mathvariant="normal">P</mi><msub><mrow><mi mathvariant="normal">C</mi></mrow><mrow><mn mathvariant="normal">5</mn></mrow></msub><mo>-</mo><mi mathvariant="normal">O</mi><mi mathvariant="normal">P</mi><msub><mrow><mi mathvariant="normal">C</mi></mrow><mrow><mn mathvariant="normal">6</mn></mrow></msub></mrow></mfenced></mrow><mrow><msup><mrow><mfenced separators="|"><mrow><mi mathvariant="normal">O</mi><mi mathvariant="normal">P</mi><msub><mrow><mi mathvariant="normal">C</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>-</mo><mi mathvariant="normal">O</mi><mi mathvariant="normal">P</mi><msub><mrow><mi mathvariant="normal">C</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>-</mo><mn mathvariant="normal">4</mn><msup><mrow><mi>L</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfrac></mtd></mtr><mtr><mtd><mfrac><mrow><mi mathvariant="normal">O</mi><mi mathvariant="normal">P</mi><msub><mrow><mi mathvariant="normal">C</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>-</mo><mi mathvariant="normal">O</mi><mi mathvariant="normal">P</mi><msub><mrow><mi mathvariant="normal">C</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow><mrow><mn mathvariant="normal">2</mn><mi>L</mi></mrow></mfrac></mtd></mtr><mtr><mtd><mfrac><mrow><mfenced separators="|"><mrow><mi mathvariant="normal">O</mi><mi mathvariant="normal">P</mi><msub><mrow><mi mathvariant="normal">C</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>-</mo><mi mathvariant="normal">O</mi><mi mathvariant="normal">P</mi><msub><mrow><mi mathvariant="normal">C</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></mfenced><mfenced separators="|"><mrow><mi mathvariant="normal">O</mi><mi mathvariant="normal">P</mi><msub><mrow><mi mathvariant="normal">C</mi></mrow><mrow><mn mathvariant="normal">5</mn></mrow></msub><mo>-</mo><mi mathvariant="normal">O</mi><mi mathvariant="normal">P</mi><msub><mrow><mi mathvariant="normal">C</mi></mrow><mrow><mn mathvariant="normal">6</mn></mrow></msub></mrow></mfenced><mo>+</mo><mn mathvariant="normal">2</mn><mi>L</mi><mfenced separators="|"><mrow><mi mathvariant="normal">O</mi><mi mathvariant="normal">P</mi><msub><mrow><mi mathvariant="normal">C</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mo>-</mo><mi mathvariant="normal">O</mi><mi mathvariant="normal">P</mi><msub><mrow><mi mathvariant="normal">C</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msub></mrow></mfenced></mrow><mrow><msup><mrow><mfenced separators="|"><mrow><mi mathvariant="normal">O</mi><mi mathvariant="normal">P</mi><msub><mrow><mi mathvariant="normal">C</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>-</mo><mi mathvariant="normal">O</mi><mi mathvariant="normal">P</mi><msub><mrow><mi mathvariant="normal">C</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>-</mo><mn mathvariant="normal">4</mn><msup><mrow><mi>L</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfrac></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412460&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412457&type= 103.12400055 33.35866928 . 得到3个姿态角后，由于沿 <math id="M158"><msub><mrow><mi>x</mi></mrow><mrow><mi>o</mi></mrow></msub><mo>,</mo><msub><mrow><mi>y</mi></mrow><mrow><mi>o</mi></mrow></msub><mo>,</mo><msub><mrow><mi>z</mi></mrow><mrow><mi>o</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412469&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412452&type= 13.03866673 4.57200003 轴平动的位移 <math id="M159"><mmultiscripts><mrow/><mprescripts/><mrow><mi>f</mi></mrow><mrow><mi>o</mi></mrow></mmultiscripts><mi mathvariant="bold-italic">t</mi><mo>=</mo><msup><mrow><mfenced separators="|"><mrow><mmultiscripts><mrow/><mprescripts/><mrow><mi>f</mi></mrow><mrow><mi>o</mi></mrow></mmultiscripts><msub><mrow><mi mathvariant="bold-italic">t</mi></mrow><mrow><mi>x</mi></mrow></msub><mo>,</mo><mmultiscripts><mrow><mi mathvariant="bold-italic">t</mi></mrow><mprescripts/><mrow><mi>f</mi></mrow><mrow><mi>o</mi></mrow></mmultiscripts><msub><mrow><mi mathvariant="bold-italic">t</mi></mrow><mrow><mi>y</mi></mrow></msub><mo>,</mo><mmultiscripts><mrow><mi mathvariant="bold-italic">t</mi></mrow><mprescripts/><mrow><mi>f</mi></mrow><mrow><mi>o</mi></mrow></mmultiscripts><msub><mrow><mi mathvariant="bold-italic">t</mi></mrow><mrow><mi>z</mi></mrow></msub></mrow></mfenced></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412466&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412471&type= 29.12533188 5.58799982 与3个轴对应的光路变化 <math id="M160"><msup><mrow><mfenced separators="|"><mrow><mmultiscripts><mrow/><mprescripts/><mrow/><mrow><mi>o</mi></mrow></mmultiscripts><msup><mrow><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">ρ</mi><mo>^</mo></mover></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup><mmultiscripts><mrow><mi mathvariant="bold-italic">t</mi></mrow><mprescripts/><mrow><mi>f</mi></mrow><mrow><mi>o</mi></mrow></mmultiscripts><mo>,</mo><mmultiscripts><mrow><mover accent="true"><mi mathvariant="bold-italic">ρ</mi><mo>^</mo></mover></mrow><mprescripts/><mrow/><mrow><mi>o</mi></mrow></mmultiscripts><msup><mrow><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">ρ</mi><mo>^</mo></mover></mrow><mrow><mn mathvariant="normal">5</mn></mrow></msub></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup><mmultiscripts><mrow><mi mathvariant="bold-italic">t</mi></mrow><mprescripts/><mrow><mi>f</mi></mrow><mrow><mi>o</mi></mrow></mmultiscripts><mo>,</mo><mmultiscripts><mrow><mover accent="true"><mi mathvariant="bold-italic">ρ</mi><mo>^</mo></mover></mrow><mprescripts/><mrow/><mrow><mi>o</mi></mrow></mmultiscripts><msup><mrow><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">ρ</mi><mo>^</mo></mover></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup><mmultiscripts><mrow><mi mathvariant="bold-italic">t</mi></mrow><mprescripts/><mrow><mi>f</mi></mrow><mrow><mi>o</mi></mrow></mmultiscripts></mrow></mfenced></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412483&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412474&type= 39.70866776 5.58799982 成正比，可以建立平移向量 <math id="M161"><mmultiscripts><mrow/><mprescripts/><mrow><mi>f</mi></mrow><mrow><mi>o</mi></mrow></mmultiscripts><mi mathvariant="bold-italic">t</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412487&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412477&type= 2.79399991 4.48733330 和光程变化 <math id="M162"><mi mathvariant="normal">O</mi><mi mathvariant="normal">P</mi><msub><mrow><mi mathvariant="normal">C</mi></mrow><mrow><mi>i</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412482&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412480&type= 8.46666718 3.72533321 读数之间的关系，即： <math display="block" id="M163"><mmultiscripts><mrow/><mprescripts/><mrow><mi>f</mi></mrow><mrow><mi>o</mi></mrow></mmultiscripts><mi mathvariant="bold-italic">t</mi><mo>=</mo><mfenced separators="|"><mrow><mtable><mtr><mtd><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><mi mathvariant="normal">O</mi><mi mathvariant="normal">P</mi><msub><mrow><mi mathvariant="normal">C</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mo>-</mo><msub><mrow><mi>B</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub></mtd></mtr><mtr><mtd><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><mi mathvariant="normal">O</mi><mi mathvariant="normal">P</mi><msub><mrow><mi mathvariant="normal">C</mi></mrow><mrow><mn mathvariant="normal">5</mn></mrow></msub><mo>-</mo><msub><mrow><mi>B</mi></mrow><mrow><mn mathvariant="normal">5</mn></mrow></msub></mtd></mtr><mtr><mtd><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><mi mathvariant="normal">O</mi><mi mathvariant="normal">P</mi><msub><mrow><mi mathvariant="normal">C</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>-</mo><msub><mrow><mi>B</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412503&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412490&type= 32.00400162 28.61733437 ， 式中： <math display="block" id="M164"><mtable><mtr><mtd><msub><mrow><mi>B</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>=</mo><mmultiscripts><mrow><mover accent="true"><mi mathvariant="bold-italic">ρ</mi><mo>^</mo></mover></mrow><mprescripts/><mrow/><mrow><mi>o</mi></mrow></mmultiscripts><msup><mrow><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">ρ</mi><mo>^</mo></mover></mrow><mrow><mi>i</mi></mrow></msub></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup><mfenced separators="|"><mrow><mmultiscripts><mrow/><mprescripts/><mrow><mi>f</mi></mrow><mrow><mi>o</mi></mrow></mmultiscripts><mi mathvariant="bold-italic">R</mi><mo>-</mo><msub><mrow><mi mathvariant="bold-italic">I</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub></mrow></mfenced><mmultiscripts><mrow><mi mathvariant="bold-italic">t</mi></mrow><mprescripts/><mrow><msub><mrow><mi>r</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow><mrow><mi>f</mi></mrow></mmultiscripts><mo>+</mo></mtd></mtr><mtr><mtd><mfrac><mrow><mi>n</mi><msup><mrow><mi mathvariant="normal">'</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mi>h</mi></mrow><mrow><mroot><mrow><mi>n</mi><msup><mrow><mi mathvariant="normal">'</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>-</mo><mn mathvariant="normal">1</mn><mo>+</mo><msup><mrow><mfenced separators="|"><mrow><msup><mrow><mover accent="true"><mi mathvariant="bold-italic">x</mi><mo>^</mo></mover></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup><mmultiscripts><mrow><mi mathvariant="bold-italic">R</mi></mrow><mprescripts/><mrow><mi>f</mi></mrow><mrow><msub><mrow><mi>r</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></mmultiscripts><mmultiscripts><mrow><mi mathvariant="bold-italic">R</mi></mrow><mprescripts/><mrow><mi>o</mi></mrow><mrow><mi>f</mi></mrow></mmultiscripts><msup><mrow><mi mathvariant="bold-italic">R</mi></mrow><mrow><mi>o</mi></mrow></msup><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">ρ</mi><mo>^</mo></mover></mrow><mrow><mi>i</mi></mrow></msub></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow><mrow/></mroot></mrow></mfrac><mo>-</mo><mi>n</mi><mi mathvariant="normal">'</mi><mi>h</mi></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412504&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412494&type= 51.90066528 19.13466644 ， 旋转矩阵 <math id="M165"><mmultiscripts><mrow/><mprescripts/><mrow><mi>f</mi></mrow><mrow><mi>o</mi></mrow></mmultiscripts><mi mathvariant="bold-italic">R</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412508&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412507&type= 4.57200003 4.48733330 可由 式（12） 式（12） 得到。 4 系统仿真及误差分析 1 4.1　系统仿真设计 2 本系统使用的是口径为2.54 cm（1 inch）的角锥棱镜，质量块边长为90 mm，共面的一组角锥棱镜中心距为35 mm。质量块的平动范围设定为±2 mm，转动范围设定为±2°，超出该范围时检测系统会自动复位。在该范围内角锥棱镜的附加光程差最大为0.030 5 mm，对应的位移误差为0.015 3 nm。 本文通过给定的姿态角 <math id="M166"><msub><mrow><mi>Q</mi></mrow><mrow><mi mathvariant="normal">r</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">f</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412512&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412510&type= 5.16466665 4.14866638 和位移 <math id="M167"><msub><mrow><mi>T</mi></mrow><mrow><mi mathvariant="normal">r</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">f</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412523&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412529&type= 4.91066647 3.72533321 仿真光程变化 <math id="M168"><mi mathvariant="normal">O</mi><mi mathvariant="normal">P</mi><msub><mrow><mi mathvariant="normal">C</mi></mrow><mrow><mi>i</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412534&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412532&type= 8.46666718 3.72533321 ，并根据第3节给出的算法计算姿态角 <math id="M169"><msub><mrow><mi>Q</mi></mrow><mrow><mi mathvariant="normal">c</mi><mi mathvariant="normal">p</mi><mi mathvariant="normal">t</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412544&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412543&type= 5.50333309 4.14866638 和位移 <math id="M170"><msub><mrow><mi>T</mi></mrow><mrow><mi mathvariant="normal">c</mi><mi mathvariant="normal">p</mi><mi mathvariant="normal">t</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412548&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412547&type= 5.16466665 3.72533321 ，通过分析误差情况验证位姿解算方法的准确性，位姿解算误差的仿真流程如 图4 图4 所示。 4.2　六自由度位姿的空间仿真 2 本文采用天舟一号主动隔振项目中搭载的石英挠性加速度计的在轨数据，将测得的航天器姿态角和相对位移作为系统质量块的给定位姿，进行算法仿真。 数值求解方法和小角度近似方法得到的姿态角和相对位移结果如 图5 图5 所示，算法的解算误差如 图6 图6 所示。其中，数值求解方法设置的牛顿迭代法误差判据为10 -10 。原始数据表明，飞行器在130~190 s以及600~700 s运行期间产生的振动较大，石英挠性加速度计检测到位置和姿态的变化较大。 图5 图5 的解算结果显示，数值计算和小角度近似解算得到的位姿都能很好地跟随给定值，但局部放大后发现，数值计算得到的位姿与给定值相当，而小角度近似求解存在一定的误差。 图6 图6 的误差结果显示，利用数值计算得到的姿态角绝对误差小于10 - 14 °，位移绝对误差小于10 - 8 nm，而利用小角度近似方法得到的姿态角误差在10 - 7 °量级，位移误差小于4 nm。数值计算的解算误差远小于小角度近似的解算误差，且计算误差不会随着飞行器振动的增大而明显变大，算法具有更高的精度和更好的稳定性。 1,2;3,4;5,6; 1,2;3,4;5,6; 4.3　六自由度位姿的模拟仿真 2 蒙特卡洛提供了一种利用随机抽样程序生成具有一定分布规律的随机数，通过概率分布完成复杂分析的方法 ［ 21 21 ］ 。因此，通过随机产生一系列平动位移范围为±2 mm，转动范围为±2°内均匀分布的位移和姿态角，可以模拟质量块在运动包络内的算法精度。 图7 图7 和 图8 图8 给出了10 000次蒙特卡洛随机仿真下，利用数值计算和小角度近似得到的位姿误差频率直方图， 表1 表1 给出了姿态角和位移解算的平均误差。另外，对于每一个随机产生的姿态角和位移点，数值计算得到的姿态角误差均小于10 - 14 °，位移误差均小于0.02 fm。 Pose Numerical analysis Small-angle approximation <math id="M171"><mrow><mrow><mi mathvariant="normal">Δ</mi><msub><mrow><mi>θ</mi></mrow><mrow><mi>x</mi></mrow></msub></mrow><mo>/</mo><mrow><mo stretchy="false">(</mo><mo>°</mo><mo stretchy="false">)</mo></mrow></mrow></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412709&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412693&type= 9.48266602 3.47133350 2.0×10 -17 3.4×10 -6 <math id="M172"><mrow><mrow><mi mathvariant="normal">Δ</mi><msub><mrow><mi>θ</mi></mrow><mrow><mi>y</mi></mrow></msub></mrow><mo>/</mo><mrow><mo stretchy="false">(</mo><mo>°</mo><mo stretchy="false">)</mo></mrow></mrow></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412698&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412696&type= 9.56733322 3.89466691 3.8×10 -17 -6.2×10 -7 <math id="M173"><mrow><mrow><mi mathvariant="normal">Δ</mi><msub><mrow><mi>θ</mi></mrow><mrow><mi>z</mi></mrow></msub></mrow><mo>/</mo><mrow><mo stretchy="false">(</mo><mo>°</mo><mo stretchy="false">)</mo></mrow></mrow></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412719&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412700&type= 9.31333351 3.47133350 5.3×10 -18 -1.2×10 -7 <math id="M174"><mrow><mrow><mi mathvariant="normal">Δ</mi><msub><mrow><mi>t</mi></mrow><mrow><mi>x</mi></mrow></msub></mrow><mo>/</mo><mrow><mi mathvariant="normal">n</mi><mi mathvariant="normal">m</mi></mrow></mrow></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412725&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412704&type= 9.39799976 3.21733332 -1.1×10 -11 10.5 <math id="M175"><mrow><mrow><mi mathvariant="normal">Δ</mi><msub><mrow><mi>t</mi></mrow><mrow><mi>y</mi></mrow></msub></mrow><mo>/</mo><mrow><mi mathvariant="normal">n</mi><mi mathvariant="normal">m</mi></mrow></mrow></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412739&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412727&type= 9.56733322 3.64066648 1.2×10 -11 16.3 <math id="M176"><mrow><mrow><mi mathvariant="normal">Δ</mi><msub><mrow><mi>t</mi></mrow><mrow><mi>z</mi></mrow></msub></mrow><mo>/</mo><mrow><mi mathvariant="normal">n</mi><mi mathvariant="normal">m</mi></mrow></mrow></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412745&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412742&type= 9.22866726 3.21733332 -4.0×10 -12 10.3 仿真数据表明，数值计算方法的最大误差远小于系统测量的纳米级精度要求，此时算法的解算精度主要取决于计算机进行浮点运算时的误差。而利用小角度近似得到姿态角误差在10 - 3 °量级，位移误差在亚微米级别，位移误差的最大值约为0.6 μm且平均误差达到了16 nm，无法确保纳米级的测量精度。因此，小角度近似求解的算法虽然简单，但精度远没有利用迭代求解的方法高。利用迭代法可以人为设置误差判据来控制迭代次数，提高算法的计算精度。 4.4　实际系统的误差分析及仿真 2 上述仿真结果表明，基于数值计算的位姿解算算法对系统测量误差的影响较小。而实际系统的位姿测量误差受各种因素的影响，这些因素会进一步影响质量块位姿解算的精度。 由于三轴的位移表达形式相似，以沿 <math id="M177"><msub><mrow><mi>z</mi></mrow><mrow><mi>o</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413022&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413019&type= 2.96333337 3.72533321 轴的平动位移为例进行分析，将 式（19） 式（19） 展开可得： <math display="block" id="M178"><mtable columnalign="left"><mtr><mtd><msub><mrow><mi>t</mi></mrow><mrow><mi>z</mi></mrow></msub><mo>=</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><mi mathvariant="normal">O</mi><mi mathvariant="normal">P</mi><msub><mrow><mi mathvariant="normal">C</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn 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id="M179"><msub><mrow><mi>z</mi></mrow><mrow><mi>o</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413022&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413019&type= 2.96333337 3.72533321 轴平动位移的误差受系统参数 <math id="M180"><mi mathvariant="normal">O</mi><mi mathvariant="normal">P</mi><msub><mrow><mi mathvariant="normal">C</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><mi>L</mi><mo>,</mo><mi>H</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412777&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412763&type= 18.28800011 4.14866638 精度的影响。由系统的误差传递公式可以得到每个因素对测量结果的影响，根据系统参数及质量块的运动范围，计算偏导数平方的最大值，则位移误差的最大值可表示为： <math display="block" id="M181"><mi>δ</mi><msub><mrow><msub><mrow><mi>t</mi></mrow><mrow><mi>z</mi></mrow></msub></mrow><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">x</mi></mrow></msub><mo>=</mo><mroot><mrow><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">4</mn></mrow></mfrac><msubsup><mrow><mi>δ</mi></mrow><mrow><mi mathvariant="normal">O</mi><mi mathvariant="normal">P</mi><msub><mrow><mi mathvariant="normal">C</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo>+</mo><msubsup><mrow><mfenced separators="|"><mrow><mfrac><mrow><mo>∂</mo><msub><mrow><mi>t</mi></mrow><mrow><mi>z</mi></mrow></msub></mrow><mrow><mo>∂</mo><mi>L</mi></mrow></mfrac></mrow></mfenced></mrow><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><msubsup><mrow><mi>δ</mi></mrow><mrow><mi>L</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo>+</mo><msubsup><mrow><mfenced separators="|"><mrow><mfrac><mrow><mo>∂</mo><msub><mrow><mi>t</mi></mrow><mrow><mi>z</mi></mrow></msub></mrow><mrow><mo>∂</mo><mi>H</mi></mrow></mfrac></mrow></mfenced></mrow><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><msubsup><mrow><mi>δ</mi></mrow><mrow><mi>H</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup></mrow><mrow/></mroot></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412789&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412785&type= 68.24134064 11.76866722 ， 其中： <math id="M182"><msubsup><mrow><mfenced separators="|"><mrow><mrow><mrow><mo>∂</mo><msub><mrow><mi>t</mi></mrow><mrow><mi>z</mi></mrow></msub></mrow><mo>/</mo><mrow><mo>∂</mo><mi>L</mi></mrow></mrow></mrow></mfenced></mrow><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo>=</mo><mn mathvariant="normal">3.04</mn><mo>×</mo><msup><mrow><mn mathvariant="normal">10</mn></mrow><mrow><mo>-</mo><mn mathvariant="normal">4</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412794&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412791&type= 40.21666718 6.18066692 ， <math id="M183"><msubsup><mrow><mfenced separators="|"><mrow><mrow><mrow><mo>∂</mo><msub><mrow><mi>t</mi></mrow><mrow><mi>z</mi></mrow></msub></mrow><mo>/</mo><mrow><mo>∂</mo><mi>H</mi></mrow></mrow></mrow></mfenced></mrow><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo>=</mo><mn mathvariant="normal">3.71</mn><mo>×</mo><msup><mrow><mn mathvariant="normal">10</mn></mrow><mrow><mo>-</mo><mn mathvariant="normal">7</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412801&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412782&type= 40.80933380 6.18066692 。质量块在加工过程中能够保证共面角锥棱镜中心距误差 <math id="M184"><mi>δ</mi><mi>L</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412815&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412813&type= 4.31799984 2.79399991 小于0.1 μm，边长误差 <math id="M185"><mi>δ</mi><mi>H</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412820&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412807&type= 5.16466665 2.79399991 小于1 μm。 激光波长作为外差干涉测量系统的测量基准，其频率不稳定会直接影响系统的测量精度 ［ 22 22 ］ 。根据参考文献［ 23 23 ］，激光频率波动引起的光程变化的误差为： <math display="block" id="M186"><mi>δ</mi><mi mathvariant="normal">O</mi><mi mathvariant="normal">P</mi><msub><mrow><mi mathvariant="normal">C</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>=</mo><mfrac><mrow><mi>δ</mi><mi>f</mi></mrow><mrow><mi>f</mi></mrow></mfrac><mfenced separators="|"><mrow><msub><mrow><mi>L</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>-</mo><msub><mrow><mi>L</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412812&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412822&type= 34.54399872 8.80533314 ， 其中： <math id="M187"><msub><mrow><mi>L</mi></mrow><mrow><mi>i</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412843&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412828&type= 3.21733332 3.72533321 为第 <math id="M188"><mi>i</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412834&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412844&type= 1.10066664 2.79399991 个测量臂的路径长度， <math id="M189"><msub><mrow><mi>L</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412847&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412837&type= 3.64066648 3.72533321 为参考臂的路径长度， <math id="M190"><mi>δ</mi><mi>f</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412851&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412848&type= 3.72533321 3.64066648 为激光频率的波动。在实际的系统设计中，激光器波长为780.246 102 nm，频率稳定度为4.7×10 - 10 。初始状态时测量臂与参考臂的长度之差为20 cm，质量块最大可产生2 mm的平动和2°的转动，此时光程变化最大为5.361 6 mm。因此，由激光频率不稳定引入的测量光程变化误差 <math id="M191"><msubsup><mrow><mi>δ</mi></mrow><mrow><mi mathvariant="normal">O</mi><mi mathvariant="normal">P</mi><msub><mrow><mi mathvariant="normal">C</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412865&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412853&type= 6.94266701 3.80999994 最大为9.7×10 - 2 nm，位移测量误差最大为4.8×10 - 2 nm。 各误差分量对 <math id="M192"><msub><mrow><mi>z</mi></mrow><mrow><mi>o</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413022&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413019&type= 2.96333337 3.72533321 轴平动位移的最大影响如 表2 表2 所示。 Parameter <math id="M193"><mi>δ</mi><msub><mrow><msub><mrow><mi>t</mi></mrow><mrow><mi>z</mi></mrow></msub></mrow><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">x</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413046&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413043&type= 7.02733374 3.21733332 <math id="M194"><mi>f</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412882&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412890&type= 2.11666679 3.13266683 4.8×10 -2 <math id="M195"><mi>L</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412896&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412893&type= 2.11666679 2.37066650 1.7 <math id="M196"><mi>H</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412903&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412897&type= 2.87866688 2.37066650 0.6 nm 针对3组角锥棱镜在安装时不完全垂直的情况，以每组角锥棱镜所在轴线建立如 图9 图9 所示的实际坐标系 <math id="M197"><mi mathvariant="normal">Σ</mi><msub><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412911&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412909&type= 4.82600021 3.80999994 。该坐标系中， <math id="M198"><msub><mrow><mi>x</mi></mrow><mrow><msub><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412914&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412919&type= 3.72533321 3.55599999 轴与 <math id="M199"><msub><mrow><mi>x</mi></mrow><mrow><mi>f</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413080&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413090&type= 2.96333337 3.72533321 轴重合， <math id="M200"><msub><mrow><mi>α</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>β</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>ψ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>χ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412955&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412953&type= 18.20333290 4.57200003 表示 <math id="M201"><mi mathvariant="normal">Σ</mi><msub><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413004&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413014&type= 4.82600021 3.80999994 坐标系各轴与质量块参考坐标系 <math id="M202"><mi mathvariant="normal">Σ</mi><mi>f</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413127&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413125&type= 4.14866638 3.64066648 各轴之间的夹角，角锥棱镜的安装垂直度反映在坐标系的夹角中。 <math id="M203"><mi mathvariant="normal">Σ</mi><mi>f</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413127&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413125&type= 4.14866638 3.64066648 坐标系相对于 <math id="M204"><mi mathvariant="normal">Σ</mi><msub><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413004&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413014&type= 4.82600021 3.80999994 坐标系的转换关系如下： <math display="block" id="M205"><mmultiscripts><mrow/><mprescripts/><mrow><mi>f</mi></mrow><mrow><msub><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></mmultiscripts><mi mathvariant="bold-italic">R</mi><mo>=</mo><mfenced separators="|"><mrow><mtable><mtr><mtd><mn mathvariant="normal">1</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd></mtr><mtr><mtd><mi mathvariant="normal">c</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">s</mi><msub><mrow><mi>α</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mtd><mtd><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><msub><mrow><mi>α</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mtd><mtd><mn mathvariant="normal">0</mn></mtd></mtr><mtr><mtd><mi mathvariant="normal">c</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">s</mi><msub><mrow><mi>β</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mtd><mtd><mi mathvariant="normal">c</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">s</mi><msub><mrow><mi>ψ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mtd><mtd><mi mathvariant="normal">c</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">s</mi><msub><mrow><mi>χ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412972&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412993&type= 46.22800064 14.39333248 . 则由角锥棱镜安装不垂直引起的光程变化误差为： <math display="block" id="M206"><mi>δ</mi><mi mathvariant="normal">O</mi><mi mathvariant="normal">P</mi><msub><mrow><mi mathvariant="normal">C</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>=</mo><mn mathvariant="normal">2</mn><mmultiscripts><mrow><mover accent="true"><mi mathvariant="bold-italic">ρ</mi><mo>^</mo></mover></mrow><mprescripts/><mrow/><mrow><mi>o</mi></mrow></mmultiscripts><msup><mrow><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">ρ</mi><mo>^</mo></mover></mrow><mrow><mi>i</mi></mrow></msub></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup><mfenced separators="|"><mrow><mmultiscripts><mrow/><mprescripts/><mrow><mi>f</mi></mrow><mrow><mi>o</mi></mrow></mmultiscripts><mi mathvariant="bold-italic">R</mi><mo>-</mo><msub><mrow><mi mathvariant="bold-italic">I</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub></mrow></mfenced><mfenced separators="|"><mrow><mmultiscripts><mrow/><mprescripts/><mrow><msub><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow><mrow><mi>f</mi></mrow></mmultiscripts><mi mathvariant="bold-italic">R</mi><mo>-</mo><msub><mrow><mi mathvariant="bold-italic">I</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub></mrow></mfenced><mmultiscripts><mrow><mi mathvariant="bold-italic">t</mi></mrow><mprescripts/><mrow><msub><mrow><mi>r</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow><mrow><mi>f</mi></mrow></mmultiscripts></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412977&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412974&type= 59.52066422 4.91066647 . 同样，以沿 <math id="M207"><msub><mrow><mi>z</mi></mrow><mrow><mi>o</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413022&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413019&type= 2.96333337 3.72533321 轴的平动位移为例分析，若 <math id="M208"><mi>δ</mi><msub><mrow><mi>α</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><mi>δ</mi><msub><mrow><mi>β</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><mi>δ</mi><msub><mrow><mi>ψ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><mi>δ</mi><msub><mrow><mi>χ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413012&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413002&type= 25.82333374 4.57200003 表示 <math id="M209"><mi mathvariant="normal">Σ</mi><msub><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413004&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413014&type= 4.82600021 3.80999994 坐标系各轴与 <math id="M210"><mi mathvariant="normal">Σ</mi><mi>f</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413127&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413125&type= 4.14866638 3.64066648 坐标系间的夹角误差， 表3 表3 给出了不同夹角误差下光程变化误差和位移误差的最大值。由 表3 表3 可知，第二组角锥棱镜（CCR3和CCR4）所在轴线与第一组角锥棱镜（CCR1和CCR2）所在轴线的垂直度对沿 <math id="M211"><msub><mrow><mi>z</mi></mrow><mrow><mi>o</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413022&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413019&type= 2.96333337 3.72533321 轴平动位移误差的影响较大。 Parameter <math id="M212"><mi>δ</mi><mi mathvariant="normal">O</mi><mi mathvariant="normal">P</mi><msub><mrow><mi mathvariant="normal">C</mi></mrow><mrow><mn>1</mn><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">x</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413040&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413025&type= 12.36133385 3.21733332 <math id="M213"><mi>δ</mi><msub><mrow><msub><mrow><mi>t</mi></mrow><mrow><mi>z</mi></mrow></msub></mrow><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">x</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413046&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413043&type= 7.02733374 3.21733332 <math id="M214"><mi>δ</mi><msub><mrow><mi>α</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>=</mo><mn>1</mn><mi mathvariant="normal">'</mi><mi mathvariant="normal">'</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413050&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413047&type= 11.34533310 3.55599999 5.9 3.0 <math id="M215"><mi>δ</mi><msub><mrow><mi>β</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>=</mo><mn>1</mn><mi mathvariant="normal">'</mi><mi mathvariant="normal">'</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413055&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413053&type= 11.34533310 4.31799984 0.2 0.1 <math id="M216"><mi>δ</mi><msub><mrow><mi>ψ</mi></mrow><mrow><msub><mrow/><mrow><mn>1</mn></mrow></msub></mrow></msub><mo>=</mo><mn>1</mn><mi mathvariant="normal">'</mi><mi mathvariant="normal">'</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413034&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413032&type= 11.26066685 3.47133350 7.2×10 -11 3.6×10 -11 <math id="M217"><mi>δ</mi><msub><mrow><mi>χ</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>=</mo><mn>1</mn><mi mathvariant="normal">'</mi><mi mathvariant="normal">'</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413068&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413036&type= 11.17599964 4.31799984 1.3×10 -6 6.4×10 -7 nm 假设质量块加工过程中存在立方体表面不完全垂直的情况，会引入系统误差。同理，以实际质量块的边建立质量块实际坐标系 <math id="M218"><mi mathvariant="normal">Σ</mi><msub><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413148&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413143&type= 4.82600021 3.80999994 ，使 <math id="M219"><msub><mrow><mi>x</mi></mrow><mrow><msub><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413089&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413073&type= 3.72533321 3.55599999 轴与 <math id="M220"><msub><mrow><mi>x</mi></mrow><mrow><mi>f</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413080&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413090&type= 2.96333337 3.72533321 轴重合，用 <math id="M221"><msub><mrow><mi>α</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>,</mo><msub><mrow><mi>β</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>,</mo><msub><mrow><mi>ψ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>,</mo><msub><mrow><mi>χ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413114&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413095&type= 18.20333290 4.57200003 表示 <math id="M222"><mi mathvariant="normal">Σ</mi><msub><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413148&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413143&type= 4.82600021 3.80999994 坐标系各轴与质量块参考坐标系 <math id="M223"><mi mathvariant="normal">Σ</mi><mi>f</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413127&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413125&type= 4.14866638 3.64066648 各轴之间的夹角， <math id="M224"><mi mathvariant="normal">Σ</mi><mi>f</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413127&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413125&type= 4.14866638 3.64066648 坐标系相对于 <math id="M225"><mi mathvariant="normal">Σ</mi><msub><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413148&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413143&type= 4.82600021 3.80999994 坐标系的旋转矩阵为 <math id="M226"><mmultiscripts><mrow/><mprescripts/><mrow><mi>f</mi></mrow><mrow><msub><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mrow></mmultiscripts><mi mathvariant="bold-italic">R</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413154&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413134&type= 4.91066647 3.97933316 ，定义同 式（23） 式（23） ，则由质量块安装不垂直引起的位移误差为： <math display="block" id="M227"><mi>δ</mi><mmultiscripts><mrow><mi mathvariant="bold-italic">t</mi></mrow><mprescripts/><mrow><mi>f</mi></mrow><mrow><mi>o</mi></mrow></mmultiscripts><mo>=</mo><mfenced separators="|"><mrow><mmultiscripts><mrow/><mprescripts/><mrow><mi>f</mi></mrow><mrow><msub><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mrow></mmultiscripts><mi mathvariant="bold-italic">R</mi><mo>-</mo><msub><mrow><mi mathvariant="bold-italic">I</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub></mrow></mfenced><mmultiscripts><mrow><mi mathvariant="bold-italic">t</mi></mrow><mprescripts/><mrow><mi>f</mi></mrow><mrow><mi>o</mi></mrow></mmultiscripts></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413159&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413139&type= 29.12533188 4.91066647 . 若 <math id="M228"><mi>δ</mi><msub><mrow><mi>α</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>=</mo><mn mathvariant="normal">1</mn><mi mathvariant="normal">'</mi><mi mathvariant="normal">'</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413163&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413160&type= 13.20800018 4.23333359 ，则由质量块的边 <math id="M229"><msub><mrow><mi>B</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><msub><mrow><mi>A</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413170&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413165&type= 8.29733276 3.72533321 不垂直于边 <math id="M230"><msub><mrow><mi>D</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><msub><mrow><mi>A</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413187&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413172&type= 8.63599968 3.72533321 所引起的位移误差最大约为9.7 nm。 角锥棱镜在轴线方向上不严格正交、共面的角锥棱镜不是中心对称等安装误差都可以反映在角锥棱镜在共面安装的过程中存在一定的角度误差和距离误差，或角锥棱镜底面与质量块的表面并没有完全重合而存在的非共面误差，这3种误差可以通过标定得到。而共面角锥棱镜的棱安装时不完全平行并不影响光线在其内部传播的光程，对系统的解算精度没有影响。为了方便分析，假设CCR1存在安装误差，其他角锥棱镜无安装误差时，角锥棱镜的安装误差对系统解算精度的影响，如 图10 图10 所示。 CCR1的共面安装误差分为角度误差和距离误差。其中，角度误差表示为角锥棱镜中心偏离质量块中心轴的角度，距离误差表示为角锥棱镜距离质量块表面中心的误差；非共面误差表示为角锥棱镜底面与质量块表面的距离。通过蒙特卡洛随机生成平动位移范围为±2 mm，转动范围为±2°的位姿数据，将误差的总体情况反映到不同安装误差下位姿误差的平均值。系统计算的姿态角误差与共面安装误差的关系如 图11 图11 （a）所示，位置误差与共面安装误差的关系如 图11 图11 （b）所示，位姿误差与非共面安装误差的关系如 图12 图12 所示。机械加工可实现的共面距离误差为100 μm，共面角度误差为10 mrad，非共面误差为2 mrad。在此加工精度下，由仿真数据可知系统的角度解算误差均值小于0.035°，位移误差均值小于0.14 μm。 1;2; 综上，系统位姿测量的误差主要有：计算机进行浮点计算时的计算误差，激光光源频率不稳定带来的误差，以及系统的加工和安装误差等。其中，系统安装误差是主要的误差来源。为保证系统位移测量误差小于10 nm，姿态角测量误差小于0.017°，根据 图11 图11 可知，角锥棱镜需要满足角度误差小于5 mrad、距离误差小于10 μm且平行度小于2 mrad的安装要求。后续可以通过传感器校正优化系统解算模型，以提高系统的精度。此外，光学元器件的非理想特性、材料热膨胀影响等因素也会影响系统测量。 5 结 论 1 本文提出了一种利用双频激光干涉进行惯性质量块六自由度解算的方法，该方法可以实现三自由度微重力的测量。通过利用搭载在天舟一号主动隔振系统上的石英挠性加速度计得到的质量块位姿进行算法仿真，以及运动范围内的蒙特卡洛随机仿真，验证了算法的准确性。仿真结果表明，基于数值计算的解算算法具有很高的精度，姿态角的绝对误差小于10 - 14 °，位移误差在10 - 8 nm数量级。本文分析了激光光源误差以及系统的加工和安装误差，并仿真了角锥棱镜安装误差对系统位姿测量精度的影响，给出了满足系统一种测量要求的安装误差约束范围。本文的六自由度测量及解算方法在精密加工和检测领域具有重要的应用前景。 参考文献 杨彪 ， 胡添元 . 空间站微重力环境研究与分析 ［J］. 载人航天 ， 2014 ， 20 （ 2 ）： 178 - 183 . doi: 10.3969/j.issn.1674-5825.2014.02.015 http://dx.doi.org/10.3969/j.issn.1674-5825.2014.02.015 空间站微重力环境研究与分析 YANG B ， HU T Y . 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