Kinematics model of articulated arm measuring machine

Lian-dong YU; Jia-ming CAO; Hui-ning ZHAO; Hua-kun JIA; Song PU

doi:10.37188/OPE.20212911.2603

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Kinematics model of articulated arm measuring machine

Micro/Nano Technology and Fine Mechanics | Updated：2022-01-25

- Kinematics model of articulated arm measuring machine
- Optics and Precision Engineering
- Affiliations：
  
  1.School of Instrument Science and Opto-electronics Engineering， Hefei University of Technology， Hefei 230009， China
  2.Anhui Province Key Laboratory of Measuring Theory and Precision Instrument， Hefei 230009， China
- Author bio：
- Funds：
- DOI：10.37188/OPE.20212911.2603 CLC： TP242.6
- Published：15 November 2021，
  
  Received：25 March 2021，
  
  Revised：07 May 2021，
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于连栋,曹家铭,赵会宁等.关节臂式坐标测量机的运动学建模[J].光学精密工程,2021,29(11):2603-2612.

YU Lian-dong,CAO Jia-ming,ZHAO Hui-ning,et al.Kinematics model of articulated arm measuring machine[J].Optics and Precision Engineering,2021,29(11):2603-2612.
于连栋,曹家铭,赵会宁等.关节臂式坐标测量机的运动学建模[J].光学精密工程,2021,29(11):2603-2612. DOI： 10.37188/OPE.20212911.2603.

YU Lian-dong,CAO Jia-ming,ZHAO Hui-ning,et al.Kinematics model of articulated arm measuring machine[J].Optics and Precision Engineering,2021,29(11):2603-2612. DOI： 10.37188/OPE.20212911.2603.

关节臂式坐标测量机的运动学建模 Kinematics model of articulated arm measuring machine 1 2 first-author 于连栋（1969-），男，山东郯城人，教授，博士生导师，1993年、1999年和2003年于合肥工业大学分别获得学士、硕士和博士学位，主要研究方向为坐标测量技术、现代测试精度理论研究和微纳米测量技术。E-mail：liandongyu@hfut.edu.cn http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24006615&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24006625&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24006620&type= 22.01336670 32.00398254 于连栋（1969-），男，山东郯城人，教授，博士生导师，1993年、1999年和2003年于合肥工业大学分别获得学士、硕士和博士学位，主要研究方向为坐标测量技术、现代测试精度理论研究和微纳米测量技术。E-mail： liandongyu@hfut.edu.cn liandongyu@hfut.edu.cn 1 2 1 2 corresp E-mail： hnzhao@mail.hfut.edu.cn E-mail： hnzhao@mail.hfut.edu.cn 赵会宁（1987-），男，河南兰考人，讲师，2011年于中原工学院获得学士学位，2014年、2018年于合肥工业大学分别获得硕士和博士学位，主要研究方向为坐标测量技术。E-mail：hnzhao@mail.hfut.edu.cn http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24006630&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24006639&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24006633&type= 22.01330566 32.00327301 赵会宁（1987-），男，河南兰考人，讲师，2011年于中原工学院获得学士学位，2014年、2018年于合肥工业大学分别获得硕士和博士学位，主要研究方向为坐标测量技术。E-mail： hnzhao@mail.hfut.edu.cn hnzhao@mail.hfut.edu.cn 1 2 1 2 合肥工业大学仪器科学与光电工程学院，安徽合肥 230009 合肥工业大学仪器科学与光电工程学院，安徽合肥 230009 School of Instrument Science and Opto-electronics Engineering， Hefei University of Technology， Hefei 230009， China School of Instrument Science and Opto-electronics Engineering， Hefei University of Technology， Hefei 230009， China 测量理论与精密仪器安徽省重点实验室，安徽合肥 230009 测量理论与精密仪器安徽省重点实验室，安徽合肥 230009 Anhui Province Key Laboratory of Measuring Theory and Precision Instrument， Hefei 230009， China Anhui Province Key Laboratory of Measuring Theory and Precision Instrument， Hefei 230009， China 关节臂式坐标测量机作为一种串联式结构的便携坐标测量仪器，其角度误差对其测量精度的影响呈放大效应。针对上述问题，在DH模型的基础上，提出了一种具有旋转轴倾斜误差运动补偿的关节臂式坐标测量机运动学建模方法，研究了旋转轴系误差运动分离方法并搭建相应的测试系统，建立了基于空间距离的结构参数误差标定模型并进行了实验。实验结果表明，相比于DH模型，采用具有旋转轴倾斜误差运动补偿的运动学模型，关节臂式坐标测量机的测量标准差由0.055 mm降低至0.037 mm。该模型能够有效提高关节臂式坐标测量机的测量精度。 The articulated arm coordinate measuring machine （AACMM） is one of the portable non-orthogonal coordinate measuring instruments with a series structure and an angle error that has an amplifying effect on the measurement accuracy， making it difficult to improve the measurement accuracy. To solve the above problems， the AACMM kinematic modeling method with a tilt motion error compensation for the rotation axis was proposed based on the DH model. The method for separating the tilt motion error for the rotating axis was studied， and the corresponding test system was built. The calibration model for the structural parameter errors was established based on the spatial distance， and the corresponding experiments were performed. The experimental results show that the measured standard deviation for the AACMM whose kinematic model included the function of compensating the tilt motion error for the rotating axis is reduced from 0.055 mm to 0.037 mm， compared to that for the DH model. The model can effectively promote the accuracy of AACMM. 精密测量关节臂式坐标测量机误差补偿倾斜误差运动运动学建模 precision measurement articulated arm coordinate measuring machine error compensation tilt error motion kinematic model 1 引　言 1 关节臂式坐标测量机是一种高精度的几何量测量仪器，相较于传统的正交式三坐标测量机，具有测量范围大、质量轻、便携性好等优点，广泛应用于汽车制造、航空航天、大型零件制造等领域［ 1 1 ］。关节臂式坐标测量机在结构上与工业机器人的相似，因此人们在机器人运动学建模方法的基础上，改进并建立其运动学模型。目前，DH模型被认为是最经典的机器人运动学建模理论［ 2 2 ］，但当相邻关节轴线相互平行时，传统的四参数DH模型无法准确地描述相邻关节之间的结构参数且存在病态性。Hayati等［ 3 3 ］提出了一种改进的四参数DH模型，以克服两个相邻关节轴线平行或接近平行时出现的病态性。在经典四参数DH模型的基础上，Veitschegger等［ 4 4 ］通过增加一项绕 <math id="M1"><mi>y</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24007383&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24007381&type= 1.86266661 3.72533321 轴结构参数，提出了一种改进的五参数DH模型。Zhuang等［ 5 5 ］以CPC（Complete and Parametrically Continuous）模型为基础，提出了一种直接从运动学模型中识别未知运动学参数的方法。He等［ 6 6 ］以POE（Product of Exponential）模型为基础，提出了一种参数校准模型；Liu等［ 7 7 ］在Local POE模型的基础上，提出了一种基于位置测量的结构参数标定模型，并以SCARA机器人和5DOF钻床为例开展了实验，通过仿真和实验验证了该校准模型的有效性。Zhao和Jiang等［ 8 8 - 9 9 8-9 ］分别采用广义几何误差模型减少非几何参数误差对关节臂式坐标测量机或者机器人精度的影响。Feng等［ 10 10 ］在利用模拟退火算法优化的BP神经网络基础上，提出了一种关节臂式坐标测量机长度误差修正模型。但上述关节臂式坐标测量机或者机器人的运动学模型中仅包含结构参数误差，未考虑旋转轴的误差运动对测量精度的影响。关节臂式坐标测量机因其串联式结构，旋转轴倾斜误差运动对其精度的影响呈放大效应。为进一步提高关节臂式坐标测量机的测量精度，本文研究了旋转轴倾斜误差运动分离方法并搭建测试系统，并在DH模型的基础上，提出了一种具有旋转轴倾斜误差运动补偿的关节臂式坐标测量机运动学建模方法，建立了基于空间距离的结构参数误差标定模型。最后，开展实验验证了该模型的有效性。 2 倾斜误差测试与分离 1 理想情况下，旋转轴只有一个绕 z 轴旋转的自由度，其他5个自由度均受到约束，即旋转轴运动不存在误差。但在实际情况中，由于旋转轴零部件的加工、装配、温度及使用磨损等误差的存在，旋转轴的运动情况与理想状态之间存在差异，有6项运动误差（径向误差 <math id="M2"><msub><mrow><mi>δ</mi></mrow><mrow><mi>x</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24006656&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24006654&type= 3.21733332 3.72533321 和 <math id="M3"><msub><mrow><mi>δ</mi></mrow><mrow><mi>y</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24006666&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24006661&type= 3.04800010 3.72533321 ，轴向误差 <math id="M4"><msub><mrow><mi>δ</mi></mrow><mrow><mi>z</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24006676&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24006671&type= 2.96333337 3.72533321 ，倾斜误差 <math id="M5"><msub><mrow><mi>ε</mi></mrow><mrow><mi>x</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24007130&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24007128&type= 2.70933342 3.72533321 和 <math id="M6"><msub><mrow><mi>ε</mi></mrow><mrow><mi>y</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24007133&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24007131&type= 2.53999996 3.72533321 和角定位误差 <math id="M7"><msub><mrow><mi>ε</mi></mrow><mrow><mi>z</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24006706&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24006701&type= 2.45533323 3.72533321 ），如图1 图1 所示。本文仅研究旋转轴倾斜误差对关节臂式坐标测量机测量精度的影响。 2.1　倾斜运动误差测试 2 对于旋转轴倾斜运动误差的测试，国家军用标准GJB1801-93《惯性技术测试设备主要性能试验方法》［ 11 11 ］中已明确规定旋转轴倾斜误差的测试方法，即水平仪法和自准直仪法。本文使用自准直仪配合平面反射镜对旋转轴的倾斜误差进行测量。旋转轴倾斜误差测试系统如图2 图2 所示，由自准直仪、平面反射镜、被测旋转轴（主轴及轴套）和支撑架组成。测试过程中，自准直仪、平面反射镜和主轴需保持共线，测试主轴与轴套之间因加工等误差而产生的倾斜误差。主轴和辅助夹具通过螺纹进行配合。平面反射镜及其调整机构固定在辅助夹具上，控制调整机构使得平面反射镜的回光十字靶标尽可能地位于自准直仪的视场中心，并保证在主轴旋转一周的情况下，十字靶标均在自准直仪的视场中。为保证数据处理效果及绘制曲线的准确性及重复性，需按照固定转动角度旋转被测旋转轴，并记录圆光栅传感器的读数 <math id="M8"><msub><mrow><mi>θ</mi></mrow><mrow><mi>i</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24006747&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24006742&type= 2.53999996 3.72533321 以及自准直仪在 <math id="M9"><mi>x</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24007380&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24007377&type= 2.03200006 2.79399991 和 <math id="M10"><mi>y</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24007383&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24007381&type= 1.86266661 3.72533321 方向的读数。 2.2　倾斜误差分离方法 2 通常情况下，自准直仪、平面反射镜和旋转轴相互之间存在安装误差，需要进行剔除。将自准直仪的二维测量数据（ <math id="M11"><mi>x</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24006775&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24006770&type= 2.03200006 2.79399991 和 <math id="M12"><mi>y</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24006781&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24006777&type= 1.86266661 3.72533321 两个方向）中的一阶项进一步分解为旋转轴运动误差产生的一阶项和平面反射镜安装误差所造成的一阶项两个部分，从而更准确地评估旋转轴的倾斜误差［ 12 12 ］。基坐标系 <math id="M13"><msub><mrow><mi>o</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><msub><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><msub><mrow><mi>z</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24006792&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24006788&type= 13.63133430 4.57200003 固定在自准直仪上， <math id="M14"><msub><mrow><mi>o</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><msub><mrow><mi>z</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24006800&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24006796&type= 6.34999990 3.72533321 方向与自准直仪的轴向方向重合，坐标系符合右手定则，轴套坐标系 <math id="M15"><msub><mrow><mi>o</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><msub><mrow><mi>z</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24006808&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24006804&type= 6.34999990 3.72533321 与 <math id="M16"><msub><mrow><mi>o</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><msub><mrow><mi>z</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24006816&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24006812&type= 6.34999990 3.72533321 的夹角为 <math id="M17"><mi mathvariant="normal">Δ</mi><mi>β</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24006824&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24006820&type= 4.40266657 3.64066648 ，即自准直仪安装误差。主轴坐标系 <math id="M18"><msub><mrow><mi>o</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><msub><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><msub><mrow><mi>z</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24006831&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24006827&type= 13.63133430 4.57200003 是在轴套坐标系 <math id="M19"><msub><mrow><mi>o</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><msub><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><msub><mrow><mi>z</mi></mrow><mrow><mn 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id="M22"><msub><mrow><mi>o</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><msub><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24006867&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24006863&type= 6.60400009 4.57200003 旋转 <math id="M23"><msub><mrow><mi>ε</mi></mrow><mrow><mi>y</mi></mrow></msub><mo stretchy="false">(</mo><mi>θ</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24006874&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24006871&type= 8.12800026 4.48733330 ，最后绕 <math id="M24"><msub><mrow><mi>o</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><msub><mrow><mi>z</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24006883&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24006878&type= 6.34999990 3.72533321 轴转动 <math id="M25"><mi>θ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24006891&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24006888&type= 1.77800000 2.79399991 。平面反射镜坐标系 <math id="M26"><msub><mrow><mi>o</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><msub><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><msub><mrow><mi>z</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24006896&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24006894&type= 13.63133430 4.57200003 是在主轴坐标系 <math id="M27"><msub><mrow><mi>o</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><msub><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><msub><mrow><mi>z</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24006901&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24006898&type= 13.63133430 4.57200003 的基础上，绕 <math id="M28"><msub><mrow><mi>o</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24006909&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24006905&type= 6.77333355 3.72533321 旋转 <math id="M29"><mi mathvariant="normal">Δ</mi><msub><mrow><mi>ϕ</mi></mrow><mrow><mi>x</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24006913&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24006910&type= 5.92666674 4.57200003 ，再绕 <math id="M30"><msub><mrow><mi>o</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><msub><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24006921&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24006918&type= 6.60400009 4.57200003 旋转 <math id="M31"><mi mathvariant="normal">Δ</mi><msub><mrow><mi>ϕ</mi></mrow><mrow><mi>y</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24006926&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24006924&type= 5.84200001 4.57200003 。根据坐标转换公式可知： <math display="block" id="M32"><msubsup><mrow><mi mathvariant="bold-italic">T</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msubsup><mo>=</mo><msubsup><mrow><mi mathvariant="bold-italic">T</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msubsup><msubsup><mrow><mi mathvariant="bold-italic">T</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msubsup><msubsup><mrow><mi mathvariant="bold-italic">T</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo>=</mo><mfenced open="[" close="]" separators="|"><mrow><mtable><mtr><mtd><msub><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">11</mn></mrow></msub></mtd><mtd><msub><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">12</mn></mrow></msub></mtd><mtd><msub><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">13</mn></mrow></msub></mtd></mtr><mtr><mtd><msub><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">21</mn></mrow></msub></mtd><mtd><msub><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">22</mn></mrow></msub></mtd><mtd><msub><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">23</mn></mrow></msub></mtd></mtr><mtr><mtd><msub><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">31</mn></mrow></msub></mtd><mtd><msub><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">32</mn></mrow></msub></mtd><mtd><msub><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">33</mn></mrow></msub></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24006931&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24006929&type= 49.19132996 13.54666710 ，式中： <math id="M33"><msubsup><mrow><mi mathvariant="bold-italic">T</mi></mrow><mrow><mi>i</mi></mrow><mrow><mi>j</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24006938&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24006935&type= 3.80999994 3.97933316 表示坐标系 <math id="M34"><mi>i</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24006944&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24006941&type= 1.10066664 2.79399991 相对于坐标系 <math id="M35"><mi>j</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24006951&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24006948&type= 1.10066664 3.64066648 的坐标转换矩阵， <math id="M36"><msubsup><mrow><mi mathvariant="bold-italic">T</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24006959&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24006955&type= 4.23333359 3.97933316 = <math id="M37"><mi mathvariant="bold-italic">R</mi><mo stretchy="false">(</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>,</mo><mi mathvariant="normal">Δ</mi><msub><mrow><mi>β</mi></mrow><mrow><mi>x</mi></mrow></msub><mo stretchy="false">)</mo><mi mathvariant="bold-italic">R</mi><mo stretchy="false">(</mo><msub><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>,</mo><mi mathvariant="normal">Δ</mi><msub><mrow><mi>β</mi></mrow><mrow><mi>y</mi></mrow></msub><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24006964&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24006961&type= 34.96733475 4.91066647 ， <math id="M38"><msubsup><mrow><mi mathvariant="bold-italic">T</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24006970&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24006967&type= 4.23333359 3.97933316 = <math id="M39"><mi mathvariant="bold-italic">R</mi><mo stretchy="false">(</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>ε</mi></mrow><mrow><mi>x</mi></mrow></msub><mo stretchy="false">(</mo><mi>θ</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mi mathvariant="bold-italic">R</mi><mo stretchy="false">(</mo><msub><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>ε</mi></mrow><mrow><mi>y</mi></mrow></msub><mo stretchy="false">(</mo><mi>θ</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mi mathvariant="bold-italic">R</mi><mo stretchy="false">(</mo><msub><mrow><mi>z</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><mi>θ</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24006977&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24006973&type= 54.10200119 4.91066647 ， <math id="M40"><msubsup><mrow><mi mathvariant="bold-italic">T</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo>=</mo><mi mathvariant="bold-italic">R</mi><mo stretchy="false">(</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>,</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24006983&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24006980&type= 19.04999924 4.65666676 <math id="M41"><mi mathvariant="normal">Δ</mi><msub><mrow><mi>ϕ</mi></mrow><mrow><mi>x</mi></mrow></msub><mo stretchy="false">)</mo><mi mathvariant="bold-italic">R</mi><mo stretchy="false">(</mo><msub><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>,</mo><mi mathvariant="normal">Δ</mi><msub><mrow><mi>ϕ</mi></mrow><mrow><mi>y</mi></mrow></msub><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24006989&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24006986&type= 25.48466682 4.91066647 。 <math id="M42"><msub><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">13</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24006995&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24006992&type= 4.06400013 3.72533321 和 <math id="M43"><msub><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">23</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24007001&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24006998&type= 4.06400013 3.72533321 为自准直仪在 <math id="M44"><mi>x</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24007380&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24007377&type= 2.03200006 2.79399991 和 <math id="M45"><mi>y</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24007383&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24007381&type= 1.86266661 3.72533321 轴方向上的读数。如果采取等间隔 <math id="M46"><mi>n</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24007010&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24007009&type= 1.86266661 2.79399991 点进行采样，则令 <math id="M47"><msub><mrow><mi>θ</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>=</mo><mrow><mrow><mn mathvariant="normal">2</mn><mi mathvariant="normal">π</mi><mi>i</mi></mrow><mo>/</mo><mrow><mi>n</mi></mrow></mrow></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24007012&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24007011&type= 15.32466698 3.72533321 。将式（1）式（1）展开，可得到： <math display="block" id="M48"><mfenced open="{" close="" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><msub><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">13</mn></mrow></msub><mo>=</mo><mi mathvariant="normal">Δ</mi><msub><mrow><mi>ϕ</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><mtext> </mtext><msub><mrow><mi>θ</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>+</mo><mi mathvariant="normal">Δ</mi><msub><mrow><mi>ϕ</mi></mrow><mrow><mi>x</mi></mrow></msub><mo>⋅</mo><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mtext> </mtext><msub><mrow><mi>θ</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>+</mo><mi mathvariant="normal">Δ</mi><msub><mrow><mi>β</mi></mrow><mrow><mi>y</mi></mrow></msub><mo>+</mo><msub><mrow><mi>ε</mi></mrow><mrow><mi>y</mi></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi>θ</mi></mrow><mrow><mi>i</mi></mrow></msub><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><msub><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">23</mn></mrow></msub><mo>=</mo><mi mathvariant="normal">Δ</mi><msub><mrow><mi>ϕ</mi></mrow><mrow><mi>y</mi></mrow></msub><mo>⋅</mo><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mtext> </mtext><msub><mrow><mi>θ</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>-</mo><mi mathvariant="normal">Δ</mi><msub><mrow><mi>ϕ</mi></mrow><mrow><mi>x</mi></mrow></msub><mo>⋅</mo><mi mathvariant="normal">c</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">s</mi><mtext> </mtext><msub><mrow><mi>θ</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>-</mo><mi mathvariant="normal">Δ</mi><msub><mrow><mi>β</mi></mrow><mrow><mi>x</mi></mrow></msub><mo>-</mo><msub><mrow><mi>ε</mi></mrow><mrow><mi>x</mi></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi>θ</mi></mrow><mrow><mi>i</mi></mrow></msub><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24007014&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24007013&type= 71.71266174 10.58333302 ，式中： <math id="M49"><mi mathvariant="normal">Δ</mi><msub><mrow><mi>β</mi></mrow><mrow><mi>y</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24007016&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24007015&type= 5.58799982 4.57200003 和 <math id="M50"><mi mathvariant="normal">Δ</mi><msub><mrow><mi>β</mi></mrow><mrow><mi>x</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24007018&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24007017&type= 5.67266655 4.57200003 为自准直仪绕着 <math id="M51"><mi>x</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24007021&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24007020&type= 2.03200006 2.79399991 和 <math id="M52"><mi>y</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24007026&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24007023&type= 1.86266661 3.72533321 轴的安装误差， <math id="M53"><mo stretchy="false">(</mo><mi mathvariant="normal">Δ</mi><msub><mrow><mi>ϕ</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><mtext> </mtext><msub><mrow><mi>θ</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>+</mo><mi mathvariant="normal">Δ</mi><msub><mrow><mi>ϕ</mi></mrow><mrow><mi>x</mi></mrow></msub><mo>⋅</mo><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mtext> </mtext><msub><mrow><mi>θ</mi></mrow><mrow><mi>i</mi></mrow></msub><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24007029&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24007027&type= 39.87799835 4.82600021 和 <math id="M54"><mo stretchy="false">(</mo><mi mathvariant="normal">Δ</mi><msub><mrow><mi>ϕ</mi></mrow><mrow><mi>y</mi></mrow></msub><mo>⋅</mo><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mtext> </mtext><msub><mrow><mi>θ</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>-</mo><mi mathvariant="normal">Δ</mi><msub><mrow><mi>ϕ</mi></mrow><mrow><mi>x</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>i</mi></mrow></msub><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24007033&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24007031&type= 39.70866776 4.82600021 为一阶项。因一阶项由部分旋转轴倾斜误差及平面反射镜安装误差共同组成，不能将一阶项全部剔除，需要将一阶项中的平面反射镜安装误差剔除，保留一阶项中的旋转轴倾斜误差。因此，将消除平面反射镜及自准直仪安装误差后获得的旋转轴倾斜误差分别记为 <math id="M55"><msub><mrow><mi>ε</mi></mrow><mrow><mi>x</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24007130&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24007128&type= 2.70933342 3.72533321 和 <math id="M56"><msub><mrow><mi>ε</mi></mrow><mrow><mi>y</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24007041&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24007038&type= 2.53999996 3.72533321 。 <math id="M57"><msub><mrow><mi>ε</mi></mrow><mrow><mi>x</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24007130&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24007128&type= 2.70933342 3.72533321 和 <math id="M58"><msub><mrow><mi>ε</mi></mrow><mrow><mi>y</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24007133&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24007131&type= 2.53999996 3.72533321 随旋转轴转动角度 <math id="M59"><msub><mrow><mi>θ</mi></mrow><mrow><mi>i</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24007087&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24007084&type= 2.53999996 3.72533321 的变化而变化，存在一一映射关系，可表示为： <math display="block" id="M60"><mfenced open="{" close="" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><msub><mrow><mi>ε</mi></mrow><mrow><mi>x</mi><mi>i</mi></mrow></msub><mo>=</mo><msub><mrow><mi>f</mi></mrow><mrow><mi>x</mi></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi>θ</mi></mrow><mrow><mi>i</mi></mrow></msub><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><msub><mrow><mi>ε</mi></mrow><mrow><mi>y</mi><mi>i</mi></mrow></msub><mo>=</mo><msub><mrow><mi>f</mi></mrow><mrow><mi>y</mi></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi>θ</mi></mrow><mrow><mi>i</mi></mrow></msub><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24007057&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24007055&type= 18.62666702 10.58333302 . 将 <math id="M61"><msub><mrow><mi>ε</mi></mrow><mrow><mi>x</mi><mi>i</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24007061&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24007059&type= 3.21733332 3.72533321 和 <math id="M62"><msub><mrow><mi>ε</mi></mrow><mrow><mi>y</mi><mi>i</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24007065&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24007063&type= 3.13266683 3.72533321 进行傅里叶级数展开： <math display="block" id="M63"><mfenced open="{" close="" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><msub><mrow><mi>ε</mi></mrow><mrow><mi>x</mi><mi>i</mi></mrow></msub><mo>=</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>x</mi><mn mathvariant="normal">0</mn></mrow></msub><mo>+</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>n</mi></mrow></munderover></mstyle><mrow><mo stretchy="false">[</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>x</mi><mi>i</mi></mrow></msub><mo>⋅</mo><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mo stretchy="false">(</mo><mi>i</mi><mo>⋅</mo><msub><mrow><mi>θ</mi></mrow><mrow><mi>i</mi></mrow></msub><mo stretchy="false">)</mo><mo>+</mo><msub><mrow><mi>b</mi></mrow><mrow><mi>x</mi><mi>i</mi></mrow></msub><mo>⋅</mo><mi mathvariant="normal">c</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">s</mi><mo stretchy="false">(</mo><mi>i</mi><mo>⋅</mo><msub><mrow><mi>θ</mi></mrow><mrow><mi>i</mi></mrow></msub><mo stretchy="false">)</mo><mo stretchy="false">]</mo></mrow></mtd></mtr><mtr><mtd><msub><mrow><mi>ε</mi></mrow><mrow><mi>y</mi><mi>i</mi></mrow></msub><mo>=</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>y</mi><mn mathvariant="normal">0</mn></mrow></msub><mo>+</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>n</mi></mrow></munderover></mstyle><mrow><mo stretchy="false">[</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>y</mi><mi>i</mi></mrow></msub><mo>⋅</mo><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mo stretchy="false">(</mo><mi>i</mi><mo>⋅</mo><msub><mrow><mi>θ</mi></mrow><mrow><mi>i</mi></mrow></msub><mo stretchy="false">)</mo><mo>+</mo><msub><mrow><mi>b</mi></mrow><mrow><mi>y</mi><mi>i</mi></mrow></msub><mo>⋅</mo><mi mathvariant="normal">c</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">s</mi><mo stretchy="false">(</mo><mi>i</mi><mo>⋅</mo><msub><mrow><mi>θ</mi></mrow><mrow><mi>i</mi></mrow></msub><mo stretchy="false">)</mo><mo stretchy="false">]</mo></mrow></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24007069&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24007066&type= 72.22066498 19.64266586 ，式中 <math id="M64"><mi>n</mi><mo>=</mo><mn mathvariant="normal">7</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24007072&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24007070&type= 8.63599968 2.79399991 。 3 新型运动学建模方法 1 3.1　DH模型 2 DH模型［ 2 2 ］是由Denavit和Hartenber于1955年提出的，广泛应用于工业机器人或者关节臂式坐标测量机的运动学建模。DH模型通过齐次坐标转换矩阵来确定相邻杆件之间的相对位姿，如图3 图3 所示。 <math display="block" id="M65"><mtable columnalign="left"><mtr><mtd><msub><mrow><mi mathvariant="bold-italic">A</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>=</mo><mi mathvariant="bold-italic">R</mi><mo stretchy="false">(</mo><msub><mrow><mi>z</mi></mrow><mrow><mi>i</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><mi>θ</mi><mo stretchy="false">)</mo><mi mathvariant="bold-italic">T</mi><mo stretchy="false">(</mo><msub><mrow><mi>z</mi></mrow><mrow><mi>i</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>d</mi></mrow><mrow><mi>i</mi></mrow></msub><mo stretchy="false">)</mo><mi mathvariant="bold-italic">T</mi><mo stretchy="false">(</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>,</mo><msub><mrow><mi>l</mi></mrow><mrow><mi>i</mi></mrow></msub><mo stretchy="false">)</mo><mi mathvariant="bold-italic">R</mi><mo stretchy="false">(</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>,</mo><msub><mrow><mi>α</mi></mrow><mrow><mi>i</mi></mrow></msub><mo stretchy="false">)</mo><mo>=</mo></mtd></mtr><mtr><mtd><mfenced open="[" close="]" separators="|"><mrow><mtable><mtr><mtd><mi mathvariant="normal">c</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">s</mi><mtext> </mtext><msub><mrow><mi>θ</mi></mrow><mrow><mi>i</mi></mrow></msub></mtd><mtd><mo>-</mo><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mtext> </mtext><msub><mrow><mi>θ</mi></mrow><mrow><mi>i</mi></mrow></msub><mi mathvariant="normal">c</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">s</mi><mtext> </mtext><msub><mrow><mi>α</mi></mrow><mrow><mi>i</mi></mrow></msub></mtd><mtd><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mtext> </mtext><msub><mrow><mi>θ</mi></mrow><mrow><mi>i</mi></mrow></msub><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mtext> </mtext><msub><mrow><mi>α</mi></mrow><mrow><mi>i</mi></mrow></msub></mtd><mtd><msub><mrow><mi>l</mi></mrow><mrow><mi>i</mi></mrow></msub><mi mathvariant="normal">c</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">s</mi><mtext> </mtext><msub><mrow><mi>θ</mi></mrow><mrow><mi>i</mi></mrow></msub></mtd></mtr><mtr><mtd><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mtext> </mtext><msub><mrow><mi>θ</mi></mrow><mrow><mi>i</mi></mrow></msub></mtd><mtd><mi mathvariant="normal">c</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">s</mi><mtext> </mtext><msub><mrow><mi>θ</mi></mrow><mrow><mi>i</mi></mrow></msub><mi mathvariant="normal">c</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">s</mi><mtext> </mtext><msub><mrow><mi>α</mi></mrow><mrow><mi>i</mi></mrow></msub></mtd><mtd><mo>-</mo><mi mathvariant="normal">c</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">s</mi><mtext> </mtext><msub><mrow><mi>θ</mi></mrow><mrow><mi>i</mi></mrow></msub><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mtext> </mtext><msub><mrow><mi>α</mi></mrow><mrow><mi>i</mi></mrow></msub></mtd><mtd><msub><mrow><mi>l</mi></mrow><mrow><mi>i</mi></mrow></msub><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mtext> </mtext><msub><mrow><mi>θ</mi></mrow><mrow><mi>i</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><mtext> </mtext><msub><mrow><mi>α</mi></mrow><mrow><mi>i</mi></mrow></msub></mtd><mtd><mi mathvariant="normal">c</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">s</mi><mtext> </mtext><msub><mrow><mi>α</mi></mrow><mrow><mi>i</mi></mrow></msub></mtd><mtd><msub><mrow><mi>d</mi></mrow><mrow><mi>i</mi></mrow></msub></mtd></mtr><mtr><mtd><mn mathvariant="normal">0</mn></mtd><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><mo>,</mo></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24007082&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24007081&type= 76.19999695 24.55333328 式中： <math id="M66"><msub><mrow><mi>θ</mi></mrow><mrow><mi>i</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24007087&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24007084&type= 2.53999996 3.72533321 表示关节变量， <math id="M67"><msub><mrow><mi>l</mi></mrow><mrow><mi>i</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24007091&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24007089&type= 1.94733346 3.72533321 表示杆件长度， <math id="M68"><msub><mrow><mi>d</mi></mrow><mrow><mi>i</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24007094&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24007092&type= 2.96333337 3.72533321 表示偏置量， <math id="M69"><msub><mrow><mi>α</mi></mrow><mrow><mi>i</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24007098&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24007096&type= 2.70933342 3.72533321 表示扭转角。 3.2　旋转轴倾斜误差运动补偿方法 2 如图4 图4 所示，对各个旋转轴的倾斜误差进行补偿［ 13 13 - 17 17 13-17 ］，分别绕着 <math id="M70"><mi>x</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24007102&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24007100&type= 2.03200006 2.79399991 和 <math id="M71"><mi>y</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24007106&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24007104&type= 1.86266661 3.72533321 方向旋转 <math id="M72"><msub><mrow><mi>ε</mi></mrow><mrow><mi>x</mi><mi>i</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24007110&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24007108&type= 3.21733332 3.72533321 和 <math id="M73"><msub><mrow><mi>ε</mi></mrow><mrow><mi>y</mi><mi>i</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24007114&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24007112&type= 3.13266683 3.72533321 ，即可对旋转轴倾斜误差运动造成的影响进行补偿。建立坐标转换方程： <math display="block" id="M74"><mtable columnalign="left"><mtr><mtd><msub><mrow><mi mathvariant="bold-italic">D</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>=</mo><mi mathvariant="bold-italic">R</mi><mo stretchy="false">(</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>,</mo><mo>-</mo><msub><mrow><mi>ε</mi></mrow><mrow><mi>x</mi><mi>i</mi></mrow></msub><mo stretchy="false">)</mo><mi mathvariant="bold-italic">R</mi><mo stretchy="false">(</mo><msub><mrow><mi>y</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>,</mo><mo>-</mo><msub><mrow><mi>ε</mi></mrow><mrow><mi>y</mi><mi>i</mi></mrow></msub><mo stretchy="false">)</mo><mo>=</mo></mtd></mtr><mtr><mtd><mfenced open="[" close="]" separators="|"><mrow><mtable><mtr><mtd><mi mathvariant="normal">c</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">s</mi><mtext> </mtext><msub><mrow><mi>ε</mi></mrow><mrow><mi>y</mi><mi>i</mi></mrow></msub></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mo>-</mo><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mtext> </mtext><msub><mrow><mi>ε</mi></mrow><mrow><mi>y</mi><mi>i</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><mtext> </mtext><msub><mrow><mi>ε</mi></mrow><mrow><mi>x</mi><mi>i</mi></mrow></msub><mo>⋅</mo><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mtext> </mtext><msub><mrow><mi>ε</mi></mrow><mrow><mi>y</mi><mi>i</mi></mrow></msub></mtd><mtd><mi mathvariant="normal">c</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">s</mi><mtext> </mtext><msub><mrow><mi>ε</mi></mrow><mrow><mi>x</mi><mi>i</mi></mrow></msub></mtd><mtd><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mtext> </mtext><msub><mrow><mi>ε</mi></mrow><mrow><mi>x</mi><mi>i</mi></mrow></msub><mo>⋅</mo><mi mathvariant="normal">c</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">s</mi><mtext> </mtext><msub><mrow><mi>ε</mi></mrow><mrow><mi>y</mi><mi>i</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><mtext> </mtext><msub><mrow><mi>ε</mi></mrow><mrow><mi>y</mi><mi>i</mi></mrow></msub><mo>⋅</mo><mi mathvariant="normal">c</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">s</mi><mtext> </mtext><msub><mrow><mi>ε</mi></mrow><mrow><mi>x</mi><mi>i</mi></mrow></msub></mtd><mtd><mo>-</mo><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mtext> </mtext><msub><mrow><mi>ε</mi></mrow><mrow><mi>x</mi><mi>i</mi></mrow></msub></mtd><mtd><mi mathvariant="normal">c</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">s</mi><mtext> </mtext><msub><mrow><mi>ε</mi></mrow><mrow><mi>x</mi><mi>i</mi></mrow></msub><mi mathvariant="normal">c</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">s</mi><mtext> </mtext><msub><mrow><mi>ε</mi></mrow><mrow><mi>y</mi><mi>i</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">0</mn></mtd><mtd><mn mathvariant="normal">1</mn></mtd></mtr></mtable></mrow></mfenced></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24007125&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24007124&type= 66.29399872 24.97666740 . 因为 <math id="M75"><msub><mrow><mi>ε</mi></mrow><mrow><mi>x</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24007130&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24007128&type= 2.70933342 3.72533321 和 <math id="M76"><msub><mrow><mi>ε</mi></mrow><mrow><mi>y</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24007133&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24007131&type= 2.53999996 3.72533321 数值较小，对式（6）式（6）进行简化得到： <math display="block" id="M77"><msub><mrow><mi mathvariant="bold-italic">D</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>≈</mo><mfenced open="[" close="]" separators="|"><mrow><mtable><mtr><mtd><mn mathvariant="normal">1</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mo>-</mo><msub><mrow><mi>ε</mi></mrow><mrow><mi>y</mi><mi>i</mi></mrow></msub></mtd><mtd><mn mathvariant="normal">0</mn></mtd></mtr><mtr><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">1</mn></mtd><mtd><msub><mrow><mi>ε</mi></mrow><mrow><mi>x</mi><mi>i</mi></mrow></msub></mtd><mtd><mn mathvariant="normal">0</mn></mtd></mtr><mtr><mtd><msub><mrow><mi>ε</mi></mrow><mrow><mi>y</mi><mi>i</mi></mrow></msub></mtd><mtd><mo>-</mo><msub><mrow><mi>ε</mi></mrow><mrow><mi>x</mi><mi>i</mi></mrow></msub></mtd><mtd><mn mathvariant="normal">1</mn></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">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=24007139&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24007136&type= 40.80933380 18.54199982 . 3.3　具有旋转轴倾斜误差补偿的运动学模型 2 在DH模型的基础上，考虑到旋转轴倾斜误差对相邻两个坐标架之间转换关系的影响，则其转换关系为： <math display="block" id="M78"><msub><mrow><mi mathvariant="bold-italic">A</mi></mrow><mrow><mi mathvariant="normal">r</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">l</mi><mi mathvariant="normal">i</mi></mrow></msub><mo>=</mo><msub><mrow><mi mathvariant="bold-italic">A</mi></mrow><mrow><mi>i</mi></mrow></msub><msub><mrow><mi mathvariant="bold-italic">D</mi></mrow><mrow><mi>i</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24007145&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24007142&type= 18.96533203 3.72533321 ，式中： <math id="M79"><msub><mrow><mi mathvariant="bold-italic">A</mi></mrow><mrow><mi mathvariant="normal">r</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">l</mi><mi mathvariant="normal">i</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24007151&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24007148&type= 6.43466663 3.72533321 为旋转轴的实际坐标系， <math id="M80"><msub><mrow><mi mathvariant="bold-italic">A</mi></mrow><mrow><mi>i</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24007157&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24007154&type= 3.55599999 3.72533321 为旋转轴的理想坐标系。因此，自主研制的关节臂式坐标测量机的坐标转换关系如图5 图5 所示。建立具有旋转轴倾斜误差补偿的关节臂式坐标测量机的运动学模型： <math display="block" id="M81"><mi mathvariant="bold-italic">T</mi><mo>=</mo><mstyle displaystyle="true"><munderover><mo>∏</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">6</mn></mrow></munderover></mstyle><mrow><msub><mrow><mi mathvariant="bold-italic">A</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow><msub><mrow><mi mathvariant="bold-italic">D</mi></mrow><mrow><mi>i</mi></mrow></msub><mi mathvariant="bold-italic">T</mi><mo stretchy="false">(</mo><mn mathvariant="normal">0</mn><mo>,</mo><mn mathvariant="normal">0</mn><mo>,</mo><msub><mrow><mi>Z</mi></mrow><mrow><mi>P</mi></mrow></msub><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24007174&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24007171&type= 38.18466568 8.55133343 . 自主研制的关节臂式坐标测量机旋转轴圆光栅的物理零位和其建模零位不一致，存在零位误差，需要进一步修正，即有： <math display="block" id="M82"><msub><mrow><mi>Θ</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>=</mo><msub><mrow><mi>θ</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>+</mo><mi mathvariant="normal">Δ</mi><msub><mrow><mi>θ</mi></mrow><mrow><mi>i</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24007181&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24007177&type= 20.74333191 3.80999994 ，式中： <math id="M83"><mi mathvariant="normal">Δ</mi><msub><mrow><mi>θ</mi></mrow><mrow><mi>i</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24007187&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24007184&type= 4.99533367 3.72533321 为关节臂式坐标测量机旋转轴圆光栅的物理零位与其建模零位之间的差值。则式（4）式（4）和式（5）式（5）分别改写为： <math display="block" id="M84"><mfenced open="{" close="" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><msub><mrow><mi>ε</mi></mrow><mrow><mi>x</mi><mi>i</mi></mrow></msub><mo>=</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>x</mi><mn mathvariant="normal">0</mn></mrow></msub><mo>+</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>n</mi></mrow></munderover></mstyle><mrow><mo stretchy="false">[</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>x</mi><mi>i</mi></mrow></msub><mo>⋅</mo><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mo stretchy="false">(</mo><mi>i</mi><mo>⋅</mo><msub><mrow><mi>Θ</mi></mrow><mrow><mi>i</mi></mrow></msub><mo stretchy="false">)</mo><mo>+</mo><msub><mrow><mi>b</mi></mrow><mrow><mi>x</mi><mi>i</mi></mrow></msub><mo>⋅</mo><mi mathvariant="normal">c</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">s</mi><mo stretchy="false">(</mo><mi>i</mi><mo>⋅</mo><msub><mrow><mi>Θ</mi></mrow><mrow><mi>i</mi></mrow></msub><mo stretchy="false">)</mo><mo stretchy="false">]</mo></mrow></mtd></mtr><mtr><mtd><msub><mrow><mi>ε</mi></mrow><mrow><mi>y</mi><mi>i</mi></mrow></msub><mo>=</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>y</mi><mn mathvariant="normal">0</mn></mrow></msub><mo>+</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>n</mi></mrow></munderover></mstyle><mrow><mo stretchy="false">[</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>y</mi><mi>i</mi></mrow></msub><mo>⋅</mo><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mo stretchy="false">(</mo><mi>i</mi><mo>⋅</mo><msub><mrow><mi>Θ</mi></mrow><mrow><mi>i</mi></mrow></msub><mo stretchy="false">)</mo><mo>+</mo><msub><mrow><mi>b</mi></mrow><mrow><mi>y</mi><mi>i</mi></mrow></msub><mo>⋅</mo><mi mathvariant="normal">c</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">s</mi><mo stretchy="false">(</mo><mi>i</mi><mo>⋅</mo><msub><mrow><mi>Θ</mi></mrow><mrow><mi>i</mi></mrow></msub><mo stretchy="false">)</mo><mo stretchy="false">]</mo></mrow></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24007193&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24007190&type= 74.50666809 19.72733307 ， <math display="block" id="M85"><mtable><mtr><mtd><msub><mrow><mi mathvariant="bold-italic">Α</mi></mrow><mrow><mi>i</mi></mrow></msub><mtext> </mtext><mo>=</mo></mtd></mtr><mtr><mtd><mfenced open="[" close="]" separators="|"><mrow><mtable><mtr><mtd><mi mathvariant="normal">c</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">s</mi><mtext> </mtext><msub><mrow><mi>Θ</mi></mrow><mrow><mi>i</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>i</mi></mrow></msub><mi mathvariant="normal">c</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">s</mi><mtext> </mtext><msub><mrow><mi>α</mi></mrow><mrow><mi>i</mi></mrow></msub></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>i</mi></mrow></msub><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mtext> </mtext><msub><mrow><mi>α</mi></mrow><mrow><mi>i</mi></mrow></msub></mtd><mtd><msub><mrow><mi>l</mi></mrow><mrow><mi>i</mi></mrow></msub><mi mathvariant="normal">c</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">s</mi><mtext> </mtext><msub><mrow><mi>Θ</mi></mrow><mrow><mi>i</mi></mrow></msub></mtd></mtr><mtr><mtd><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mtext> </mtext><msub><mrow><mi>Θ</mi></mrow><mrow><mi>i</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>i</mi></mrow></msub><mi mathvariant="normal">c</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">s</mi><mtext> </mtext><msub><mrow><mi>α</mi></mrow><mrow><mi>i</mi></mrow></msub></mtd><mtd><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>i</mi></mrow></msub><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mtext> </mtext><msub><mrow><mi>α</mi></mrow><mrow><mi>i</mi></mrow></msub></mtd><mtd><msub><mrow><mi>l</mi></mrow><mrow><mi>i</mi></mrow></msub><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mtext> </mtext><msub><mrow><mi>Θ</mi></mrow><mrow><mi>i</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><mtext> </mtext><msub><mrow><mi>α</mi></mrow><mrow><mi>i</mi></mrow></msub></mtd><mtd><mi mathvariant="normal">c</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">s</mi><mtext> </mtext><msub><mrow><mi>α</mi></mrow><mrow><mi>i</mi></mrow></msub></mtd><mtd><msub><mrow><mi>d</mi></mrow><mrow><mi>i</mi></mrow></msub></mtd></mtr><mtr><mtd><mn mathvariant="normal">0</mn></mtd><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></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24007197&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24007194&type= 76.36933136 23.53733444 . 4 距离误差标定模型 1 因关节臂式坐标测量机的零部件加工、装配等误差的存在，其结构参数的实际值和名义值之间存在差异，故关节臂式坐标测量机的整机装配完成后，需对其参数误差进行精确标定［ 18 18 - 19 19 18-19 ］。为获得关节臂式坐标测量机精确的结构参数，以自制长度标准件作为距离误差标定的长度标准量，建立基于空间距离的参数误差标定模型。自制长度标准件如图6 图6 所示，由石英棒及两端的支撑座等零部件组成。石英棒的热膨胀系数较小，热稳定性较高，受温度变化的影响较小，被选作长度标准件主体。另外，支撑座上锥窝孔与关节臂式坐标测量机校准测头相匹配，采用自制长度标准件进行关节臂式坐标测量机的结构参数误差标定实验数据采集。假设两点 <math id="M86"><msub><mrow><mi>P</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24007217&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24007213&type= 3.64066648 3.72533321 和 <math id="M87"><msub><mrow><mi>P</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24007225&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24007221&type= 3.64066648 3.72533321 相对于关节臂式坐标测量机的基坐标系 <math id="M88"><mo stretchy="false">(</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>,</mo><msub><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>,</mo><msub><mrow><mi>z</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24007232&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24007229&type= 16.42533302 4.91066647 的空间坐标为 <math id="M89"><msub><mrow><mi>P</mi></mrow><mrow><mn mathvariant="normal">1</mn><mi>i</mi></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn><mi>i</mi></mrow></msub><mo>,</mo><msub><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">1</mn><mi>i</mi></mrow></msub><mo>,</mo><msub><mrow><mi>z</mi></mrow><mrow><mn mathvariant="normal">1</mn><mi>i</mi></mrow></msub><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24007240&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24007236&type= 23.02933502 4.91066647 ， <math id="M90"><msub><mrow><mi>P</mi></mrow><mrow><mn mathvariant="normal">2</mn><mi>i</mi></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn><mi>i</mi></mrow></msub><mo>,</mo><msub><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">2</mn><mi>i</mi></mrow></msub><mo>,</mo><msub><mrow><mi>z</mi></mrow><mrow><mn mathvariant="normal">2</mn><mi>i</mi></mrow></msub><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24007245&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24007242&type= 23.02933502 4.91066647 ，则两点之间的测量距离 <math id="M91"><msub><mrow><mi>d</mi></mrow><mrow><mi>i</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24007251&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24007248&type= 2.96333337 3.72533321 为： <math display="block" id="M92"><msub><mrow><mi>d</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>=</mo><mroot><mrow><mo stretchy="false">(</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn><mi>i</mi></mrow></msub><mo>-</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn><mi>i</mi></mrow></msub><msup><mrow><mo stretchy="false">)</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><mo stretchy="false">(</mo><msub><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">1</mn><mi>i</mi></mrow></msub><mo>-</mo><msub><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">2</mn><mi>i</mi></mrow></msub><msup><mrow><mo stretchy="false">)</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><mo stretchy="false">(</mo><msub><mrow><mi>z</mi></mrow><mrow><mn mathvariant="normal">1</mn><mi>i</mi></mrow></msub><mo>-</mo><msub><mrow><mi>z</mi></mrow><mrow><mn mathvariant="normal">2</mn><mi>i</mi></mrow></msub><msup><mrow><mo stretchy="false">)</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow><mrow/></mroot></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24007257&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24007254&type= 69.51132965 5.33400011 . 假设 <math id="M93"><msub><mrow><mi>P</mi></mrow><mrow><mn mathvariant="normal">1</mn><mi>i</mi></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn><mi mathvariant="normal">i</mi></mrow></msub><mo>,</mo><msub><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">1</mn><mi>i</mi></mrow></msub><mo>,</mo><msub><mrow><mi>z</mi></mrow><mrow><mn mathvariant="normal">1</mn><mi>i</mi></mrow></msub><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24007263&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24007259&type= 22.77533340 4.91066647 ， <math id="M94"><msub><mrow><mi>P</mi></mrow><mrow><mn mathvariant="normal">2</mn><mi>i</mi></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn><mi>i</mi></mrow></msub><mo>,</mo><msub><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">2</mn><mi>i</mi></mrow></msub><mo>,</mo><msub><mrow><mi>z</mi></mrow><mrow><mn mathvariant="normal">2</mn><mi mathvariant="normal">i</mi></mrow></msub><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24007269&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24007266&type= 22.77533340 4.91066647 之间的空间距离名义值为 <math id="M95"><mi>l</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24007275&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24007272&type= 1.18533325 2.79399991 ，可知测量距离和实际距离之间的差值 <math id="M96"><msub><mrow><mi>v</mi></mrow><mrow><mi>i</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24007279&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24007277&type= 2.62466669 3.72533321 为： <math display="block" id="M97"><msub><mrow><mi>v</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>=</mo><msub><mrow><mi>d</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>-</mo><mi>l</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24007288&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24007283&type= 16.34066582 3.72533321 . 根据最小二乘原理，目标函数为： <math display="block" id="M98"><mtable columnalign="left"><mtr><mtd><mi>M</mi><mi>i</mi><mi>n</mi><mtext> </mtext><mi>Q</mi><mo stretchy="false">(</mo><mi mathvariant="bold-italic">b</mi><mo stretchy="false">)</mo><mo>=</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>m</mi></mrow></munderover></mstyle><mrow><msubsup><mrow><mi>v</mi></mrow><mrow><mi>i</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo>=</mo></mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>m</mi></mrow></munderover></mstyle><mrow><mo stretchy="false">(</mo><msub><mrow><mi>d</mi></mrow><mrow><mi>i</mi></mrow></msub><msup><mrow><mo>-</mo><mi>l</mi><mo stretchy="false">)</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow><mo>=</mo></mtd></mtr><mtr><mtd><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>m</mi></mrow></munderover></mstyle><mrow><msup><mrow><mo stretchy="false">[</mo><mroot><mrow><mo stretchy="false">(</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn><mi>i</mi></mrow></msub><mo>-</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn><mi>i</mi></mrow></msub><msup><mrow><mo stretchy="false">)</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><mo stretchy="false">(</mo><msub><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">1</mn><mi>i</mi></mrow></msub><mo>-</mo><msub><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">2</mn><mi>i</mi></mrow></msub><msup><mrow><mo stretchy="false">)</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><mo stretchy="false">(</mo><msub><mrow><mi>z</mi></mrow><mrow><mn mathvariant="normal">1</mn><mi>i</mi></mrow></msub><mo>-</mo><msub><mrow><mi>z</mi></mrow><mrow><mn mathvariant="normal">2</mn><mi>i</mi></mrow></msub><msup><mrow><mo stretchy="false">)</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow><mrow/></mroot><mo>-</mo><mi>l</mi><mo stretchy="false">]</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24007295&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24007292&type= 73.74466705 18.88066673 . 式（9）式（9）中共包含25项待标定参数，除去冗余参数，将其中 <math id="M99"><mi mathvariant="normal">Δ</mi><msub><mrow><mi>θ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24007301&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24007298&type= 5.41866684 3.72533321 ~ <math id="M100"><mi mathvariant="normal">Δ</mi><msub><mrow><mi>θ</mi></mrow><mrow><mn mathvariant="normal">6</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24007308&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24007305&type= 5.41866684 3.72533321 ， <math id="M101"><mi mathvariant="normal">Δ</mi><msub><mrow><mi>α</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24007315&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24007311&type= 5.58799982 3.72533321 ~ <math id="M102"><mi mathvariant="normal">Δ</mi><msub><mrow><mi>α</mi></mrow><mrow><mn mathvariant="normal">5</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24007322&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24007319&type= 5.58799982 3.72533321 ， <math id="M103"><msub><mrow><mi>l</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24007327&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24007324&type= 2.37066650 3.72533321 ~ <math id="M104"><msub><mrow><mi>l</mi></mrow><mrow><mn mathvariant="normal">5</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24007333&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24007329&type= 2.37066650 3.72533321 ， <math id="M105"><msub><mrow><mi>d</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24007338&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24007336&type= 3.30200005 3.72533321 ~ <math id="M106"><msub><mrow><mi>d</mi></mrow><mrow><mn mathvariant="normal">5</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24007345&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24007341&type= 3.30200005 3.72533321 ， <math id="M107"><msub><mrow><mi>Z</mi></mrow><mrow><mi mathvariant="normal">p</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24007351&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24007348&type= 3.47133350 3.72533321 这21项线性无关的参数记为 <math id="M108"><mi mathvariant="bold-italic">b</mi><mo>=</mo><mfenced separators="|"><mrow><msub><mrow><mi>b</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>b</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>b</mi></mrow><mrow><mn mathvariant="normal">21</mn></mrow></msub></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24007357&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24007355&type= 29.63333130 4.82600021 ， <math id="M109"><mi mathvariant="bold-italic">b</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24007420&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24007418&type= 1.86266661 2.87866688 为参数向量，其初值为 <math id="M110"><msup><mrow><mi mathvariant="bold-italic">b</mi></mrow><mrow><mo stretchy="false">(</mo><mn mathvariant="normal">0</mn><mo stretchy="false">)</mo></mrow></msup><mo>=</mo><mfenced separators="|"><mrow><msubsup><mrow><mi>b</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mo stretchy="false">(</mo><mn mathvariant="normal">0</mn><mo stretchy="false">)</mo></mrow></msubsup><mo>,</mo><msubsup><mrow><mi>b</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mo stretchy="false">(</mo><mn mathvariant="normal">0</mn><mo stretchy="false">)</mo></mrow></msubsup><mo>,</mo><mo>⋯</mo><mo>,</mo><msubsup><mrow><mi>b</mi></mrow><mrow><mn mathvariant="normal">21</mn></mrow><mrow><mo stretchy="false">(</mo><mn mathvariant="normal">0</mn><mo stretchy="false">)</mo></mrow></msubsup></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24007370&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24007367&type= 35.98332977 4.74133301 ［ 20 20 ］。利用L-M算法即可求得参数向量 <math id="M111"><mi mathvariant="bold-italic">b</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24007420&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24007418&type= 1.86266661 2.87866688 的最佳估计值。 5 实　验 1 5.1　旋转轴倾斜误差测试 2 利用图2 图2 所示测试系统，分别对关节臂式坐标测量机6个旋转轴的倾斜误差进行实际测试，并在式（11）式（11）的基础上，对各个旋转轴的倾斜误差测量结果进行拟合。每个轴系进行3次倾斜误差测试实验，以保证实验数据的重复性。因旋转轴3和5，4和6的结构相同，仅列出部分旋转轴测试数据，旋转轴1~4的倾斜误差测量结果、傅里叶级数拟合情况及残差如图7 图7 所示。各个旋转轴在 <math id="M112"><mi>x</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24007380&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24007377&type= 2.03200006 2.79399991 和 <math id="M113"><mi>y</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24007383&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24007381&type= 1.86266661 3.72533321 方向的倾斜误差的原始数据和拟合后残差如表1 表1 所示。 1;2; 2 倾斜误差峰峰值 2 原始数据 2 拟合残差 <math id="M114"><mi>x</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24007412&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24007411&type= 1.77800000 2.37066650 方向 <math id="M115"><mi>y</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24007415&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24007413&type= 1.60866666 3.13266683 方向 <math id="M116"><mi>x</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24007412&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24007411&type= 1.77800000 2.37066650 方向 <math id="M117"><mi>y</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24007415&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24007413&type= 1.60866666 3.13266683 方向旋转轴1 1.40 2.08 0.42 0.30 旋转轴2 2.24 2.64 0.45 0.48 旋转轴3 0.58 1.08 0.17 0.16 旋转轴4 1.41 1.53 0.21 0.27 旋转轴5 0.64 0.68 0.15 0.27 旋转轴6 1.21 0.62 0.45 0.19 ″ 由图7 图7 和表1 表1 可知，各轴系旋转轴的倾斜误差拟合残差结果均在0.50″以内，原始数据和拟合曲线的贴合程度较高。 5.2　标定实验 2 在关节臂式坐标测量机结构参数误差的标定过程中，为更加准确地获得参数向量 <math id="M118"><mi mathvariant="bold-italic">b</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24007420&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24007418&type= 1.86266661 2.87866688 的最佳估计值，在其各关节处的采样数据应尽可能地覆盖整个关节空间［ 21 21 ］。自制长度标准件两锥窝孔之间的距离为749.481 mm，具体采样策略如图8 图8 所示：以关节臂式坐标测量机的基坐标系 o - xyz 为中心，将自制长度标准件分别放置于3个位置，位置1与 <math id="M119"><mi>o</mi><mi>x</mi><mi>y</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24007427&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24007426&type= 5.67266655 3.72533321 平面的夹角为45°、位置2平行于 <math id="M120"><mi>o</mi><mi>x</mi><mi>y</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24007427&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24007426&type= 5.67266655 3.72533321 平面、位置3与 <math id="M121"><mi>o</mi><mi>x</mi><mi>y</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24007427&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24007426&type= 5.67266655 3.72533321 平面的夹角为90°。为更加准确地获得关节式坐标测量机的结构参数，在其各关节处的采样数据尽可能地覆盖整个关节空间。参考ASME B4.22关节式坐标测量机性能评价标准中的单点精度测试方法，每个位置采集800组数据，共采集2 400组数据。以具有旋转轴倾斜误差补偿的运动学模型作为整机建模方法，结合距离误差标定模型，利用L-M算法求解出关节臂式坐标测量机的结构参数。关节臂式坐标测量机标定后的结构参数如表2 表2 所示。分别使用标定前后的关节臂式坐标测量机对自制长度标准件进行测量，测量误差如图9 图9 所示。标定前测量标准差为0.345 mm，标定后测量标准差为0.037 mm，对比可知关节臂式坐标测量机的测量误差减小了近9/10。注： * 表示冗余参数 <math id="M122"><mi mathvariant="normal">Δ</mi><mi>θ</mi><mo>/</mo><mi mathvariant="normal">r</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">d</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24007436&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24007434&type= 9.22866726 2.79399991 <math id="M123"><mi>α</mi><mo>/</mo><mi mathvariant="normal">r</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">d</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24007440&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24007438&type= 7.28133297 2.79399991 <math id="M124"><mi>l</mi><mo>/</mo><mi mathvariant="normal">m</mi><mi mathvariant="normal">m</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24007446&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24007443&type= 7.61999989 2.79399991 <math id="M125"><mi>d</mi><mo>/</mo><mi mathvariant="normal">m</mi><mi mathvariant="normal">m</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24007450&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24007448&type= 8.38199997 2.79399991 <math id="M127"><msub><mrow><mi mathvariant="bold-italic">A</mi></mrow><mrow><mn>1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24007457&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24007455&type= 3.47133350 3.21733332 0.000 000 * -1.570 277 42.149 902 300.000 * <math id="M128"><msub><mrow><mi mathvariant="bold-italic">A</mi></mrow><mrow><mn>2</mn></mrow></msub></math> 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mathvariant="bold-italic">A</mi></mrow><mrow><mn>5</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24007475&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24007473&type= 3.47133350 3.21733332 0.605 352 -1.570 139 28.855 103 448.369 32 <math id="M132"><msub><mrow><mi mathvariant="bold-italic">A</mi></mrow><mrow><mn>6</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24007479&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24007476&type= 3.47133350 3.21733332 2.080 948 1.570 796 * 28.855 103 * -0.103 186 5 <math id="M126"><msub><mrow><mi>Z</mi></mrow><mrow><mi mathvariant="normal">p</mi></mrow></msub><mo>=</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24007454&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24007452&type= 6.51933336 3.21733332 200.859 24 5.3　对比实验 2 为验证具有旋转轴倾斜误差补偿的运动学模型的有效性，分别使用DH模型及具有旋转轴倾斜误差补偿的运动学模型作为整机建模方法，以关节臂式坐标测量机作为圆心，将自制长度标准件均布放置于4个不同的位置，每个位置测量50组长度距离［ 22 22 ］，计算得到测量误差，对比实验结果如图10 图10 所示。最后，分别计算基于两种不同运动学模型的测量标准差。结果表明，具有旋转轴倾斜误差补偿的运动学建模方法相比DH模型，标准件的测量标准差由0.055 mm减少到0.037 mm。 6 结　论 1 针对旋转轴系倾斜误差对关节臂式坐标测量机测量精度的影响，本文在DH模型的基础上提出了一种具有旋转轴倾斜误差补偿的运动学模型。以具有旋转轴倾斜误差补偿的运动学模型作为整机建模方法，使用距离误差标定模型求解出关节臂式坐标测量机的结构参数。标定实验结果表明，测量误差的标准差从0.345 mm（标定前）降低至0.037 mm（标定后）。对比实验结果表明，相较于DH模型，使用该模型作为整机建模方法的测量标准差从0.055 mm减少到0.037 mm，有效地提高了关节臂式坐标测量机的测量精度。实验结果充分证明了具有旋转轴倾斜误差补偿的运动学模型的有效性。参考文献于连栋，赵会宁 . 关节类坐标测量机关键技术及进展［J］. 仪器仪表学报， 2017 ， 38 （ 8 ）： 1879 - 1888 . doi: 10.3969/j.issn.0254-3087.2017.08.006 http://dx.doi.org/10.3969/j.issn.0254-3087.2017.08.006 关节类坐标测量机关键技术及进展 YU L D ， ZHAO H N . 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