基于刚度预测-神经网络的调姿平台误差补偿

刘毅; 马小腾; 丰宗强; 姚建涛; 赵永生

doi:10.37188/OPE.20223024.3139

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基于刚度预测-神经网络的调姿平台误差补偿

微纳技术与精密机械 | 更新时间：2022-12-27

- 基于刚度预测-神经网络的调姿平台误差补偿
- Stiffness prediction-neural network based error compensation for attitude adjustment platform
- 光学精密工程
- 作者机构：
  
  1.燕山大学河北省并联机器人与机电系统实验室，河北秦皇岛 066004
- 作者简介：
  
  [ "刘毅（1991-），男，山西省大同市人，博士研究生，2014年于燕山大学获得学士学位，主要从事机器人技术、机构学理论及其应用技术、高精度调姿装备设计理论。E-mail： liuyi415@stumail.ysu.edu.cn" ]
  
  刘毅（1991-），男，山西省大同市人，博士研究生，2014年于燕山大学获得学士学位，主要从事机器人技术、机构学理论及其应用技术、高精度调姿装备设计理论。E-mail： liuyi415@stumail.ysu.edu.cn
  [ "姚建涛（1980-），男，河北保定人，博士，教授，博士生导师，2004年、2009年于燕山大学分别获得学士、博士学位，主要研究方向为多维力传感器技术、机器人技术、并联机构理论及应用。E-mail：jtyao@ysu.edu.cn" ]
  
  姚建涛（1980-），男，河北保定人，博士，教授，博士生导师，2004年、2009年于燕山大学分别获得学士、博士学位，主要研究方向为多维力传感器技术、机器人技术、并联机构理论及应用。E-mail：jtyao@ysu.edu.cn
- 基金信息：
  
  国家自然科学基金资助项目(U2037202;52075466)
- DOI：10.37188/OPE.20223024.3139 中图分类号： TH115
- 纸质出版日期：2022-12-25，
  
  收稿日期：2022-06-23，
  
  修回日期：2022-08-02，
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引用本文
刘毅,马小腾,丰宗强等.基于刚度预测-神经网络的调姿平台误差补偿[J].光学精密工程,2022,30(24):3139-3158.

LIU Yi,MA Xiaoteng,FENG Zongqiang,et al.Stiffness prediction-neural network based error compensation for attitude adjustment platform[J].Optics and Precision Engineering,2022,30(24):3139-3158.
刘毅,马小腾,丰宗强等.基于刚度预测-神经网络的调姿平台误差补偿[J].光学精密工程,2022,30(24):3139-3158. DOI： 10.37188/OPE.20223024.3139.

LIU Yi,MA Xiaoteng,FENG Zongqiang,et al.Stiffness prediction-neural network based error compensation for attitude adjustment platform[J].Optics and Precision Engineering,2022,30(24):3139-3158. DOI： 10.37188/OPE.20223024.3139.

基于刚度预测-神经网络的调姿平台误差补偿 Stiffness prediction-neural network based error compensation for attitude adjustment platform first-author 刘毅（1991-），男，山西省大同市人，博士研究生，2014年于燕山大学获得学士学位，主要从事机器人技术、机构学理论及其应用技术、高精度调姿装备设计理论。E-mail： liuyi415@stumail.ysu.edu.cn http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115331&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115336&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115333&type= 22.01332855 32.00399780 刘毅（1991-），男，山西省大同市人，博士研究生，2014年于燕山大学获得学士学位，主要从事机器人技术、机构学理论及其应用技术、高精度调姿装备设计理论。E-mail： liuyi415@stumail.ysu.edu.cn liuyi415@stumail.ysu.edu.cn corresp E-mail： jtyao@ysu.edu.cn E-mail： jtyao@ysu.edu.cn 姚建涛（1980-），男，河北保定人，博士，教授，博士生导师，2004年、2009年于燕山大学分别获得学士、博士学位，主要研究方向为多维力传感器技术、机器人技术、并联机构理论及应用。E-mail：jtyao@ysu.edu.cn http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115339&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115344&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115341&type= 22.01333618 32.00399780 姚建涛（1980-），男，河北保定人，博士，教授，博士生导师，2004年、2009年于燕山大学分别获得学士、博士学位，主要研究方向为多维力传感器技术、机器人技术、并联机构理论及应用。E-mail： jtyao@ysu.edu.cn jtyao@ysu.edu.cn 燕山大学河北省并联机器人与机电系统实验室，河北秦皇岛 066004 燕山大学河北省并联机器人与机电系统实验室，河北秦皇岛 066004 Laboratory of Parallel Robotics and Mechatronic Systems in Hebei Province， Yanshan University， Qinhuangdao 066004， China Laboratory of Parallel Robotics and Mechatronic Systems in Hebei Province， Yanshan University， Qinhuangdao 066004， China 变形误差对高精度调姿装备末端定位精度的影响不可忽视。为提高2RRPU/2RPU/U两轴并联调姿平台精度，提出了基于刚度模型预测误差趋势与神经网络算法提升预测精度的误差补偿模型。首先，在调姿平台全雅克比矩阵和弹性变形矩阵的基础上建立理论刚度模型，与Ansys数据刚度模型预测对比验证受载变形趋势预测的有效性。然后，搭建基于Simulink-Adams-Ansys-OPC的模拟仿真环境，随机载荷下采集平台全姿态仿真数据，基于刚度模型和速度雅克比矩阵对姿态误差和驱动误差趋势进行预测，基于速度雅可比实现末端误差到驱动补偿的映射，进一步利用神经网络算法提升误差预测的精度。仿真实验结果表明采用误差补偿模型后调姿平台获得位姿精度提高9%，验证了“刚度预测-神经网络”模型对平台姿态精度提升的有效性。 The impact of deformation error on the end positioning accuracy of high-precision attitude adjustment equipment cannot be ignored. To improve the accuracy of a 2RRPU/2RPU/U two-axis parallel attitude platform， an error compensation model is proposed based on a stiffness model to predict the error trend and a neural network algorithm to improve the prediction accuracy. The theoretical stiffness model is first established based on the full Jacobi and elastic deformation matrices of the attitude-adjusting platform. The validity of the prediction of the loaded deformation trend is verified by comparing it with the prediction by the Ansys data stiffness model. Then， a Simulink-Adams-Ansys-OPC-based simulation environment is built， and the platform full attitude simulation data is collected under random load. Next， the attitude and drive error trends are predicted based on the stiffness model and velocity Jacobi matrix， and the mapping from end error to drive compensation is realized based on the velocity Jacobi. The accuracy of the error prediction is further improved by using a neural network algorithm. The simulation results show that the attitude accuracy of the platform is improved by 9% after adopting the error compensation model， which verifies the effectiveness of the “stiffness prediction-neural network” model in the improvement of the platform attitude accuracy. 并联调姿平台刚度建模神经网络误差补偿 attitude adjustment platform stiffness modeling neural network error compensation 1 引言 1 并联机构的主要误差来源于几何误差与变形误差，变形误差是指由重载、变载引起的各分支的弹性变形导致的动平台位置姿态误差［ 1 1 - 5 5 1-5 ］。现阶段学者对于机构的误差研究基于运动学建立误差模型，进行误差分析，采用智能算法基于误差数据进行末端的补偿。田威等人［ 6 6 ］提出一种基于遗传粒子群算法优化深度神经网络（Genetic Particle Swarm Optimization Deep Neural Networks， GPSO-DNN）的定位误差预测模型的误差补偿方法，精度提高了77.57%。张吉旺等人［ 7 7 ］利用并联机构的正解建立机构的误差模型，分析误差来源最后通过粒子群算法补偿末端误差。侯雨雷等人［ 8 8 ］利用模糊神经网络搭建并联机构末端的综合误差预测模型，利用运动学误差模型和粒子群算法把误差补偿到驱动参数。Ruibo He等人［ 9 9 ］介绍了基于指数乘积（Product of Exponential， POE）公式的误差模型，用于串联机器人校准。DA-YONG YU等人［ 10 10 ］构造了反向传播神经网络补偿器预测腿长误差。Hoai-Nhan Nguyen等人［ 11 11 ］使用扩展的卡尔曼滤波（Extended Kalman Filter， EKF）算法建模并识别机器人几何参数。马立等人［ 12 12 ］提出了基于BP神经网络在精密定位的局部工作空间内对机器人关节空间进行误差补偿的方法。廖昭洋等［ 13 13 ］提出一种通过构建时-空混合的图卷积神经网络，将机器人定位误差补偿问题转化为带约束下的高维寻优问题，利用遗传算法进行误差补偿。周炜等［ 14 14 - 15 15 14-15 ］提出了一种基于空间插值的精度补偿方法。王龙飞等［ 16 16 ］通过ELM（Extreme Learning Machine）算法建立目标点实际坐标与理论坐标间映射模型，使机器人绝对定位精度提高了45%。Nguyen等［ 11 11 ］提出利用人工神经网络补偿非几何误差的方法。许允斗等人［ 17 17 ］提出了基于激光跟踪仪的位姿测量原理，零点标定对机器人定位精度的提升非常明显。李玉昆等人［ 18 18 ］通过建立3-UPS/S并联调姿平台的变形误差，预测末端轨迹的误差。陈守欢等人［ 19 19 ］基于刚度模型研究了机器人在铣削加工过程中的误差补偿方法，证明了基于刚度模型的误差补偿方法的有效性。综上，稳定平台的误差包含多种误差来源，其中对于大承载环境，变形误差尤其突出，并且现有传统的标定手段难以实现有效的补偿，基于数据补偿的也多为综合误差的补偿，并没有细分误差的来源，基于刚度模型和神经网络算法［ 20 20 ］针对变形引起的误差进行补偿的研究较少。本文针对基于2RRPU/2RPU/U的两轴并联调姿平台变形误差补偿问题，提出了基于刚度模型预测误差趋势与神经网络算法提升预测精度的误差补偿模型。基于全雅可比矩阵与虚功原理建立了2RRPU/2RPU/U的两轴并联调姿平台的完整刚度模型，并通过有限元分析验证了建模方法的准确性。在此基础上，搭建Simulink-Adams-Ansys刚柔耦合仿真系统，随机采集调姿平台受载运行的全姿态仿真数据。通过刚度模型预测平台的微变形趋势，采用实际姿态角度和理论姿态角度的差值作为姿态误差量，并把预测的姿态误差量与实际误差量进行对比，预测的角度偏差值通过速度雅可比矩阵映射到驱动杆长的误差量；基于仿真实验数据，计算实际驱动杆长与理论驱动杆长的误差量作为补偿实际需要的补偿量。从而对比预测的驱动补偿量与实际需要的驱动补偿量的关系趋势。进一步，利用神经网络算法对初步预测的驱动补偿量进一步预测，完成预测结果精度的提升。 2 平台刚度模型分析 1 本文研究对象是基于2RRPU/2RPU/U机构含U约束分支两轴并联调姿平台，调姿平台由上平台、下平台、RRPU主动支链、RRPU从动支链、RPU主动支链、RPU从动支链和U支链组成，用于雷达、激光测角系统、电子对抗等设备架设。各支链驱动和约束分布如图1 图1 所示。 2.1　建立全雅可比矩阵 2 2 . 1 . 1　RRPU主动支链 3 a.驱动力 F 1 映射解析 RRPU主动支链与末端平台连接点 M 1 的速度为： <math display="block" id="M1"><msubsup><mrow><mi mathvariant="bold-italic">v</mi></mrow><mrow><mi>p</mi><mn mathvariant="normal">1</mn></mrow><mrow><mi>a</mi></mrow></msubsup><mo>=</mo><mi mathvariant="bold-italic">v</mi><mo>+</mo><mi mathvariant="bold-italic">ω</mi><mo>×</mo><msub><mrow><mi mathvariant="bold-italic">e</mi></mrow><mrow><mi>p</mi><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115353&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115352&type= 26.92399979 4.06400013 ，其中： <math id="M2"><mi mathvariant="bold-italic">v</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115357&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115355&type= 2.03200006 2.79399991 为上平台的质心速度矢量（mm/s）； <math id="M3"><mi mathvariant="bold-italic">ω</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115364&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115361&type= 2.62466669 2.79399991 为上平台的质心角速度矢量（rad/s）； <math id="M4"><msub><mrow><mi mathvariant="bold-italic">e</mi></mrow><mrow><mi>p</mi><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115370&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115367&type= 3.72533321 3.72533321 为上平台坐标原点 P 指向 <math id="M5"><msub><mrow><mi mathvariant="bold-italic">P</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115376&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115374&type= 3.72533321 3.72533321 点矢量（mm）。上平台在驱动电动缸所在矢量方向投影速度大小等于驱动电动缸线速度： <math display="block" id="M6"><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">l</mi><mo>˙</mo></mover></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>=</mo><msubsup><mrow><mi mathvariant="bold-italic">Q</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msubsup><mo>·</mo><msubsup><mrow><mi mathvariant="bold-italic">v</mi></mrow><mrow><mi>p</mi><mn mathvariant="normal">1</mn></mrow><mrow><mi>a</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115382&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115380&type= 17.18733406 5.07999992 . 由式（1）式（1）和式（2）式（2）联立整理得： <math display="block" id="M7"><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">l</mi><mo>˙</mo></mover></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>=</mo><mo stretchy="false">[</mo><msubsup><mrow><mi mathvariant="bold-italic">Q</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msubsup><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">e</mi></mrow><mrow><mi>p</mi><mn mathvariant="normal">1</mn></mrow></msub><mo>×</mo><msub><mrow><mi mathvariant="bold-italic">Q</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><msup><mrow><mo stretchy="false">)</mo></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup><mo stretchy="false">]</mo><mfenced open="[" close="]" separators="|"><mrow><mtable><mtr><mtd><mi mathvariant="bold-italic">v</mi></mtd></mtr><mtr><mtd><mi mathvariant="bold-italic">ω</mi></mtd></mtr></mtable></mrow></mfenced><mo>=</mo><mo stretchy="false">[</mo><msub><mrow><mi mathvariant="bold-italic">J</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">]</mo><mfenced open="[" close="]" separators="|"><mrow><mtable><mtr><mtd><mi mathvariant="bold-italic">v</mi></mtd></mtr><mtr><mtd><mi mathvariant="bold-italic">ω</mi></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115388&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115385&type= 58.58933258 8.55133343 ，其中， <math id="M8"><msub><mrow><mi mathvariant="bold-italic">J</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115393&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115390&type= 2.96333337 3.72533321 为驱动力 <math id="M9"><msub><mrow><mi mathvariant="bold-italic">F</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115399&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115396&type= 3.80999994 3.72533321 与上平台的速度映射关系。 b.约束力 <math id="M10"><msubsup><mrow><mi mathvariant="bold-italic">F</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>r</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115405&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115401&type= 3.89466691 3.97933316 映射解析上平台在约束方向不存在自由度，即平台速度与约束力的功率为零。根据虚功原理： <math display="block" id="M11"><msubsup><mrow><mi>f</mi></mrow><mrow><mi>r</mi><mn mathvariant="normal">1</mn></mrow><mrow/></msubsup><msub><mrow><mi mathvariant="bold-italic">C</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>·</mo><mi mathvariant="bold-italic">v</mi><mo>+</mo><mo stretchy="false">(</mo><msubsup><mrow><mi>f</mi></mrow><mrow><mi>r</mi><mn mathvariant="normal">1</mn></mrow><mrow/></msubsup><msub><mrow><mi mathvariant="bold-italic">C</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>×</mo><msub><mrow><mi mathvariant="bold-italic">d</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">)</mo><mo>·</mo><mi>ω</mi><mo>=</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115411&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115408&type= 45.04266739 5.07999992 ，其中： <math id="M12"><msub><mrow><mi>f</mi></mrow><mrow><mi>r</mi><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115416&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115413&type= 3.72533321 4.57200003 为 <math id="M13"><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=38115422&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115419&type= 2.37066650 3.72533321 杆上的约束力大小（N）； <math id="M14"><msub><mrow><mi mathvariant="bold-italic">C</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115428&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115425&type= 3.72533321 3.72533321 为 <math id="M15"><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=38115443&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115441&type= 2.37066650 3.72533321 杆上的约束力单位矢量方向； <math id="M16"><msub><mrow><mi mathvariant="bold-italic">d</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115438&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115436&type= 3.47133350 3.72533321 为 <math id="M17"><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=38115443&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115441&type= 2.37066650 3.72533321 杆上的约束力到U支链质心的矢量距离（mm）。由式（4）式（4）化简可得： <math display="block" id="M18"><mn mathvariant="normal">0</mn><mo>=</mo><mo stretchy="false">[</mo><msub><mrow><mi mathvariant="bold-italic">C</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mtext> </mtext><mn mathvariant="normal">0</mn><mo stretchy="false">]</mo><mfenced open="[" close="]" separators="|"><mrow><mtable><mtr><mtd><mi mathvariant="bold-italic">v</mi></mtd></mtr><mtr><mtd><mi mathvariant="bold-italic">ω</mi></mtd></mtr></mtable></mrow></mfenced><mo>=</mo><mo stretchy="false">[</mo><msub><mrow><mi mathvariant="bold-italic">J</mi></mrow><mrow><mn mathvariant="normal">1</mn><mi mathvariant="normal">r</mi><mi mathvariant="normal">f</mi></mrow></msub><mo stretchy="false">]</mo><mfenced open="[" close="]" separators="|"><mrow><mtable><mtr><mtd><mi mathvariant="bold-italic">v</mi></mtd></mtr><mtr><mtd><mi mathvariant="bold-italic">ω</mi></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115451&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115448&type= 44.70399857 8.55133343 ，其中， <math id="M19"><msub><mrow><mi mathvariant="bold-italic">J</mi></mrow><mrow><mn mathvariant="normal">1</mn><mi mathvariant="normal">r</mi><mi mathvariant="normal">f</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115455&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115452&type= 4.23333359 3.72533321 为约束力 <math id="M20"><msubsup><mrow><mi mathvariant="bold-italic">F</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>r</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115462&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115459&type= 3.89466691 3.97933316 与上平台的速度映射关系。 2 . 1 . 2　RPU主动支链 3 a.驱动力 <math id="M21"><msub><mrow><mi mathvariant="bold-italic">F</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115467&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115464&type= 3.80999994 3.72533321 映射解析采用与RRPU主动支链中驱动力 <math id="M22"><msub><mrow><mi mathvariant="bold-italic">F</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115472&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115470&type= 3.80999994 3.72533321 相同的求解方法，可得： <math display="block" id="M23"><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">l</mi><mo>˙</mo></mover></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>=</mo><mo stretchy="false">[</mo><msubsup><mrow><mi mathvariant="bold-italic">Q</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msubsup><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">e</mi></mrow><mrow><mi>p</mi><mn mathvariant="normal">2</mn></mrow></msub><mo>×</mo><msub><mrow><mi mathvariant="bold-italic">Q</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><msup><mrow><mo stretchy="false">)</mo></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup><mo stretchy="false">]</mo><mfenced open="[" close="]" separators="|"><mrow><mtable><mtr><mtd><mi mathvariant="bold-italic">v</mi></mtd></mtr><mtr><mtd><mi mathvariant="bold-italic">ω</mi></mtd></mtr></mtable></mrow></mfenced><mo>=</mo><mo stretchy="false">[</mo><msub><mrow><mi mathvariant="bold-italic">J</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">]</mo><mfenced open="[" close="]" separators="|"><mrow><mtable><mtr><mtd><mi mathvariant="bold-italic">v</mi></mtd></mtr><mtr><mtd><mi mathvariant="bold-italic">ω</mi></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115476&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115474&type= 58.58933258 8.55133343 ，其中： <math id="M24"><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">l</mi><mo>˙</mo></mover></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115479&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115477&type= 2.37066650 4.48733330 为 <math id="M25"><msub><mrow><mi>l</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115492&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115490&type= 2.37066650 3.72533321 杆驱动标量速度大小（mm）； <math id="M26"><msub><mrow><mi mathvariant="bold-italic">Q</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115486&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115485&type= 4.06400013 4.06400013 为 <math id="M27"><msub><mrow><mi>l</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115492&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115490&type= 2.37066650 3.72533321 杆单位矢量方向； <math id="M28"><msub><mrow><mi mathvariant="bold-italic">J</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115498&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115495&type= 2.96333337 3.72533321 为约束力 <math id="M29"><msub><mrow><mi mathvariant="bold-italic">F</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115504&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115500&type= 3.80999994 3.72533321 与末端平台速度映射关系。 b.约束力 <math id="M30"><msubsup><mrow><mi mathvariant="bold-italic">F</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>r</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115507&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115505&type= 3.89466691 3.97933316 映射解析根据虚功原理： <math display="block" id="M31"><msub><mrow><mi>f</mi></mrow><mrow><mi>r</mi><mn mathvariant="normal">2</mn></mrow></msub><msub><mrow><mi mathvariant="bold-italic">C</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>·</mo><mi mathvariant="bold-italic">v</mi><mo>+</mo><mo stretchy="false">(</mo><msub><mrow><mi>f</mi></mrow><mrow><mi>r</mi><mn mathvariant="normal">2</mn></mrow></msub><mi mathvariant="bold-italic">C</mi><mmultiscripts><mrow><mo>×</mo></mrow><mprescripts/><mrow><mn mathvariant="normal">2</mn></mrow><mrow/></mmultiscripts><msub><mrow><mi mathvariant="bold-italic">d</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">)</mo><mo>·</mo><mi mathvariant="bold-italic">ω</mi><mo>=</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115511&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115509&type= 45.46599960 4.91066647 ，其中： <math id="M32"><msub><mrow><mi>f</mi></mrow><mrow><mi>r</mi><mn mathvariant="normal">2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115515&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115513&type= 3.72533321 4.57200003 为 <math id="M33"><msub><mrow><mi>l</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115519&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115517&type= 2.37066650 3.72533321 杆上的约束力大小（N）； <math id="M34"><msub><mrow><mi mathvariant="bold-italic">C</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115523&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115521&type= 3.72533321 3.72533321 为 <math id="M35"><msub><mrow><mi>l</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115541&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115538&type= 2.37066650 3.72533321 杆上的约束力单位矢量方向； <math id="M36"><msub><mrow><mi mathvariant="bold-italic">d</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115535&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115532&type= 3.47133350 3.72533321 为 <math id="M37"><msub><mrow><mi>l</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115541&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115538&type= 2.37066650 3.72533321 杆上的约束力到U支链质心的矢量距离（mm）。由式（7）式（7）化简，可得： <math display="block" id="M38"><mn mathvariant="normal">0</mn><mo>=</mo><mo stretchy="false">[</mo><msub><mrow><mi>f</mi></mrow><mrow><mi>r</mi><mn mathvariant="normal">2</mn></mrow></msub><msub><mrow><mi mathvariant="bold-italic">C</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mtext> </mtext><mn mathvariant="normal">0</mn><mo stretchy="false">]</mo><mfenced open="[" close="]" separators="|"><mrow><mtable><mtr><mtd><mi mathvariant="bold-italic">v</mi></mtd></mtr><mtr><mtd><mi mathvariant="bold-italic">w</mi></mtd></mtr></mtable></mrow></mfenced><mo>=</mo><mo stretchy="false">[</mo><msub><mrow><mi mathvariant="bold-italic">J</mi></mrow><mrow><mn mathvariant="normal">2</mn><mi>r</mi><mi>f</mi></mrow></msub><mo stretchy="false">]</mo><mfenced open="[" close="]" separators="|"><mrow><mtable><mtr><mtd><mi mathvariant="bold-italic">v</mi></mtd></mtr><mtr><mtd><mi mathvariant="bold-italic">w</mi></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115546&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115543&type= 50.71533585 8.55133343 ，其中， <math id="M39"><msub><mrow><mi mathvariant="bold-italic">J</mi></mrow><mrow><mn mathvariant="normal">2</mn><mi>r</mi><mi>f</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115550&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115548&type= 4.31799984 3.72533321 为约束力 <math id="M40"><msubsup><mrow><mi mathvariant="bold-italic">J</mi></mrow><mrow><mi>w</mi></mrow><mrow><mi>r</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115553&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115551&type= 3.55599999 3.97933316 与末端平台的速度映射关系。 c.约束力偶 <math id="M41"><msubsup><mrow><mi mathvariant="bold-italic">M</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>r</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115557&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115555&type= 5.16466665 4.06400013 映射解析约束力偶只会约束平台的姿态变化，根据虚功原理： <math display="block" id="M42"><msub><mrow><mi mathvariant="bold-italic">M</mi></mrow><mrow><mi>r</mi></mrow></msub><msub><mrow><mi mathvariant="bold-italic">C</mi></mrow><mrow><mi>c</mi><mn mathvariant="normal">2</mn></mrow></msub><mo>·</mo><mi mathvariant="bold-italic">ω</mi><mo>=</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115561&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115559&type= 19.81200027 3.80999994 ，其中： <math id="M43"><msub><mrow><mi mathvariant="bold-italic">M</mi></mrow><mrow><mi>r</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115565&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115563&type= 4.65666676 3.80999994 为约束力偶 <math id="M44"><msubsup><mrow><mi>M</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>r</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115570&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115567&type= 4.99533367 3.97933316 的大小（N·mm）； <math id="M45"><msub><mrow><mi mathvariant="bold-italic">C</mi></mrow><mrow><mi>c</mi><mn mathvariant="normal">2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115575&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115572&type= 4.48733330 3.72533321 为约束力偶 <math id="M46"><msubsup><mrow><mi mathvariant="bold-italic">M</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>r</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115581&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115577&type= 5.16466665 4.06400013 的矢量方向。由式（9）式（9）化简，可得： <math display="block" id="M47"><mn mathvariant="normal">0</mn><mo>=</mo><mo stretchy="false">[</mo><mn mathvariant="normal">0</mn><mtext> </mtext><msub><mrow><mi mathvariant="bold-italic">M</mi></mrow><mrow><mi>r</mi></mrow></msub><msub><mrow><mi mathvariant="bold-italic">C</mi></mrow><mrow><mi>c</mi><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">]</mo><mfenced open="[" close="]" separators="|"><mrow><mtable><mtr><mtd><mi mathvariant="bold-italic">v</mi></mtd></mtr><mtr><mtd><mi mathvariant="bold-italic">w</mi></mtd></mtr></mtable></mrow></mfenced><mo>=</mo><mo stretchy="false">[</mo><msub><mrow><mi mathvariant="bold-italic">J</mi></mrow><mrow><mn mathvariant="normal">2</mn><mi>r</mi><mi>m</mi></mrow></msub><mo stretchy="false">]</mo><mfenced open="[" close="]" separators="|"><mrow><mtable><mtr><mtd><mi mathvariant="bold-italic">v</mi></mtd></mtr><mtr><mtd><mi mathvariant="bold-italic">w</mi></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115585&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115583&type= 52.83200073 8.55133343 ，其中， <math id="M48"><msub><mrow><mi mathvariant="bold-italic">J</mi></mrow><mrow><mn mathvariant="normal">2</mn><mi>r</mi><mi>m</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115589&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115587&type= 5.24933338 3.72533321 为约束力偶 <math id="M49"><msubsup><mrow><mi mathvariant="bold-italic">M</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>r</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115594&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115592&type= 5.16466665 4.06400013 与上平台的速度映射关系。 2 . 1 . 3　恰约束U支链 3 a.约束力 <math id="M50"><msub><mrow><mi mathvariant="bold-italic">M</mi></mrow><mrow><mi mathvariant="normal">U</mi><mi mathvariant="normal">X</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115597&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115596&type= 6.77333355 3.80999994 ， <math id="M51"><msub><mrow><mi mathvariant="bold-italic">M</mi></mrow><mrow><mi mathvariant="normal">U</mi><mi mathvariant="normal">Y</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115601&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115599&type= 6.85799980 3.80999994 和 <math id="M52"><msub><mrow><mi mathvariant="bold-italic">M</mi></mrow><mrow><mi mathvariant="normal">U</mi><mi mathvariant="normal">Z</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115605&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115603&type= 6.51933336 3.80999994 映射解析 U支链约束力方向始终保持不变，并且约束力作用点为动坐标系原点，约束力与上平台之间的力矩始终为零，根据虚功原理可得： <math display="block" id="M53"><msub><mrow><mi>f</mi></mrow><mrow><mi mathvariant="normal">U</mi><mi>i</mi></mrow></msub><msub><mrow><mi mathvariant="bold-italic">C</mi></mrow><mrow><mi mathvariant="normal">U</mi><mi>i</mi></mrow></msub><mo>·</mo><mi mathvariant="bold-italic">v</mi><mo>+</mo><mo stretchy="false">(</mo><msub><mrow><mi>f</mi></mrow><mrow><mi mathvariant="normal">U</mi><mi>i</mi></mrow></msub><msub><mrow><mi mathvariant="bold-italic">C</mi></mrow><mrow><mi mathvariant="normal">U</mi><mi>i</mi></mrow></msub><mo>×</mo><msub><mrow><mi mathvariant="bold-italic">d</mi></mrow><mrow><mi mathvariant="normal">U</mi><mi>i</mi></mrow></msub><mo stretchy="false">)</mo><mo>·</mo><mi mathvariant="bold-italic">ω</mi><mo>=</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115609&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115607&type= 49.19132996 4.91066647 ，其中： <math id="M54"><msub><mrow><mi>f</mi></mrow><mrow><mi>U</mi><mi>i</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115613&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115611&type= 4.14866638 4.57200003 为U支链约束力的大小， i 取 x ， y ， z 用于区分具体约束力（N）； <math id="M55"><msubsup><mrow><mi mathvariant="bold-italic">C</mi></mrow><mrow><mi mathvariant="normal">U</mi></mrow><mrow><mi>i</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115617&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115615&type= 4.23333359 3.97933316 为约束力的单位矢量方向； <math id="M56"><msubsup><mrow><mi mathvariant="bold-italic">d</mi></mrow><mrow><mi mathvariant="normal">U</mi></mrow><mrow><mi>i</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115621&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115619&type= 3.97933316 3.97933316 为约束力作用点到上平台质心距离矢量（mm）。由式（11）式（11）化简可得： <math display="block" id="M57"><mn mathvariant="normal">0</mn><mo>=</mo><mo stretchy="false">[</mo><msub><mrow><mi mathvariant="bold-italic">C</mi></mrow><mrow><mi mathvariant="normal">U</mi><mn mathvariant="normal">1</mn></mrow></msub><mtext> </mtext><mn mathvariant="normal">0</mn><mo stretchy="false">]</mo><mfenced open="[" close="]" separators="|"><mrow><mtable><mtr><mtd><mi mathvariant="bold-italic">v</mi></mtd></mtr><mtr><mtd><mi mathvariant="bold-italic">ω</mi></mtd></mtr></mtable></mrow></mfenced><mo>=</mo><mo stretchy="false">[</mo><msub><mrow><mi mathvariant="bold-italic">J</mi></mrow><mrow><mi mathvariant="normal">U</mi><mi>i</mi></mrow></msub><mo stretchy="false">]</mo><mfenced open="[" close="]" separators="|"><mrow><mtable><mtr><mtd><mi mathvariant="bold-italic">v</mi></mtd></mtr><mtr><mtd><mi mathvariant="bold-italic">ω</mi></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115624&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115622&type= 46.14333344 8.55133343 ，其中， <math id="M58"><msub><mrow><mi mathvariant="bold-italic">J</mi></mrow><mrow><mi mathvariant="normal">U</mi><mi>i</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115628&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115626&type= 4.14866638 3.72533321 为U支链3个约束力与上平台的速度映射关系， i 取 x ， y ， z 。 b.约束力偶 <math id="M59"><msub><mrow><mi>M</mi></mrow><mrow><mi mathvariant="normal">U</mi><mi mathvariant="normal">Z</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115632&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115630&type= 6.43466663 3.72533321 映射解析 U支链约束力偶只与上平台的转动存在映射关系，根据虚功原理： <math display="block" id="M60"><msub><mrow><mi mathvariant="bold-italic">M</mi></mrow><mrow><mi mathvariant="normal">U</mi><mi mathvariant="normal">Z</mi></mrow></msub><msubsup><mrow><mi>C</mi></mrow><mrow><mi mathvariant="normal">U</mi><mi mathvariant="normal">Z</mi></mrow><mrow><mi>m</mi></mrow></msubsup><mo>·</mo><mi>ω</mi><mo>=</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115636&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115634&type= 22.69066620 4.06400013 ，其中， <math id="M61"><msubsup><mrow><mi mathvariant="bold-italic">C</mi></mrow><mrow><mi mathvariant="normal">U</mi><mi mathvariant="normal">Z</mi></mrow><mrow><mi>m</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115640&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115638&type= 5.41866684 3.97933316 为约束力偶 <math id="M62"><msub><mrow><mi mathvariant="bold-italic">M</mi></mrow><mrow><mi mathvariant="normal">U</mi><mi mathvariant="normal">Z</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115644&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115642&type= 6.51933336 3.80999994 矢量方向。由式（13）式（13）化简可得： <math display="block" id="M63"><mn mathvariant="normal">0</mn><mo>=</mo><mo stretchy="false">[</mo><mn mathvariant="normal">0</mn><mtext> </mtext><msub><mrow><mi mathvariant="bold-italic">M</mi></mrow><mrow><mi mathvariant="normal">U</mi><mi mathvariant="normal">Z</mi></mrow></msub><msubsup><mrow><mi mathvariant="bold-italic">C</mi></mrow><mrow><mi mathvariant="normal">U</mi><mi mathvariant="normal">Z</mi></mrow><mrow><mi>m</mi></mrow></msubsup><mo stretchy="false">]</mo><mfenced open="[" close="]" separators="|"><mrow><mtable><mtr><mtd><mi mathvariant="bold-italic">v</mi></mtd></mtr><mtr><mtd><mi mathvariant="bold-italic">ω</mi></mtd></mtr></mtable></mrow></mfenced><mo>=</mo><mo stretchy="false">[</mo><msub><mrow><mi mathvariant="bold-italic">J</mi></mrow><mrow><mi mathvariant="normal">U</mi><mi mathvariant="normal">M</mi><mi mathvariant="normal">Z</mi></mrow></msub><mo stretchy="false">]</mo><mfenced open="[" close="]" separators="|"><mrow><mtable><mtr><mtd><mi mathvariant="bold-italic">v</mi></mtd></mtr><mtr><mtd><mi mathvariant="bold-italic">ω</mi></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115649&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115647&type= 55.11800003 8.55133343 ，其中， <math id="M64"><msub><mrow><mi mathvariant="bold-italic">J</mi></mrow><mrow><mi mathvariant="normal">U</mi><mi mathvariant="normal">M</mi><mi mathvariant="normal">Z</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115653&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115651&type= 6.51933336 3.72533321 为U支链约束力偶 <math id="M65"><msub><mrow><mi mathvariant="bold-italic">M</mi></mrow><mrow><mi mathvariant="normal">U</mi><mi mathvariant="normal">Z</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115658&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115656&type= 6.51933336 3.80999994 与上平台的速度映射关系。 2 . 1 . 4　全雅可比矩阵 3 汇总各分支驱动、约束与上平台速度的映射关系获得调姿平台全雅可比矩阵为： <math display="block" id="M66"><msub><mrow><mfenced open="[" close="]" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">l</mi><mo>˙</mo></mover></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mtd></mtr><mtr><mtd><mn mathvariant="normal">0</mn></mtd></mtr><mtr><mtd><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">l</mi><mo>˙</mo></mover></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mtd></mtr><mtr><mtd><mn mathvariant="normal">0</mn></mtd></mtr><mtr><mtd><mn mathvariant="normal">0</mn></mtd></mtr><mtr><mtd><mn mathvariant="normal">0</mn></mtd></mtr><mtr><mtd><mn mathvariant="normal">0</mn></mtd></mtr><mtr><mtd><mn mathvariant="normal">0</mn></mtd></mtr><mtr><mtd><mn mathvariant="normal">0</mn></mtd></mtr></mtable></mrow></mfenced></mrow><mrow><mn mathvariant="normal">9</mn><mo>×</mo><mn mathvariant="normal">1</mn></mrow></msub><mo>=</mo><msub><mrow><mfenced open="[" close="]" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><msub><mrow><mi mathvariant="bold-italic">J</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mtd></mtr><mtr><mtd><msub><mrow><mi mathvariant="bold-italic">J</mi></mrow><mrow><mn mathvariant="normal">1</mn><mi>r</mi><mi>f</mi></mrow></msub></mtd></mtr><mtr><mtd><msub><mrow><mi mathvariant="bold-italic">J</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mtd></mtr><mtr><mtd><msub><mrow><mi mathvariant="bold-italic">J</mi></mrow><mrow><mn mathvariant="normal">2</mn><mi>r</mi><mi>f</mi></mrow></msub></mtd></mtr><mtr><mtd><msub><mrow><mi mathvariant="bold-italic">J</mi></mrow><mrow><mn mathvariant="normal">2</mn><mi>r</mi><mi>m</mi></mrow></msub></mtd></mtr><mtr><mtd><msub><mrow><mi mathvariant="bold-italic">J</mi></mrow><mrow><mi mathvariant="normal">U</mi><mi mathvariant="normal">X</mi></mrow></msub></mtd></mtr><mtr><mtd><msub><mrow><mi mathvariant="bold-italic">J</mi></mrow><mrow><mi mathvariant="normal">U</mi><mi mathvariant="normal">Y</mi></mrow></msub></mtd></mtr><mtr><mtd><msub><mrow><mi mathvariant="bold-italic">J</mi></mrow><mrow><mi mathvariant="normal">U</mi><mi mathvariant="normal">Z</mi></mrow></msub></mtd></mtr><mtr><mtd><msub><mrow><mi mathvariant="bold-italic">J</mi></mrow><mrow><mi mathvariant="normal">U</mi><mi mathvariant="normal">M</mi><mi mathvariant="normal">Z</mi></mrow></msub></mtd></mtr></mtable></mrow></mfenced></mrow><mrow><mn mathvariant="normal">9</mn><mo>×</mo><mn mathvariant="normal">6</mn></mrow></msub><mo>·</mo><msub><mrow><mfenced open="[" close="]" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><mi mathvariant="bold-italic">v</mi></mtd></mtr><mtr><mtd><mi mathvariant="bold-italic">ω</mi></mtd></mtr></mtable></mrow></mfenced></mrow><mrow><mn mathvariant="normal">6</mn><mo>×</mo><mn mathvariant="normal">1</mn></mrow></msub><mo>=</mo><mo stretchy="false">[</mo><msup><mrow><mi mathvariant="bold-italic">J</mi></mrow><mrow><mi>a</mi></mrow></msup><msub><mrow><mo stretchy="false">]</mo></mrow><mrow><mn mathvariant="normal">9</mn><mo>×</mo><mn mathvariant="normal">6</mn></mrow></msub><msub><mrow><mfenced open="[" close="]" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><mi mathvariant="bold-italic">v</mi></mtd></mtr><mtr><mtd><mi mathvariant="bold-italic">ω</mi></mtd></mtr></mtable></mrow></mfenced></mrow><mrow><mn mathvariant="normal">6</mn><mo>×</mo><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115662&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115660&type= 69.17266846 45.88933182 . 2.2　弹性变形与刚度模型矩阵 2 2 . 2 . 1　RRPU主动支链 3 （1）驱动力 <math id="M67"><msub><mrow><mi mathvariant="bold-italic">F</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115670&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115668&type= 3.80999994 3.72533321 对应支链弹性变形驱动力 <math id="M68"><msub><mrow><mi mathvariant="bold-italic">F</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115670&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115668&type= 3.80999994 3.72533321 沿RRPU支链电动缸矢量方向，其受力变形如图2 图2 ，支链的合变形需要考虑两轴变形组合。 a.电动缸驱动杆的变形量： <math display="block" id="M69"><mi mathvariant="normal">Δ</mi><msub><mrow><mi>λ</mi></mrow><mrow><mn mathvariant="normal">11</mn></mrow></msub><mo>=</mo><mfrac><mrow><msub><mrow><mi mathvariant="bold-italic">F</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi>l</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>-</mo><msub><mrow><mi>l</mi></mrow><mrow><mn mathvariant="normal">1</mn><mi>g</mi></mrow></msub><mo stretchy="false">)</mo></mrow><mrow><msub><mrow><mi>E</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><msub><mrow><mi>A</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115685&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115683&type= 29.97200012 8.63599968 ，其中： <math id="M70"><mi mathvariant="normal">Δ</mi><msub><mrow><mi>λ</mi></mrow><mrow><mn mathvariant="normal">11</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115691&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115688&type= 6.34999990 3.72533321 为电动缸驱动杆拉压变形量（mm）； <math id="M71"><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=38115700&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115697&type= 2.37066650 3.72533321 为RRPU支链总的长度（mm）； <math id="M72"><msub><mrow><mi>l</mi></mrow><mrow><mn mathvariant="normal">1</mn><mi>g</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115705&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115703&type= 3.38666677 3.72533321 为电动缸缸体固定长度（mm）； <math id="M73"><msub><mrow><mi>E</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115715&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115711&type= 3.80999994 3.72533321 为电动缸驱动杆弹性模量（MPa）； <math id="M74"><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=38115723&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115719&type= 4.23333359 3.72533321 为电动缸驱动杆的界面面积（mm 2 ）。 b.电动缸缸体的变形量： <math display="block" id="M75"><mi mathvariant="normal">Δ</mi><msub><mrow><mi>λ</mi></mrow><mrow><mn mathvariant="normal">12</mn></mrow></msub><mo>=</mo><mfrac><mrow><msub><mrow><mi mathvariant="bold-italic">F</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><msub><mrow><mi>l</mi></mrow><mrow><mn mathvariant="normal">1</mn><mi>g</mi></mrow></msub></mrow><mrow><msub><mrow><mi>E</mi></mrow><mrow><mn mathvariant="normal">1</mn><mi>g</mi></mrow></msub><msub><mrow><mi>A</mi></mrow><mrow><mn mathvariant="normal">1</mn><mi>g</mi></mrow></msub></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115729&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115725&type= 22.18266678 8.80533314 ，其中： <math id="M76"><mi mathvariant="normal">Δ</mi><msub><mrow><mi>λ</mi></mrow><mrow><mn mathvariant="normal">12</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115736&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115733&type= 6.34999990 3.72533321 为电动缸缸体拉压变形量（mm）； <math id="M77"><msub><mrow><mi>E</mi></mrow><mrow><mn mathvariant="normal">1</mn><mi>g</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115745&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115739&type= 4.82600021 3.72533321 为电动缸缸体弹性模量（MPa）； <math id="M78"><msub><mrow><mi>A</mi></mrow><mrow><mn mathvariant="normal">1</mn><mi>g</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115753&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115748&type= 5.16466665 3.72533321 为电动缸缸体截面面积（mm 2 ）。 c.偏置连杆拉压变形：偏置连杆拉压变形是由驱动力在沿偏置连杆方向分力导致，所以首先计算分力的大小： <math display="block" id="M79"><msub><mrow><mi>F</mi></mrow><mrow><mn mathvariant="normal">1</mn><mi>p</mi></mrow></msub><mo>=</mo><mfrac><mrow><msub><mrow><mi mathvariant="bold-italic">F</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><msub><mrow><mi mathvariant="bold-italic">Q</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>·</mo><mi mathvariant="bold-italic">R</mi></mrow><mrow><mfenced open="|" close="|" separators="|"><mrow><mi mathvariant="bold-italic">R</mi></mrow></mfenced></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115760&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115757&type= 22.26733208 9.14400005 ，其中： <math id="M80"><msub><mrow><mi>F</mi></mrow><mrow><mn mathvariant="normal">1</mn><mi>p</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115768&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115763&type= 4.74133301 3.72533321 为驱动力 <math id="M81"><msub><mrow><mi mathvariant="bold-italic">F</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115776&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115772&type= 3.80999994 3.72533321 沿偏置连杆方向分力大小（N）； <math id="M82"><msub><mrow><mi mathvariant="bold-italic">Q</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115783&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115778&type= 4.06400013 4.06400013 为 <math id="M83"><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=38115790&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115787&type= 2.37066650 3.72533321 单位矢量方向； <math id="M84"><mi mathvariant="bold-italic">R</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115798&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115794&type= 2.87866688 2.79399991 为偏移连杆矢量方向（mm）。由式（18）式（18）进一步计算载荷所对应的弹性变形，所对应的弹性变形采用拉压变形公式为： <math display="block" id="M85"><mi mathvariant="normal">Δ</mi><msub><mrow><mi>λ</mi></mrow><mrow><mn mathvariant="normal">13</mn></mrow></msub><mo>=</mo><mfrac><mrow><msub><mrow><mi>F</mi></mrow><mrow><mn mathvariant="normal">1</mn><mi>p</mi></mrow></msub><msub><mrow><mi>l</mi></mrow><mrow><mn mathvariant="normal">1</mn><mi>R</mi></mrow></msub></mrow><mrow><msub><mrow><mi>E</mi></mrow><mrow><mn mathvariant="normal">1</mn><mi>R</mi></mrow></msub><msub><mrow><mi>A</mi></mrow><mrow><mn mathvariant="normal">1</mn><mi>R</mi></mrow></msub></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115805&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115800&type= 23.02933502 8.38199997 ，其中： <math id="M86"><mi mathvariant="normal">Δ</mi><msub><mrow><mi>λ</mi></mrow><mrow><mn mathvariant="normal">13</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115813&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115809&type= 6.34999990 3.72533321 为载荷引起的偏置连杆拉压变形量（mm）； <math id="M87"><msub><mrow><mi>l</mi></mrow><mrow><mn mathvariant="normal">1</mn><mi>R</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115821&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115818&type= 3.72533321 3.72533321 表示偏置连杆长度（mm）； <math id="M88"><msub><mrow><mi>E</mi></mrow><mrow><mn mathvariant="normal">1</mn><mi>R</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115828&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115824&type= 5.24933338 3.72533321 为偏置连杆弹性模量（MPa）； <math id="M89"><msub><mrow><mi>A</mi></mrow><mrow><mn mathvariant="normal">1</mn><mi>R</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115836&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115831&type= 5.58799982 3.72533321 为偏置连杆截面面积（mm 2 ）。 d.偏置连杆弯曲变形：偏置连杆弯曲变形是由驱动力在垂直偏置连杆方向分力导致，所以需要首先计算分力的大小： <math display="block" id="M90"><msub><mrow><mi>F</mi></mrow><mrow><mn mathvariant="normal">1</mn><mi>v</mi></mrow></msub><mo>=</mo><mfrac><mrow><msub><mrow><mi mathvariant="bold-italic">F</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">Q</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>×</mo><mi mathvariant="bold-italic">R</mi><mo stretchy="false">)</mo></mrow><mrow><mfenced open="|" close="|" separators="|"><mrow><mi mathvariant="bold-italic">R</mi></mrow></mfenced></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115844&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115839&type= 29.29466629 9.39799976 ，其中， <math id="M91"><msub><mrow><mi>F</mi></mrow><mrow><mn mathvariant="normal">1</mn><mi>v</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115852&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115846&type= 4.74133301 3.72533321 为驱动力 <math id="M92"><msub><mrow><mi mathvariant="bold-italic">F</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115857&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115854&type= 3.80999994 3.72533321 沿偏置连杆方向分力大小（N）。由式（20）式（20）进一步计算载荷所对应的弹性变形： <math display="block" id="M93"><mi mathvariant="normal">Δ</mi><msub><mrow><mi>λ</mi></mrow><mrow><mn mathvariant="normal">14</mn></mrow></msub><mo>=</mo><mfrac><mrow><msub><mrow><mi>F</mi></mrow><mrow><mn mathvariant="normal">1</mn><mi>v</mi></mrow></msub><msubsup><mrow><mi>l</mi></mrow><mrow><mn mathvariant="normal">1</mn><mi>R</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msubsup></mrow><mrow><mn mathvariant="normal">3</mn><msub><mrow><mi>E</mi></mrow><mrow><mn mathvariant="normal">1</mn><mi>R</mi></mrow></msub><msub><mrow><mi>I</mi></mrow><mrow><mn mathvariant="normal">1</mn><mi>R</mi></mrow></msub></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115862&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115859&type= 23.19866753 8.63599968 ，其中： <math id="M94"><mi mathvariant="normal">Δ</mi><msub><mrow><mi>λ</mi></mrow><mrow><mn mathvariant="normal">14</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115868&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115865&type= 6.34999990 3.72533321 为载荷引起的偏置连杆弯曲变形量（mm）； <math id="M95"><msub><mrow><mi>I</mi></mrow><mrow><mn mathvariant="normal">1</mn><mi>R</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115870&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115869&type= 3.89466691 3.72533321 为偏置连杆的抗弯截面模量（mm 4 ）。 e.偏置连杆组合变量 <math id="M96"><mi mathvariant="normal">Δ</mi><msub><mrow><mi>λ</mi></mrow><mrow><mi>R</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115875&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115872&type= 5.75733376 3.72533321 ： <math display="block" id="M97"><mi mathvariant="normal">Δ</mi><msub><mrow><mi>λ</mi></mrow><mrow><mi>R</mi></mrow></msub><mo>=</mo><mroot><mrow><mi mathvariant="normal">Δ</mi><msubsup><mrow><mi>λ</mi></mrow><mrow><mn mathvariant="normal">13</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo>+</mo><mi mathvariant="normal">Δ</mi><msubsup><mrow><mi>λ</mi></mrow><mrow><mn mathvariant="normal">14</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup></mrow><mrow/></mroot></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115880&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115877&type= 31.49600029 4.74133301 . f.RRPU主动支链总变形量 <math id="M98"><mi mathvariant="normal">Δ</mi><msub><mrow><mi>λ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115887&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115884&type= 5.41866684 3.72533321 ： <math display="block" id="M99"><mi mathvariant="normal">Δ</mi><msub><mrow><mi>λ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>=</mo><mi mathvariant="normal">Δ</mi><msub><mrow><mi>λ</mi></mrow><mrow><mn mathvariant="normal">11</mn></mrow></msub><mo>+</mo><mi mathvariant="normal">Δ</mi><msub><mrow><mi>λ</mi></mrow><mrow><mn mathvariant="normal">12</mn></mrow></msub><mo>+</mo><mi mathvariant="normal">Δ</mi><msub><mrow><mi>λ</mi></mrow><mrow><mi>R</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115894&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115891&type= 38.35400009 3.80999994 . g.RRPU主动支链刚度系数 <math id="M100"><msub><mrow><mi>k</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115898&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115897&type= 2.79399991 3.72533321 ： <math display="block" id="M101"><msub><mrow><mi>k</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>=</mo><mfrac><mrow><msub><mrow><mi>F</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow><mrow><mi mathvariant="normal">Δ</mi><msub><mrow><mi>λ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></mfrac><mo>+</mo><mfrac><mrow><msub><mrow><mi>F</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow><mrow><mi mathvariant="normal">Δ</mi><msub><mrow><mi>λ</mi></mrow><mrow><mn mathvariant="normal">11</mn></mrow></msub><mo>+</mo><mi mathvariant="normal">Δ</mi><msub><mrow><mi>λ</mi></mrow><mrow><mn mathvariant="normal">12</mn></mrow></msub><mo>+</mo><mi mathvariant="normal">Δ</mi><msub><mrow><mi>λ</mi></mrow><mrow><mi>R</mi></mrow></msub></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115904&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115902&type= 47.41333389 8.38199997 . 由式（16）式（16）、式（17）式（17）和式（24）式（24）联立，获得RRPU主动支链刚度系数最终表达式形式为： <math display="block" id="M102"><msub><mrow><mi>k</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>=</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mfrac><mrow><mo stretchy="false">(</mo><msub><mrow><mi>l</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>-</mo><msubsup><mrow><mi>l</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>g</mi></mrow></msubsup><mo stretchy="false">)</mo></mrow><mrow><msub><mrow><mi>E</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><msub><mrow><mi>A</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></mfrac><mo>+</mo><mfrac><mrow><msubsup><mrow><mi>l</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>g</mi></mrow></msubsup></mrow><mrow><msubsup><mrow><mi>E</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>g</mi></mrow></msubsup><msubsup><mrow><mi>A</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>g</mi></mrow></msubsup></mrow></mfrac><mo>+</mo><mroot><mrow><msup><mrow><mfenced separators="|"><mrow><mfrac><mrow><mfenced open="|" close="|" separators="|"><mrow><msub><mrow><mi mathvariant="bold-italic">Q</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>·</mo><mi mathvariant="bold-italic">R</mi></mrow></mfenced><msubsup><mrow><mi>l</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>R</mi></mrow></msubsup></mrow><mrow><mfenced open="|" close="|" separators="|"><mrow><mi mathvariant="bold-italic">R</mi></mrow></mfenced><msubsup><mrow><mi>E</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>R</mi></mrow></msubsup><msubsup><mrow><mi>A</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>R</mi></mrow></msubsup></mrow></mfrac></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><msup><mrow><mfenced separators="|"><mrow><mfrac><mrow><mfenced open="|" close="|" separators="|"><mrow><msub><mrow><mi mathvariant="bold-italic">Q</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>×</mo><mi mathvariant="bold-italic">R</mi></mrow></mfenced><mo stretchy="false">(</mo><msubsup><mrow><mi>l</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>R</mi></mrow></msubsup><msup><mrow><mo stretchy="false">)</mo></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msup></mrow><mrow><mn mathvariant="normal">3</mn><msubsup><mrow><mi>E</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>R</mi></mrow></msubsup><msubsup><mrow><mi>I</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>R</mi></mrow></msubsup><mfenced open="|" close="|" separators="|"><mrow><mi mathvariant="bold-italic">R</mi></mrow></mfenced></mrow></mfrac></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow><mrow/></mroot></mrow></mfrac><mi mathvariant="normal">Δ</mi><msub><mrow><mi>λ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115908&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115906&type= 103.29333496 17.61066628 . （2）约束力 <math id="M103"><msubsup><mrow><mi mathvariant="bold-italic">F</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>r</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115912&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115910&type= 3.89466691 3.97933316 对应支链弹性变形约束力 <math id="M104"><msubsup><mrow><mi mathvariant="bold-italic">F</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>r</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115919&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115917&type= 3.89466691 3.97933316 作用在U副的质心点，弹性变形主要分析对应支链的微变形量，所以需要平移到支链末端，约束力 <math id="M105"><msubsup><mrow><mi mathvariant="bold-italic">F</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>r</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115919&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115917&type= 3.89466691 3.97933316 平移过程会产生约束力偶 <math id="M106"><msub><mrow><mi mathvariant="bold-italic">M</mi></mrow><mrow><mn mathvariant="normal">1</mn><mi>a</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115923&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115921&type= 5.84200001 3.80999994 ，约束力偶 <math id="M107"><msub><mrow><mi mathvariant="bold-italic">M</mi></mrow><mrow><mn mathvariant="normal">1</mn><mi>a</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115927&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115926&type= 5.84200001 3.80999994 矢量方向按照矢量进行偏移，约束力偶 <math id="M108"><msub><mrow><mi mathvariant="bold-italic">M</mi></mrow><mrow><mn mathvariant="normal">1</mn><mi>a</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115929&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115928&type= 5.84200001 3.80999994 会使支链产生弯曲变形和扭转变形，因扭转变形对支链末端位移影响较小，所以这里不考虑扭转变形，约束力 <math id="M109"><msubsup><mrow><mi mathvariant="bold-italic">F</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>r</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115932&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115931&type= 3.89466691 3.97933316 载荷分布见图3 图3 。 a.约束力引起支链的弯曲变形量 <math id="M111"><mi mathvariant="normal">Δ</mi><msubsup><mrow><mi>λ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>r</mi><mi>f</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115946&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115945&type= 5.75733376 3.97933316 ： <math display="block" id="M112"><mi mathvariant="normal">Δ</mi><msubsup><mrow><mi>λ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>r</mi><mi>f</mi></mrow></msubsup><mo>=</mo><mfrac><mrow><msubsup><mrow><mi mathvariant="bold-italic">F</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>r</mi></mrow></msubsup><msubsup><mrow><mi>l</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msubsup></mrow><mrow><mn mathvariant="normal">3</mn><msub><mrow><mi>E</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><msub><mrow><mi>I</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115950&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115948&type= 19.72733307 8.63599968 ，其中， <math id="M113"><msubsup><mrow><mi>F</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>r</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115953&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115951&type= 3.80999994 3.97933316 为约束力矢量 <math id="M114"><msubsup><mrow><mi mathvariant="bold-italic">F</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>r</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115958&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115956&type= 3.89466691 3.97933316 的大小（N）。 b.平移约束力 <math id="M115"><msubsup><mrow><mi mathvariant="bold-italic">F</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>r</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115961&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115960&type= 3.89466691 3.97933316 产生的附加力偶 <math id="M116"><msubsup><mrow><mi mathvariant="bold-italic">M</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>a</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115965&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115963&type= 5.33400011 4.06400013 ： <math display="block" id="M117"><msubsup><mrow><mi mathvariant="bold-italic">M</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>a</mi></mrow></msubsup><mo>=</mo><mfrac><mrow><msubsup><mrow><mi mathvariant="bold-italic">F</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>r</mi></mrow></msubsup><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">S</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>×</mo><msub><mrow><mi mathvariant="bold-italic">e</mi></mrow><mrow><mi>p</mi><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">)</mo></mrow><mrow><mfenced open="|" close="|" separators="|"><mrow><msub><mrow><mi mathvariant="bold-italic">S</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mrow></mfenced></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115969&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115967&type= 30.64933395 9.39799976 ，其中， <math id="M118"><msub><mrow><mi mathvariant="bold-italic">S</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115973&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115971&type= 3.47133350 3.72533321 为约束力的矢量方向，与支链第二个旋转副轴线重合。 c.引起弯曲变形的力偶大小 <math id="M119"><msubsup><mrow><mi mathvariant="bold-italic">M</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>a</mi><mi>p</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115977&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115975&type= 6.26533365 4.06400013 ： <math display="block" id="M120"><msubsup><mrow><mi mathvariant="bold-italic">M</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>a</mi><mi>p</mi></mrow></msubsup><mo>=</mo><mfenced open="|" close="|" separators="|"><mrow><msubsup><mrow><mi mathvariant="bold-italic">M</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>a</mi></mrow></msubsup><mo>×</mo><msub><mrow><mi mathvariant="bold-italic">Q</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115981&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115979&type= 27.60133362 4.48733330 . d.弯曲变形约束力偶 <math id="M121"><msubsup><mrow><mi mathvariant="bold-italic">M</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>a</mi><mi>p</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115985&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115983&type= 6.26533365 4.06400013 产生的变形量 <math id="M122"><mi mathvariant="normal">Δ</mi><msubsup><mrow><mi>λ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>a</mi><mi>m</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115989&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115987&type= 6.85799980 3.97933316 ： <math display="block" id="M123"><mi mathvariant="normal">Δ</mi><msubsup><mrow><mi>λ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>a</mi><mi>m</mi></mrow></msubsup><mo>=</mo><mfrac><mrow><msubsup><mrow><mi mathvariant="bold-italic">M</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>a</mi><mi>p</mi></mrow></msubsup><msubsup><mrow><mi>l</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup></mrow><mrow><mn mathvariant="normal">2</mn><msub><mrow><mi>E</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><msub><mrow><mi>I</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115993&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115991&type= 21.33600044 8.63599968 . e.组合变形： <math display="block" id="M124"><mi mathvariant="normal">Δ</mi><msubsup><mrow><mi>λ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>r</mi></mrow></msubsup><mo>=</mo><mroot><mrow><mo stretchy="false">(</mo><mi mathvariant="normal">Δ</mi><msubsup><mrow><mi>λ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>r</mi><mi>f</mi></mrow></msubsup><msup><mrow><mo stretchy="false">)</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><mo stretchy="false">(</mo><mi mathvariant="normal">Δ</mi><msubsup><mrow><mi>λ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>a</mi><mi>m</mi></mrow></msubsup><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=38115997&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115995&type= 39.11600113 4.82600021 ，其中， <math id="M125"><mi mathvariant="normal">Δ</mi><msubsup><mrow><mi>λ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>r</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116001&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38115999&type= 5.41866684 3.97933316 为RPU支链受载总变形量（mm）。 f.刚度系数： <math display="block" id="M126"><msubsup><mrow><mi>k</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>r</mi></mrow></msubsup><mo>=</mo><mfrac><mrow><msub><mrow><mi>F</mi></mrow><mrow><mi>r</mi></mrow></msub></mrow><mrow><mi mathvariant="normal">Δ</mi><msubsup><mrow><mi>λ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>r</mi></mrow></msubsup></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116005&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116003&type= 13.80066681 8.63599968 ，其中， <math id="M127"><msubsup><mrow><mi>k</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>r</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116008&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116007&type= 2.79399991 3.97933316 为RRPU主动支链约束力 <math id="M128"><msubsup><mrow><mi mathvariant="bold-italic">F</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>r</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116012&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116010&type= 3.89466691 3.97933316 对应的总刚度系数（N/mm）。由式（26）式（26）至式（31）式（31）联立可得： <math display="block" id="M129"><msubsup><mrow><mi>k</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>r</mi></mrow></msubsup><mo>=</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mroot><mrow><msup><mrow><mfenced separators="|"><mrow><mfrac><mrow><msubsup><mrow><mi>l</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msubsup></mrow><mrow><mn mathvariant="normal">3</mn><msub><mrow><mi>E</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><msub><mrow><mi>I</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></mfrac></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><msup><mrow><mfenced separators="|"><mrow><mfrac><mrow><mfenced open="|" close="|" separators="|"><mrow><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">S</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>×</mo><msub><mrow><mi mathvariant="bold-italic">e</mi></mrow><mrow><mi>p</mi><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">)</mo><mo>×</mo><msub><mrow><mi mathvariant="bold-italic">Q</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></mfenced><msubsup><mrow><mi>l</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup></mrow><mrow><mn mathvariant="normal">2</mn><mfenced open="|" close="|" separators="|"><mrow><msub><mrow><mi mathvariant="bold-italic">S</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mrow></mfenced><msub><mrow><mi>E</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><msub><mrow><mi>I</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></mfrac></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow><mrow/></mroot></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116016&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116014&type= 64.85466003 18.62666702 . 2 . 2 . 2　RPU主动支链 3 （1）驱动力 <math id="M130"><msub><mrow><mi mathvariant="bold-italic">F</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116019&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116018&type= 3.80999994 3.72533321 对应支链弹性变形 RPU主动支链驱动力 <math id="M131"><msub><mrow><mi mathvariant="bold-italic">F</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116023&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116021&type= 3.80999994 3.72533321 沿驱动电动缸方向，作用力主要变形包含支链电动缸推杆和电动缸缸体拉压变形，支链变形如图4 图4 所示。 a.电动缸驱动杆的变形量 <math id="M133"><mi mathvariant="normal">Δ</mi><msubsup><mrow><mi>λ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>p</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116037&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116034&type= 5.41866684 3.97933316 ： <math display="block" id="M134"><mi mathvariant="normal">Δ</mi><msubsup><mrow><mi>λ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>p</mi></mrow></msubsup><mo>=</mo><mfrac><mrow><msub><mrow><mi mathvariant="bold-italic">F</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi>l</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>-</mo><msubsup><mrow><mi>l</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>g</mi></mrow></msubsup><mo stretchy="false">)</mo></mrow><mrow><msub><mrow><mi>E</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><msub><mrow><mi>A</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116040&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116039&type= 28.27866554 8.63599968 ，其中： <math id="M135"><msub><mrow><mi>l</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116045&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116043&type= 2.37066650 3.72533321 为RPU主动支链总长度（mm）； <math id="M136"><msubsup><mrow><mi>l</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>g</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116049&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116048&type= 2.62466669 3.97933316 为RPU主动支链电动缸体长度（mm）； <math id="M137"><msub><mrow><mi>E</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116053&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116051&type= 3.80999994 3.72533321 为RPU主动支链电动缸推杆弹性模量（MPa）； <math id="M138"><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=38116057&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116055&type= 4.23333359 3.72533321 为RPU主动支链电动缸推杆截面面积（mm 2 ）。 b.电动缸缸体的变形量： <math display="block" id="M139"><mi mathvariant="normal">Δ</mi><msubsup><mrow><mi>λ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>g</mi></mrow></msubsup><mo>=</mo><mfrac><mrow><msub><mrow><mi mathvariant="bold-italic">F</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><msubsup><mrow><mi>l</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>g</mi></mrow></msubsup></mrow><mrow><msubsup><mrow><mi>E</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>g</mi></mrow></msubsup><msubsup><mrow><mi>A</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>g</mi></mrow></msubsup></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116062&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116059&type= 19.55800056 8.80533314 ，其中： <math id="M140"><mi mathvariant="normal">Δ</mi><msubsup><mrow><mi>λ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>g</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116065&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116063&type= 5.41866684 3.97933316 为RPU主动支链电动缸缸体变形量（mm）； <math id="M141"><msubsup><mrow><mi>E</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>g</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116069&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116067&type= 4.06400013 3.97933316 为RPU主动支链电动缸缸体弹性模量（MPa）； <math id="M142"><msubsup><mrow><mi>A</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>g</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116073&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116071&type= 4.23333359 3.97933316 为RPU主动支链电动缸缸体截面面积（mm 2 ）； <math id="M143"><msubsup><mrow><mi>l</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>g</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116076&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116075&type= 2.62466669 3.97933316 为RPU主动支链电动缸缸体固定长度（mm）。 c.支链的总变形： <math display="block" id="M144"><mi mathvariant="normal">Δ</mi><msub><mrow><mi>λ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>=</mo><mi mathvariant="normal">Δ</mi><msubsup><mrow><mi>λ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>p</mi></mrow></msubsup><mo>+</mo><mi mathvariant="normal">Δ</mi><msubsup><mrow><mi>λ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>g</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116080&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116078&type= 25.73866653 4.06400013 . d.刚度系数： <math display="block" id="M145"><msub><mrow><mi>k</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>=</mo><mfrac><mrow><msub><mrow><mi mathvariant="bold-italic">F</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mrow><mrow><mi mathvariant="normal">Δ</mi><msub><mrow><mi>λ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116085&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116083&type= 13.71599960 8.38199997 . 由式（33）式（33）至式（36）式（36）联立可得： <math display="block" id="M146"><msub><mrow><mi>k</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>=</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mfrac><mrow><mo stretchy="false">(</mo><msub><mrow><mi>l</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>-</mo><msubsup><mrow><mi>l</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>g</mi></mrow></msubsup><mo stretchy="false">)</mo></mrow><mrow><msub><mrow><mi>E</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><msub><mrow><mi>A</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mrow></mfrac><mo>+</mo><mfrac><mrow><msubsup><mrow><mi>l</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>g</mi></mrow></msubsup></mrow><mrow><msubsup><mrow><mi>E</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>g</mi></mrow></msubsup><msubsup><mrow><mi>A</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>g</mi></mrow></msubsup></mrow></mfrac></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116089&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116087&type= 36.15266418 13.97000027 . （2）约束力 <math id="M147"><msubsup><mrow><mi mathvariant="bold-italic">F</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>r</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116093&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116091&type= 3.89466691 3.97933316 对应支链弹性变形约束力 <math id="M148"><msubsup><mrow><mi mathvariant="bold-italic">F</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>r</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116107&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116106&type= 3.89466691 3.97933316 平移到RPU支链虎克铰副中心，会产生附加的空间力偶 <math id="M149"><msubsup><mrow><mi mathvariant="bold-italic">M</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>a</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116101&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116099&type= 5.33400011 4.06400013 ，空间力偶 <math id="M150"><msubsup><mrow><mi mathvariant="bold-italic">M</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>a</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116111&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116109&type= 5.33400011 4.06400013 会造成支链的弯曲变形和扭转变形，因扭转变形对末端位移量影响较小，所以不考虑扭转变形，所以支链末端变形由约束力 <math id="M151"><msubsup><mrow><mi mathvariant="bold-italic">F</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>r</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116107&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116106&type= 3.89466691 3.97933316 引起的弯曲变形和附加力偶 <math id="M152"><msubsup><mrow><mi mathvariant="bold-italic">M</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>a</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116111&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116109&type= 5.33400011 4.06400013 引起的弯曲变形组成。 a.平移后的约束力 <math id="M154"><msubsup><mrow><mi mathvariant="bold-italic">F</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>r</mi><mi>t</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116125&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116123&type= 4.48733330 3.97933316 造成的变形量： <math display="block" id="M155"><mi mathvariant="normal">Δ</mi><msubsup><mrow><mi>λ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>r</mi></mrow></msubsup><mo>=</mo><mfrac><mrow><msubsup><mrow><mi mathvariant="bold-italic">F</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>r</mi><mi>t</mi></mrow></msubsup><msubsup><mrow><mi>l</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msubsup></mrow><mrow><mn mathvariant="normal">3</mn><msub><mrow><mi>E</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><msub><mrow><mi>I</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116130&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116127&type= 19.38866615 8.63599968 ，其中， <math id="M156"><mi mathvariant="normal">Δ</mi><msubsup><mrow><mi>λ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>r</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116133&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116131&type= 5.41866684 3.97933316 为RPU主动支链的约束力 <math id="M157"><msubsup><mrow><mi mathvariant="bold-italic">F</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>r</mi><mi>t</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116137&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116135&type= 4.48733330 3.97933316 对应的变形量（mm）； <math id="M158"><msub><mrow><mi>I</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116142&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116139&type= 2.53999996 3.72533321 为RPU主动支链的抗弯截面模量（mm4）。 b.附加力偶 <math id="M159"><msubsup><mrow><mi mathvariant="bold-italic">M</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>a</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116145&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116144&type= 5.33400011 4.06400013 的数值大小： <math display="block" id="M160"><msubsup><mrow><mi mathvariant="bold-italic">M</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>a</mi></mrow></msubsup><mo>=</mo><msubsup><mrow><mi mathvariant="bold-italic">F</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>r</mi></mrow></msubsup><mo>×</mo><msub><mrow><mi mathvariant="bold-italic">e</mi></mrow><mrow><mi>p</mi><mn mathvariant="normal">2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116148&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116147&type= 22.60600090 4.06400013 ，其中， <math id="M161"><msubsup><mrow><mi mathvariant="bold-italic">M</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>a</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116179&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116178&type= 5.33400011 4.06400013 为RPU主动支链平移约束力 <math id="M162"><msubsup><mrow><mi mathvariant="bold-italic">F</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>r</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116152&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116151&type= 3.89466691 3.97933316 产生的附加约束力偶（N·mm）。 c.附加约束力偶 <math id="M163"><msubsup><mrow><mi mathvariant="bold-italic">M</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>a</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116179&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116178&type= 5.33400011 4.06400013 在垂直支链方向上投影： <math display="block" id="M164"><msubsup><mrow><mi mathvariant="bold-italic">M</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>a</mi></mrow></msubsup><mo>=</mo><mfenced open="|" close="|" separators="|"><mrow><msubsup><mrow><mi mathvariant="bold-italic">M</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>a</mi></mrow></msubsup><mo>×</mo><msub><mrow><mi mathvariant="bold-italic">Q</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116156&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116155&type= 26.58533096 4.48733330 ，其中， <math id="M165"><msubsup><mrow><mi mathvariant="bold-italic">M</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>r</mi><mi>v</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116158&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116157&type= 6.09600019 4.06400013 为RPU主动支链附加约束力偶 <math id="M166"><msubsup><mrow><mi mathvariant="bold-italic">M</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>a</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116179&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116178&type= 5.33400011 4.06400013 在垂直支链方向大小（N·mm）。 d.附加力偶 <math id="M167"><msubsup><mrow><mi mathvariant="bold-italic">M</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>a</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116173&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116172&type= 5.33400011 4.06400013 造成弯曲变形的变形量： <math display="block" id="M168"><mi mathvariant="normal">Δ</mi><msubsup><mrow><mi>λ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>a</mi></mrow></msubsup><mo>=</mo><mfrac><mrow><msubsup><mrow><mi mathvariant="bold-italic">M</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>r</mi><mi>v</mi></mrow></msubsup><msubsup><mrow><mi>l</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup></mrow><mrow><mn mathvariant="normal">2</mn><msub><mrow><mi>E</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><msub><mrow><mi>I</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116169&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116167&type= 19.72733307 8.63599968 ，其中， <math id="M169"><mi mathvariant="normal">Δ</mi><msubsup><mrow><mi>λ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>a</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116171&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116170&type= 5.41866684 3.97933316 为RPU主动支链附加约束力偶 <math id="M170"><msubsup><mrow><mi mathvariant="bold-italic">M</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>a</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116173&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116172&type= 5.33400011 4.06400013 造成弯曲变形量（mm）。 e.组合变形： <math display="block" id="M171"><mi mathvariant="normal">Δ</mi><msubsup><mrow><mi>λ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>r</mi><mi>f</mi></mrow></msubsup><mo>=</mo><mroot><mrow><mo stretchy="false">(</mo><mi mathvariant="normal">Δ</mi><msubsup><mrow><mi>λ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>r</mi></mrow></msubsup><msup><mrow><mo stretchy="false">)</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><mo stretchy="false">(</mo><mi mathvariant="normal">Δ</mi><msubsup><mrow><mi>λ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>a</mi></mrow></msubsup><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=38116175&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116174&type= 37.59199905 4.82600021 ，其中， <math id="M172"><mi mathvariant="normal">Δ</mi><msubsup><mrow><mi>λ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>r</mi><mi>f</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116177&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116176&type= 5.75733376 3.97933316 为RPU主动支链附加约束力偶 <math id="M173"><msubsup><mrow><mi mathvariant="bold-italic">M</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>a</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116179&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116178&type= 5.33400011 4.06400013 造成总变形量（mm）。 f.刚度系数： <math display="block" id="M174"><msubsup><mrow><mi>k</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>r</mi><mi>f</mi></mrow></msubsup><mo>=</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi mathvariant="normal">Δ</mi><msubsup><mrow><mi>λ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>r</mi><mi>f</mi></mrow></msubsup></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116181&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116180&type= 14.73200035 8.55133343 ，其中， <math id="M175"><msubsup><mrow><mi>k</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>r</mi><mi>f</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116183&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116182&type= 3.38666677 3.97933316 为RPU主动支链约束力刚度系数（N/mm）。由式（38）式（38）至式（43）式（43）联立可得： <math display="block" id="M176"><msubsup><mrow><mi>k</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>r</mi><mi>f</mi></mrow></msubsup><mo>=</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mroot><mrow><msup><mrow><mfenced separators="|"><mrow><mfrac><mrow><msubsup><mrow><mi>l</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msubsup></mrow><mrow><mn mathvariant="normal">3</mn><msub><mrow><mi>E</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><msub><mrow><mi>I</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mrow></mfrac></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><msup><mrow><mfenced separators="|"><mrow><mfrac><mrow><mfenced open="|" close="|" separators="|"><mrow><msub><mrow><mi mathvariant="bold-italic">S</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>×</mo><msub><mrow><mi mathvariant="bold-italic">e</mi></mrow><mrow><mi>p</mi><mn mathvariant="normal">2</mn></mrow></msub><mo>×</mo><msub><mrow><mi mathvariant="bold-italic">Q</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mrow></mfenced><msubsup><mrow><mi>l</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup></mrow><mrow><mn mathvariant="normal">2</mn><msub><mrow><mi>E</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><msub><mrow><mi>I</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mrow></mfrac></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow><mrow/></mroot></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116185&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116184&type= 62.90732956 18.62666702 ，其中， <math id="M177"><msub><mrow><mi mathvariant="bold-italic">S</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116187&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116186&type= 3.47133350 3.72533321 为约束力 <math id="M178"><msubsup><mrow><mi mathvariant="bold-italic">F</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>r</mi><mi>t</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116189&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116188&type= 4.48733330 3.97933316 单位矢量方向。（3）约束力偶 <math id="M179"><msubsup><mrow><mi mathvariant="bold-italic">M</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>r</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116191&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116190&type= 5.16466665 4.06400013 对应支链弹性变形约束力偶 <math id="M180"><msubsup><mrow><mi mathvariant="bold-italic">M</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>r</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116197&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116196&type= 5.16466665 4.06400013 方向垂直于RPU主动支链虎克铰两个转轴轴线形成的平面。该约束力偶同样会造成支链的弯曲变形和扭转变形，只考虑约束力偶 <math id="M181"><msubsup><mrow><mi mathvariant="bold-italic">M</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>r</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116197&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116196&type= 5.16466665 4.06400013 在垂直于支链的方向会造成支链的弯曲变形。 a.约束力偶 <math id="M182"><msubsup><mrow><mi mathvariant="bold-italic">M</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>r</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116197&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116196&type= 5.16466665 4.06400013 的矢量方向： <math display="block" id="M183"><msubsup><mrow><mi mathvariant="bold-italic">M</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>r</mi></mrow></msubsup><mo>=</mo><mfrac><mrow><msubsup><mrow><mi mathvariant="bold-italic">M</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>r</mi></mrow></msubsup><mo>·</mo><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">e</mi></mrow><mrow><mi>p</mi><mn mathvariant="normal">2</mn></mrow></msub><mo>×</mo><msub><mrow><mi mathvariant="bold-italic">U</mi></mrow><mrow><mi>a</mi><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">)</mo></mrow><mrow><mfenced open="|" close="|" separators="|"><mrow><msub><mrow><mi mathvariant="bold-italic">e</mi></mrow><mrow><mi>p</mi><mn mathvariant="normal">2</mn></mrow></msub></mrow></mfenced></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116199&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116198&type= 34.03599930 10.32933331 ，其中： <math id="M184"><msubsup><mrow><mi mathvariant="bold-italic">M</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>r</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116201&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116200&type= 5.16466665 4.06400013 为RPU主动支链约束力偶矢量； <math id="M185"><msub><mrow><mi mathvariant="bold-italic">U</mi></mrow><mrow><mi>a</mi><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116203&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116202&type= 5.07999992 3.72533321 为RPU主动支链旋转副轴线单位矢量方向。 b.约束力偶 <math id="M186"><msubsup><mrow><mi mathvariant="bold-italic">M</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>r</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116226&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116225&type= 5.16466665 4.06400013 在垂直 <math id="M187"><msub><mrow><mi>l</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116207&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116206&type= 2.37066650 3.72533321 的方向投影： <math display="block" id="M188"><msubsup><mrow><mi mathvariant="bold-italic">M</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>r</mi><mi>v</mi></mrow></msubsup><mo>=</mo><msubsup><mrow><mi mathvariant="bold-italic">M</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>r</mi></mrow></msubsup><mo>×</mo><msub><mrow><mi mathvariant="bold-italic">Q</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116210&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116208&type= 24.72266769 4.31799984 ，其中， <math id="M189"><msub><mrow><mi mathvariant="bold-italic">Q</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116212&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116211&type= 4.06400013 4.06400013 为RPU主动支链滑动副的单位矢量方向。 c.约束力偶 <math id="M190"><msubsup><mrow><mi mathvariant="bold-italic">M</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>r</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116226&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116225&type= 5.16466665 4.06400013 产生的弯曲变形量： <math display="block" id="M191"><mi mathvariant="normal">Δ</mi><msubsup><mrow><mi>λ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>r</mi><mi>m</mi><mi>v</mi></mrow></msubsup><mo>=</mo><mfrac><mrow><mfenced open="|" close="|" separators="|"><mrow><msubsup><mrow><mi mathvariant="bold-italic">M</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>r</mi><mi>v</mi></mrow></msubsup></mrow></mfenced><msubsup><mrow><mi>l</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup></mrow><mrow><mn mathvariant="normal">2</mn><msub><mrow><mi>E</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><msub><mrow><mi>I</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116216&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116215&type= 24.72266769 9.14400005 ，其中， <math id="M192"><mi mathvariant="normal">Δ</mi><msubsup><mrow><mi>λ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>r</mi><mi>m</mi><mi>v</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116218&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116217&type= 7.61999989 3.97933316 为RPU主动支链约束力偶 <math id="M193"><msubsup><mrow><mi mathvariant="bold-italic">M</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>r</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116226&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116225&type= 5.16466665 4.06400013 产生的弯曲变形（mm）。 d.刚度系数： <math display="block" id="M194"><msubsup><mrow><mi>k</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>r</mi><mi>m</mi></mrow></msubsup><mo>=</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi mathvariant="normal">Δ</mi><msubsup><mrow><mi>λ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>r</mi><mi>m</mi><mi>v</mi></mrow></msubsup></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116222&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116221&type= 17.44133186 8.55133343 ，其中， <math id="M195"><msubsup><mrow><mi>k</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>r</mi><mi>m</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116224&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116223&type= 4.31799984 3.97933316 为RPU主动支链约束力偶 <math id="M196"><msubsup><mrow><mi mathvariant="bold-italic">M</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>r</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116226&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116225&type= 5.16466665 4.06400013 刚度系数（N/mm）。由式（45）式（45）至式（48）式（48）联立可得： <math display="block" id="M197"><msubsup><mrow><mi>k</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>r</mi><mi>m</mi></mrow></msubsup><mo>=</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mfrac><mrow><mfenced open="|" close="|" separators="|"><mrow><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">e</mi></mrow><mrow><mi>p</mi><mn mathvariant="normal">2</mn></mrow></msub><mo>×</mo><msub><mrow><mi mathvariant="bold-italic">S</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">)</mo><mo>×</mo><msub><mrow><mi mathvariant="bold-italic">Q</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mrow></mfenced><msubsup><mrow><mi>l</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup></mrow><mrow><mn mathvariant="normal">2</mn><msub><mrow><mi>E</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><msub><mrow><mi>I</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mfenced open="|" close="|" separators="|"><mrow><msub><mrow><mi mathvariant="bold-italic">e</mi></mrow><mrow><mi>p</mi><mn mathvariant="normal">2</mn></mrow></msub></mrow></mfenced></mrow></mfrac></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116228&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116227&type= 39.53932953 16.25600052 . 2 . 2 . 3　恰约束U支链 3 （1）约束力 <math id="M198"><msub><mrow><mi mathvariant="bold-italic">F</mi></mrow><mrow><mi mathvariant="normal">U</mi><mi mathvariant="normal">X</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116230&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116229&type= 5.75733376 3.72533321 对应支链弯曲变形： <math display="block" id="M199"><mi mathvariant="normal">Δ</mi><msub><mrow><mi>λ</mi></mrow><mrow><mi mathvariant="normal">U</mi><mi mathvariant="normal">X</mi></mrow></msub><mo>=</mo><mfrac><mrow><msub><mrow><mi mathvariant="bold-italic">F</mi></mrow><mrow><mi mathvariant="normal">U</mi><mi mathvariant="normal">X</mi></mrow></msub><msubsup><mrow><mi>l</mi></mrow><mrow><mi mathvariant="normal">U</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msubsup></mrow><mrow><mn mathvariant="normal">3</mn><msub><mrow><mi>E</mi></mrow><mrow><mi mathvariant="normal">U</mi></mrow></msub><msub><mrow><mi>I</mi></mrow><mrow><mi mathvariant="normal">U</mi></mrow></msub></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116232&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116231&type= 22.43666649 8.63599968 ，其中： <math id="M200"><msub><mrow><mi>F</mi></mrow><mrow><mi mathvariant="normal">U</mi><mi mathvariant="normal">X</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116234&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116233&type= 5.67266655 3.72533321 为约束力 <math id="M201"><msub><mrow><mi mathvariant="bold-italic">F</mi></mrow><mrow><mi mathvariant="normal">U</mi><mi mathvariant="normal">X</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116236&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116235&type= 5.75733376 3.72533321 的数值大小（N）； <math id="M202"><msub><mrow><mi>l</mi></mrow><mrow><mi mathvariant="normal">U</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116238&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116237&type= 2.87866688 3.72533321 为恰约束U支链中心的高度（mm）； <math id="M203"><msub><mrow><mi>E</mi></mrow><mrow><mi mathvariant="normal">U</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116241&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116240&type= 4.40266657 3.72533321 为恰约束U支链弹性模量（MPa）； <math id="M204"><msub><mrow><mi>I</mi></mrow><mrow><mi mathvariant="normal">U</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116243&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116242&type= 3.04800010 3.72533321 为恰约束U支链的抗弯界面模量（mm 4 ）。由式（50）式（50）计算刚度系数： <math display="block" id="M205"><msub><mrow><mi>k</mi></mrow><mrow><mi mathvariant="normal">U</mi><mi mathvariant="normal">X</mi></mrow></msub><mo>=</mo><mfrac><mrow><msub><mrow><mi>F</mi></mrow><mrow><mi mathvariant="normal">U</mi><mi mathvariant="normal">X</mi></mrow></msub></mrow><mrow><mi mathvariant="normal">Δ</mi><msub><mrow><mi>λ</mi></mrow><mrow><mi mathvariant="normal">U</mi><mi mathvariant="normal">X</mi></mrow></msub></mrow></mfrac><mo>=</mo><mfrac><mrow><mn mathvariant="normal">3</mn><msub><mrow><mi>E</mi></mrow><mrow><mi mathvariant="normal">U</mi></mrow></msub><msub><mrow><mi>I</mi></mrow><mrow><mi mathvariant="normal">U</mi></mrow></msub></mrow><mrow><msubsup><mrow><mi>l</mi></mrow><mrow><mi mathvariant="normal">U</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msubsup></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116245&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116244&type= 32.76599884 8.63599968 ，其中， <math id="M206"><msub><mrow><mi>k</mi></mrow><mrow><mi mathvariant="normal">U</mi><mi mathvariant="normal">X</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116247&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116246&type= 4.74133301 3.72533321 为恰约束U支链约束 <math id="M207"><msub><mrow><mi mathvariant="bold-italic">F</mi></mrow><mrow><mi mathvariant="normal">U</mi><mi mathvariant="normal">X</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116249&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116248&type= 5.75733376 3.72533321 刚度系数（N/mm）。（2）约束力 <math id="M208"><msub><mrow><mi mathvariant="bold-italic">F</mi></mrow><mrow><mi mathvariant="normal">U</mi><mi mathvariant="normal">Y</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116251&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116250&type= 5.84200001 3.72533321 对应支链弯曲变形： <math display="block" id="M209"><mi mathvariant="normal">Δ</mi><msub><mrow><mi>λ</mi></mrow><mrow><mi mathvariant="normal">U</mi><mi mathvariant="normal">Y</mi></mrow></msub><mo>=</mo><mfrac><mrow><msub><mrow><mi>F</mi></mrow><mrow><mi mathvariant="normal">U</mi><mi mathvariant="normal">Y</mi></mrow></msub><msubsup><mrow><mi>l</mi></mrow><mrow><mi mathvariant="normal">U</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msubsup></mrow><mrow><mn mathvariant="normal">3</mn><msub><mrow><mi>E</mi></mrow><mrow><mi mathvariant="normal">U</mi></mrow></msub><msub><mrow><mi>I</mi></mrow><mrow><mi mathvariant="normal">U</mi></mrow></msub></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116253&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116252&type= 22.52133369 8.63599968 ，其中： <math id="M210"><mi mathvariant="normal">Δ</mi><msub><mrow><mi>λ</mi></mrow><mrow><mi mathvariant="normal">U</mi><mi mathvariant="normal">Y</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116255&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116254&type= 7.36600018 3.72533321 为约束力 <math id="M211"><msub><mrow><mi mathvariant="bold-italic">F</mi></mrow><mrow><mi mathvariant="normal">U</mi><mi mathvariant="normal">Y</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116257&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116256&type= 5.84200001 3.72533321 造成的U支链的弯曲变形量（mm）； <math id="M212"><msub><mrow><mi>F</mi></mrow><mrow><mi mathvariant="normal">U</mi><mi mathvariant="normal">Y</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116259&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116258&type= 5.75733376 3.72533321 为约束力 <math id="M213"><msub><mrow><mi mathvariant="bold-italic">F</mi></mrow><mrow><mi mathvariant="normal">U</mi><mi mathvariant="normal">Y</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116261&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116260&type= 5.84200001 3.72533321 的数值大小（N）。由式（52）式（52）计算刚度系数： <math display="block" id="M214"><msub><mrow><mi>k</mi></mrow><mrow><mi mathvariant="normal">U</mi><mi mathvariant="normal">Y</mi></mrow></msub><mo>=</mo><mfrac><mrow><msub><mrow><mi mathvariant="bold-italic">F</mi></mrow><mrow><mi mathvariant="normal">U</mi><mi mathvariant="normal">Y</mi></mrow></msub></mrow><mrow><mi mathvariant="normal">Δ</mi><msub><mrow><mi>λ</mi></mrow><mrow><mi mathvariant="normal">U</mi><mi mathvariant="normal">Y</mi></mrow></msub></mrow></mfrac><mo>=</mo><mfrac><mrow><mn mathvariant="normal">3</mn><msub><mrow><mi>E</mi></mrow><mrow><mi mathvariant="normal">U</mi></mrow></msub><msub><mrow><mi>I</mi></mrow><mrow><mi mathvariant="normal">U</mi></mrow></msub></mrow><mrow><msubsup><mrow><mi>l</mi></mrow><mrow><mi mathvariant="normal">U</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msubsup></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116263&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116262&type= 32.93532944 8.63599968 ，其中， <math id="M215"><msub><mrow><mi>k</mi></mrow><mrow><mi mathvariant="normal">U</mi><mi mathvariant="normal">Y</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116265&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116264&type= 4.82600021 3.72533321 为恰约束U支链约束 <math id="M216"><msub><mrow><mi mathvariant="bold-italic">F</mi></mrow><mrow><mi mathvariant="normal">U</mi><mi mathvariant="normal">Y</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116267&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116266&type= 5.84200001 3.72533321 刚度系数（N/mm）。（3）约束力 <math id="M217"><msub><mrow><mi mathvariant="bold-italic">F</mi></mrow><mrow><mi mathvariant="normal">U</mi><mi mathvariant="normal">Z</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116269&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116268&type= 5.50333309 3.72533321 对应支链拉压变形量： <math display="block" id="M218"><mi mathvariant="normal">Δ</mi><msub><mrow><mi>λ</mi></mrow><mrow><mi mathvariant="normal">U</mi><mi mathvariant="normal">Z</mi></mrow></msub><mo>=</mo><mfrac><mrow><msub><mrow><mi>F</mi></mrow><mrow><mi mathvariant="normal">U</mi><mi mathvariant="normal">Z</mi></mrow></msub><msub><mrow><mi>l</mi></mrow><mrow><mi mathvariant="normal">U</mi></mrow></msub></mrow><mrow><msub><mrow><mi>E</mi></mrow><mrow><mi mathvariant="normal">U</mi></mrow></msub><msub><mrow><mi>A</mi></mrow><mrow><mi mathvariant="normal">U</mi></mrow></msub></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116271&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116270&type= 22.01333237 8.38199997 ，其中： <math id="M219"><mi mathvariant="normal">Δ</mi><msub><mrow><mi>λ</mi></mrow><mrow><mi mathvariant="normal">U</mi><mi mathvariant="normal">Z</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116273&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116272&type= 7.11199999 3.72533321 为约束力 <math id="M220"><msub><mrow><mi>F</mi></mrow><mrow><mi mathvariant="normal">U</mi><mi mathvariant="normal">Z</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116277&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116276&type= 5.50333309 3.72533321 造成的拉压变形量（mm）； <math id="M221"><msub><mrow><mi>F</mi></mrow><mrow><mi mathvariant="normal">U</mi><mi mathvariant="normal">Z</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116277&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116276&type= 5.50333309 3.72533321 为约束力 <math id="M222"><msub><mrow><mi mathvariant="bold-italic">F</mi></mrow><mrow><mi mathvariant="normal">U</mi><mi mathvariant="normal">Z</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116279&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116278&type= 5.50333309 3.72533321 的数值大小（N）； <math id="M223"><msub><mrow><mi>A</mi></mrow><mrow><mi mathvariant="normal">U</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116281&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116280&type= 4.74133301 3.72533321 为U支链支柱横截面积（mm 2 ）。由式（54）式（54）进一步计算刚度系数： <math display="block" id="M224"><msub><mrow><mi>k</mi></mrow><mrow><mi mathvariant="normal">U</mi><mi mathvariant="normal">Z</mi></mrow></msub><mo>=</mo><mfrac><mrow><msub><mrow><mi mathvariant="bold-italic">F</mi></mrow><mrow><mi mathvariant="normal">U</mi><mi mathvariant="normal">Z</mi></mrow></msub></mrow><mrow><mi mathvariant="normal">Δ</mi><msub><mrow><mi>λ</mi></mrow><mrow><mi mathvariant="normal">U</mi><mi mathvariant="normal">Z</mi></mrow></msub></mrow></mfrac><mo>=</mo><mfrac><mrow><msub><mrow><mi>E</mi></mrow><mrow><mi mathvariant="normal">U</mi></mrow></msub><msub><mrow><mi>A</mi></mrow><mrow><mi mathvariant="normal">U</mi></mrow></msub></mrow><mrow><msub><mrow><mi>l</mi></mrow><mrow><mi mathvariant="normal">U</mi></mrow></msub></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116283&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116282&type= 32.08866501 8.38199997 ，其中， <math id="M225"><msub><mrow><mi>k</mi></mrow><mrow><mi mathvariant="normal">U</mi><mi mathvariant="normal">Z</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116285&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116284&type= 4.48733330 3.72533321 为恰约束U支链约束 <math id="M226"><msub><mrow><mi mathvariant="bold-italic">F</mi></mrow><mrow><mi mathvariant="normal">U</mi><mi mathvariant="normal">Z</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116287&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116286&type= 5.50333309 3.72533321 刚度系数（N/mm）。（4）约束力偶 <math id="M227"><msub><mrow><mi mathvariant="bold-italic">M</mi></mrow><mrow><mi mathvariant="normal">U</mi><mi mathvariant="normal">Z</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116289&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116288&type= 6.51933336 3.80999994 对应支链弯曲变形只考虑约束力偶 <math id="M228"><msub><mrow><mi mathvariant="bold-italic">M</mi></mrow><mrow><mi mathvariant="normal">U</mi><mi mathvariant="normal">Z</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116291&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116290&type= 6.51933336 3.80999994 引起的弯曲变形，约束力偶 <math id="M229"><msub><mrow><mi mathvariant="bold-italic">M</mi></mrow><mrow><mi mathvariant="normal">U</mi><mi mathvariant="normal">Z</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116293&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116292&type= 6.51933336 3.80999994 矢量方向沿恰约束U支链虎克铰两个轴线形成平面的法线方向。 a.求解约束力偶 <math id="M230"><msub><mrow><mi mathvariant="bold-italic">M</mi></mrow><mrow><mi mathvariant="normal">U</mi><mi mathvariant="normal">Z</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116295&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116294&type= 6.51933336 3.80999994 的矢量形式： <math display="block" id="M231"><msub><mrow><mi mathvariant="bold-italic">M</mi></mrow><mrow><mi mathvariant="normal">U</mi><mi mathvariant="normal">Z</mi></mrow></msub><mo>=</mo><msubsup><mrow><mi mathvariant="bold-italic">M</mi></mrow><mrow><mi mathvariant="normal">U</mi></mrow><mrow><mi>r</mi><mi>m</mi></mrow></msubsup><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">R</mi></mrow><mrow><mi>x</mi></mrow></msub><mo>×</mo><msub><mrow><mi mathvariant="bold-italic">R</mi></mrow><mrow><mi>y</mi></mrow></msub><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116297&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116296&type= 34.37466812 4.65666676 ，其中： <math id="M232"><msubsup><mrow><mi mathvariant="bold-italic">M</mi></mrow><mrow><mi mathvariant="normal">U</mi></mrow><mrow><mi>r</mi><mi>m</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116299&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116298&type= 6.60400009 4.06400013 为U支链约束力偶（N·mm）； <math id="M233"><msub><mrow><mi mathvariant="bold-italic">R</mi></mrow><mrow><mi>x</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116301&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116300&type= 4.06400013 3.72533321 为U支链旋转副1的单位矢量方向； <math id="M234"><msub><mrow><mi mathvariant="bold-italic">R</mi></mrow><mrow><mi>y</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116303&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116302&type= 3.97933316 3.72533321 为U支链旋转副2的单位矢量方向。 b.约束力偶 <math id="M235"><msub><mrow><mi mathvariant="bold-italic">M</mi></mrow><mrow><mi mathvariant="normal">U</mi><mi mathvariant="normal">Z</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116305&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116304&type= 6.51933336 3.80999994 在垂直与支链的方向投影： <math display="block" id="M236"><msubsup><mrow><mi mathvariant="bold-italic">M</mi></mrow><mrow><mi mathvariant="normal">U</mi></mrow><mrow><mi>r</mi><mi>m</mi><mi>z</mi></mrow></msubsup><mo>=</mo><mfenced open="|" close="|" separators="|"><mrow><msubsup><mrow><mi mathvariant="bold-italic">M</mi></mrow><mrow><mi mathvariant="normal">U</mi></mrow><mrow><mi>r</mi><mi>m</mi></mrow></msubsup><mo>×</mo><mi mathvariant="bold-italic">Z</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116307&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116306&type= 28.53266525 4.48733330 ，其中， <math id="M237"><msubsup><mrow><mi mathvariant="bold-italic">M</mi></mrow><mrow><mi mathvariant="normal">U</mi></mrow><mrow><mi>r</mi><mi>m</mi><mi>z</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116309&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116308&type= 7.45066643 4.06400013 为U支链约束力偶沿单位 Z 轴方向分量大小（N·mm）。 c.造成U支链弯曲变形量： <math display="block" id="M238"><mi mathvariant="normal">Δ</mi><msubsup><mrow><mi>λ</mi></mrow><mrow><mi mathvariant="normal">U</mi></mrow><mrow><mi>r</mi><mi>m</mi></mrow></msubsup><mo>=</mo><mfrac><mrow><msubsup><mrow><mi>M</mi></mrow><mrow><mi mathvariant="normal">U</mi></mrow><mrow><mi>r</mi><mi>m</mi><mi>z</mi></mrow></msubsup><msubsup><mrow><mi>l</mi></mrow><mrow><mi mathvariant="normal">U</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup></mrow><mrow><mn mathvariant="normal">2</mn><msub><mrow><mi>E</mi></mrow><mrow><mi mathvariant="normal">U</mi></mrow></msub><msub><mrow><mi>A</mi></mrow><mrow><mi mathvariant="normal">U</mi></mrow></msub></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116311&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116310&type= 23.45266724 8.63599968 ，其中， <math id="M239"><mi mathvariant="normal">Δ</mi><msubsup><mrow><mi>λ</mi></mrow><mrow><mi mathvariant="normal">U</mi></mrow><mrow><mi>r</mi><mi>m</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116313&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116312&type= 6.68866634 3.97933316 为约束力偶 <math id="M240"><msub><mrow><mi mathvariant="bold-italic">M</mi></mrow><mrow><mi mathvariant="normal">U</mi><mi mathvariant="normal">Z</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116315&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116314&type= 6.51933336 3.80999994 造成U支链的弯曲变形量（mm）。 d.约束力偶 <math id="M241"><msub><mrow><mi mathvariant="bold-italic">M</mi></mrow><mrow><mi mathvariant="normal">U</mi><mi mathvariant="normal">Z</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116317&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116316&type= 6.51933336 3.80999994 刚度系数 <math display="block" id="M242"><msubsup><mrow><mi>k</mi></mrow><mrow><mi mathvariant="normal">U</mi></mrow><mrow><mi>r</mi><mi>m</mi><mi>z</mi></mrow></msubsup><mo>=</mo><mfrac><mrow><msubsup><mrow><mi mathvariant="bold-italic">M</mi></mrow><mrow><mi mathvariant="normal">U</mi></mrow><mrow><mi>r</mi><mi>m</mi></mrow></msubsup></mrow><mrow><mi mathvariant="normal">Δ</mi><msubsup><mrow><mi>λ</mi></mrow><mrow><mi mathvariant="normal">U</mi></mrow><mrow><mi>r</mi><mi>m</mi></mrow></msubsup></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116319&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116318&type= 17.35666656 8.89000034 . （59）由式（56）式（56）至式（59）联立可得刚度系数 <math id="M243"><msubsup><mrow><mi>k</mi></mrow><mrow><mi mathvariant="normal">U</mi></mrow><mrow><mi>r</mi><mi>m</mi><mi>z</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116321&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116320&type= 5.16466665 3.97933316 为： <math display="block" id="M244"><msubsup><mrow><mi>k</mi></mrow><mrow><mi mathvariant="normal">U</mi></mrow><mrow><mi>r</mi><mi>m</mi><mi>z</mi></mrow></msubsup><mo>=</mo><mfrac><mrow><mn mathvariant="normal">2</mn><msub><mrow><mi>E</mi></mrow><mrow><mi mathvariant="normal">U</mi></mrow></msub><msub><mrow><mi>I</mi></mrow><mrow><mi mathvariant="normal">U</mi></mrow></msub></mrow><mrow><mfenced open="|" close="|" separators="|"><mrow><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">R</mi></mrow><mrow><mi>x</mi></mrow></msub><mo>×</mo><msub><mrow><mi mathvariant="bold-italic">R</mi></mrow><mrow><mi>y</mi></mrow></msub><mo stretchy="false">)</mo><mo>×</mo><mi mathvariant="bold-italic">Z</mi></mrow></mfenced><msubsup><mrow><mi>l</mi></mrow><mrow><mi mathvariant="normal">U</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116323&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116322&type= 39.11600113 10.07533360 . 汇总求得弹性变形矩阵： <math display="block" id="M245"><msub><mrow><mi mathvariant="bold-italic">K</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>=</mo><mi>d</mi><mi>i</mi><mi>g</mi><mi>a</mi><mo stretchy="false">(</mo><msub><mrow><mi>k</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msubsup><mrow><mi>k</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>r</mi></mrow></msubsup><mo>,</mo><msub><mrow><mi>k</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>,</mo><msubsup><mrow><mi>k</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>r</mi><mi>f</mi></mrow></msubsup><mo>,</mo><msubsup><mrow><mi>k</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>r</mi><mi>m</mi></mrow></msubsup><mo>,</mo><msubsup><mrow><mi>k</mi></mrow><mrow><mi mathvariant="normal">U</mi></mrow><mrow><mi>x</mi></mrow></msubsup><mo>,</mo><msubsup><mrow><mi>k</mi></mrow><mrow><mi mathvariant="normal">U</mi></mrow><mrow><mi>y</mi></mrow></msubsup><mo>,</mo><msubsup><mrow><mi>k</mi></mrow><mrow><mi mathvariant="normal">U</mi></mrow><mrow><mi>z</mi></mrow></msubsup><mo>,</mo><msubsup><mrow><mi>k</mi></mrow><mrow><mi mathvariant="normal">U</mi></mrow><mrow><mi>m</mi><mi>z</mi></mrow></msubsup><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116325&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116324&type= 65.44733429 5.07999992 ，其中， <math id="M246"><msub><mrow><mi mathvariant="bold-italic">K</mi></mrow><mrow><mi>p</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116327&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116326&type= 4.14866638 3.72533321 为2RRPU/RPU/U调姿平台弹性变形矩阵；弹性变形方程： <math display="block" id="M247"><msub><mrow><mi mathvariant="bold-italic">τ</mi></mrow><mrow><mn mathvariant="normal">9</mn><mo>×</mo><mn mathvariant="normal">1</mn></mrow></msub><mo>=</mo><mo stretchy="false">[</mo><msub><mrow><mi mathvariant="bold-italic">K</mi></mrow><mrow><mi>p</mi></mrow></msub><msub><mrow><mo stretchy="false">]</mo></mrow><mrow><mn mathvariant="normal">9</mn><mo>×</mo><mn mathvariant="normal">9</mn></mrow></msub><mi>δ</mi><msub><mrow><mi mathvariant="bold-italic">q</mi></mrow><mrow><mn mathvariant="normal">9</mn><mo>×</mo><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116329&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116328&type= 32.17333603 4.82600021 ，其中： <math id="M248"><msub><mrow><mi mathvariant="bold-italic">τ</mi></mrow><mrow><mn mathvariant="normal">9</mn><mo>×</mo><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116331&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116330&type= 6.43466663 3.72533321 为各关节的力矩矢量（N）； <math id="M249"><mi>δ</mi><msub><mrow><mi mathvariant="bold-italic">q</mi></mrow><mrow><mn mathvariant="normal">9</mn><mo>×</mo><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116333&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116332&type= 8.63599968 4.57200003 为个关节的虚位移（mm）。根据虚功原理，平台在平衡情况下，各关节所做虚功之和等于末端载荷虚功之和： <math display="block" id="M250"><msup><mrow><mi mathvariant="bold-italic">τ</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup><mo>·</mo><mi>δ</mi><mi mathvariant="bold-italic">q</mi><mo>=</mo><msup><mrow><mi mathvariant="bold-italic">F</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup><mo>·</mo><mi mathvariant="bold-italic">D</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116335&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116334&type= 22.01333237 3.89466691 ，其中： <math id="M251"><mi mathvariant="bold-italic">τ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116337&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116336&type= 1.77800000 2.79399991 为关节力矩矢量（N）； <math id="M252"><mi>δ</mi><mi mathvariant="bold-italic">q</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116345&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116344&type= 3.89466691 3.64066648 为各关节的虚位移（mm）； <math id="M253"><mi mathvariant="bold-italic">F</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116341&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116340&type= 2.79399991 2.79399991 为末端载荷的力矩矢量（N）； <math id="M254"><mi mathvariant="bold-italic">D</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116343&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116342&type= 3.04800010 2.79399991 为末端的虚位移（mm）。 <math id="M255"><mi>δ</mi><mi mathvariant="bold-italic">q</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116345&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116344&type= 3.89466691 3.64066648 和 D 之间非独立，应该满足几何约束条件，两者之间的几何约束由雅可比所规定，应满足： <math display="block" id="M256"><mi mathvariant="bold-italic">D</mi><mo>=</mo><mo stretchy="false">[</mo><mi mathvariant="bold-italic">J</mi><mo stretchy="false">]</mo><mi>δ</mi><mi mathvariant="bold-italic">q</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116347&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116346&type= 18.28800011 3.97933316 . 由式（63）式（63）和式（64）式（64）联立得： <math display="block" id="M257"><mi mathvariant="bold-italic">F</mi><mo>=</mo><mi mathvariant="bold-italic">K</mi><mi>δ</mi><mi mathvariant="bold-italic">p</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116349&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116348&type= 13.97000027 3.64066648 ，其中， <math id="M258"><mi mathvariant="bold-italic">K</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116351&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116350&type= 3.13266683 2.79399991 为整个模型的刚度矩阵： 2.3　刚度模型验证 2 平台姿态角度 <math id="M259"><mi>α</mi><mo>=</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116353&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116352&type= 8.63599968 2.79399991 ， <math id="M260"><mi>β</mi><mo>=</mo><mo>-</mo><mn mathvariant="normal">5</mn><mo>°</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116355&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116354&type= 13.54666710 3.89466691 ， <math id="M261"><mi>α</mi><mo>=</mo><mo>-</mo><mn mathvariant="normal">5</mn><mo>°</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116357&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116356&type= 13.29266548 3.04800010 ， <math id="M262"><mi>β</mi><mo>=</mo><mn mathvariant="normal">5</mn><mo>°</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116359&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116358&type= 9.99066734 3.89466691 姿态下，平台各支链的受力变形云图如图6 图6 。随机生成50组载荷数据，载荷数据的范围为： <math id="M263"><mo>±</mo><mo stretchy="false">(</mo><mn mathvariant="normal">100</mn><mo>~</mo><mn mathvariant="normal">1</mn><mtext> </mtext><mn mathvariant="normal">000</mn><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116364&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116363&type= 22.86000061 3.55599999 N。分别通过Ansys仿真模型和理论刚度模型对上平台中心的微位移进行预测。两种方式预测结果对比如图7 图7 ~ 图8 图8 所示， α =0°， β =0°姿态下，拟合直线线性度为， x 方向0.89， y 方向0.58， z 方向0.96，合方向0.73； α = - 5°， β =5°姿态下，拟合直线线性度为， x 方向0.85， y 方向0.75， z 方向0.70，合方向0.77（越接近1，线性度越好），即在 x ， y ， z 合方向上，Ansys仿真模型与理论刚度模型的预测数据结果在变化趋势上是一致，刚度模型能够反应平台所受的外界载荷与末端形成的变形之间的关系。 1;2; 3 仿真模型与数据采集 1 通过Simulink-Adams-Ansys软件联合搭建图9 图9 所示的仿真模拟环境。其中Adams软件搭建刚性全物理仿真环境；采用Ansys对支链的主要构件进行柔性化；Simulink作为Adams-Ansys柔性仿真环境的控制系统，完成指令发出以及关键数据的采集。Simulink环境仿真的控制系统基于实时接受Adams运行的反馈数据，发送平台仿真运行的指令，Simulink与Adams联合仿真数据传输信号如表1 表1 。输入信号类型信号说明 L1_drive_input 输入位置模式驱动RRPU支链电动缸 L2_drive_input 输入位置模式驱动RPU支链电动缸 revolve_acc_input 输入加速度模式驱动跟踪平台回转电机 Pitch_acc_input 输入加速度模式驱动跟踪平台俯仰电机 L1_length_output 输出 RRPU支链电动缸的实际长度 L2_length_output 输出 RPU支链电动缸的实际长度 alpha_real_output 输出调姿平台绕 x 轴 <math id="M266"><mi>α</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116382&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116381&type= 1.69333339 2.37066650 角度 beta_real_output 输出调姿平台绕 y 轴 <math id="M267"><mi>β</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116384&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116383&type= 1.94733346 3.13266683 角度 revolve_angle_output 输出跟踪平台实时回转角度 Pitch_angle_output 输出跟踪平台实时俯仰角度 Fe_x_output 输出上平台沿 z 轴方向的受力 Fe_y_output 输出上平台沿 y 轴方向的受力 Fe_z_output 输出上平台沿 z 轴方向的受力 Me_x_output 输出上平台沿 x 轴方向的力偶 Me_y_output 输出上平台沿 y 轴方向的力偶 Me_z_output 输出上平台沿 z 轴方向的力偶 Time_output 输出仿真过程Adams的实时时间仿真采用调姿平台与跟踪平台联动的形式模拟平台实际的运载环境。跟踪平台采用“加速度”模式，模拟实际运转过程中跟踪目标产生的随机运动载荷；调姿平台进行“全空间姿态扫描”模式，模拟平台在任意调整姿态下受载情况。通过仿真实验，采集15 720组实验数据，用于预测模型的训练，其中10组实验数据见表2 表2 和表3 表3 。编号 <math id="M268"><msup><mrow><mi>α</mi></mrow><mrow><mi>r</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116388&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116387&type= 2.87866688 3.04800010 /rad <math id="M269"><msup><mrow><mi>β</mi></mrow><mrow><mi>r</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116390&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116389&type= 3.21733332 3.89466691 /rad <math id="M270"><msubsup><mrow><mi>L</mi></mrow><mrow><mn>1</mn></mrow><mrow><mi>r</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116392&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116391&type= 3.64066648 3.97933316 /mm <math id="M271"><msubsup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow><mrow><mi>r</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116394&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116393&type= 3.64066648 3.97933316 /mm <math id="M272"><msup><mrow><mi>α</mi></mrow><mrow><mi>t</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116396&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116395&type= 2.62466669 3.04800010 /rad <math id="M273"><msup><mrow><mi>β</mi></mrow><mrow><mi>t</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116399&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116398&type= 3.04800010 3.89466691 /rad <math id="M274"><msubsup><mrow><mi>L</mi></mrow><mrow><mn>1</mn></mrow><mrow><mi>t</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116401&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116400&type= 3.64066648 3.97933316 /mm <math id="M275"><msubsup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow><mrow><mi>t</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116403&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116402&type= 3.64066648 3.97933316 /mm 1 -0.140 -0.084 701.106 614.398 -0.131 -0.093 704.847 617.847 2 0.175 0.047 650.004 737.750 0.182 0.038 653.847 740.847 3 -0.052 0.182 598.746 648.001 -0.044 0.174 602.097 651.097 4 0.175 0.154 610.246 737.753 0.182 0.144 614.097 740.847 5 0.175 -0.042 684.484 737.766 0.182 -0.052 688.347 740.847 6 -0.122 0.103 628.500 621.250 -0.114 0.095 631.847 624.347 7 0.175 0.099 630.491 737.757 0.182 0.089 634.347 740.847 8 -0.174 0.122 622.127 601.850 -0.167 0.111 626.097 604.597 9 0.018 0.132 617.243 675.256 0.025 0.122 621.097 678.347 10 0.123 -0.209 750.723 717.028 0.130 -0.217 754.097 719.847 编号 <math id="M276"><msubsup><mrow><mi>F</mi></mrow><mrow><mi>e</mi><mi>x</mi></mrow><mrow><mi>r</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116407&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116406&type= 4.06400013 3.38666677 /N <math id="M277"><msubsup><mrow><mi>F</mi></mrow><mrow><mi>e</mi><mi>y</mi></mrow><mrow><mi>r</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116409&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116408&type= 3.97933316 3.38666677 /N <math id="M278"><msubsup><mrow><mi>F</mi></mrow><mrow><mi>e</mi><mi>z</mi></mrow><mrow><mi>r</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116411&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116410&type= 3.89466691 3.38666677 /N <math id="M279"><msubsup><mrow><mi>M</mi></mrow><mrow><mi>e</mi><mi>x</mi></mrow><mrow><mi>r</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116413&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116412&type= 4.91066647 3.47133350 /（N·m -1 ） <math id="M280"><msubsup><mrow><mi>M</mi></mrow><mrow><mi>e</mi><mi>y</mi></mrow><mrow><mi>r</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116415&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116414&type= 4.82600021 3.47133350 /（N·m -1 ） <math id="M281"><msubsup><mrow><mi>M</mi></mrow><mrow><mi>e</mi><mi>z</mi></mrow><mrow><mi>r</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116417&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116416&type= 4.74133301 3.47133350 /（N·m -1 ） 1 -36.200 34.446 -5.415 -23 103.390 -25 070.880 2 506.712 2 -1.222 0.613 -3.934 -105.952 -803.991 -110.028 3 0.553 -0.695 -5.230 275.456 773.382 -96.211 4 0.950 -0.612 -4.032 700.720 909.381 46.906 5 5.849 -5.323 -4.907 3 856.979 3 889.783 723.520 6 -0.305 0.018 -2.468 -174.750 -120.763 31.912 7 2.894 -2.041 -5.466 1 748.686 2 200.867 187.711 8 39.159 -37.624 -13.199 22 967.717 27 074.169 -7 718.717 9 2.222 -2.072 -5.071 1 350.870 1 801.997 -157.752 10 9.564 -11.141 -1.386 7 316.494 6 425.502 1 919.056 由主动支链驱动数据，通过正解计算姿态数据与仿真实际测量姿态数据对比计算偏差值，绘制出各姿态下偏差出现的幅值数据，如图10 图10 ，偏差值随两个姿态角呈现带状区域分布，在各姿态下，误差整体具备随机性，误差本身没有明显的变化趋势。 1,2; 4 预测模型搭建与训练 1 采用神经网络算法进行姿态误差的最终预测。 4.1　基于刚度模型预测误差趋势 2 由于调姿平台主要作用是补偿转角上的误差，因此只参考刚度模型预测的 <math id="M282"><msub><mrow><mi>δ</mi></mrow><mrow><mi>α</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116429&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116428&type= 3.13266683 3.72533321 和 <math id="M283"><msub><mrow><mi>δ</mi></mrow><mrow><mi>β</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116431&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116430&type= 3.13266683 3.72533321 。如图11 图11 ，采用实际姿态角度和理论姿态角度差值作为姿态误差量，并把预测的姿态误差量与实际误差量进行对比。然后预测的角度偏差值通过调姿平台速度Jacobian矩阵映射到驱动杆长的误差量；同样，基于采集的仿真实验数据，计算实际驱动杆长与理论驱动杆长误差量作为补偿实际需要补偿量。可进一步对比预测的驱动补偿量与实际需要的驱动补偿量的关系趋势。 4 . 1 . 1　基于刚度矩阵预测位姿误差 3 基于外部载荷和平台姿态数据，利用刚度模型对采集的数据进行初步预测，核心映射关系为： <math display="block" id="M284"><mfenced open="[" close="]" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><mi>δ</mi><mi mathvariant="bold-italic">r</mi></mtd></mtr><mtr><mtd><mi>δ</mi><mi mathvariant="bold-italic">θ</mi></mtd></mtr></mtable></mrow></mfenced><mo>=</mo><mo stretchy="false">[</mo><msub><mrow><mi mathvariant="bold-italic">K</mi></mrow><mrow><mi>α</mi><mo>,</mo><mi>β</mi></mrow></msub><mo stretchy="false">]</mo><mfenced open="[" close="]" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><msub><mrow><mi mathvariant="bold-italic">F</mi></mrow><mrow><mi mathvariant="normal">e</mi></mrow></msub></mtd></mtr><mtr><mtd><msub><mrow><mi mathvariant="bold-italic">M</mi></mrow><mrow><mi mathvariant="normal">e</mi></mrow></msub></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116436&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116435&type= 30.56466866 9.05933285 ，其中： <math id="M285"><mi>δ</mi><mi mathvariant="bold-italic">r</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116438&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116437&type= 3.64066648 2.79399991 为上平台的中心点变形位移矢量（m）； <math id="M286"><mi>δ</mi><mi mathvariant="bold-italic">θ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116440&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116439&type= 3.89466691 2.79399991 为上平台中心点角度微偏移矢量，［ <math id="M287"><mi>δ</mi><msub><mrow><mi mathvariant="bold-italic">θ</mi></mrow><mrow><mi>α</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116442&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116441&type= 4.99533367 3.72533321 <math id="M288"><mi>δ</mi><msub><mrow><mi mathvariant="bold-italic">θ</mi></mrow><mrow><mi>β</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116444&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116443&type= 4.99533367 3.72533321 <math id="M289"><mi>δ</mi><msub><mrow><mi mathvariant="bold-italic">θ</mi></mrow><mrow><mi>γ</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116446&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116445&type= 4.99533367 3.72533321 ］；［ <math id="M290"><msub><mrow><mi mathvariant="bold-italic">K</mi></mrow><mrow><mi>α</mi><mo>,</mo><mi>β</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116448&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116447&type= 6.09600019 3.72533321 ］为考虑平台姿态的刚度矩阵。 4 . 1 . 2　姿态映射关系 3 并联机构包含绕 X 轴与 Y 轴的转动自由度，因此主要分析刚度模型在该方向变形。刚度模型预测的变形量主要是在绝对坐标系下，上平台姿态的偏转量，利用速度雅可比矩阵把变形映射到U支链两个旋转副的偏转量，微变形的关系式为： <math display="block" id="M291"><mi>δ</mi><mi mathvariant="bold-italic">θ</mi><mo>=</mo><mi>δ</mi><mi>α</mi><msub><mrow><mi>ω</mi></mrow><mrow><mi>α</mi></mrow></msub><mo>+</mo><mi>δ</mi><mi>β</mi><msub><mrow><mi>ω</mi></mrow><mrow><mi>β</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116450&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116449&type= 28.53266525 4.65666676 ，其中： <math id="M292"><mi>δ</mi><mi>α</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116452&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116451&type= 3.89466691 2.79399991 ， <math id="M293"><mi>δ</mi><mi>β</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116454&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116453&type= 3.89466691 3.64066648 为绕U支链 X ， Y 轴的角度微偏移量（rad）； <math id="M294"><msub><mrow><mi>ω</mi></mrow><mrow><mi>α</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116456&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116455&type= 3.72533321 3.72533321 ， <math id="M295"><msub><mrow><mi>ω</mi></mrow><mrow><mi>β</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116458&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116457&type= 3.72533321 3.72533321 为U支链 X ， Y 轴线的单位矢量方向。联立式（66）式（66）和式（67）式（67）得预测转角与驱动关节映射关系： <math display="block" id="M296"><mfenced open="{" close="" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><mi>δ</mi><mi>α</mi><mo>=</mo><mi>δ</mi><msub><mrow><mi mathvariant="bold-italic">θ</mi></mrow><mrow><mi>x</mi></mrow></msub></mtd></mtr><mtr><mtd><mi>δ</mi><mi>β</mi><mo>=</mo><mfrac><mrow><mi>δ</mi><msub><mrow><mi mathvariant="bold-italic">θ</mi></mrow><mrow><mi>y</mi></mrow></msub></mrow><mrow><mi mathvariant="normal">c</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">s</mi><mi>α</mi></mrow></mfrac></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116460&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116459&type= 18.88066673 13.71599960 . 4 . 1 . 3　姿态误差预测对比 3 由式（66）式（66）对仿真实验数据预测平台姿态角度误差，再由式（68）式（68）把预测的误差角度映射到U支链两转轴对应的角度误差。如图12 图12 ，由 <math id="M297"><mi>α</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116462&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116461&type= 1.94733346 2.79399991 转角预测效果图可知，当预测偏差值 <math id="M298"><mi>α</mi><mo><</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116464&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116463&type= 8.63599968 2.79399991 时，存在上升映射趋势；当预测偏差值 <math id="M299"><mi>α</mi><mo>></mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116466&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116465&type= 8.63599968 2.79399991 时，存在下降映射趋势。对于 <math id="M300"><mi>β</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116468&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116467&type= 2.28600001 3.64066648 转角同样具备映射的趋势。由数值分析，刚度模型预测偏差值具有正负变化，是由于理论刚度模型预测姿态误差时，没有考虑上平台刚度，仿真模型中上平台受到跟踪平台的重力载荷会导致支链的变形在特定方向始终存在一定偏移量。 1;2; 对比刚度模型预测结果与仿真数据的实际误差值，其中10组数据见表4 表4 。编号 <math id="M301"><mi mathvariant="normal">Δ</mi><msup><mrow><mi>α</mi></mrow><mrow><mi>r</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116478&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116477&type= 4.57200003 2.62466669 /rad <math id="M302"><mi mathvariant="normal">Δ</mi><msup><mrow><mi>β</mi></mrow><mrow><mi>r</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116480&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116479&type= 4.74133301 3.38666677 /rad <math id="M303"><mi>δ</mi><msubsup><mrow><mi>α</mi></mrow><mrow><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">f</mi><mi mathvariant="normal">f</mi></mrow><mrow><mi mathvariant="normal">p</mi><mi mathvariant="normal">r</mi><mi mathvariant="normal">e</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116482&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116481&type= 5.75733376 3.38666677 /rad <math id="M304"><mi>δ</mi><msubsup><mrow><mi>β</mi></mrow><mrow><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">f</mi><mi mathvariant="normal">f</mi></mrow><mrow><mi mathvariant="normal">p</mi><mi mathvariant="normal">r</mi><mi mathvariant="normal">e</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116484&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116483&type= 5.92666674 4.14866638 /rad 1 -0.330 88 -0.144 78 -0.009 33 0.009 10 2 0.114 59 -0.056 31 -0.006 59 0.007 98 3 0.048 08 0.073 82 -0.006 73 0.009 60 4 0.018 12 -0.025 92 -0.007 28 0.008 05 5 0.014 69 -0.015 57 -0.007 27 0.009 38 6 0.297 89 -0.018 13 -0.007 01 0.008 25 7 -0.068 59 -0.308 67 -0.007 90 0.008 00 8 0.094 53 -0.063 64 -0.006 80 0.009 39 9 0.007 54 0.009 02 -0.007 18 0.009 37 10 -0.008 56 -0.013 48 -0.007 94 0.008 99 4 . 1 . 4　驱动误差预测对比 3 由雅可比矩阵得转角偏差与驱动偏差映射关系： <math display="block" id="M305"><mfenced open="[" close="]" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><mi>δ</mi><msub><mrow><mi>l</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mtd></mtr><mtr><mtd><mi>δ</mi><msub><mrow><mi>l</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mtd></mtr></mtable></mrow></mfenced><mo>=</mo><mo stretchy="false">[</mo><mi mathvariant="bold-italic">J</mi><mo stretchy="false">]</mo><mfenced open="[" close="]" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><mi>δ</mi><mi>α</mi></mtd></mtr><mtr><mtd><mi>δ</mi><mi>β</mi></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116488&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116487&type= 26.33133125 9.05933285 . 基于预测姿态误差，由式（69）式（69）进一步预测出驱动量误差值。如图13 图13 所示，理论刚度模型驱动误差预测值与实际驱动误差值之间存在正相关变化趋势。 1;2; 部分预测数据结果见表5 表5 。编号 <math id="M306"><mi mathvariant="normal">Δ</mi><msubsup><mrow><mi>L</mi></mrow><mrow><mn>1</mn></mrow><mrow><mi>r</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116498&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116497&type= 5.24933338 3.38666677 /mm <math id="M307"><mi mathvariant="normal">Δ</mi><msubsup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow><mrow><mi>r</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116500&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116499&type= 5.24933338 3.38666677 /mm <math id="M308"><mi>δ</mi><msubsup><mrow><mi>L</mi></mrow><mrow><mn>1</mn><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">f</mi><mi mathvariant="normal">f</mi></mrow><mrow><mi mathvariant="normal">p</mi><mi mathvariant="normal">r</mi><mi mathvariant="normal">e</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116502&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116501&type= 6.94266701 3.38666677 /mm <math id="M309"><mi>δ</mi><msubsup><mrow><mi>L</mi></mrow><mrow><mn>2</mn><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">f</mi><mi mathvariant="normal">f</mi></mrow><mrow><mi mathvariant="normal">p</mi><mi mathvariant="normal">r</mi><mi mathvariant="normal">e</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116504&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116503&type= 6.94266701 3.38666677 /mm 1 2.626 54 -6.271 16 -3.740 94 -3.449 43 2 0.626 43 0.527 63 -3.843 02 -3.096 76 3 -0.114 18 0.319 11 -3.350 20 -3.095 95 4 -0.133 53 0.426 58 -3.850 36 -3.093 61 5 -0.358 79 -0.147 79 -3.862 59 -3.081 32 6 0.184 46 0.073 65 -3.346 78 -3.097 22 7 -0.457 10 0.336 42 -3.855 70 -3.089 64 8 -10.757 58 6.424 19 -3.969 88 -2.747 18 9 -0.316 61 0.423 31 -3.853 94 -3.091 10 10 0.589 86 0.990 93 -3.373 54 -2.818 80 4.2　BP神经网络训练 2 利用神经网络算法对理论刚度模型初步预测的驱动补偿量进一步预测，完成预测结果精度提升。为说明理论刚度模型初步预测方法的优越性，搭建3组神经网络模型，如表6 表6 所示。model01：理论姿态角 <math id="M310"><msup><mrow><mi>α</mi></mrow><mrow><mi>t</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116508&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116507&type= 2.62466669 3.04800010 和理论姿态角 <math id="M311"><msup><mrow><mi>β</mi></mrow><mrow><mi>t</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116514&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116513&type= 3.04800010 3.89466691 为输入；model02：理论姿态角 <math id="M312"><msup><mrow><mi>α</mi></mrow><mrow><mi>t</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116529&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116528&type= 2.62466669 3.04800010 、理论姿态角 <math id="M313"><msup><mrow><mi>β</mi></mrow><mrow><mi>t</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116514&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116513&type= 3.04800010 3.89466691 、外力 <math id="M314"><msub><mrow><mi mathvariant="bold-italic">F</mi></mrow><mrow><mi mathvariant="normal">e</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116516&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116515&type= 3.72533321 3.72533321 、外力矩 <math id="M315"><msub><mrow><mi mathvariant="bold-italic">M</mi></mrow><mrow><mi mathvariant="normal">e</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116518&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116517&type= 4.74133301 3.80999994 为输入；model03：刚度模型预测补偿量 <math id="M316"><mi mathvariant="normal">Δ</mi><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=38116520&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116519&type= 6.09600019 3.72533321 和 <math id="M317"><mi mathvariant="normal">Δ</mi><msub><mrow><mi>L</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116522&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116521&type= 6.09600019 3.72533321 为输入。采用统一基本结构避免神经网络结构不同引起训练效果差异。激活函数：atan函数；反向传递函数：Adams函数；损失函数：均值损失函数；model01结构如图14 图14 。注： “结构”列为输入个数、隐藏层个数、每层神经元个数、输出个数。序号名称输入输出结构 1 model01 α t ， β t δL 1 ， δL 2 （2，20，10，2） 2 model02 α t ， β t ， F e ， M e δL 1 ， δL 2 （8，20，10，2） 3 model03 Δ L 1 ，Δ L 2 δL 1 ， δL 2 （2，20，10，2） 4 . 2 . 1　model01基于姿态数据训练 3 利用仿真实验数据训练神经网络，将数据分为训练数据和测试数据，总数量为15 720组， <math id="M318"><msup><mrow><mi>α</mi></mrow><mrow><mi>t</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116529&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116528&type= 2.62466669 3.04800010 ， <math id="M319"><msup><mrow><mi>β</mi></mrow><mrow><mi>t</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116531&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116530&type= 3.04800010 3.89466691 与 <math id="M320"><mi>δ</mi><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=38116533&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116532&type= 5.50333309 3.72533321 ， <math id="M321"><mi>δ</mi><msub><mrow><mi>L</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116535&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116534&type= 5.50333309 3.72533321 量级相差较大，做归一化处理，逐步增加训练次数，记录逐次训练后的损失值。损失值随训练次数变化曲线如图15 图15 ，初期网络预测损失值有所下降，随训练次数增加损失值在一定范围内震荡，没有趋于收敛。原因是由于输入与输出数据之间没有良好映射关系，神经网络拟合过程中，同一组输入对应多组不同的输出，数据本身没有良好指向性。 4 . 2 . 2　model02基于姿态与外部载荷数据训练 3 model02神经网络输入中外力 F e 和外力偶 M e 表示受到的外部六维力载荷，投影在定坐标中表示为［ F e x F e y F e z ］和［ M e x M e y M e z ］。损失值随训练次数增加变化曲线如图17 图17 ，随训练次数增加，损失值逐步下降，当训练次数到达100次以后下降开始迟缓。虽考虑了驱动误差外部载诱因，但这个复杂映射关系需要神经网络拟合，并且一般实验数据不可避免的存在不同程度的随机误差，导致数据本身指向性不稳定，训练后期神经网络损失值下降逐渐变缓。 4 . 2 . 3　model03基于刚度模型数据训练 3 采用刚度模型预测的Δ α 和Δ β 作为神经网络的输入，可以通过刚度模型预处理数据提取了数据中包含的变形误差的数据特征，有效预测实际变形的趋势，减小神经网络复杂程度，提升整体模型的预测精度。如图17 图17 所示，神经网络损失值迅速收敛，并且随训练次数增加，损失值具有逐渐变小的趋势。随机选取50数据进行模型预测对比验证，利用搭建的3组神经网络模型进行预测并记录结果数据，对比计算预测补偿量与实际补偿量的偏差值，求解出50组偏差值的均值，并将其作为预测效果对比的指标，50组预测偏差值分布情况见图18 图18 。 model01预测效果在两个驱动上均表现出较大误差，model02和model03在两个驱动上都有较好补偿效果，说明基于外部载荷的训练模式能有效提升神经网络训练效果，基于外载荷的训练方式是必要的。综合比较，model03神经网络训练后预测的精度更加准确。其中预测结果部分数据结果见表7 表7 。 2 编号 2 实际补偿值/mm 2 model01补偿值/mm 2 model02补偿值/mm 2 model03补偿值/mm <math id="M322"><mi mathvariant="normal">Δ</mi><msubsup><mrow><mi>L</mi></mrow><mrow><mn>1</mn></mrow><mrow><mi>r</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116549&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116548&type= 5.24933338 3.38666677 <math id="M323"><mi mathvariant="normal">Δ</mi><msubsup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow><mrow><mi>r</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116551&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116550&type= 5.24933338 3.38666677 <math id="M324"><mi mathvariant="normal">Δ</mi><msubsup><mrow><mi>L</mi></mrow><mrow><mn>1</mn></mrow><mrow><mi mathvariant="normal">p</mi><mi mathvariant="normal">r</mi><mi mathvariant="normal">e</mi><mn>1</mn></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116553&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116552&type= 7.45066643 3.38666677 <math id="M325"><mi mathvariant="normal">Δ</mi><msubsup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow><mrow><mi mathvariant="normal">p</mi><mi mathvariant="normal">r</mi><mi mathvariant="normal">e</mi><mn>1</mn></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116555&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116554&type= 7.45066643 3.38666677 <math id="M326"><mi mathvariant="normal">Δ</mi><msubsup><mrow><mi>L</mi></mrow><mrow><mn>1</mn></mrow><mrow><mi mathvariant="normal">p</mi><mi mathvariant="normal">r</mi><mi mathvariant="normal">e</mi><mn>2</mn></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116557&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116556&type= 7.45066643 3.38666677 <math id="M327"><mi mathvariant="normal">Δ</mi><msubsup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow><mrow><mi mathvariant="normal">p</mi><mi mathvariant="normal">r</mi><mi mathvariant="normal">e</mi><mn>2</mn></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116559&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116558&type= 7.45066643 3.38666677 <math id="M328"><mi mathvariant="normal">Δ</mi><msubsup><mrow><mi>L</mi></mrow><mrow><mn>1</mn></mrow><mrow><mi mathvariant="normal">p</mi><mi mathvariant="normal">r</mi><mi mathvariant="normal">e</mi><mn>3</mn></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116561&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116560&type= 7.45066643 3.38666677 <math id="M329"><mi mathvariant="normal">Δ</mi><msubsup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow><mrow><mi mathvariant="normal">p</mi><mi mathvariant="normal">r</mi><mi mathvariant="normal">e</mi><mn>3</mn></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116563&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116562&type= 7.45066643 3.38666677 1 -3.740 9 -3.449 4 -3.601 8 -3.609 2 -3.570 9 -4.012 6 -3.416 6 -3.180 1 2 -3.843 0 -3.096 8 -3.601 8 -3.609 2 -3.850 7 -3.597 8 -3.660 7 -3.105 6 3 -3.350 2 -3.095 9 -3.602 9 -3.607 9 -3.380 6 -3.624 0 -3.589 5 -3.123 3 4 -3.850 4 -3.093 6 -3.602 1 -3.608 9 -3.835 4 -3.597 8 -3.593 9 -3.122 2 5 -3.862 6 -3.081 3 -3.601 8 -3.609 2 -3.853 0 -3.597 7 -3.576 8 -3.126 5 6 -3.346 8 -3.097 2 -3.601 9 -3.609 1 -3.392 5 -3.629 4 -3.589 9 -3.123 2 7 -3.855 7 -3.089 6 -3.601 9 -3.609 1 -3.834 4 -3.597 8 -3.588 2 -3.123 6 8 -3.969 9 -2.747 2 -3.601 9 -3.609 1 -3.851 9 -3.196 5 -3.543 1 -2.783 6 9 -3.853 9 -3.091 1 -3.601 9 -3.609 1 -3.850 2 -3.598 7 -3.592 6 -3.122 5 10 -3.373 5 -2.818 8 -3.593 8 -3.341 6 -3.525 3 -3.594 2 -3.685 2 -3.099 2 5 仿真验证模型补偿效果 1 随机生成10组姿态参数作为仿真实验验证目标数据。对比采用理论数据驱动与“刚度-神经网络”模型预测补偿数据驱动效果。理论数据驱动通过反解解析式即可获得所对应驱动值。基于“刚度-神经网络”模型预测，计算出对应姿态下外部理论静载荷，采用理论刚度模型进行驱动偏差值的初步预测，采用训练完成的model03神经网络进行最终驱动误差的补偿量，各组预测补偿量还需要累加上理论驱动数据作为最终仿真实验的驱动量，最后仿真验证对比驱动补偿效果。补偿流程图如图19 图19 所示。 2 编号 2 姿态角 3 外部力 3 外力偶 α /rad β /rad F ex /N F ey /N F ez /N M ex /（N·mm -1 ） M ey /（N·mm -1 ） M ez /（N·mm -1 ） 1 -0.120 -0.176 0 0 -4 151.79 -301 051.28 -459 749.94 0 2 -0.033 0.101 0 0 -4 151.79 -106 502.24 176 054.63 0 3 -0.047 -0.120 0 0 -4 151.79 -137 849.22 -332 487.49 0 4 -0.085 -0.191 0 0 -4 151.79 -221 843.27 -493 609.78 0 5 -0.084 -0.023 0 0 -4 151.79 -223 862.99 -109 816.38 0 6 0.174 -0.166 0 0 -4 151.79 362 026.06 -437 118.77 0 7 0.111 0.147 0 0 -4 151.79 223 223.07 281 649.50 0 8 0.080 0.052 0 0 -4 151.79 153 782.74 63 168.79 0 9 -0.008 0.165 0 0 -4 151.79 -48 893.74 322 819.34 0 10 0.188 -0.006 0 0 -4 151.79 400 907.45 -70 625.68 0 基于理论数据作为仿真实验的驱动量。根据设定的10组目标姿态，由反解解析式计算获得理论驱动量。对驱动量进行整理排序，逐步进行仿真验证。基于Simulink控制系统，实验仿真获得平台运行的实时数据，并汇总处理出姿态误差量，见表9 表9 。 2 编号 2 目标姿态 2 理论驱动杆长 2 实际姿态角度 2 误差值（×10 -3 ） α /rad β /rad L 1 /mm L 2 /mm α r /rad β r /rad α e /rad β e /rad 1 -0.120 -0.176 737.852 737.852 -0.128 1 -0.168 1 8.212 2 -7.895 3 2 -0.033 0.101 629.250 629.250 -0.041 1 0.109 4 8.029 1 -8.498 8 3 -0.047 -0.120 715.875 715.875 -0.054 7 -0.112 1 7.833 5 -7.830 9 4 -0.085 -0.191 744.061 744.061 -0.093 0 -0.183 1 8.046 2 -7.857 9 5 -0.084 -0.023 677.366 677.366 -0.092 2 -0.015 0 8.101 7 -8.120 0 6 0.174 -0.166 733.401 733.401 0.166 2 -0.157 7 7.773 8 -8.298 0 7 0.111 0.147 612.244 612.244 0.103 1 0.155 6 7.894 4 -8.619 0 8 0.080 0.052 648.127 648.127 0.072 4 0.060 2 7.645 5 -8.127 6 9 -0.008 0.165 605.170 605.170 -0.016 0 0.173 8 8.069 9 -8.828 5 10 0.188 -0.006 670.662 670.662 0.180 2 0.002 3 7.738 1 -8.266 8 基于预测数据作为仿真实验的驱动量。根据外部载荷和平台目标姿态数据，初步预测出平台驱动量的误差值，再基于神经网络模型model03预测出驱动需要的具体补偿量。同样进行仿真实验采集运行过程的姿态数据，具体数据见表10 表10 。 2 编号 2 姿态角 2 补偿后的驱动 2 实际姿态角 2 误差量（×10 -3 ） α /rad β /rad <math id="M330"><msubsup><mrow><mi>L</mi></mrow><mrow><mn>1</mn></mrow><mrow><mi>c</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116574&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116573&type= 3.13266683 3.38666677 /mm <math id="M331"><msubsup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow><mrow><mi>c</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116576&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116575&type= 3.13266683 3.38666677 /mm α r /rad β r /rad α e /rad β e /rad 1 -0.120 -0.176 741.467 740.943 -0.119 9 -0.177 0 -8.173 5 7.882 3 2 -0.033 0.101 632.897 632.332 -0.033 1 0.099 7 -7.987 1 8.429 6 3 -0.047 -0.120 719.610 718.931 -0.046 8 -0.121 5 -7.811 1 7.823 0 4 -0.085 -0.191 747.663 747.156 -0.084 9 -0.192 0 -8.018 0 7.855 6 5 -0.084 -0.023 681.101 680.422 -0.084 2 -0.024 5 -8.058 1 8.074 8 6 0.174 -0.166 737.029 736.488 0.173 8 -0.167 0 -7.795 3 8.347 1 7 0.111 0.147 615.880 615.329 0.110 9 0.145 9 -7.880 0 8.577 6 8 0.080 0.052 651.776 651.208 0.080 2 0.050 8 -7.634 0 8.107 9 9 -0.008 0.165 608.817 608.251 -0.008 2 0.163 8 -8.031 4 8.745 3 10 0.188 -0.006 674.326 673.739 0.187 8 -0.007 1 -7.752 6 8.289 3 如图20 图20 所示，由表9 表9 和表10 表10 对比误差值，理论驱动数据的平均误差量为 <math id="M332"><msubsup><mrow><mover accent="true"><mi>α</mi><mo>¯</mo></mover></mrow><mrow><mi>t</mi></mrow><mrow><mi>e</mi></mrow></msubsup><mo>=</mo><mn mathvariant="normal">0.008</mn><mtext> </mtext><mn mathvariant="normal">70</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116580&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116579&type= 20.91266632 4.06400013 rad， <math id="M333"><msubsup><mrow><mover accent="true"><mi>β</mi><mo>¯</mo></mover></mrow><mrow><mi>t</mi></mrow><mrow><mi>e</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116582&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116581&type= 3.47133350 5.24933338 = - 0.008 06 rad，补偿驱动数据的平均误差量为 <math id="M334"><msubsup><mrow><mover accent="true"><mi>α</mi><mo>¯</mo></mover></mrow><mrow><mi>c</mi></mrow><mrow><mi>e</mi></mrow></msubsup><mo>=</mo><mo>-</mo><mn mathvariant="normal">0.007</mn><mtext> </mtext><mn mathvariant="normal">91</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116584&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116583&type= 25.23066711 4.06400013 rad， <math id="M335"><msubsup><mrow><mover accent="true"><mi>β</mi><mo>¯</mo></mover></mrow><mrow><mi>c</mi></mrow><mrow><mi>e</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116586&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=38116585&type= 3.47133350 5.24933338 =0.008 21 rad。采用“刚度-神经网络”补偿驱动量平均误差降低约9%。分析仿真实验结果表明：基于刚度模型的误差预测模型可以有效提升调姿平台驱动精度。 1,2; 6 结论 1 以2RRPU/2RPU/U调姿平台为研究对象，提出了基于刚度模型与神经网络算法结合的误差驱动补偿方法。基于调姿平台的全雅克比矩阵和弹性变形矩阵，建立了调姿平台的理论刚度模型，与Ansys数据刚度模型预测对比验证其对平台受载变形趋势预测的有效性。搭建了基于Simulink-Adams-Ansys-OPC的模拟仿真环境，随机载荷下采集平台全姿态仿真数据。基于刚度模型和雅克比矩阵完成了平台姿态误差和驱动误差趋势的预测，利用神经网络算法对理论刚度模型初步预测的驱动补偿量进一步预测，完成预测结果精度的提升。得到以下结论：（1）理论刚度模型可以有效预测模型受载的变形趋势；（2）在仿真环境下，排除了结构参数等因素的线性误差影响，突出研究变形误差对精度影响以及补偿方法，验证了基于刚度的误差补偿方法的必要性和有效性。（3）3组BP神经网络训练结果对比表明，引入外载荷作为网络的输入，使网络的训练效果趋于收敛，获得更好的训练效果。基于模型补偿的平台精度提升约9%，同时也说明对于新的载荷情况，补偿模型仍具备良好的鲁棒性，对原始训练数据依赖性更小。（4）以外载荷数据、刚度模型数据训练补偿驱动量，适用于大负载，变载荷工况条件，有益于在实践中进一步提高大负载，变载调姿设备的定位精度。主要创新点：提出了基于刚度模型预测误差趋势与神经网络算法提升预测精度的误差补偿模型，降低了预测模型对数据本身的依赖性，使预测模型本身能够一定程度上反应误差的来源，提高了预测模型对多变环境的鲁棒性。本文取得进展部分，仍有进一步提升的空间：刚度模型基于载荷预测变形并没有考虑平台运动的惯性力，但实际本身平台变速运行具有一定的惯性力，需要基于动力学模型修正刚度模型的预测。并且所研究的误差补偿模型目前只在仿真阶段取得一定的进展，有待在船舰实际环境进行模拟实验验证。参考文献 BELCHIOR J ， GUILLO M ， COURTEILLE E ， et al . 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