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Attitude planning and control method of low-orbit optical satellite along-track stereoscopic imaging
Micro/Nano Technology and Fine Mechanics | Updated：2022-08-08
|
Attitude planning and control method of low-orbit optical satellite along-track stereoscopic imaging
- Optics and Precision Engineering
- Affiliations：
1.Changchun Institute of Optics， Fine Mechanics and Physics， Chinese Academy of Sciences， Changchun 130033， China
2.University of Chinese Academy of Sciences， Beijing 100049， China
3.Chang Guang Satellite Technology Limited Company， Changchun 130102， China
- Author bio：
- Funds：
- DOI：10.37188/OPE.20223014.1682 CLC： TP391.4;TH691.9
Published：25 July 2022，
Received：23 November 2021，
Revised：17 January 2022，
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Cite this article
李峰,万秋华,刘萌萌等.低轨光学卫星同轨立体成像姿态规划与控制方法[J].光学精密工程,2022,30(14):1682-1693.
LI Feng,WAN Qiuhua,LIU Mengmeng,et al.Attitude planning and control method of low-orbit optical satellite along-track stereoscopic imaging[J].Optics and Precision Engineering,2022,30(14):1682-1693.
李峰,万秋华,刘萌萌等.低轨光学卫星同轨立体成像姿态规划与控制方法[J].光学精密工程,2022,30(14):1682-1693. DOI： 10.37188/OPE.20223014.1682.
LI Feng,WAN Qiuhua,LIU Mengmeng,et al.Attitude planning and control method of low-orbit optical satellite along-track stereoscopic imaging[J].Optics and Precision Engineering,2022,30(14):1682-1693. DOI： 10.37188/OPE.20223014.1682.
Attitude planning and control method of low-orbit optical satellite along-track stereoscopic imaging
低轨光学卫星同轨立体成像姿态规划与控制方法 Attitude planning and control method of low-orbit optical satellite along-track stereoscopic imaging 1 2 3 first-author 李 峰（1988-），男，河北乐亭人，博士研究生，分别于2011年、2013年于哈尔滨工程大学获得学士、硕士学位，主要从光学遥感卫星控制系统设计与任务规划等方面的研究。 E-mail： lifeng@charmingglobe.com http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32995765&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32995771&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32995768&type= 22.01332855 32.00399780 李 峰 （1988-），男，河北乐亭人，博士研究生，分别于2011年、2013年于哈尔滨工程大学获得学士、硕士学位，主要从光学遥感卫星控制系统设计与任务规划等方面的研究。 E-mail： lifeng@charmingglobe.com lifeng@charmingglobe.com 1 2 corresp E-mail： wanqh@ciomp.ac.cn E-mail： wanqh@ciomp.ac.cn 万秋华（1962-），女，吉林长春人，博士，研究员，博士生导师，主要从事光电位移精密测量技术和规划优化等方面的研究。E-mail： wanqh@ciomp.ac.cn http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32995774&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32995780&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32995777&type= 22.01332855 32.00399780 万秋华 （1962-），女，吉林长春人，博士，研究员，博士生导师，主要从事光电位移精密测量技术和规划优化等方面的研究。E-mail： wanqh@ciomp.ac.cn wanqh@ciomp.ac.cn 3 3 1 2 3 中国科学院 长春光学精密机械与物理研究所，吉林 长春 130033 中国科学院 长春光学精密机械与物理研究所，吉林 长春 130033 Changchun Institute of Optics， Fine Mechanics and Physics， Chinese Academy of Sciences， Changchun 130033， China Changchun Institute of Optics， Fine Mechanics and Physics， Chinese Academy of Sciences， Changchun 130033， China 中国科学院大学，北京 100049 中国科学院大学，北京 100049 University of Chinese Academy of Sciences， Beijing 100049， China University of Chinese Academy of Sciences， Beijing 100049， China 长光卫星技术股份有限公司，吉林 长春 130102 长光卫星技术股份有限公司，吉林 长春 130102 Chang Guang Satellite Technology Limited Company， Changchun 130102， China Chang Guang Satellite Technology Limited Company， Changchun 130102， China 为了提高低轨光学卫星同轨立体成像中多次成像对目标区域指向精度，从而获取更大的重叠覆盖面积，即立体像对有效面积。首先，设计了一种采用地球椭球模型，并考虑地球自转以及卫星滚动方向机动的同轨双视/三视立体成像姿态规划方法。其次，为了缩减机动时间，以提高卫星单次成像可拍摄的区域长度，考虑执行机构力矩与角动量约束，设计了一种路径规划快速机动控制算法（Path planning Fast Maneuver Control，PFMC），即基于旋转轴不变约束的三轴机动的最短路径和角加速度连续路径规划以及结合角加速度前馈与内外环控制的快速机动算法。最后，以“吉林一号”卫星参数进行数学仿真，控制精度优于0.02°，稳定度优于0.001（°） / s，成像时长不低于10 s。在轨测试期间，获得了乌鲁木齐市的立体像对与数字表面模型，有效覆盖面积大于1 600 km 2 ，双视成像的重叠率超过97%，验证了同轨双视立体成像规划与控制方法的可行性和有效性。 To improve the point accuracy of along-track stereoscopic imaging， a larger coverage area must be obtained； that is， the effective area for stereoscopic imaging must be improved. Herein， first， a method for attitude planning of along-track dual/triple perspective stereoscopic imaging is designed， which adopts the earth ellipsoid model and considers the earth rotation and satellite rolling direction maneuver. Next， to reduce the maneuver time and to increase the area that can be imaged by the satellite in a single shot， a method called path planning fast maneuver control is designed considering the actuator moment and angular momentum constraints， including two parts. According to a situation wherein the angular velocity of the satellite is limited owing to the torque and angular momentum constraints of the actuator， a three-axis maneuver attitude planner with the shortest path and continuous angular acceleration is designed based on the invariant rotation axis constraint. An attitude maneuver algorithm combining angular acceleration feed-forward and internal–external loop control is further designed to improve the dynamic performance of attitude tracking. Finally， a mathematical simulation is conducted with the parameters of the Jilin-1 satellite， whereby the control accuracy is better than 0.02°， the stability is better than 0.001 （°）/s， and the imaging duration is no less than 10 s. During the in-orbit test， the stereo image pairs and digital surface models of Urumqi city are obtained， with an effective coverage area of 1600 km2 and an overlap rate of dual-view imaging exceeding 97%， which verifies the feasibility and effectiveness of the along-track dual-view stereo imaging planning and control method. 同轨立体成像 三轴姿态规划 快速机动 姿态控制 along-track stereoscopic imaging tree-axial attitude planning fast maneuver attitude control 1 引 言 1 随着航天技术的发展，光学卫星的总体设计、姿态确定方法、控制以及成像参数优化等技术与方法得到深入的研究与长足的进步，并使得利用光学卫星获得遥感影像成为大范围地理测绘的重要手段。随着遥感与测绘技术的发展，高分辨率立体影像得到广泛应用 ［ 1 1 - 4 4 1-4 ］ 。 卫星多次对同一地面目标进行不同角度观测的成像方式可实现对目标的立体测绘，获取三维影像。专用立体成像测绘卫星采用多线阵相机进行连续长条带立体成像 ［ 5 5 ］ ，经济成本高、研制难度大。微小卫星体积小、重量轻，可采用单相机通过卫星俯仰轴的快速姿态机动实现同轨立体成像或通过多次过境进行异轨立体成像。其中，同轨立体成像缩短了立体像对的获取时间间隔，影像环境差异小，方便测图处理 ［ 6 6 - 7 7 6-7 ］ ，成为获取立体像对的趋势。王中果等推导出基于圆形地球模型星下点轨迹下的立体成像条带长度通用公式和最小机动能力需求，为简化分析并不考虑地球椭率、自转以及滚动方向的姿态机动 ［ 5 5 ］ 。张新伟等采用平面地球模型，对同轨立体成像时不同重叠条带长度对所需姿态机动能力进行了分析 ［ 8 8 ］ ，未考虑地球曲率的影响，导致卫星成像姿态求解结果存在偏差。田原等从立体像对后续处理角度出发，利用基高比和几何分辨率差异两方面构建同轨立体成像规划目标函数，提出了一种基于步进搜索策略的成像规划方法，决策多次拍摄的时间窗口 ［ 6 6 ］ ，为已确定成像参数的同轨立体成像时间窗口计算提供研究思路。 在卫星幅宽确定的情况下，条带长度决定了成像覆盖能力，因此要求卫星具有三轴大角度姿态快速机动能力，现有研究主要面向姿态规划与控制算法两个方向。针对侧摆方向大角度机动的姿态规划研究，资源三号与高分三号卫星分别设计了三段梯形和五段梯形角速度的姿态规划 ［ 9 9 - 10 10 9-10 ］ ，但角速度存在突变。申晓宁等针对加速度突变问题提出了基于S型角速度曲线的机动姿态规划方法 ［ 11 11 ］ 。Ha和Femiano等介绍了角加速度为连续曲线的姿态机动规划方法 ［ 12 12 ］ 。面向控制算法的研究如：Li和Bainum等学者认为经典的欧拉定轴转动是最优控制或接近最优控制 ［ 13 13 ］ ；华冰等设计了基于欧拉轴转动的递阶饱和姿态控制律 ［ 14 14 ］ ，但机动过程存在快速性与稳定性的矛盾。汪礼成等的研究表明引入加速度前馈能够改善控制性能 ［ 15 15 ］ 。韩京清在自抗扰控制中详细说明了通过姿态规划可以解决传统控制存在的快速性与超调性矛盾问题 ［ 16 16 ］ 。 本文考虑地球椭率、自转以及滚动方向的姿态机动，进行同轨立体成像姿态规划和成像时间窗口确定，提高规划的精度；进一步考虑微小卫星执行机构输出力矩和角动量约束，设计了路径规划快速机动控制（Path planning Fast Maneuver Control，PFMC）算法，基于旋转轴不变约束设计了三轴机动的最短路径和角加速度连续的姿态引导路径规划器，同时设计了结合角加速度前馈与内外环控制的快速机动算法。最后通过仿真与在轨试验验证了同轨双视立体成像的姿态规划与PFMC算法的可行性和有效性。 2 同轨立体成像方案 1 低轨光学微小卫星进行同轨立体成像时，在同一轨道圈次内通过卫星三轴快速机动可对同一目标进行不同角度的多次观测，如 图1 图1 所示。双视立体成像通过前视和后视两次成像实现，三视立体成像则须增加一次正视成像，但都要求重复覆盖，才可生产出立体影像。 立体成像方案分为成像姿态规划和PFMC两部分。其中成像姿态规划用于高精度的成像姿态与成像时间计算，PFMC分为三轴路径规划设计与快速机动控制，卫星同轨立体成像控制流图如 图2 图2 所示。 成像姿态规划中，为了提高立体成像姿态求解的精度，采用地球椭球模型确定成像点的位置，并考虑地球的自转确定成像点与卫星的相对位置与速度信息，进行含有侧摆机动的立体成像姿态求解。通过对沿卫星飞行方向的特性分析，获取成像窗口时间及相应的成像姿态四元数，详细计算过程如下。 2.1 卫星成像姿态求解 2 采用椭球形地球模型，考虑地球椭率的影响，由成像点的地理经度 <math id="M1"><msub><mrow><mi>θ</mi></mrow><mrow><mi mathvariant="normal">l</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">n</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32995805&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32995801&type= 4.48733330 3.72533321 、地理纬度 <math id="M2"><msub><mrow><mi>θ</mi></mrow><mrow><mi mathvariant="normal">l</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">t</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32995812&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32995808&type= 3.97933316 3.72533321 和高度 <math 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</mo><mtext> </mtext></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32995837&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32995835&type= 73.15200043 41.82533264 ， 其中： <math id="M6"><mi>r</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32995842&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32995838&type= 1.52400005 2.79399991 为赤道半径， <math id="M7"><mi>e</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32995846&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32995843&type= 1.52400005 2.79399991 为偏心率。 通过卫星导航数据和轨道递推公式得到地固系下卫星的实时位置向量 <math id="M8"><mi mathvariant="bold-italic">S</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32995995&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32995992&type= 2.37066650 2.79399991 ，对应的沿轨道坐标系 <math id="M9"><mo stretchy="false">(</mo><msub><mrow><mi>O</mi></mrow><mrow><mi mathvariant="normal">o</mi></mrow></msub><msub><mrow><mi>X</mi></mrow><mrow><mi mathvariant="normal">o</mi></mrow></msub><msub><mrow><mi>Y</mi></mrow><mrow><mi mathvariant="normal">o</mi></mrow></msub><msub><mrow><mi>Z</mi></mrow><mrow><mi mathvariant="normal">o</mi></mrow></msub><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32995859&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32995856&type= 19.21933365 4.48733330 中 <math id="M10"><msub><mrow><mi>Z</mi></mrow><mrow><mi mathvariant="normal">o</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32995864&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32995861&type= 3.47133350 3.72533321 轴反方向的单位向量为： <math display="block" id="M11"><msub><mrow><mi mathvariant="bold-italic">e</mi></mrow><mrow><mi>S</mi></mrow></msub><mo>=</mo><mfrac><mrow><mi mathvariant="bold-italic">S</mi></mrow><mrow><mfenced open="‖" close="‖" separators="|"><mrow><mi mathvariant="bold-italic">S</mi></mrow></mfenced></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32995870&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32995868&type= 14.22399998 9.14400005 ， 其中， <math id="M12"><mfenced open="‖" close="‖" separators="|"><mrow><mi mathvariant="bold-italic">S</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32995876&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32995873&type= 5.75733376 3.97933316 向量 <math id="M13"><mi mathvariant="bold-italic">S</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32995882&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32995879&type= 2.37066650 2.79399991 的模。 地固系下地球自转角速度为： <math display="block" id="M14"><msub><mrow><mi mathvariant="bold-italic">ω</mi></mrow><mrow><mi mathvariant="normal">E</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">r</mi><mi mathvariant="normal">t</mi><mi mathvariant="normal">h</mi></mrow></msub><mo>=</mo><msup><mrow><mfenced open="[" close="]" separators="|"><mrow><mtable><mtr><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><msub><mrow><mi>ω</mi></mrow><mrow><mi mathvariant="normal">E</mi><mi mathvariant="normal">e</mi></mrow></msub></mtd></mtr></mtable></mrow></mfenced></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32995888&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32995885&type= 33.02000046 4.74133301 ， 其中， <math id="M15"><msub><mrow><mi>ω</mi></mrow><mrow><mi mathvariant="normal">E</mi><mi mathvariant="normal">e</mi></mrow></msub><mo>=</mo><mn mathvariant="normal">7.292</mn><mtext> </mtext><mn mathvariant="normal">115</mn><mo>×</mo><msup><mrow><mn mathvariant="normal">10</mn></mrow><mrow><mo>-</mo><mn mathvariant="normal">5</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32995894&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32995891&type= 35.98332977 4.06400013 rad/s。 考虑地球自转的影响，若卫星速度向量为 <math id="M16"><mi mathvariant="bold-italic">V</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32995900&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32995897&type= 2.96333337 2.96333337 ，卫星相对成像点的相对速度为： <math display="block" id="M17"><msub><mrow><mi mathvariant="bold-italic">V</mi></mrow><mrow><mi mathvariant="normal">s</mi></mrow></msub><mo>=</mo><mi mathvariant="bold-italic">V</mi><mo>+</mo><msub><mrow><mi mathvariant="bold-italic">ω</mi></mrow><mrow><mi mathvariant="normal">E</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">r</mi><mi mathvariant="normal">t</mi><mi mathvariant="normal">h</mi></mrow></msub><mo>×</mo><mi mathvariant="bold-italic">S</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32995906&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32995903&type= 30.90333366 3.89466691 ， <math display="block" id="M18"><msub><mrow><mi mathvariant="bold-italic">e</mi></mrow><mrow><msub><mrow><mi>V</mi></mrow><mrow><mi mathvariant="normal">s</mi></mrow></msub></mrow></msub><mo>=</mo><mfrac><mrow><msub><mrow><mi mathvariant="bold-italic">V</mi></mrow><mrow><mi mathvariant="normal">s</mi></mrow></msub></mrow><mrow><mfenced open="‖" close="‖" separators="|"><mrow><msub><mrow><mi mathvariant="bold-italic">V</mi></mrow><mrow><mi mathvariant="normal">s</mi></mrow></msub></mrow></mfenced></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32995910&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32995908&type= 16.51000023 9.14400005 . 以卫星的位置、速度和成像点的经度、纬度、高度以及地球自转角速度作为输入，进行卫星的姿态求解，求解流程如 图4 图4 所示。 <math id="M19"><msub><mrow><mi>C</mi></mrow><mrow><mi mathvariant="normal">e</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32995920&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32995918&type= 3.64066648 3.72533321 ， <math id="M20"><mi>P</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32995923&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32995922&type= 2.45533323 2.79399991 ， <math id="M21"><mi>S</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32995928&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32995925&type= 2.20133328 2.79399991 三点共面，卫星成像时相机指向的单位向量 <math id="M22"><msub><mrow><mi mathvariant="bold-italic">e</mi></mrow><mrow><mi>k</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32995932&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32995930&type= 2.62466669 3.72533321 为： <math display="block" id="M23"><msub><mrow><mi mathvariant="bold-italic">e</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>=</mo><mfrac><mrow><mi mathvariant="bold-italic">S</mi><mo>-</mo><mi mathvariant="bold-italic">P</mi></mrow><mrow><mfenced open="‖" close="‖" separators="|"><mrow><mi mathvariant="bold-italic">S</mi><mo>-</mo><mi mathvariant="bold-italic">P</mi></mrow></mfenced></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32995939&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32995935&type= 21.42066574 9.14400005 . 平面 <math id="M24"><msub><mrow><mi>C</mi></mrow><mrow><mi mathvariant="normal">e</mi></mrow></msub><mi>P</mi><mi>S</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32995945&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32995941&type= 8.63599968 3.72533321 的法线向量 <math id="M25"><msub><mrow><mi mathvariant="bold-italic">e</mi></mrow><mrow><mi>n</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32995951&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32995949&type= 2.79399991 3.72533321 在地固系的表示为： <math display="block" id="M26"><msub><mrow><mi mathvariant="bold-italic">e</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>=</mo><mfrac><mrow><msub><mrow><mi mathvariant="bold-italic">e</mi></mrow><mrow><mi mathvariant="normal">S</mi></mrow></msub><mo>×</mo><msub><mrow><mi mathvariant="bold-italic">e</mi></mrow><mrow><mi>k</mi></mrow></msub></mrow><mrow><mfenced open="‖" close="‖" separators="|"><mrow><msub><mrow><mi mathvariant="bold-italic">e</mi></mrow><mrow><mi mathvariant="normal">S</mi></mrow></msub><mo>×</mo><msub><mrow><mi mathvariant="bold-italic">e</mi></mrow><mrow><mi>k</mi></mrow></msub></mrow></mfenced></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32995957&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32995954&type= 22.26733208 9.14400005 . 由欧拉旋转定理知，向量 <math id="M27"><msub><mrow><mi mathvariant="bold-italic">e</mi></mrow><mrow><mi mathvariant="normal">S</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32995963&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32995960&type= 2.96333337 3.72533321 以向量 <math id="M28"><msub><mrow><mi mathvariant="bold-italic">e</mi></mrow><mrow><mi>n</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32995968&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32995966&type= 2.79399991 3.72533321 为旋转轴旋转角度 <math id="M29"><mi>φ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996077&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996074&type= 2.37066650 3.72533321 得到向量 <math id="M30"><msub><mrow><mi mathvariant="bold-italic">e</mi></mrow><mrow><mi>k</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32995982&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32995978&type= 2.62466669 3.72533321 ，旋转角度为： <math display="block" id="M31"><mi>φ</mi><mo>=</mo><mi mathvariant="normal">a</mi><mi mathvariant="normal">r</mi><mi mathvariant="normal">c</mi><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mo stretchy="false">(</mo><mfenced open="‖" close="‖" separators="|"><mrow><msub><mrow><mi mathvariant="bold-italic">e</mi></mrow><mrow><mi mathvariant="normal">S</mi></mrow></msub><mo>×</mo><msub><mrow><mi mathvariant="bold-italic">e</mi></mrow><mrow><mi>k</mi></mrow></msub></mrow></mfenced><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32995990&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32995984&type= 34.62866592 4.06400013 . 由 <math id="M32"><mi mathvariant="bold-italic">S</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32995995&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32995992&type= 2.37066650 2.79399991 ， <math id="M33"><mi mathvariant="bold-italic">k</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996002&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32995998&type= 1.86266661 2.79399991 ， <math id="M34"><mi mathvariant="bold-italic">n</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996012&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996008&type= 2.11666679 2.79399991 ， <math id="M35"><msub><mrow><mi mathvariant="bold-italic">V</mi></mrow><mrow><mi mathvariant="normal">s</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996019&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996015&type= 3.97933316 3.80999994 计算轨道坐标系三轴在地固系表示的单位向量 <math id="M36"><msub><mrow><mi mathvariant="bold-italic">e</mi></mrow><mrow><mi>o</mi><mi>x</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996025&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996022&type= 3.80999994 3.72533321 ， <math id="M37"><msub><mrow><mi mathvariant="bold-italic">e</mi></mrow><mrow><mi>o</mi><mi>y</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996031&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996028&type= 3.72533321 3.72533321 ， <math id="M38"><msub><mrow><mi mathvariant="bold-italic">e</mi></mrow><mrow><mi>o</mi><mi>z</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996037&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996034&type= 3.64066648 3.72533321 分别为： <math display="block" id="M39"><mfenced open="{" close="" separators="|"><mrow><mtable><mtr><mtd><msub><mrow><mi mathvariant="bold-italic">e</mi></mrow><mrow><mi>o</mi><mi>y</mi></mrow></msub><mo>=</mo><mfrac><mrow><msub><mrow><mi mathvariant="bold-italic">e</mi></mrow><mrow><msub><mrow><mi>V</mi></mrow><mrow><mi mathvariant="normal">s</mi></mrow></msub></mrow></msub><mo>×</mo><msub><mrow><mi mathvariant="bold-italic">e</mi></mrow><mrow><mi mathvariant="normal">S</mi></mrow></msub></mrow><mrow><mfenced open="‖" close="‖" separators="|"><mrow><msub><mrow><mi mathvariant="bold-italic">e</mi></mrow><mrow><msub><mrow><mi>V</mi></mrow><mrow><mi mathvariant="normal">s</mi></mrow></msub></mrow></msub><mo>×</mo><msub><mrow><mi mathvariant="bold-italic">e</mi></mrow><mrow><mi mathvariant="normal">S</mi></mrow></msub></mrow></mfenced></mrow></mfrac></mtd></mtr><mtr><mtd><msub><mrow><mi mathvariant="bold-italic">e</mi></mrow><mrow><mi>o</mi><mi>z</mi></mrow></msub><mo>=</mo><mo>-</mo><msub><mrow><mi mathvariant="bold-italic">e</mi></mrow><mrow><mi mathvariant="normal">S</mi></mrow></msub><mo> </mo><mo> </mo></mtd></mtr><mtr><mtd><msub><mrow><mi mathvariant="bold-italic">e</mi></mrow><mrow><mi>o</mi><mi>x</mi></mrow></msub><mo>=</mo><mfrac><mrow><msub><mrow><mi mathvariant="bold-italic">e</mi></mrow><mrow><mi>o</mi><mi>y</mi></mrow></msub><mo>×</mo><msub><mrow><mi mathvariant="bold-italic">e</mi></mrow><mrow><mi>o</mi><mi>z</mi></mrow></msub></mrow><mrow><mfenced open="‖" close="‖" separators="|"><mrow><msub><mrow><mi mathvariant="bold-italic">e</mi></mrow><mrow><mi>o</mi><mi>y</mi></mrow></msub><mo>×</mo><msub><mrow><mi mathvariant="bold-italic">e</mi></mrow><mrow><mi>o</mi><mi>z</mi></mrow></msub></mrow></mfenced></mrow></mfrac></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996043&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996040&type= 26.92399979 27.77066803 . 旋转轴 <math id="M40"><msub><mrow><mi mathvariant="bold-italic">e</mi></mrow><mrow><mi>n</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996048&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996046&type= 2.79399991 3.72533321 在轨道系下表示 <math id="M41"><msub><mrow><mi>e</mi></mrow><mrow><msub><mrow><mi>n</mi></mrow><mrow><mi mathvariant="normal">o</mi></mrow></msub><mi>x</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996054&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996051&type= 4.74133301 3.64066648 ， <math id="M42"><msub><mrow><mi>e</mi></mrow><mrow><msub><mrow><mi>n</mi></mrow><mrow><mi mathvariant="normal">o</mi></mrow></msub><mi>y</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996060&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996057&type= 4.65666676 3.72533321 ， <math id="M43"><msub><mrow><mi>e</mi></mrow><mrow><msub><mrow><mi>n</mi></mrow><mrow><mi mathvariant="normal">o</mi></mrow></msub><mi>z</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996065&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996063&type= 4.48733330 3.64066648 为： <math display="block" id="M44"><mfenced open="{" close="" separators="|"><mrow><mtable><mtr><mtd><msub><mrow><mi>e</mi></mrow><mrow><msub><mrow><mi>n</mi></mrow><mrow><mi mathvariant="normal">o</mi></mrow></msub><mi>x</mi></mrow></msub><mo>=</mo><mfenced open="‖" close="‖" separators="|"><mrow><msub><mrow><mi mathvariant="bold-italic">e</mi></mrow><mrow><mi>o</mi><mi>x</mi></mrow></msub><msub><mrow><mi mathvariant="bold-italic">e</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow></mfenced></mtd></mtr><mtr><mtd><msub><mrow><mi>e</mi></mrow><mrow><msub><mrow><mi>n</mi></mrow><mrow><mi mathvariant="normal">o</mi></mrow></msub><mi>y</mi></mrow></msub><mo>=</mo><mfenced open="‖" close="‖" separators="|"><mrow><msub><mrow><mi mathvariant="bold-italic">e</mi></mrow><mrow><mi>o</mi><mi>y</mi></mrow></msub><msub><mrow><mi mathvariant="bold-italic">e</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow></mfenced></mtd></mtr><mtr><mtd><msub><mrow><mi>e</mi></mrow><mrow><msub><mrow><mi>n</mi></mrow><mrow><mi mathvariant="normal">o</mi></mrow></msub><mi>z</mi></mrow></msub><mo>=</mo><mfenced open="‖" close="‖" separators="|"><mrow><msub><mrow><mi mathvariant="bold-italic">e</mi></mrow><mrow><mi>o</mi><mi>z</mi></mrow></msub><msub><mrow><mi mathvariant="bold-italic">e</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow></mfenced></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996071&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996068&type= 21.84399986 18.03400040 . 旋转角 <math id="M45"><mi>φ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996077&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996074&type= 2.37066650 3.72533321 在轨道系下表示 <math id="M46"><msub><mrow><mi>φ</mi></mrow><mrow><msub><mrow><mi>n</mi></mrow><mrow><mi mathvariant="normal">o</mi></mrow></msub><mi>x</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996082&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996080&type= 5.58799982 3.72533321 ， <math id="M47"><msub><mrow><mi>φ</mi></mrow><mrow><msub><mrow><mi>n</mi></mrow><mrow><mi mathvariant="normal">o</mi></mrow></msub><mi>y</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996087&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996084&type= 5.50333309 3.72533321 ， <math id="M48"><msub><mrow><mi>φ</mi></mrow><mrow><msub><mrow><mi>n</mi></mrow><mrow><mi mathvariant="normal">o</mi></mrow></msub><mi>z</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996092&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996089&type= 5.33400011 3.72533321 为： <math display="block" id="M49"><mfenced open="{" close="" separators="|"><mrow><mtable><mtr><mtd><msub><mrow><mi>φ</mi></mrow><mrow><msub><mrow><mi>n</mi></mrow><mrow><mi mathvariant="normal">o</mi></mrow></msub><mi>x</mi></mrow></msub><mo>=</mo><msub><mrow><mi>e</mi></mrow><mrow><msub><mrow><mi>n</mi></mrow><mrow><mi mathvariant="normal">o</mi></mrow></msub><mi>x</mi></mrow></msub><mi>φ</mi></mtd></mtr><mtr><mtd><msub><mrow><mi>φ</mi></mrow><mrow><msub><mrow><mi>n</mi></mrow><mrow><mi mathvariant="normal">o</mi></mrow></msub><mi>y</mi></mrow></msub><mo>=</mo><msub><mrow><mi>e</mi></mrow><mrow><msub><mrow><mi>n</mi></mrow><mrow><mi mathvariant="normal">o</mi></mrow></msub><mi>y</mi></mrow></msub><mi>φ</mi></mtd></mtr><mtr><mtd><msub><mrow><mi>φ</mi></mrow><mrow><msub><mrow><mi>n</mi></mrow><mrow><mi mathvariant="normal">o</mi></mrow></msub><mi>z</mi></mrow></msub><mo>=</mo><msub><mrow><mi>e</mi></mrow><mrow><msub><mrow><mi>n</mi></mrow><mrow><mi mathvariant="normal">o</mi></mrow></msub><mi>z</mi></mrow></msub><mi>φ</mi></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996098&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996095&type= 19.64266586 16.25600052 . 成像时卫星本体姿态相对轨道系旋转四元数： <math display="block" id="M50"><msub><mrow><mi mathvariant="bold-italic">q</mi></mrow><mrow><mi mathvariant="normal">A</mi></mrow></msub><mo>=</mo><msup><mrow><mfenced open="[" close="]" separators="|"><mrow><mtable><mtr><mtd><mi mathvariant="normal">c</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">s</mi><mtext> </mtext><mfenced separators="|"><mrow><mfrac><mrow><mi>φ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac></mrow></mfenced></mtd><mtd><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mtext> </mtext><mfenced separators="|"><mrow><mfrac><mrow><mi>φ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac></mrow></mfenced><msub><mrow><mi mathvariant="bold-italic">e</mi></mrow><mrow><msub><mrow><mi>n</mi></mrow><mrow><mi mathvariant="normal">o</mi></mrow></msub></mrow></msub></mtd></mtr></mtable></mrow></mfenced></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996104&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996101&type= 44.61933517 10.66800022 . 卫星成像时本体姿态相对惯性系旋转四元数： <math display="block" id="M51"><msub><mrow><mi mathvariant="bold-italic">q</mi></mrow><mrow><mi mathvariant="normal">f</mi></mrow></msub><mo>=</mo><msub><mrow><mi mathvariant="bold-italic">q</mi></mrow><mrow><mi mathvariant="normal">O</mi><mi mathvariant="normal">I</mi></mrow></msub><mo>⊗</mo><msub><mrow><mi mathvariant="bold-italic">q</mi></mrow><mrow><mi mathvariant="normal">A</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996109&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996107&type= 20.40466690 4.91066647 ， 其中， <math id="M52"><msub><mrow><mi mathvariant="bold-italic">q</mi></mrow><mrow><mi mathvariant="normal">O</mi><mi mathvariant="normal">I</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996115&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996112&type= 4.31799984 4.57200003 为轨道系相对惯性系旋转四元数。 2.2 成像时间窗口确定 2 以同轨立体双视和三视为例，通过对沿卫星飞行方向的特性进行分析，获取成像时间窗口。沿轨方向的旋转角度为 <math id="M53"><msub><mrow><mi>φ</mi></mrow><mrow><msub><mrow><mi>n</mi></mrow><mrow><mi mathvariant="normal">o</mi></mrow></msub><mi>y</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996120&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996118&type= 5.50333309 3.72533321 ，立体成像前后视角度为 <math id="M54"><mi>θ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996126&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996123&type= 1.77800000 2.79399991 ，即前视成像要求 <math id="M55"><msub><mrow><mi>φ</mi></mrow><mrow><msub><mrow><mi>n</mi></mrow><mrow><mi mathvariant="normal">o</mi></mrow></msub><mi>y</mi></mrow></msub><mo>=</mo><mi>θ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996132&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996129&type= 12.10733318 3.72533321 ，后视成像要求 <math id="M56"><msub><mrow><mi>φ</mi></mrow><mrow><msub><mrow><mi>n</mi></mrow><mrow><mi mathvariant="normal">o</mi></mrow></msub><mi>y</mi></mrow></msub><mo>=</mo><mo>-</mo><mi>θ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996138&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996135&type= 15.66333294 3.72533321 。 2 . 2 . 1 双视模式 3 双视模式下卫星成像过程分为前视与后视两部分，如 图5 图5 所示。 双视成像的成像与机动时间关系表如 表1 表1 。当观测角度 <math id="M57"><mi>θ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996153&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996150&type= 1.77800000 2.79399991 确定后，时间 <math id="M58"><msub><mrow><mi>T</mi></mrow><mrow><mi mathvariant="normal">t</mi><mi mathvariant="normal">r</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">n</mi><mi mathvariant="normal">s</mi></mrow></msub><mo>+</mo><msub><mrow><mi>T</mi></mrow><mrow><mi mathvariant="normal">i</mi><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">g</mi><mi mathvariant="normal">e</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996159&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996156&type= 18.96533203 3.80999994 为定值，通过提高卫星的机动能力， <math id="M59"><msub><mrow><mi>T</mi></mrow><mrow><mi mathvariant="normal">t</mi><mi mathvariant="normal">r</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">n</mi><mi mathvariant="normal">s</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996165&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996162&type= 6.68866634 3.72533321 减小为 <math id="M60"><msub><mrow><mi>T</mi></mrow><mrow><mi mathvariant="normal">t</mi><mi mathvariant="normal">r</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">n</mi><mi mathvariant="normal">s</mi></mrow></msub><mo>-</mo><mi mathvariant="normal">Δ</mi><mi>T</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996171&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996168&type= 16.51000023 3.72533321 ，则成像时间 <math id="M61"><msub><mrow><mi>T</mi></mrow><mrow><mi mathvariant="normal">i</mi><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">g</mi><mi mathvariant="normal">e</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996177&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996174&type= 7.53533268 3.72533321 增加为 <math id="M62"><msub><mrow><mi>T</mi></mrow><mrow><mi mathvariant="normal">i</mi><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">g</mi><mi mathvariant="normal">e</mi></mrow></msub><mo>+</mo><mi mathvariant="normal">Δ</mi><mi>T</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996184&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996182&type= 17.35666656 3.80999994 。 Item Time quantum Imaging time Maneuver time Front view （ <math id="M63"><mi>θ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996260&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996258&type= 1.52400005 2.37066650 ） <math id="M64"><msub><mrow><mi>A</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>~</mo><msub><mrow><mi>A</mi></mrow><mrow><mn>2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996267&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996263&type= 10.07533360 3.21733332 <math id="M65"><msub><mrow><mi>T</mi></mrow><mrow><mi mathvariant="normal">i</mi><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">g</mi><mi mathvariant="normal">e</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996333&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996330&type= 6.60400009 3.21733332 / Transform <math id="M66"><msub><mrow><mi>A</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>~</mo><msub><mrow><mi>B</mi></mrow><mrow><mn>1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996281&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996278&type= 9.73666668 3.21733332 / <math id="M67"><msub><mrow><mi>T</mi></mrow><mrow><mi mathvariant="normal">t</mi><mi mathvariant="normal">r</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">n</mi><mi mathvariant="normal">s</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996216&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996214&type= 5.84200001 3.21733332 Rear view（ <math id="M68"><mo>-</mo><mi>θ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996222&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996219&type= 4.48733330 2.37066650 ） <math id="M69"><msub><mrow><mi>B</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>~</mo><msub><mrow><mi>B</mi></mrow><mrow><mn>2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996293&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996291&type= 9.39799976 3.21733332 <math id="M70"><msub><mrow><mi>T</mi></mrow><mrow><mi mathvariant="normal">i</mi><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">g</mi><mi mathvariant="normal">e</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996333&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996330&type= 6.60400009 3.21733332 / 2 . 2 . 2 三视模式 3 三视模式可对目标区域进行三次成像，以获得三个视角的影像。三视成像过程分为前视、正视与后视三部分，其中前视与后视成像角度大小通常相等，如 图6 图6 所示。 三视成像的成像与机动时间关系如 表2 表2 ，对比双视成像可知三视成像要求卫星具有更强的机动能力。 Item Time quantum Imaging time Maneuver time Front view （ <math id="M71"><mi>θ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996260&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996258&type= 1.52400005 2.37066650 ） <math id="M72"><msub><mrow><mi>A</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>~</mo><msub><mrow><mi>A</mi></mrow><mrow><mn>2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996267&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996263&type= 10.07533360 3.21733332 <math id="M73"><msub><mrow><mi>T</mi></mrow><mrow><mi mathvariant="normal">i</mi><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">g</mi><mi mathvariant="normal">e</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996333&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996330&type= 6.60400009 3.21733332 / Transform1 <math id="M74"><msub><mrow><mi>A</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>~</mo><msub><mrow><mi>B</mi></mrow><mrow><mn>1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996281&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996278&type= 9.73666668 3.21733332 / <math id="M75"><msub><mrow><mi>T</mi></mrow><mrow><mi mathvariant="normal">t</mi><mi mathvariant="normal">r</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">n</mi><mi mathvariant="normal">s</mi><mn>1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996288&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996284&type= 6.68866634 3.21733332 Nadir view <math id="M76"><msub><mrow><mi>B</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>~</mo><msub><mrow><mi>B</mi></mrow><mrow><mn>2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996293&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996291&type= 9.39799976 3.21733332 <math id="M77"><msub><mrow><mi>T</mi></mrow><mrow><mi mathvariant="normal">i</mi><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">g</mi><mi mathvariant="normal">e</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996333&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996330&type= 6.60400009 3.21733332 / Transform2 <math id="M78"><msub><mrow><mi>B</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>~</mo><msub><mrow><mi>C</mi></mrow><mrow><mn>1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996304&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996301&type= 9.48266602 3.21733332 / <math id="M79"><msub><mrow><mi>T</mi></mrow><mrow><mi mathvariant="normal">t</mi><mi mathvariant="normal">r</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">n</mi><mi mathvariant="normal">s</mi><mn>2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996311&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996308&type= 6.68866634 3.21733332 Rear view （ <math id="M80"><mo>-</mo><mi>θ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996320&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996316&type= 4.48733330 2.37066650 ） <math id="M81"><msub><mrow><mi>C</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>~</mo><msub><mrow><mi>C</mi></mrow><mrow><mn>2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996327&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996323&type= 9.56733322 3.21733332 <math id="M82"><msub><mrow><mi>T</mi></mrow><mrow><mi mathvariant="normal">i</mi><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">g</mi><mi mathvariant="normal">e</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996333&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996330&type= 6.60400009 3.21733332 / 由于成像时前视和后视角度的存在，卫星的成像时间并非成像点的过境时间窗口，需要通过轨道递推并根据实时旋转角度的分量与前后视角度的关系进行反向求解获取。三视成像时间窗口确定流程图如 图7 图7 所示。 具体获取过程为： Step 1.设卫星前、正、后视成像时间初值 <math id="M83"><msub><mrow><mi>T</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>=</mo><msub><mrow><mi>T</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>=</mo><msub><mrow><mi>T</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mo>=</mo><msup><mrow><mn mathvariant="normal">10</mn></mrow><mrow><mn mathvariant="normal">20</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996353&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996350&type= 31.49600029 3.97933316 ，初始时间 <math id="M84"><mi>T</mi><mo>=</mo><msub><mrow><mi>T</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996359&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996356&type= 11.17599964 3.72533321 。 Step 2.轨道递推得到时间 <math id="M85"><mi>T</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996371&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996368&type= 2.53999996 2.79399991 对应的卫星位置、速度信息。 Step 3.由卫星位置和速度信息、成像点的经纬高以及地球自转角速度，经过卫星成像姿态求解，获得时间 <math id="M86"><mi>T</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996371&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996368&type= 2.53999996 2.79399991 对应的旋转角度在卫星飞行方向的分量 <math id="M87"><msub><mrow><mi>φ</mi></mrow><mrow><msub><mrow><mi>n</mi></mrow><mrow><mi mathvariant="normal">o</mi></mrow></msub><mi>y</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996378&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996375&type= 5.50333309 3.72533321 。 Step 4.判定 T ，若 <math id="M88"><mi>T</mi><mo><</mo><msub><mrow><mi>T</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996384&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996380&type= 11.17599964 3.72533321 ，转至Step 5判前视成像；若 <math id="M89"><msub><mrow><mi>T</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo><</mo><mi>T</mi><mo><</mo><msub><mrow><mi>T</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996388&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996386&type= 19.72733307 3.72533321 ，转至Step 6判正视成像；若 <math id="M90"><msub><mrow><mi>T</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo><</mo><mi>T</mi><mo><</mo><msub><mrow><mi>T</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996395&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996391&type= 19.72733307 3.72533321 转至Step 7判后视成像。 Step 5.判断前视角度 <math id="M91"><msub><mrow><mi>φ</mi></mrow><mrow><msub><mrow><mi>n</mi></mrow><mrow><mi mathvariant="normal">o</mi></mrow></msub><mi>y</mi></mrow></msub><mo>=</mo><mi>θ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996403&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996399&type= 12.10733318 3.72533321 ，如果成立，获取对应的前视成像时间 <math id="M92"><msub><mrow><mi>T</mi></mrow><mrow><mn mathvariant="normal">01</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996409&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996406&type= 4.74133301 3.72533321 与成像姿态 <math id="M93"><msub><mrow><mi mathvariant="bold-italic">q</mi></mrow><mrow><mi mathvariant="normal">f</mi><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996415&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996412&type= 3.72533321 4.57200003 ，修改成像时间值 <math id="M94"><msub><mrow><mi>T</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>=</mo><msub><mrow><mi>T</mi></mrow><mrow><mn mathvariant="normal">01</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996421&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996418&type= 13.29266548 3.72533321 ，转至Step 6。否则直接转至Step 8。 Step 6.判断正视角度 <math id="M95"><msub><mrow><mi>φ</mi></mrow><mrow><msub><mrow><mi>n</mi></mrow><mrow><mi mathvariant="normal">o</mi></mrow></msub><mi>y</mi></mrow></msub><mo>=</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996426&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996424&type= 12.19200039 3.72533321 ，如果成立，获取对应的正视成像时间 <math id="M96"><msub><mrow><mi>T</mi></mrow><mrow><mn mathvariant="normal">02</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996432&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996429&type= 4.74133301 3.72533321 与成像姿态 <math id="M97"><msub><mrow><mi mathvariant="bold-italic">q</mi></mrow><mrow><mi mathvariant="normal">f</mi><mn mathvariant="normal">2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996438&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996435&type= 3.72533321 4.57200003 ，修改成像时间值 <math id="M98"><msub><mrow><mi>T</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>=</mo><msub><mrow><mi>T</mi></mrow><mrow><mn mathvariant="normal">02</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996444&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996440&type= 13.29266548 3.72533321 ，转至Step 5。否则直接转至Step 8。 Step 7.判断后视角度 <math id="M99"><msub><mrow><mi>φ</mi></mrow><mrow><msub><mrow><mi>n</mi></mrow><mrow><mi mathvariant="normal">o</mi></mrow></msub><mi>y</mi></mrow></msub><mo>=</mo><mo>-</mo><mi>θ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996451&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996448&type= 15.66333294 3.72533321 ，如果成立，获取对应的后视成像时间 <math id="M100"><msub><mrow><mi>T</mi></mrow><mrow><mn mathvariant="normal">03</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996456&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996454&type= 4.74133301 3.72533321 与成像姿态 <math id="M101"><msub><mrow><mi mathvariant="bold-italic">q</mi></mrow><mrow><mi mathvariant="normal">f</mi><mn mathvariant="normal">3</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996464&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996460&type= 3.72533321 4.57200003 ，转至Step 9。否则转至Step 8。 Step 8.轨道进行递推，递推时间增加 <math id="M102"><mi mathvariant="normal">Δ</mi><mi>T</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996470&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996467&type= 5.07999992 2.79399991 ， <math id="M103"><mi>T</mi><mo>=</mo><mi>T</mi><mo>+</mo><mi mathvariant="normal">Δ</mi><mi>T</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996475&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996472&type= 19.81200027 2.96333337 ，转至Step 2。 Step 9.姿态求解结束，输出结果。 对于双视成像，前视成像时间为 <math id="M104"><msub><mrow><mi>T</mi></mrow><mrow><mn mathvariant="normal">04</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996481&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996479&type= 4.74133301 3.72533321 ，成像姿态为 <math id="M105"><msub><mrow><mi mathvariant="bold-italic">q</mi></mrow><mrow><mi mathvariant="normal">f</mi><mn mathvariant="normal">4</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996487&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996484&type= 3.72533321 4.57200003 ，后视成像时间为 <math id="M106"><msub><mrow><mi>T</mi></mrow><mrow><mn mathvariant="normal">05</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996493&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996490&type= 4.74133301 3.72533321 ，成像姿态为 <math id="M107"><msub><mrow><mi mathvariant="bold-italic">q</mi></mrow><mrow><mi mathvariant="normal">f</mi><mn mathvariant="normal">5</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996497&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996495&type= 3.72533321 4.57200003 。 3 PFMC三轴路径规划设计 1 同轨立体成像要求卫星快速进行三轴大角度姿态机动，同时通过提高卫星的机动性能可延长同轨立体成像的成像时长，提高成像覆盖能力。而微小卫星的执行机构力矩和角动量的大小是有限制的，导致卫星机动角速度受限。首先，考虑执行机构的约束设计一种基于旋转轴不变约束的最短路径方法，然后设计一种角加速度连续的八段式路径规划器，最终，设计了角加速度前馈结合内外环控制的快速机动算法，提高姿态跟踪的动态性能。 3.1 旋转轴/角计算 2 目标坐标系 <math id="M108"><mo> </mo><mi>O</mi><msup><mrow><mi>x</mi></mrow><mrow><mi>'</mi></mrow></msup><msup><mrow><mi>y</mi></mrow><mrow><mi>'</mi></mrow></msup><msup><mrow><mi>z</mi></mrow><mrow><mi>'</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996503&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996500&type= 11.34533310 3.89466691 相对初始坐标系 <math id="M109"><mo> </mo><mi>O</mi><mi>x</mi><mi>y</mi><mi>z</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996507&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996505&type= 8.63599968 3.72533321 的姿态转换，如 图8 图8 所示，用四元数形式表示为： <math display="block" id="M110"><mo/><mi mathvariant="bold-italic">q</mi><mo>=</mo><msup><mrow><mo stretchy="false">[</mo><mtable><mtr><mtd><msub><mrow><mi>q</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mtd><mtd><mover accent="true"><mi>q</mi><mo>¯</mo></mover></mtd></mtr></mtable><mo> </mo><mo stretchy="false">]</mo></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup><mo>=</mo><msup><mrow><mo stretchy="false">[</mo><mtable><mtr><mtd><mi mathvariant="normal">c</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">s</mi><mtext> </mtext><mfrac><mrow><mi>φ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><mo> </mo></mtd><mtd><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mtext> </mtext><mfrac><mrow><mi>φ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><msup><mrow><msub><mrow><mi mathvariant="bold-italic">e</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup></mtd></mtr></mtable><mo> </mo><mo stretchy="false">]</mo></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996510&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996509&type= 60.02866364 7.87400007 ， 其中： <math id="M111"><msubsup><mrow><mi>q</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo>+</mo><msup><mrow><mover accent="true"><mi>q</mi><mo>¯</mo></mover></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>=</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996514&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996512&type= 17.52599907 4.82600021 ， <math id="M112"><msub><mrow><mi mathvariant="bold-italic">e</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>=</mo><msup><mrow><mfenced open="[" close="]" separators="|"><mrow><mtable><mtr><mtd><mi mathvariant="bold-italic">i</mi></mtd><mtd><mi mathvariant="bold-italic">j</mi><mo> </mo></mtd><mtd><mi mathvariant="bold-italic">k</mi></mtd></mtr></mtable></mrow></mfenced></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996518&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996516&type= 24.29933357 5.50333309 ， <math id="M113"><mi mathvariant="bold-italic">i</mi><mo> </mo><msup><mrow><mo> </mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><msup><mrow><mi mathvariant="bold-italic">j</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo> </mo><mo> </mo><mo>+</mo><mo> </mo><mo> </mo><mo> </mo><msup><mrow><mi mathvariant="bold-italic">k</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>=</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996522&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996520&type= 26.07733345 3.89466691 。 卫星的目标姿态相对初始姿态的期望四元数： <math display="block" id="M114"><msub><mrow><mi mathvariant="bold-italic">q</mi></mrow><mrow><mi mathvariant="normal">Q</mi></mrow></msub><mo>=</mo><msubsup><mrow><mi mathvariant="bold-italic">q</mi></mrow><mrow><mi mathvariant="normal">C</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msubsup><mo>⊗</mo><msub><mrow><mi mathvariant="bold-italic">q</mi></mrow><mrow><mi mathvariant="normal">F</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996533&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996531&type= 21.67466736 5.16466665 ， 其中：初始四元数 <math id="M115"><msub><mrow><mi mathvariant="bold-italic">q</mi></mrow><mrow><mi mathvariant="normal">C</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996585&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996583&type= 3.55599999 4.57200003 为卫星的初始姿态相对惯性系的旋转四元数，目标四元数 <math id="M116"><msub><mrow><mi mathvariant="bold-italic">q</mi></mrow><mrow><mi mathvariant="normal">F</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996545&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996542&type= 3.47133350 4.57200003 为卫星的目标姿态相对惯性系的旋转四元数。 由 <math id="M117"><msub><mrow><mi mathvariant="bold-italic">q</mi></mrow><mrow><mi mathvariant="normal">Q</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996594&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996592&type= 3.64066648 4.57200003 反向求得旋转角 <math id="M118"><msub><mrow><mi>θ</mi></mrow><mrow><mi mathvariant="normal">Q</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996789&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996787&type= 3.47133350 3.72533321 和旋转轴 <math id="M119"><msub><mrow><mi mathvariant="bold-italic">e</mi></mrow><mrow><mi>n</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996598&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996596&type= 2.79399991 3.72533321 ： <math display="block" id="M120"><mfenced open="{" close="" separators="|"><mrow><mtable><mtr><mtd><msub><mrow><mi>θ</mi></mrow><mrow><mi mathvariant="normal">A</mi></mrow></msub><mo>=</mo><mn mathvariant="normal">2</mn><mi mathvariant="normal">a</mi><mi mathvariant="normal">r</mi><mi mathvariant="normal">c</mi><mi mathvariant="normal">c</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">s</mi><mo stretchy="false">(</mo><msub><mrow><mi>q</mi></mrow><mrow><mi mathvariant="normal">Q</mi><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><msub><mrow><mi mathvariant="bold-italic">e</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>=</mo><mrow><mrow><msub><mrow><mover accent="true"><mi>q</mi><mo>¯</mo></mover></mrow><mrow><mi mathvariant="normal">Q</mi></mrow></msub></mrow><mo>/</mo><mrow><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mtext> </mtext><mfrac><mrow><msub><mrow><mi>θ</mi></mrow><mrow><mi mathvariant="normal">Q</mi></mrow></msub></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac></mrow></mrow><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo>,</mo><msub><mrow><mi>θ</mi></mrow><mrow><mi mathvariant="normal">Q</mi></mrow></msub><mo>≠</mo><mn mathvariant="normal">0</mn></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996562&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996560&type= 38.86199951 15.23999977 . 当 <math id="M121"><msub><mrow><mi>θ</mi></mrow><mrow><mi mathvariant="normal">Q</mi></mrow></msub><mo>=</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996565&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996563&type= 10.15999985 3.72533321 时，对应的四元数为 <math id="M122"><msub><mrow><mi mathvariant="bold-italic">q</mi></mrow><mrow><mi mathvariant="normal">Q</mi></mrow></msub><mo>=</mo><msup><mrow><mfenced open="[" close="]" separators="|"><mrow><mtable><mtr><mtd><mn mathvariant="normal">1</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd></mtr></mtable></mrow></mfenced></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996570&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996568&type= 32.08866501 4.74133301 ，目标姿态与初始姿态重合，不需进行姿态转换。 3.2 旋转轴不变约束求解 2 在卫星姿态机动的过程中，角加速度、角速度以及控制力矩受执行机构性能的限制。考虑卫星的性能约束通过路径规划器对卫星的机动过程进行实时规划，提高卫星的机动能力。 在满足卫星转动惯量 <math id="M123"><mi mathvariant="bold-italic">I</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996574&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996571&type= 1.43933344 2.79399991 、反作用飞轮力矩 <math id="M124"><mi mathvariant="bold-italic">T</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996578&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996577&type= 2.62466669 2.79399991 和角动量 <math id="M125"><mi mathvariant="bold-italic">H</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996582&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996581&type= 3.04800010 2.79399991 约束下，为了实现初始四元数 <math id="M126"><msub><mrow><mi mathvariant="bold-italic">q</mi></mrow><mrow><mi mathvariant="normal">C</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996585&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996583&type= 3.55599999 4.57200003 至目标四元数 <math id="M127"><msub><mrow><mi mathvariant="bold-italic">q</mi></mrow><mrow><mi mathvariant="normal">F</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996590&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996588&type= 3.47133350 4.57200003 的最短路径旋转，即整个期望四元数 <math id="M128"><msub><mrow><mi mathvariant="bold-italic">q</mi></mrow><mrow><mi mathvariant="normal">Q</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996594&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996592&type= 3.64066648 4.57200003 的机动过程中均保证旋转轴 <math id="M129"><msub><mrow><mi mathvariant="bold-italic">e</mi></mrow><mrow><mi>n</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996598&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996596&type= 2.79399991 3.72533321 不变，在进行路径规划时，需要对三轴角加速度、角速度进行限幅和约束设置。 路径规划器的输入为期望旋转角度 <math id="M130"><msub><mrow><mi>θ</mi></mrow><mrow><mi mathvariant="normal">Q</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996602&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996600&type= 3.47133350 3.72533321 ，角加速度限值 <math id="M131"><msub><mrow><mi>α</mi></mrow><mrow><mi mathvariant="normal">L</mi><mi mathvariant="normal">G</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996777&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996775&type= 4.91066647 3.72533321 ，角速度限值 <math id="M132"><msub><mrow><mi>ω</mi></mrow><mrow><mi mathvariant="normal">L</mi><mi mathvariant="normal">G</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996610&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996607&type= 5.41866684 3.72533321 ，输出为实时角度，实时角速度，PFMC三轴路径规划如 图9 图9 所示。 角加速度限值 <math id="M133"><msub><mrow><mi>α</mi></mrow><mrow><mi mathvariant="normal">L</mi><mi mathvariant="normal">G</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996777&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996775&type= 4.91066647 3.72533321 的计算如下： <math display="block" id="M134"><msub><mrow><mover accent="true"><mi>α</mi><mo>¯</mo></mover></mrow><mrow><mi mathvariant="normal">i</mi></mrow></msub><mo>=</mo><mfenced open="{" close="" separators="|"><mrow><mtable><mtr><mtd><mrow><mrow><mo stretchy="false">(</mo><msup><mrow><mi mathvariant="bold-italic">I</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><msub><mrow><mi mathvariant="bold-italic">T</mi><mo stretchy="false">)</mo></mrow><mrow><mi mathvariant="normal">i</mi></mrow></msub></mrow><mo>/</mo><mrow><msub><mrow><mi mathvariant="bold-italic">e</mi></mrow><mrow><mi mathvariant="normal">i</mi></mrow></msub><mo> </mo><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">e</mi></mrow><mrow><mi mathvariant="normal">i</mi></mrow></msub><mo>≠</mo><mn mathvariant="normal">0</mn></mrow></mrow></mtd></mtr><mtr><mtd><mo> </mo><msub><mrow><mi>M</mi></mrow><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">x</mi></mrow></msub><mtext> </mtext><mo> </mo><mtext> </mtext><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">e</mi></mrow><mrow><mi mathvariant="normal">i</mi></mrow></msub><mo>=</mo><mn mathvariant="normal">0</mn></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996623&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996621&type= 39.53932953 10.15999985 ， 其中， <math id="M135"><msub><mrow><mi>M</mi></mrow><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">x</mi></mrow></msub><mo>=</mo><msup><mrow><mn mathvariant="normal">10</mn></mrow><mrow><mn mathvariant="normal">20</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996627&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996625&type= 17.61066628 3.97933316 ，为一个较大的数。 <math display="block" id="M136"><msub><mrow><mi mathvariant="bold-italic">α</mi></mrow><mrow><mi mathvariant="normal">L</mi><mi mathvariant="normal">G</mi></mrow></msub><mo>=</mo><msubsup><mrow><mfenced open="[" close="]" separators="|"><mrow><mtable><mtr><mtd><msub><mrow><mover accent="true"><mi>α</mi><mo>¯</mo></mover></mrow><mrow><mi mathvariant="bold-italic">x</mi></mrow></msub></mtd><mtd><msub><mrow><mover accent="true"><mi>α</mi><mo>¯</mo></mover></mrow><mrow><mi mathvariant="bold-italic">y</mi></mrow></msub></mtd><mtd><msub><mrow><mover accent="true"><mi>α</mi><mo>¯</mo></mover></mrow><mrow><mi mathvariant="bold-italic">z</mi></mrow></msub></mtd></mtr></mtable></mrow></mfenced></mrow><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msubsup><msub><mrow><mi mathvariant="bold-italic">e</mi></mrow><mrow><mi>n</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996631&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996629&type= 36.32199860 5.24933338 ， <math display="block" id="M137"><msub><mrow><mi>α</mi></mrow><mrow><mi mathvariant="normal">L</mi><mi mathvariant="normal">G</mi></mrow></msub><mo>=</mo><mfenced open="‖" close="‖" separators="|"><mrow><msub><mrow><mi mathvariant="bold-italic">α</mi></mrow><mrow><mi mathvariant="normal">L</mi><mi mathvariant="normal">G</mi></mrow></msub></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996635&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996633&type= 18.20333290 3.97933316 ， 其中， <math id="M138"><msubsup><mrow><mfenced open="[" close="]" separators="|"><mrow><mtable><mtr><mtd><msub><mrow><mover accent="true"><mi>α</mi><mo>¯</mo></mover></mrow><mrow><mi mathvariant="bold-italic">x</mi></mrow></msub></mtd><mtd><msub><mrow><mover accent="true"><mi>α</mi><mo>¯</mo></mover></mrow><mrow><mi mathvariant="bold-italic">y</mi></mrow></msub></mtd><mtd><msub><mrow><mover accent="true"><mi>α</mi><mo>¯</mo></mover></mrow><mrow><mi mathvariant="bold-italic">z</mi></mrow></msub></mtd></mtr></mtable></mrow></mfenced></mrow><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996640&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996638&type= 23.45266724 5.24933338 为求向量 <math id="M139"><msubsup><mrow><mfenced open="[" close="]" separators="|"><mrow><mtable><mtr><mtd><msub><mrow><mover accent="true"><mi>α</mi><mo>¯</mo></mover></mrow><mrow><mi mathvariant="bold-italic">x</mi></mrow></msub></mtd><mtd><msub><mrow><mover accent="true"><mi>α</mi><mo>¯</mo></mover></mrow><mrow><mi mathvariant="bold-italic">y</mi></mrow></msub></mtd><mtd><msub><mrow><mover accent="true"><mi>α</mi><mo>¯</mo></mover></mrow><mrow><mi mathvariant="bold-italic">z</mi></mrow></msub></mtd></mtr></mtable></mrow></mfenced></mrow><mrow/><mrow><mi mathvariant="normal">T</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996644&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996642&type= 21.84399986 5.24933338 的最小值。 角速度限值 <math id="M140"><msub><mrow><mi>ω</mi></mrow><mrow><mi mathvariant="normal">L</mi><mi mathvariant="normal">G</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996648&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996646&type= 5.41866684 3.72533321 的计算如下： <math display="block" id="M141"><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">ω</mi><mo>¯</mo></mover></mrow><mrow><mi mathvariant="normal">i</mi></mrow></msub><mo>=</mo><mfenced open="{" close="" separators="|"><mrow><mtable><mtr><mtd><mfrac><mrow><mo stretchy="false">[</mo><mo stretchy="false">(</mo><msup><mrow><mi mathvariant="bold-italic">I</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><msub><mrow><mi mathvariant="bold-italic">H</mi><mo stretchy="false">)</mo></mrow><mrow><mi mathvariant="normal">i</mi></mrow></msub><mo>,</mo><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">ω</mi></mrow><mrow><mi mathvariant="normal">L</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">m</mi></mrow></msub><msub><mrow><mo stretchy="false">)</mo></mrow><mrow><mi mathvariant="normal">i</mi></mrow></msub><msub><mrow><mo stretchy="false">]</mo></mrow><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi></mrow></msub></mrow><mrow><msub><mrow><mi mathvariant="bold-italic">e</mi></mrow><mrow><mi mathvariant="normal">i</mi></mrow></msub></mrow></mfrac><mo>,</mo></mtd><mtd><msub><mrow><mi mathvariant="bold-italic">e</mi></mrow><mrow><mi mathvariant="normal">i</mi></mrow></msub><mo>≠</mo><mn mathvariant="normal">0</mn></mtd></mtr><mtr><mtd><msub><mrow><mi>M</mi></mrow><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">x</mi></mrow></msub><mo>,</mo></mtd><mtd><msub><mrow><mi mathvariant="bold-italic">e</mi></mrow><mrow><mi mathvariant="normal">i</mi></mrow></msub><mo>=</mo><mn mathvariant="normal">0</mn></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996652&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996650&type= 56.81133270 14.81666565 ， 其中， <math id="M142"><msub><mrow><mi mathvariant="bold-italic">ω</mi></mrow><mrow><mi mathvariant="normal">L</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">m</mi></mrow></msub><mo>=</mo><msup><mrow><mfenced open="[" close="]" separators="|"><mrow><mtable><mtr><mtd><msub><mrow><mi>ω</mi></mrow><mrow><mi mathvariant="normal">L</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">m</mi><mi>x</mi></mrow></msub></mtd><mtd><msub><mrow><mi>ω</mi></mrow><mrow><mi mathvariant="normal">L</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">m</mi><mi>y</mi></mrow></msub></mtd><mtd><msub><mrow><mi>ω</mi></mrow><mrow><mi mathvariant="normal">L</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">m</mi><mi>z</mi></mrow></msub></mtd></mtr></mtable></mrow></mfenced></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996656&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996654&type= 44.19599915 4.74133301 。 <math display="block" id="M143"><msub><mrow><mi mathvariant="bold-italic">ω</mi></mrow><mrow><mi mathvariant="normal">L</mi><mi mathvariant="normal">G</mi></mrow></msub><mo>=</mo><msubsup><mrow><mfenced open="[" close="]" separators="|"><mrow><mtable><mtr><mtd><msub><mrow><mover accent="true"><mi>ω</mi><mo>¯</mo></mover></mrow><mrow><mi>x</mi></mrow></msub></mtd><mtd><msub><mrow><mover accent="true"><mi>ω</mi><mo>¯</mo></mover></mrow><mrow><mi>y</mi></mrow></msub></mtd><mtd><msub><mrow><mover accent="true"><mi>ω</mi><mo>¯</mo></mover></mrow><mrow><mi>z</mi></mrow></msub></mtd></mtr></mtable></mrow></mfenced></mrow><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msubsup><msub><mrow><mi mathvariant="bold-italic">e</mi></mrow><mrow><mi>n</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996660&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996658&type= 38.26933289 5.24933338 ， <math display="block" id="M144"><msub><mrow><mi>ω</mi></mrow><mrow><mi mathvariant="normal">L</mi><mi mathvariant="normal">G</mi></mrow></msub><mo>=</mo><mfenced open="‖" close="‖" separators="|"><mrow><msub><mrow><mi mathvariant="bold-italic">ω</mi></mrow><mrow><mi mathvariant="normal">L</mi><mi mathvariant="normal">G</mi></mrow></msub></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996664&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996662&type= 19.21933365 3.97933316 ， 其中，飞轮角动量 <math id="M145"><mi mathvariant="bold-italic">H</mi><mo>=</mo><msup><mrow><mfenced open="[" close="]" separators="|"><mrow><mtable><mtr><mtd><msub><mrow><mi>H</mi></mrow><mrow><mi>x</mi></mrow></msub></mtd><mtd><msub><mrow><mi>H</mi></mrow><mrow><mi>y</mi></mrow></msub></mtd><mtd><msub><mrow><mi>H</mi></mrow><mrow><mi>z</mi></mrow></msub></mtd></mtr></mtable></mrow></mfenced></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996668&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996666&type= 32.59666443 4.74133301 。 3.3 路径规划器设计 2 为规避角加速度突变的问题，实现飞轮力矩的平稳变化。借鉴文献［ 12 12 ］中角加速度为连续曲线的思路，增加角加速度总的平稳段，设计一种角加速度连续的八段式路径规划器，角加速度生成函数 <math id="M146"><mfrac><mrow><mi mathvariant="normal">d</mi></mrow><mrow><mi mathvariant="normal">d</mi><mi>t</mi></mrow></mfrac><mi>f</mi><mo stretchy="false">(</mo><mi>t</mi><mo>,</mo><mi mathvariant="normal">Δ</mi><msub><mrow><mi>t</mi></mrow><mrow><mi mathvariant="normal">A</mi></mrow></msub><mo>,</mo><mi mathvariant="normal">Δ</mi><msub><mrow><mi>t</mi></mrow><mrow><mi mathvariant="normal">B</mi></mrow></msub><mo>,</mo><mi mathvariant="normal">Δ</mi><msub><mrow><mi>t</mi></mrow><mrow><mi mathvariant="normal">C</mi></mrow></msub><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996672&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996670&type= 32.68133163 7.87400007 简记为 <math id="M147"><mfrac><mrow><mi mathvariant="normal">d</mi><mi>f</mi></mrow><mrow><mi mathvariant="normal">d</mi><mi>t</mi></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996676&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996674&type= 3.72533321 7.95866632 定义如下： <math display="block" id="M148"><mfrac><mrow><mi mathvariant="normal">d</mi><mi>f</mi></mrow><mrow><mi mathvariant="normal">d</mi><mi>t</mi></mrow></mfrac><mo>=</mo><mfenced open="{" close="" separators="|"><mrow><mtable><mtr><mtd><msup><mrow><mfenced separators="|"><mrow><mfrac><mrow><mi>t</mi><mo>-</mo><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow><mrow><mi mathvariant="normal">Δ</mi><msub><mrow><mi>t</mi></mrow><mrow><mi mathvariant="normal">A</mi></mrow></msub></mrow></mfrac></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mfenced separators="|"><mrow><mn mathvariant="normal">3</mn><mo>-</mo><mn mathvariant="normal">2</mn><mfenced separators="|"><mrow><mfrac><mrow><mi>t</mi><mo>-</mo><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow><mrow><mi mathvariant="normal">Δ</mi><msub><mrow><mi>t</mi></mrow><mrow><mi mathvariant="normal">A</mi></mrow></msub></mrow></mfrac></mrow></mfenced></mrow></mfenced><mo>,</mo></mtd><mtd><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo><</mo><mi>t</mi><mo>≤</mo><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mtext> </mtext><mo>=</mo><mtext> </mtext><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>+</mo><mi mathvariant="normal">Δ</mi><msub><mrow><mi>t</mi></mrow><mrow><mi>A</mi></mrow></msub></mtd></mtr><mtr><mtd><mn mathvariant="normal">1</mn><mo>,</mo></mtd><mtd><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo><</mo><mi>t</mi><mo>≤</mo><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>=</mo><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><mi mathvariant="normal">Δ</mi><msub><mrow><mi>t</mi></mrow><mrow><mi>B</mi></mrow></msub></mtd></mtr><mtr><mtd><msup><mrow><mfenced separators="|"><mrow><mfrac><mrow><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mo>-</mo><mi>t</mi></mrow><mrow><mi mathvariant="normal">Δ</mi><msub><mrow><mi>t</mi></mrow><mrow><mi mathvariant="normal">A</mi></mrow></msub></mrow></mfrac></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mfenced separators="|"><mrow><mn mathvariant="normal">3</mn><mo>-</mo><mn mathvariant="normal">2</mn><mfenced separators="|"><mrow><mfrac><mrow><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mo>-</mo><mi>t</mi></mrow><mrow><mi mathvariant="normal">Δ</mi><msub><mrow><mi>t</mi></mrow><mrow><mi mathvariant="normal">A</mi></mrow></msub></mrow></mfrac></mrow></mfenced></mrow></mfenced><mo>,</mo></mtd><mtd><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo><</mo><mi>t</mi><mo>≤</mo><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mo>=</mo><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>+</mo><mi mathvariant="normal">Δ</mi><msub><mrow><mi>t</mi></mrow><mrow><mi>A</mi></mrow></msub></mtd></mtr><mtr><mtd><mn mathvariant="normal">0</mn><mo>,</mo></mtd><mtd><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mo><</mo><mi>t</mi><mo>≤</mo><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msub><mo>=</mo><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mo>+</mo><mi mathvariant="normal">Δ</mi><msub><mrow><mi>t</mi></mrow><mrow><mi>C</mi></mrow></msub></mtd></mtr><mtr><mtd><mo>-</mo><msup><mrow><mfenced separators="|"><mrow><mfrac><mrow><mi>t</mi><mo>-</mo><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msub></mrow><mrow><mi mathvariant="normal">Δ</mi><msub><mrow><mi>t</mi></mrow><mrow><mi mathvariant="normal">A</mi></mrow></msub></mrow></mfrac></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mfenced separators="|"><mrow><mn mathvariant="normal">3</mn><mo>-</mo><mn mathvariant="normal">2</mn><mfenced separators="|"><mrow><mfrac><mrow><mi>t</mi><mo>-</mo><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msub></mrow><mrow><mi mathvariant="normal">Δ</mi><msub><mrow><mi>t</mi></mrow><mrow><mi mathvariant="normal">A</mi></mrow></msub></mrow></mfrac></mrow></mfenced></mrow></mfenced><mo>,</mo></mtd><mtd><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msub><mo><</mo><mi>t</mi><mo>≤</mo><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">5</mn></mrow></msub><mtext> </mtext><mo>=</mo><mtext> </mtext><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msub><mo>+</mo><mi mathvariant="normal">Δ</mi><msub><mrow><mi>t</mi></mrow><mrow><mi>A</mi></mrow></msub></mtd></mtr><mtr><mtd><mo>-</mo><mn mathvariant="normal">1</mn><mo>,</mo></mtd><mtd><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">5</mn></mrow></msub><mo><</mo><mi>t</mi><mo>≤</mo><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">6</mn></mrow></msub><mo>=</mo><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">5</mn></mrow></msub><mo>+</mo><mi mathvariant="normal">Δ</mi><msub><mrow><mi>t</mi></mrow><mrow><mi>B</mi></mrow></msub></mtd></mtr><mtr><mtd><msup><mrow><mfenced separators="|"><mrow><mfrac><mrow><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">7</mn></mrow></msub><mo>-</mo><mi>t</mi></mrow><mrow><mi mathvariant="normal">Δ</mi><msub><mrow><mi>t</mi></mrow><mrow><mi mathvariant="normal">A</mi></mrow></msub></mrow></mfrac></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mfenced separators="|"><mrow><mn mathvariant="normal">3</mn><mo>-</mo><mn mathvariant="normal">2</mn><mfenced separators="|"><mrow><mfrac><mrow><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">7</mn></mrow></msub><mo>-</mo><mi>t</mi></mrow><mrow><mi mathvariant="normal">Δ</mi><msub><mrow><mi>t</mi></mrow><mrow><mi mathvariant="normal">A</mi></mrow></msub></mrow></mfrac></mrow></mfenced></mrow></mfenced><mo>,</mo></mtd><mtd><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">6</mn></mrow></msub><mo><</mo><mi>t</mi><mo>≤</mo><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">7</mn></mrow></msub><mtext> </mtext><mo>=</mo><mtext> </mtext><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">6</mn></mrow></msub><mo>+</mo><mi mathvariant="normal">Δ</mi><msub><mrow><mi>t</mi></mrow><mrow><mi>A</mi></mrow></msub></mtd></mtr><mtr><mtd><mn mathvariant="normal">0</mn><mo>,</mo></mtd><mtd><mi>t</mi><mo>></mo><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">7</mn></mrow></msub></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996680&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996678&type= 91.44000244 75.94599915 ， 其中： <math id="M149"><mi mathvariant="normal">Δ</mi><msub><mrow><mi>t</mi></mrow><mrow><mi mathvariant="normal">A</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996684&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996682&type= 5.24933338 3.72533321 为设定值，限定了角加速度的上升时间，可根据卫星执行机构的动态性能合理选择； <math id="M150"><mi mathvariant="normal">Δ</mi><msub><mrow><mi>t</mi></mrow><mrow><mi mathvariant="normal">B</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996688&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996686&type= 5.07999992 3.72533321 ， <math id="M151"><mi mathvariant="normal">Δ</mi><msub><mrow><mi>t</mi></mrow><mrow><mi mathvariant="normal">C</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996692&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996690&type= 5.16466665 3.72533321 的值与期望角度 <math id="M152"><msub><mrow><mi>θ</mi></mrow><mrow><mi mathvariant="normal">Q</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996789&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996787&type= 3.47133350 3.72533321 的大小有关， <math id="M153"><mi mathvariant="normal">Δ</mi><msub><mrow><mi>t</mi></mrow><mrow><mi mathvariant="normal">B</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996699&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996697&type= 5.07999992 3.72533321 ， <math id="M154"><mi mathvariant="normal">Δ</mi><msub><mrow><mi>t</mi></mrow><mrow><mi mathvariant="normal">C</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996704&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996701&type= 5.16466665 3.72533321 ， <math id="M155"><mover><mrow><mi>θ</mi></mrow><mrow><mo>⋅</mo></mrow></mover></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996707&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996705&type= 2.20133328 5.16466665 具体计算过程如下： （1）当期望旋转角度 <math id="M156"><msub><mrow><mi>θ</mi></mrow><mrow><mi mathvariant="normal">Q</mi></mrow></msub><mo>></mo><mfrac><mrow><msubsup><mrow><mi>ω</mi></mrow><mrow><mi mathvariant="normal">L</mi><mi mathvariant="normal">G</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup></mrow><mrow><msub><mrow><mi>α</mi></mrow><mrow><mi mathvariant="normal">L</mi><mi mathvariant="normal">G</mi></mrow></msub></mrow></mfrac><mo>+</mo><mi mathvariant="normal">Δ</mi><msub><mrow><mi>t</mi></mrow><mrow><mi mathvariant="normal">A</mi></mrow></msub><msub><mrow><mi>ω</mi></mrow><mrow><mi mathvariant="normal">L</mi><mi mathvariant="normal">G</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996711&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996709&type= 30.22599983 8.63599968 时，规划角加速度 <math id="M157"><mover accent="true"><mrow><mi>θ</mi></mrow><mi>¨</mi></mover></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996773&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996771&type= 1.77800000 3.55599999 的最大值达到 <math id="M158"><msub><mrow><mi>α</mi></mrow><mrow><mi mathvariant="normal">L</mi><mi mathvariant="normal">G</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996777&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996775&type= 4.91066647 3.72533321 ，规划角速度 <math id="M159"><mover accent="true"><mi>θ</mi><mo>˙</mo></mover></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996834&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996832&type= 1.77800000 3.55599999 的最大值达到 <math id="M160"><msub><mrow><mi>ω</mi></mrow><mrow><mi mathvariant="normal">L</mi><mi mathvariant="normal">G</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996785&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996783&type= 5.41866684 3.72533321 。 <math display="block" id="M161"><mfenced open="{" close="" separators="|"><mrow><mtable><mtr><mtd><mi mathvariant="normal">Δ</mi><msub><mrow><mi>t</mi></mrow><mrow><mi mathvariant="normal">B</mi></mrow></msub><mo>=</mo><mfrac><mrow><msub><mrow><mi>ω</mi></mrow><mrow><mi mathvariant="normal">L</mi><mi mathvariant="normal">G</mi></mrow></msub></mrow><mrow><msub><mrow><mi>α</mi></mrow><mrow><mi mathvariant="normal">L</mi><mi mathvariant="normal">G</mi></mrow></msub></mrow></mfrac><mo>-</mo><mi mathvariant="normal">Δ</mi><msub><mrow><mi>t</mi></mrow><mrow><mi mathvariant="normal">A</mi></mrow></msub></mtd></mtr><mtr><mtd><mi mathvariant="normal">Δ</mi><msub><mrow><mi>t</mi></mrow><mrow><mi mathvariant="normal">C</mi></mrow></msub><mo>=</mo><mfrac><mrow><msub><mrow><mi>θ</mi></mrow><mrow><mi mathvariant="normal">Q</mi></mrow></msub></mrow><mrow><msub><mrow><mi>ω</mi></mrow><mrow><mi mathvariant="normal">d</mi></mrow></msub></mrow></mfrac><mo>-</mo><mn mathvariant="normal">2</mn><mi mathvariant="normal">Δ</mi><msub><mrow><mi>t</mi></mrow><mrow><mi mathvariant="normal">A</mi></mrow></msub><mo>-</mo><mi mathvariant="normal">Δ</mi><msub><mrow><mi>t</mi></mrow><mrow><mi mathvariant="normal">B</mi></mrow></msub></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996733&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996731&type= 38.18466568 19.64266586 ， <math display="block" id="M162"><mover accent="true"><mrow><mi>θ</mi></mrow><mi>¨</mi></mover><mo>=</mo><msub><mrow><mi>α</mi></mrow><mrow><mi mathvariant="normal">L</mi><mi mathvariant="normal">G</mi></mrow></msub><mfrac><mrow><mi>d</mi></mrow><mrow><mi mathvariant="normal">d</mi><mi mathvariant="normal">t</mi></mrow></mfrac><mi>f</mi><mfenced separators="|"><mrow><mi>t</mi><mo>,</mo><mi mathvariant="normal">Δ</mi><msub><mrow><mi>t</mi></mrow><mrow><mi mathvariant="normal">A</mi></mrow></msub><mo>,</mo><mi mathvariant="normal">Δ</mi><msub><mrow><mi>t</mi></mrow><mrow><mi mathvariant="normal">B</mi></mrow></msub><mo>,</mo><mi mathvariant="normal">Δ</mi><msub><mrow><mi>t</mi></mrow><mrow><mi mathvariant="normal">C</mi></mrow></msub></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996764&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996762&type= 44.95800018 7.95866632 . （2）当 <math id="M163"><mn mathvariant="normal">2</mn><mo> </mo><mo> </mo><msub><mrow><mi>α</mi></mrow><mrow><mi mathvariant="normal">L</mi><mi mathvariant="normal">G</mi></mrow></msub><mo> </mo><mo> </mo><mi mathvariant="normal">Δ</mi><msubsup><mrow><mi>t</mi></mrow><mrow><mi mathvariant="normal">A</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo>≤</mo><msub><mrow><mi>θ</mi></mrow><mrow><mi mathvariant="normal">Q</mi></mrow></msub><mo>≤</mo><mfrac><mrow><msubsup><mrow><mi>ω</mi></mrow><mrow><mi mathvariant="normal">L</mi><mi mathvariant="normal">G</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup></mrow><mrow><msub><mrow><mi>α</mi></mrow><mrow><mi mathvariant="normal">L</mi><mi mathvariant="normal">G</mi></mrow></msub></mrow></mfrac><mo>+</mo><mi mathvariant="normal">Δ</mi><msub><mrow><mi>t</mi></mrow><mrow><mi mathvariant="normal">A</mi></mrow></msub><msub><mrow><mi>ω</mi></mrow><mrow><mi mathvariant="normal">L</mi><mi mathvariant="normal">G</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996741&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996739&type= 48.68333054 8.63599968 时，规划角加速度 <math id="M164"><mover accent="true"><mrow><mi>θ</mi></mrow><mi>¨</mi></mover></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996773&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996771&type= 1.77800000 3.55599999 的最大值达到 <math id="M165"><msub><mrow><mi>α</mi></mrow><mrow><mi mathvariant="normal">L</mi><mi mathvariant="normal">G</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996777&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996775&type= 4.91066647 3.72533321 ，规划角速度 <math id="M166"><mover accent="true"><mi>θ</mi><mo>˙</mo></mover></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996834&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996832&type= 1.77800000 3.55599999 的最大值未达到 <math id="M167"><msub><mrow><mi>ω</mi></mrow><mrow><mi mathvariant="normal">L</mi><mi mathvariant="normal">G</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996785&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996783&type= 5.41866684 3.72533321 。 <math display="block" id="M168"><mfenced open="{" close="" separators="|"><mrow><mtable><mtr><mtd><mn mathvariant="normal">2</mn><mi mathvariant="normal">Δ</mi><msubsup><mrow><mi>t</mi></mrow><mrow><mi mathvariant="normal">A</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo>+</mo><mn mathvariant="normal">3</mn><mi mathvariant="normal">Δ</mi><msub><mrow><mi>t</mi></mrow><mrow><mi mathvariant="normal">A</mi></mrow></msub><mi mathvariant="normal">Δ</mi><msub><mrow><mi>t</mi></mrow><mrow><mi mathvariant="normal">B</mi></mrow></msub><mo>+</mo><mi mathvariant="normal">Δ</mi><msubsup><mrow><mi>t</mi></mrow><mrow><mi mathvariant="normal">B</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo>=</mo><mfrac><mrow><msub><mrow><mi>θ</mi></mrow><mrow><mi mathvariant="normal">Q</mi></mrow></msub></mrow><mrow><msub><mrow><mi>α</mi></mrow><mrow><mi mathvariant="normal">L</mi><mi mathvariant="normal">G</mi></mrow></msub></mrow></mfrac></mtd></mtr><mtr><mtd><mi mathvariant="normal">Δ</mi><msub><mrow><mi>t</mi></mrow><mrow><mi mathvariant="normal">C</mi></mrow></msub><mo>=</mo><mn mathvariant="normal">0</mn></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996761&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996759&type= 46.73600006 14.13933277 ， <math display="block" id="M169"><mover accent="true"><mrow><mi>θ</mi></mrow><mi>¨</mi></mover><mo>=</mo><msub><mrow><mi>α</mi></mrow><mrow><mi mathvariant="normal">L</mi><mi mathvariant="normal">G</mi></mrow></msub><mfrac><mrow><mi>d</mi></mrow><mrow><mi mathvariant="normal">d</mi><mi mathvariant="normal">t</mi></mrow></mfrac><mi>f</mi><mfenced separators="|"><mrow><mi>t</mi><mo>,</mo><mi mathvariant="normal">Δ</mi><msub><mrow><mi>t</mi></mrow><mrow><mi mathvariant="normal">A</mi></mrow></msub><mo>,</mo><mi mathvariant="normal">Δ</mi><msub><mrow><mi>t</mi></mrow><mrow><mi mathvariant="normal">B</mi></mrow></msub><mo>,</mo><mi mathvariant="normal">Δ</mi><msub><mrow><mi>t</mi></mrow><mrow><mi mathvariant="normal">C</mi></mrow></msub></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996764&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996762&type= 44.95800018 7.95866632 . （3）当 <math id="M170"><mi>θ</mi><mo><</mo><mn mathvariant="normal">2</mn><mo> </mo><mo> </mo><msub><mrow><mi>α</mi></mrow><mrow><mi mathvariant="normal">L</mi><mi mathvariant="normal">G</mi></mrow></msub><mo> </mo><mo> </mo><mi mathvariant="normal">Δ</mi><msubsup><mrow><mi>t</mi></mrow><mrow><mi mathvariant="normal">A</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996768&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996766&type= 20.23533440 3.97933316 时，规划角加速度 <math id="M171"><mover accent="true"><mrow><mi>θ</mi></mrow><mi>¨</mi></mover></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996773&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996771&type= 1.77800000 3.55599999 的最大值未达到 <math id="M172"><msub><mrow><mi>α</mi></mrow><mrow><mi mathvariant="normal">L</mi><mi mathvariant="normal">G</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996777&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996775&type= 4.91066647 3.72533321 ，规划角速度 <math id="M173"><mover accent="true"><mi>θ</mi><mo>˙</mo></mover></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996834&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996832&type= 1.77800000 3.55599999 的最大值未达到 <math id="M174"><msub><mrow><mi>ω</mi></mrow><mrow><mi mathvariant="normal">L</mi><mi mathvariant="normal">G</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996785&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996783&type= 5.41866684 3.72533321 。机动角度 <math id="M175"><msub><mrow><mi>θ</mi></mrow><mrow><mi mathvariant="normal">Q</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996789&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996787&type= 3.47133350 3.72533321 对应的时间均为 <math id="M176"><mn mathvariant="normal">4</mn><mi mathvariant="normal">Δ</mi><msub><mrow><mi>t</mi></mrow><mrow><mi mathvariant="normal">A</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996793&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996791&type= 7.11199999 3.72533321 。 <math display="block" id="M177"><mfenced open="{" close="" separators="|"><mrow><mtable><mtr><mtd><mi mathvariant="normal">Δ</mi><msub><mrow><mi>t</mi></mrow><mrow><mi mathvariant="normal">B</mi></mrow></msub><mo>=</mo><mn mathvariant="normal">0</mn></mtd></mtr><mtr><mtd><mi mathvariant="normal">Δ</mi><msub><mrow><mi>t</mi></mrow><mrow><mi mathvariant="normal">C</mi></mrow></msub><mo>=</mo><mn mathvariant="normal">0</mn></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996797&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996795&type= 13.37733269 8.55133343 ， <math display="block" id="M178"><mover accent="true"><mrow><mi>θ</mi></mrow><mi>¨</mi></mover><mo>=</mo><mfrac><mrow><msub><mrow><mi>θ</mi></mrow><mrow><mi mathvariant="normal">Q</mi></mrow></msub></mrow><mrow><mn mathvariant="normal">2</mn><mo> </mo><mo> </mo><msub><mrow><mi>α</mi></mrow><mrow><mi mathvariant="normal">L</mi><mi mathvariant="normal">G</mi></mrow></msub><mo> </mo><mo> </mo><mi mathvariant="normal">Δ</mi><msubsup><mrow><mi>t</mi></mrow><mrow><mi mathvariant="normal">A</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup></mrow></mfrac><mfrac><mrow><mi>d</mi></mrow><mrow><mi mathvariant="normal">d</mi><mi mathvariant="normal">t</mi></mrow></mfrac><mi>f</mi><mfenced separators="|"><mrow><mi>t</mi><mo>,</mo><mi mathvariant="normal">Δ</mi><msub><mrow><mi>t</mi></mrow><mrow><mi mathvariant="normal">A</mi></mrow></msub><mo>,</mo><mi mathvariant="normal">Δ</mi><msub><mrow><mi>t</mi></mrow><mrow><mi mathvariant="normal">B</mi></mrow></msub><mo>,</mo><mi mathvariant="normal">Δ</mi><msub><mrow><mi>t</mi></mrow><mrow><mi mathvariant="normal">C</mi></mrow></msub></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996803&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996800&type= 54.77933884 8.63599968 . 以 <math id="M179"><msub><mrow><mi>θ</mi></mrow><mrow><mi mathvariant="normal">Q</mi></mrow></msub><mo>=</mo><msup><mrow><mn mathvariant="normal">60</mn></mrow><mrow><mi mathvariant="normal">o</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996807&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996805&type= 13.12333298 3.97933316 ， <math id="M180"><mi mathvariant="normal">Δ</mi><msub><mrow><mi>t</mi></mrow><mrow><mi mathvariant="normal">A</mi></mrow></msub><mo> </mo><mo>=</mo><mo> </mo><mo> </mo><mn mathvariant="normal">5</mn><mtext> </mtext><mi mathvariant="normal">s</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996811&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996809&type= 15.57866764 3.80999994 ， <math id="M181"><msub><mrow><mi>ω</mi></mrow><mrow><mi mathvariant="normal">L</mi><mi mathvariant="normal">G</mi></mrow></msub><mo>=</mo><mn mathvariant="normal">1.5</mn><mtext> </mtext><msup><mrow><mo stretchy="false">(</mo></mrow><mrow><mi mathvariant="normal">o</mi></mrow></msup><mo stretchy="false">)</mo><mo>/</mo><mi mathvariant="normal">s</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996816&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996814&type= 23.28333473 4.65666676 ， <math id="M182"><msub><mrow><mi>α</mi></mrow><mrow><mi mathvariant="normal">L</mi><mi mathvariant="normal">G</mi></mrow></msub><mo>=</mo><mn mathvariant="normal">0.116</mn><mtext> </mtext><mn mathvariant="normal">1</mn><mtext> </mtext><msup><mrow><mo stretchy="false">(</mo></mrow><mrow><mi mathvariant="normal">o</mi></mrow></msup><mo stretchy="false">)</mo><mo>/</mo><msup><mrow><mi mathvariant="normal">s</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996820&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996818&type= 30.05666542 4.65666676 ，作为输入，产生规划角加速度、角速度和角度，如 图10 图10 所示。 加速度共8段，分为上升段2段，平稳段4段，下降段2段。角加速度的上升段1、平稳段1和下降段1，对应角速度的上升段；角加速度的平稳段2对应角速度的平稳段；角加速度的上升段2、平稳段3和下降段2，对应角速度的下降段；角加速度的平稳段4的值为零，对应角速度的值也为零，角度值达到期望角度。 由路径规划器产生的实时规划角度 <math id="M183"><mi>θ</mi><mo>∈</mo><mfenced open="[" close="]" separators="|"><mrow><mn mathvariant="normal">0</mn><mo>,</mo><msub><mrow><mi>θ</mi></mrow><mrow><mi mathvariant="normal">Q</mi></mrow></msub></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996831&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996829&type= 15.66333294 4.40266657 、角速度 <math id="M184"><mover accent="true"><mi>θ</mi><mo>˙</mo></mover></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996834&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996832&type= 1.77800000 3.55599999 和角加速度 <math id="M185"><mover accent="true"><mrow><mi>θ</mi></mrow><mi>¨</mi></mover></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996837&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996836&type= 1.77800000 3.55599999 求解得到规划四元数 <math id="M186"><msub><mrow><mi mathvariant="bold-italic">q</mi></mrow><mrow><mi mathvariant="normal">G</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996842&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996840&type= 3.64066648 4.57200003 、本体系三轴规划角速度 <math id="M187"><msub><mrow><mi mathvariant="bold-italic">ω</mi></mrow><mrow><mi mathvariant="normal">G</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996847&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996845&type= 4.23333359 3.72533321 、三轴规划角加速度 <math id="M188"><msub><mrow><mi mathvariant="bold-italic">α</mi></mrow><mrow><mi mathvariant="normal">G</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996851&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996849&type= 3.80999994 3.72533321 分别为： <math display="block" id="M189"><msub><mrow><mi mathvariant="bold-italic">q</mi></mrow><mrow><mi mathvariant="normal">G</mi></mrow></msub><mo>=</mo><msup><mrow><mfenced open="[" close="]" separators="|"><mrow><mi mathvariant="normal">c</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">s</mi><mtext> </mtext><mi>θ</mi><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mtext> </mtext><mi>θ</mi><mo> </mo><mo> </mo><msub><mrow><mi mathvariant="bold-italic">e</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow></mfenced></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996856&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996854&type= 33.18933105 4.74133301 ， <math display="block" id="M190"><msub><mrow><mi mathvariant="bold-italic">ω</mi></mrow><mrow><mi mathvariant="normal">G</mi></mrow></msub><mo>=</mo><mover accent="true"><mi>θ</mi><mo>˙</mo></mover><msub><mrow><mi mathvariant="bold-italic">e</mi></mrow><mrow><mi>n</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996861&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996858&type= 13.63133430 4.48733330 ， <math display="block" id="M191"><msub><mrow><mi mathvariant="bold-italic">α</mi></mrow><mrow><mi mathvariant="normal">G</mi></mrow></msub><mo>=</mo><mover accent="true"><mrow><mi>θ</mi></mrow><mi>¨</mi></mover><msub><mrow><mi mathvariant="bold-italic">e</mi></mrow><mrow><mi>n</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996865&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996862&type= 13.12333298 4.48733330 . 4 PFMC快速机动控制设计 1 刚体卫星的动力学与运动学方程描述如下 ［ 17 17 ］ ： <math display="block" id="M192"><mover accent="true"><mi>q</mi><mo>˙</mo></mover><mo>=</mo><mfenced open="[" close="]" separators="|"><mrow><mtable><mtr><mtd><msub><mrow><mover accent="true"><mi>q</mi><mo>˙</mo></mover></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mtd></mtr><mtr><mtd><mover accent="true"><mrow><mover accent="true"><mi>q</mi><mo>¯</mo></mover></mrow><mo>˙</mo></mover></mtd></mtr></mtable></mrow></mfenced><mo>=</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><mfenced open="[" close="]" separators="|"><mrow><mtable><mtr><mtd><mo>-</mo><msup><mrow><mover accent="true"><mi mathvariant="bold-italic">q</mi><mo>¯</mo></mover></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup></mtd></mtr><mtr><mtd><mi mathvariant="bold-italic">S</mi><mfenced separators="|"><mrow><mover accent="true"><mi mathvariant="bold-italic">q</mi><mo>¯</mo></mover></mrow></mfenced><mo>+</mo><msub><mrow><mi>q</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><msub><mrow><mi mathvariant="bold-italic">E</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub></mtd></mtr></mtable></mrow></mfenced><mi mathvariant="bold-italic">ω</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996870&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996868&type= 47.24399948 11.34533310 ， <math display="block" id="M193"><mi mathvariant="bold-italic">I</mi><mover accent="true"><mi mathvariant="bold-italic">ω</mi><mo>˙</mo></mover><mo>=</mo><mo>-</mo><mi mathvariant="bold-italic">S</mi><mo stretchy="false">(</mo><mi mathvariant="bold-italic">ω</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi mathvariant="bold-italic">I</mi><mi mathvariant="bold-italic">ω</mi><mo>+</mo><mi mathvariant="bold-italic">H</mi><mo stretchy="false">)</mo><mo>+</mo><mi mathvariant="bold-italic">u</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996874&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996872&type= 43.34933472 3.55599999 ， 其中： <math id="M194"><mi mathvariant="bold-italic">u</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996879&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996877&type= 2.11666679 2.79399991 为控制力矩， <math id="M195"><mi mathvariant="bold-italic">S</mi><mfenced separators="|"><mrow><mi mathvariant="bold-italic">x</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996883&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996881&type= 8.46666718 3.97933316 为反对称矩阵， <math id="M196"><mi mathvariant="bold-italic">S</mi><mfenced separators="|"><mrow><mi mathvariant="bold-italic">x</mi></mrow></mfenced><mo>=</mo><mfenced open="[" close="]" separators="|"><mrow><mtable><mtr><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mo>-</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub></mtd><mtd><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mtd></mtr><mtr><mtd><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mo>-</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mtd></mtr><mtr><mtd><mo>-</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mtd><mtd><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mtd><mtd><mn mathvariant="normal">0</mn></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996886&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996885&type= 43.43399811 13.54666710 ， <math id="M197"><msub><mrow><mi mathvariant="bold-italic">E</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mo>=</mo><mfenced open="[" close="]" separators="|"><mrow><mtable><mtr><mtd><mn mathvariant="normal">1</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd></mtr><mtr><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">1</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd></mtr><mtr><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">1</mn></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996890&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996888&type= 24.63800049 13.54666710 。 对双视和三视成像中的初始姿态 <math id="M198"><msub><mrow><mi mathvariant="bold-italic">q</mi></mrow><mrow><mi mathvariant="normal">C</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996894&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996892&type= 3.55599999 4.57200003 和目标姿态 <math id="M199"><msub><mrow><mi mathvariant="bold-italic">q</mi></mrow><mrow><mi mathvariant="normal">F</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996898&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996896&type= 3.47133350 4.57200003 定义如下： 双视成像的姿态机动控制中，前视的成像姿态为初始姿态，后视的成像姿态为目标姿态： <math display="block" id="M200"><mfenced open="{" close="" separators="|"><mrow><mtable><mtr><mtd><msub><mrow><mi mathvariant="bold-italic">q</mi></mrow><mrow><mi mathvariant="normal">C</mi></mrow></msub><mo>=</mo><msub><mrow><mi mathvariant="bold-italic">q</mi></mrow><mrow><mi mathvariant="normal">f</mi><mn mathvariant="normal">4</mn></mrow></msub></mtd></mtr><mtr><mtd><msub><mrow><mi mathvariant="bold-italic">q</mi></mrow><mrow><mi mathvariant="normal">F</mi></mrow></msub><mo>=</mo><msub><mrow><mi mathvariant="bold-italic">q</mi></mrow><mrow><mi mathvariant="normal">f</mi><mn mathvariant="normal">5</mn></mrow></msub></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996902&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996900&type= 13.97000027 10.24466610 . 同理，三视成像中，前视至正视的姿态机动对应： <math display="block" id="M201"><mfenced open="{" close="" separators="|"><mrow><mtable><mtr><mtd><msub><mrow><mi mathvariant="bold-italic">q</mi></mrow><mrow><mi mathvariant="normal">C</mi></mrow></msub><mo>=</mo><msub><mrow><mi mathvariant="bold-italic">q</mi></mrow><mrow><mi mathvariant="normal">f</mi><mn mathvariant="normal">1</mn></mrow></msub></mtd></mtr><mtr><mtd><msub><mrow><mi mathvariant="bold-italic">q</mi></mrow><mrow><mi mathvariant="normal">F</mi></mrow></msub><mo>=</mo><msub><mrow><mi mathvariant="bold-italic">q</mi></mrow><mrow><mi mathvariant="normal">f</mi><mn mathvariant="normal">2</mn></mrow></msub></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996906&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996904&type= 13.97000027 10.24466610 . 正视至后视的姿态机动对应： <math display="block" id="M202"><mfenced open="{" close="" separators="|"><mrow><mtable><mtr><mtd><msub><mrow><mi mathvariant="bold-italic">q</mi></mrow><mrow><mi mathvariant="normal">C</mi></mrow></msub><mo>=</mo><msub><mrow><mi mathvariant="bold-italic">q</mi></mrow><mrow><mi mathvariant="normal">f</mi><mn mathvariant="normal">2</mn></mrow></msub></mtd></mtr><mtr><mtd><msub><mrow><mi mathvariant="bold-italic">q</mi></mrow><mrow><mi mathvariant="normal">F</mi></mrow></msub><mo>=</mo><msub><mrow><mi mathvariant="bold-italic">q</mi></mrow><mrow><mi mathvariant="normal">f</mi><mn mathvariant="normal">3</mn></mrow></msub></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996910&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996908&type= 13.97000027 10.24466610 . 卫星的机动与成像过程在多个坐标系下的姿态与角速度转换示意图如 图11 图11 所示。 轨道系相对卫星的初始姿态的旋转四元数： <math display="block" id="M203"><msub><mrow><mi mathvariant="bold-italic">q</mi></mrow><mrow><mi mathvariant="normal">A</mi><mn mathvariant="normal">1</mn></mrow></msub><mo>=</mo><msubsup><mrow><mi mathvariant="bold-italic">q</mi></mrow><mrow><mi mathvariant="normal">C</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msubsup><mo>⊗</mo><msub><mrow><mi mathvariant="bold-italic">q</mi></mrow><mrow><mi mathvariant="normal">O</mi><mi mathvariant="normal">I</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996921&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996919&type= 23.53733444 5.16466665 . 路径规划后实时目标姿态 <math id="M204"><msub><mrow><mi mathvariant="bold-italic">q</mi></mrow><mrow><mi mathvariant="normal">d</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996925&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996923&type= 3.13266683 4.57200003 为： <math display="block" id="M205"><msub><mrow><mi mathvariant="bold-italic">q</mi></mrow><mrow><mi mathvariant="normal">d</mi></mrow></msub><mo>=</mo><msub><mrow><mi mathvariant="bold-italic">q</mi></mrow><mrow><mi mathvariant="normal">C</mi></mrow></msub><mo>⊗</mo><msub><mrow><mi mathvariant="bold-italic">q</mi></mrow><mrow><mi mathvariant="normal">G</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996929&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996927&type= 19.98133469 4.91066647 . 偏差四元数 <math id="M206"><msub><mrow><mi mathvariant="bold-italic">q</mi></mrow><mrow><mi mathvariant="normal">E</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996933&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996931&type= 3.47133350 4.57200003 为： <math display="block" id="M207"><msub><mrow><mi mathvariant="bold-italic">q</mi></mrow><mrow><mi mathvariant="normal">E</mi></mrow></msub><mo>=</mo><msubsup><mrow><mi mathvariant="bold-italic">q</mi></mrow><mrow><mi mathvariant="normal">d</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msubsup><mo>⊗</mo><mi mathvariant="bold-italic">q</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996937&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996935&type= 20.06599998 5.16466665 ， 其中，卫星惯性系下四元数 <math id="M208"><mi mathvariant="bold-italic">q</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996941&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996939&type= 1.94733346 3.64066648 为卫星本体坐标系相对惯性系的旋转四元数。 路径规划后实时目标角速度 <math id="M209"><msub><mrow><mi mathvariant="bold-italic">ω</mi></mrow><mrow><mi mathvariant="normal">d</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996945&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996943&type= 3.72533321 3.72533321 为： <math display="block" id="M210"><msub><mrow><mi mathvariant="bold-italic">ω</mi></mrow><mrow><mi mathvariant="normal">d</mi></mrow></msub><mo>=</mo><mi mathvariant="bold-italic">R</mi><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">q</mi></mrow><mrow><mi mathvariant="normal">A</mi><mn mathvariant="normal">1</mn></mrow></msub><mo>⊗</mo><msub><mrow><mi mathvariant="bold-italic">q</mi></mrow><mrow><mi mathvariant="normal">G</mi></mrow></msub><mo stretchy="false">)</mo><msub><mrow><mi mathvariant="bold-italic">ω</mi></mrow><mrow><mi mathvariant="normal">O</mi><mi mathvariant="normal">I</mi></mrow></msub><mo>+</mo><msub><mrow><mi mathvariant="bold-italic">ω</mi></mrow><mrow><mi mathvariant="normal">G</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996950&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996948&type= 42.41799927 4.91066647 ， 其中， <math id="M211"><msub><mrow><mi mathvariant="bold-italic">ω</mi></mrow><mrow><mi mathvariant="normal">O</mi><mi mathvariant="normal">I</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996953&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996952&type= 4.99533367 3.72533321 为轨道角速度， <math id="M212"><mi mathvariant="bold-italic">R</mi><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">q</mi></mrow><mrow><mi mathvariant="normal">A</mi><mn mathvariant="normal">1</mn></mrow></msub><mo>⊗</mo><msub><mrow><mi mathvariant="bold-italic">q</mi></mrow><mrow><mi mathvariant="normal">G</mi></mrow></msub><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996957&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996955&type= 19.30400085 4.91066647 为 <math id="M213"><msub><mrow><mi mathvariant="bold-italic">q</mi></mrow><mrow><mi mathvariant="normal">A</mi><mn mathvariant="normal">1</mn></mrow></msub><mo>⊗</mo><msub><mrow><mi mathvariant="bold-italic">q</mi></mrow><mrow><mi mathvariant="normal">G</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996961&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996959&type= 13.12333298 4.91066647 对应的旋转矩阵。 偏差角速度 <math id="M214"><msub><mrow><mi mathvariant="bold-italic">ω</mi></mrow><mrow><mi mathvariant="normal">E</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996966&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996963&type= 4.06400013 3.72533321 为： <math display="block" id="M215"><msub><mrow><mi mathvariant="bold-italic">ω</mi></mrow><mrow><mi mathvariant="normal">E</mi></mrow></msub><mo>=</mo><mi mathvariant="bold-italic">ω</mi><mo>-</mo><mi mathvariant="bold-italic">R</mi><mfenced separators="|"><mrow><msub><mrow><mi mathvariant="bold-italic">q</mi></mrow><mrow><mi mathvariant="normal">E</mi></mrow></msub></mrow></mfenced><msub><mrow><mi mathvariant="bold-italic">ω</mi></mrow><mrow><mi mathvariant="normal">d</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996970&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996968&type= 30.56466866 4.82600021 ， 其中，卫星角速度 <math id="M216"><mi mathvariant="bold-italic">ω</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996974&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996972&type= 2.62466669 2.79399991 为卫星本体系相对惯性系的转动角速度， <math id="M217"><mi mathvariant="bold-italic">R</mi><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">q</mi></mrow><mrow><mi mathvariant="normal">E</mi></mrow></msub><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996978&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996976&type= 9.65200043 4.82600021 为 <math id="M218"><msub><mrow><mi mathvariant="bold-italic">q</mi></mrow><mrow><mi mathvariant="normal">E</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996981&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996980&type= 3.47133350 4.57200003 对应的旋转矩阵。 为了进一步提高卫星机动的快速性，设计内外环控制的同时引入规划角加速度作为前馈输入，控制器设计为： 外环控制： <math display="block" id="M219"><msub><mrow><mi mathvariant="bold-italic">u</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>=</mo><mo>-</mo><msub><mrow><mi mathvariant="bold-italic">K</mi></mrow><mrow><mi mathvariant="normal">P</mi></mrow></msub><msub><mrow><mi mathvariant="bold-italic">q</mi></mrow><mrow><mi mathvariant="normal">E</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996986&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996984&type= 19.55800056 4.57200003 ， 其中， <math id="M220"><msub><mrow><mi mathvariant="bold-italic">K</mi></mrow><mrow><mi mathvariant="normal">P</mi></mrow></msub><mo>=</mo><msub><mrow><mi>K</mi></mrow><mrow><mi mathvariant="normal">P</mi></mrow></msub><mi mathvariant="bold-italic">I</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996990&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996988&type= 15.15533447 3.72533321 ，增益矩阵系数 <math id="M221"><msub><mrow><mi>K</mi></mrow><mrow><mi mathvariant="normal">P</mi></mrow></msub><mo>></mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996993&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996991&type= 10.92199993 3.72533321 。 内环控制： <math display="block" id="M222"><msub><mrow><mi mathvariant="bold-italic">u</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>=</mo><msub><mrow><mi mathvariant="bold-italic">K</mi></mrow><mrow><mi mathvariant="normal">q</mi></mrow></msub><msub><mrow><mi mathvariant="bold-italic">α</mi></mrow><mrow><mi mathvariant="normal">G</mi></mrow></msub><mo>-</mo><msub><mrow><mi mathvariant="bold-italic">K</mi></mrow><mrow><mi mathvariant="normal">d</mi></mrow></msub><msub><mrow><mi mathvariant="bold-italic">ω</mi></mrow><mrow><mi mathvariant="normal">E</mi></mrow></msub><mo>-</mo><mi mathvariant="bold-italic">S</mi><mfenced separators="|"><mrow><mi mathvariant="bold-italic">ω</mi></mrow></mfenced><mfenced separators="|"><mrow><mi mathvariant="bold-italic">I</mi><mi mathvariant="bold-italic">ω</mi><mo>+</mo><mi mathvariant="bold-italic">H</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996995&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996994&type= 58.67399979 4.06400013 ， 其中， <math id="M223"><msub><mrow><mi mathvariant="bold-italic">K</mi></mrow><mrow><mi mathvariant="normal">q</mi></mrow></msub><mo>=</mo><msub><mrow><mi>K</mi></mrow><mrow><mi mathvariant="normal">q</mi></mrow></msub><mi mathvariant="bold-italic">I</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996997&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996996&type= 14.73200035 3.72533321 ， <math id="M224"><msub><mrow><mi mathvariant="bold-italic">K</mi></mrow><mrow><mi mathvariant="normal">d</mi></mrow></msub><mo>=</mo><msub><mrow><mi>K</mi></mrow><mrow><mi mathvariant="normal">d</mi></mrow></msub><mi mathvariant="bold-italic">I</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32997000&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32996999&type= 14.73200035 3.72533321 ，增益矩阵系数 <math id="M225"><msub><mrow><mi>K</mi></mrow><mrow><mi mathvariant="normal">q</mi></mrow></msub><mo>></mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32997002&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32997001&type= 10.66800022 3.72533321 ， <math id="M226"><msub><mrow><mi>K</mi></mrow><mrow><mi mathvariant="normal">d</mi></mrow></msub><mo>></mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32997004&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32997003&type= 10.66800022 3.72533321 。 <math display="block" id="M227"><mi mathvariant="bold-italic">u</mi><mo>=</mo><msub><mrow><mi mathvariant="bold-italic">u</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><msub><mrow><mi mathvariant="bold-italic">u</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32997006&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32997005&type= 18.20333290 3.80999994 . 5 数学仿真与在轨测试 1 5.1 数学仿真 2 采用轨道高度535 km，降交点地方时10：00 am的“吉林一号卫星”对本文PFMC算法与文献［ 14 14 ］中递阶饱和模糊控制算法（Hierarchical Saturated Fuzzy Control，HSFC）以及传统比例微分（Proportion Differentiation， PD）控制进行对比仿真。 5 . 1 . 1 卫星参数 3 卫星转动惯量 <math id="M228"><mi mathvariant="bold-italic">I</mi><mo>=</mo><mfenced open="[" close="]" separators="|"><mrow><mtable><mtr><mtd><mn mathvariant="normal">54.6</mn></mtd><mtd><mn mathvariant="normal">0.69</mn></mtd><mtd><mo>-</mo><mn mathvariant="normal">0.17</mn></mtd></mtr><mtr><mtd><mn mathvariant="normal">0.69</mn></mtd><mtd><mn mathvariant="normal">49.2</mn></mtd><mtd><mn mathvariant="normal">0.02</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn mathvariant="normal">0.17</mn></mtd><mtd><mn mathvariant="normal">0.02</mn></mtd><mtd><mn mathvariant="normal">28.7</mn></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32997009&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32997007&type= 43.51866913 13.54666710 ，飞轮角动量 <math id="M229"><mi mathvariant="bold-italic">H</mi><mo>=</mo><msup><mrow><mfenced open="[" close="]" separators="|"><mrow><mtable><mtr><mtd><mn mathvariant="normal">2</mn></mtd><mtd><mn mathvariant="normal">2</mn></mtd><mtd><mn mathvariant="normal">2</mn></mtd></mtr></mtable></mrow></mfenced></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32997012&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32997010&type= 25.90800095 4.74133301 ，飞轮力矩 <math id="M230"><mi mathvariant="bold-italic">T</mi><mo>=</mo><msup><mrow><mfenced open="[" close="]" separators="|"><mrow><mtable><mtr><mtd><mn mathvariant="normal">0.1</mn></mtd><mtd><mn mathvariant="normal">0.1</mn></mtd><mtd><mn mathvariant="normal">0.1</mn></mtd></mtr></mtable></mrow></mfenced></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32997016&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32997014&type= 34.29000092 4.74133301 。 5 . 1 . 2 控制参数 3 输入角速度限值 <math id="M231"><msub><mrow><mi mathvariant="bold-italic">ω</mi></mrow><mrow><mi mathvariant="normal">L</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">m</mi></mrow></msub><mo>=</mo><msup><mrow><mfenced open="[" close="]" separators="|"><mrow><mtable><mtr><mtd><mn mathvariant="normal">1.5</mn></mtd><mtd><mn mathvariant="normal">1.5</mn></mtd><mtd><mn mathvariant="normal">1.5</mn></mtd></mtr></mtable></mrow></mfenced></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32997019&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32997018&type= 37.76133347 4.74133301 ，前馈控制增益矩阵系数 <math id="M232"><msub><mrow><mi>K</mi></mrow><mrow><mi mathvariant="normal">q</mi></mrow></msub><mo>=</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32997021&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32997020&type= 7.95866632 3.72533321 0.85，比例控制增益矩阵系数 <math id="M233"><msub><mrow><mi>K</mi></mrow><mrow><mi mathvariant="normal">p</mi></mrow></msub><mo>=</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32997023&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32997022&type= 7.95866632 3.72533321 1.5，微分控制增益矩阵系数 <math id="M234"><msub><mrow><mi>K</mi></mrow><mrow><mi mathvariant="normal">d</mi></mrow></msub><mo>=</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32997025&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=32997024&type= 7.95866632 3.72533321 1.5。 由于卫星的转动惯量与飞轮力矩和角动量约束，该卫星只能进行同轨双视立体成像，针对前后视25°的立体成像进行仿真分析。 5 . 1 . 3 仿真结果 3 PFMC算法通过姿态规划获取到的三轴规划角加速度、角速度、角度如 图12 图12 所示。 PFMC、HSFC和PD三种控制在轨道系下的角速度与姿态角分别如 图13 图13 和 图14 图14 所示，其中PD在机动阶段较快，但在收敛段的收敛速度减慢，通过设计变参数的数值，HSFC与PFMC的收敛性可达到相似效果，但PFMC的稳定段的收敛速度更快。如图 12 12 ~ 14 14 所示，三轴均存在机动， Y 轴旋转角为50°，角速度达到限幅1.5 （°）/s。 5.2 在轨飞行测试 2 采用具有双相机的“吉林一号卫星”对乌鲁木齐市进行同轨双视立体成像的在轨验证，考虑卫星数据生产过程中单景数据的生成及成像区域的重叠率，实际任务规划时后视成像相比前视成像提前开机2 s，延迟关机1 s，每次成像可生成4景图像。前、后视成像结果分别如 图15 图15 和 图16 图16 所示，基于前后视立体像对合成的数字表面模型如 图17 图17 所示，两次成像区域顶点的经纬度如 表3 表3 所示，两次成像覆盖区域重叠率大于97%。 Item Front view 25° Rear view -25° Top left （87.373，44.003） （87.391，44.057） Bottom left （87.253，43.621） （87.244，43.602） Top right （87.939，43.910） （87.958，43.970） Bottom right （87.815，43.533） （87.809，43.515） 6 结 论 1 本文首先采用地球椭球模型确定成像点的位置，并考虑地球的自转确定成像点与卫星的相对位置与速度信息，进行含有侧摆机动的立体成像姿态规划算法计算。通过对沿卫星飞行方向的特性分析，获取成像窗口时间及相应的成像姿态四元数，提高了立体成像姿态规划的精度。然后，考虑执行机构力矩和角动量性能设计了一种基于旋转轴不变约束的最短路径规划方法；规避角加速度的突变问题，设计了一种角加速度连续的八段式路径规划器。之后，设计了角加速度前馈结合内外环控制的快速机动算法，实现卫星三轴大角度姿态机动，提高了姿态跟踪的动态性能和成像覆盖能力。最后，对“吉林一号”卫星仿真并在轨验证了姿态规划方案与路径规划快速机动控制算法的可行性。对乌鲁木齐进行在轨双视立体成像，成像姿态控制精度优于0.02°，稳定度优于0.001 （°）/s，单次覆盖面积约1 600 km 2 ，重叠率约为97%，进一步验证了算法的有效性。 参考文献 李强 ， 孔林 ， 张雷 ， 等 . 多光谱大幅宽光学遥感卫星的热设计及验证 ［J］. 光学 精密工程 ， 2020 ， 28 （ 4 ）： 904 - 913 . doi: 10.3788/OPE.20202804.0904 http://dx.doi.org/10.3788/OPE.20202804.0904 多光谱大幅宽光学遥感卫星的热设计及验证 LI Q ， KONG L ， ZHANG L ， et al . 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