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Article(id=1228653355640222029, tenantId=1146029695717560320, journalId=1225147924628267009, issueId=1228653350485422347, articleNumber=null, orderNo=null, doi=10.16385/j.cnki.issn.1004-4523.2024.10.018, pmid=null, cstr=null, oa=null, hot=null, price=null, onlineType=0, articleFormat=0, articleType=null, articleTypeStr=null, receivedDate=1716048000000, receivedDateStr=2024-05-19, revisedDate=1720972800000, revisedDateStr=2024-07-15, acceptedDate=null, acceptedDateStr=null, onlineDate=1770863387260, onlineDateStr=2026-02-12, pubDate=1730044800000, pubDateStr=2024-10-28, doiRegisterDate=null, doiRegisterDateStr=null, onlineIssueDate=1770863387260, onlineIssueDateStr=2026-02-12, onlineJustAcceptDate=null, onlineJustAcceptDateStr=null, onlineFirstDate=null, onlineFirstDateStr=null, sourceXml=null, magXml=null, createTime=1770863387260, creator=13701087609, updateTime=1770863387260, updator=13701087609, issue=Issue{id=1228653350485422347, tenantId=1146029695717560320, journalId=1225147924628267009, year='2024', volume='37', issue='10', pageStart='1625', pageEnd='1802', issueExtLink='null', onlineDate='null', pubDate='null', beforeIssueId=null, nextIssueId=null, price=null, status=1, issueComplete=1, articleOrder=1, issueType=-1, specialIssue=null, createTime=1770863386031, creator=13701087609, updateTime=1770863862999, updator=13701087609, preIssue=null, nextIssue=null, ext={EN=IssueExt(id=1228655351092936954, tenantId=1146029695717560320, journalId=1225147924628267009, issueId=1228653350485422347, language=EN, specialIssueTitle=, coverIllustrator=null, specialIssueEditor=, specialIssueAbout=), CN=IssueExt(id=1228655351092936955, tenantId=1146029695717560320, journalId=1225147924628267009, issueId=1228653350485422347, language=CN, specialIssueTitle=, coverIllustrator=null, specialIssueEditor=, specialIssueAbout=)}, issueFiles=null}, startPage=1792, endPage=1802, ext={EN=ArticleExt(id=1228653355992543576, articleId=1228653355640222029, tenantId=1146029695717560320, journalId=1225147924628267009, language=EN, title=Dynamics modeling and control design method for long-term deorbiting of an electrodynamic tether system, columnId=null, journalTitle=Journal of Vibration Engineering, columnName=null, runingTitle=null, highlight=null, articleAbstract=

This paper investigates the dynamics and control problems of the long-term deorbiting of an electrodynamic tethered satellite system. The dynamics modeling of the system is carried out based on a dumbbell model assumption. To improve the accuracy of the system model，the orbital dynamics is described using a set of modified equinoctial elements，involving the effects of Lorentz force，atmospheric drag and J2 perturbation force. Three current control strategies are proposed to regulate the electrodynamic forces for achieving a stable long-term deorbiting process，namely，the constant current input，the directionally variable current input，and the optimal control strategies. In the design of the optimal control strategy，the long-term deorbiting problem is formulated as an inverse problem of dynamics with nonlinear constraints，which is further solved via a nonlinear programming method to obtain the optimal reference trajectories. The deorbiting of the system is then achieved using the modified current control input obtained from a tracking feedback control law. Additionally，an energy-based current switch control strategy is adopted to ensure the stability of system and the efficient utilization of Lorentz force. Case studies of the system with designed physical parameters are conducted to analyze the deorbiting efficiency and to validate the effectiveness of the proposed control strategies.

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本文针对电动力绳系卫星系统的长期离轨问题，研究离轨过程中系统的非线性动力学建模以及离轨控制设计方法。基于哑铃模型假设，进行系统的姿态动力学建模。为提高动力学建模精度，计入电动力、大气阻力和地球J2摄动等动力学载荷对系统轨道运动的影响，采用修正的非奇异轨道要素描述系统的轨道动力学。针对离轨的动力学控制问题，提出了常值电流输入、方向可变的电流输入和最优控制三种电流控制设计方法，通过系绳电流设计调节电动力载荷实现离轨机动。在最优控制设计方法中，构建了带非线性约束的动力学反问题，基于非线性规划算法求解该反问题，得到最优控制参考轨迹，通过闭环反馈跟踪修正电流进行离轨。另外，提出了一种基于能量的电流开关控制策略辅助离轨，以保持系统姿态稳定和高效地利用电动力。为高效离轨对系统进行了动力学设计，并通过仿真算例对比分析了控制方法的离轨效率，验证了控制方法的有效性。

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罗操群（1993―），男，博士，助理教授。E-mail： 。
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颜家慧（2000—），女，硕士研究生。E-mail： 。

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颜家慧（2000—），女，硕士研究生。E-mail： 。

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颜家慧（2000—），女，硕士研究生。E-mail： 。

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strateies 2 and 3 under restriction of 18°, figureFileSmall=ER072w6qJAmhAkQ9e4qJeg==, figureFileBig=sDEM/x7Y7PGc1rD0AdP6bA==, tableContent=null), ArticleFig(id=1228697682009588079, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228653355640222029, language=CN, label=图18, caption=18°摆角限制下控制方案2和3的摆角变化, figureFileSmall=ER072w6qJAmhAkQ9e4qJeg==, figureFileBig=sDEM/x7Y7PGc1rD0AdP6bA==, tableContent=null), ArticleFig(id=1228697682072502640, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228653355640222029, language=EN, label=Tab.1, caption=

Physical parameters of the system

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主星质量子星质量系绳长度系绳密度主星迎风面面积子星迎风面面积阻力系数主星质量
600.0 kg0.8 kg1.0 km1.0×10-3 kg/m2.8 m×1.6 m0.5 m×0.3 m2.2600.0 kg
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系统物理参数

, figureFileSmall=null, figureFileBig=null, tableContent=
主星质量子星质量系绳长度系绳密度主星迎风面面积子星迎风面面积阻力系数主星质量
600.0 kg0.8 kg1.0 km1.0×10-3 kg/m2.8 m×1.6 m0.5 m×0.3 m2.2600.0 kg
), ArticleFig(id=1228697682223497586, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228653355640222029, language=EN, label=Tab.2, caption=

The initial orbital parameters of the system

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工况轨道半长轴/km轨道偏心率升交点赤经/(°)轨道倾角/(°)真近点角/(°)近地点辐角/(°)轨道半长轴/km
工况1500004400500
工况2500006500500
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系统初始轨道参数

, figureFileSmall=null, figureFileBig=null, tableContent=
工况轨道半长轴/km轨道偏心率升交点赤经/(°)轨道倾角/(°)真近点角/(°)近地点辐角/(°)轨道半长轴/km
工况1500004400500
工况2500006500500
), ArticleFig(id=1228697682374492532, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228653355640222029, language=EN, label=Tab.3, caption=

Deorbiting time under different restriction of libration angles of control strategies 1 and 2

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摆角限制/(°)控制方案1效率控制方案2效率
12144.91天基准144.68天+0.16%
15135.87天+6.24%135.90天+6.22%
18128.24天+11.50%127.03天+12.34%
21--120.96天+16.53%
25--116.65天+19.50%
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不同摆角限制下控制方案1与控制方案2的离轨时间

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摆角限制/(°)控制方案1效率控制方案2效率
12144.91天基准144.68天+0.16%
15135.87天+6.24%135.90天+6.22%
18128.24天+11.50%127.03天+12.34%
21--120.96天+16.53%
25--116.65天+19.50%
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Deorbiting time under different restriction of libration angles of control strategies 1 and 2

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摆角限制控制方案1效率控制方案2效率
12°230.86天基准174.24天+24.53%
15°--158.29天+31.43%
18°--150.61天+34.76%
21°--153.68天+33.43%
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不同摆角限制下控制方案1与控制方案2的离轨时间

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摆角限制控制方案1效率控制方案2效率
12°230.86天基准174.24天+24.53%
15°--158.29天+31.43%
18°--150.61天+34.76%
21°--153.68天+33.43%
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电动力绳系卫星系统长期离轨动力学建模与控制设计方法
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颜家慧 1 , 陈提 1 , 文浩 1 , 罗操群 1 , 彭文轩 2 , 恽卫东 2
振动工程学报 | 2024,37(10): 1792-1802
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振动工程学报 | 2024, 37(10): 1792-1802
电动力绳系卫星系统长期离轨动力学建模与控制设计方法
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颜家慧1 , 陈提1, 文浩1, 罗操群1 , 彭文轩2, 恽卫东2
作者信息
  • 1南京航空航天大学航空航天结构力学及控制全国重点实验室,江苏　南京　210016
  • 2上海宇航系统工程研究所,上海　201108

    • 颜家慧（2000—），女，硕士研究生。E-mail： 。

通讯作者:

罗操群（1993―），男，博士，助理教授。E-mail： 。
Dynamics modeling and control design method for long-term deorbiting of an electrodynamic tether system
Jia-hui YAN1 , Ti CHEN1, Hao WEN1, Cao-qun LUO1 , Wen-xuan PENG2, Wei-dong YUN2
Affiliations
  • 1State Key Laboratory of Mechanics and Control for Aerospace Structures，Nanjing University of Aeronautics and Astronautics，Nanjing 210016，China
  • 2Shanghai Institute of Aerospace System Engineering，Shanghai 201108，China
出版时间: 2024-10-28 doi: 10.16385/j.cnki.issn.1004-4523.2024.10.018
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摘要
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本文针对电动力绳系卫星系统的长期离轨问题，研究离轨过程中系统的非线性动力学建模以及离轨控制设计方法。基于哑铃模型假设，进行系统的姿态动力学建模。为提高动力学建模精度，计入电动力、大气阻力和地球J2摄动等动力学载荷对系统轨道运动的影响，采用修正的非奇异轨道要素描述系统的轨道动力学。针对离轨的动力学控制问题，提出了常值电流输入、方向可变的电流输入和最优控制三种电流控制设计方法，通过系绳电流设计调节电动力载荷实现离轨机动。在最优控制设计方法中，构建了带非线性约束的动力学反问题，基于非线性规划算法求解该反问题，得到最优控制参考轨迹，通过闭环反馈跟踪修正电流进行离轨。另外，提出了一种基于能量的电流开关控制策略辅助离轨，以保持系统姿态稳定和高效地利用电动力。为高效离轨对系统进行了动力学设计，并通过仿真算例对比分析了控制方法的离轨效率，验证了控制方法的有效性。

绳系卫星系统  /  电动力  /  最优控制  /  长期离轨

This paper investigates the dynamics and control problems of the long-term deorbiting of an electrodynamic tethered satellite system. The dynamics modeling of the system is carried out based on a dumbbell model assumption. To improve the accuracy of the system model，the orbital dynamics is described using a set of modified equinoctial elements，involving the effects of Lorentz force，atmospheric drag and J2 perturbation force. Three current control strategies are proposed to regulate the electrodynamic forces for achieving a stable long-term deorbiting process，namely，the constant current input，the directionally variable current input，and the optimal control strategies. In the design of the optimal control strategy，the long-term deorbiting problem is formulated as an inverse problem of dynamics with nonlinear constraints，which is further solved via a nonlinear programming method to obtain the optimal reference trajectories. The deorbiting of the system is then achieved using the modified current control input obtained from a tracking feedback control law. Additionally，an energy-based current switch control strategy is adopted to ensure the stability of system and the efficient utilization of Lorentz force. Case studies of the system with designed physical parameters are conducted to analyze the deorbiting efficiency and to validate the effectiveness of the proposed control strategies.

tethered satellite system  /  electrodynamic  /  optimal control  /  long-term deorbiting
颜家慧, 陈提, 文浩, 罗操群, 彭文轩, 恽卫东. 电动力绳系卫星系统长期离轨动力学建模与控制设计方法. 振动工程学报, 2024 , 37 (10) : 1792 -1802 . DOI: 10.16385/j.cnki.issn.1004-4523.2024.10.018
Jia-hui YAN, Ti CHEN, Hao WEN, Cao-qun LUO, Wen-xuan PENG, Wei-dong YUN. Dynamics modeling and control design method for long-term deorbiting of an electrodynamic tether system[J]. Journal of Vibration Engineering, 2024 , 37 (10) : 1792 -1802 . DOI: 10.16385/j.cnki.issn.1004-4523.2024.10.018
正文
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引言
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近年来，由于太空技术发展与其成本的下降，世界各国进行了多次太空任务，产生的太空垃圾飞速增长，其对在轨航天器和地球环境的威胁不可忽视。为清除太空垃圾，学者们提出了多种解决方案，如太阳帆、机械臂和绳网捕捉等，以实现太空环境的清理，降低在轨航天器的碰撞风险［1-3］。其中，电动力绳系卫星技术因无需消耗推进剂、绿色环保、易于实现和成本低等优点受到广泛关注。不同于其他需要额外推力的清理方式，电动力绳系卫星系统利用导电系绳切割地磁场，产生电动力作为动力实现空间碎片的离轨机动［4-7］。然而，当电动力绳系卫星系统在倾斜轨道上离轨时，电动力会不断向系统注入能量，导致系绳产生大幅度的摆动，影响运动稳定性［8-9］。另外，考虑到电-磁耦合、环境摄动影响以及姿态运动和轨道运动间的耦合作用［10-12］，电动力绳系卫星系统的在轨运动呈现出强非线性，给系统的稳定离轨机动带来困难。
为解决这些问题，国内外学者针对电动力绳系卫星系统的离轨动力学与控制进行了深入研究。Corsi等［13］曾提出一种基于李雅普诺夫/哈密顿函数的电流开关控制方法来调节电流以限制系绳摆动。Williams［14-15］基于能量率反馈机制设计了一种跟踪控制方法，基于系绳实际的摆动能量速率与参考轨迹进行反馈控制，实现系绳摆动的长期稳定；随后，又提出了时域差分速度最优控制方法以及时滞摆动速率预测反馈电流控制，结合Chebyshev-Gauss-Lobatto伪谱法解决最优轨道转移过程中系统电流的快速调节问题。Kojima等［16］研究了电动力绳系系统在椭圆轨道上的混沌摆动，基于系统的局部稳定性提出了时滞速率反馈电流控制方法，在增强系统鲁棒性的同时减少了控制功耗。张健等［17］针对电动力绳系卫星系统的轨道机动问题，设计了常值电流控制、Bang-Bang式可变电流控制和傅里叶形式电流控制三种不同的电流控制方法。Li等［18］采用退步控制与滑模控制相结合的非线性控制方法来减小系统的摆角，提高系统的鲁棒性。Wen等［19］针对电动力绳系卫星系统的离轨问题，提出了系绳电流的模型预测控制方法和自适应增益策略来稳定系绳摆动；此外，基于电动力绳系卫星非奇异动力学模型，设计了一种基于能量方法的摆动抑制控制方法［20］。Luo等［21］提出了一种基于能量率的闭环跟踪控制方法，通过跟踪轨道周期解来降低计算难度，从而实现系统的长期离轨。上述研究大多关注离轨过程系绳摆动的稳定控制，未考虑长期离轨过程中控制方案的离轨效率。
本文关注电动力绳系系统长期高效离轨过程中的动力学与控制问题。为研究电动力绳系卫星长期离轨过程的动力学问题，本文基于哑铃模型假设建立系统的姿态动力学模型，采用非奇异轨道参数描述系统轨道运动，同时计入电动力、大气阻力和地球J2摄动的影响。针对系统的长期离轨控制问题，基于电流开关控制和最优控制提出了多种控制设计方法，并通过对比分析研究了不同控制设计方法下电动力绳系卫星系统的离轨效率。在保证离轨稳定性的同时，通过设计控制参数，提高离轨效率，实现系统的长期稳定离轨。
1　动力学模型
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本文关注的电动力绳系卫星系统由主星M、子星S以及连接主星和子星的电动力系绳t组成。假设系统质心运行在开普勒倾斜轨道上。主星、子星和系绳的质量分别为，和，且系绳长度为。基于哑铃模型假设建立系统动力学模型，忽略卫星姿态和系绳柔性。在长期离轨过程中，系统的运动可分为慢变的轨道运动及快变的姿态运动，下面从这两方面建立系统的动力学模型。
1.1　轨道运动
采用地球地心惯性坐标系来描述系统的轨道运动。图1为简化后的系统示意图，其中，表示地球质心，X轴指向春分点方向，Z轴为地球自转方向，Y轴方向垂直于X-Z平面。
由于基于经典轨道根数的轨道方程在偏心率为0以及轨道倾角为0°或90°时会产生奇异现象，故在描述轨道运动时，采用修正的非奇异轨道参数建立系统的轨道运动方程。非奇异轨道参数为：
式中　轨道半长轴a、轨道倾角i、轨道偏心率e、升交点赤经、近地点辐角和真近点角ν为经典轨道根数。
用高斯变化方程表示系统轨道根数随时间的变化［19］：
其中：
式中　“”表示对时间t的导数；为摄动力引起的系统摄动加速度，其中，为电动力，为大气阻力摄动，为地球引力场不均匀性导致的摄动，各摄动力的具体描述见附录，为系统总质量；，，分别为摄动加速度在轨道坐标系三个主轴上的投影分量；为地球引力常数。
1.2　姿态运动
采用参考轨道坐标系来描述系统的姿态运动，如图2所示。表示系统质心，轴和轴分别指向天顶和系统飞行方向，轴与轴和轴形成右手坐标系。系统相对于轨道坐标系的瞬时姿态可由俯仰角以及滚转角描述。
系统的动能由系统牵连运动动能、系统绕质心相对转动动能以及系统相对质心径向运动动能组成，总动能表示为：
其中：
式中　 为轨道坐标系角速率；为系统质心相对于地球质心的运行半径。
系统的势能主要是指重力势能Vg。假设系统由无数质点组成，其中任一质点相对于系统质心的位置向量为 ，则系统势能为：
式中　 表示向量的二范数算子。由于绳长远小于轨道半径，即，对式（6）展开并省略，，的二次以上项，得到系统势能的简化公式：
根据第二类拉格朗日Lagrange方程，建立系统的姿态运动控制方程［22］：
式中　 Qe和Qφ分别表示电动力与大气阻力在俯仰角方向和滚动角方向产生的广义力炬，且有：
表示轨道坐标系下的局部地磁强度，通过虚功原理推导出由洛伦兹力引起的广义扭矩与：
式中
其中，为从主星M指向子星S的系绳方向单位矢量，可表示为：
大气阻力引起的广义扭矩与可分别表示为：
式中　 ，与分别为主星、子星与系绳各部分所受大气阻力；s1，s2可分别表示为：
基于上述内容，式（2）和（8）描述了系统的完整动力学模型。
2　控制设计方法
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本节针对电动力绳系卫星系统长期离轨给出三种不同的控制设计方案，分别是：包括常值电流输入与方向可变的电流输入的直接控制设计方案，以及最优控制设计方案。
2.1　直接控制设计方案
在直接控制设计方案中，给出一种基于能量的电流开关控制策略，其原理是通过主动控制电动力系绳上产生的电动力来实现系绳上的阻尼注入，控制系绳摆动，同时最大化洛伦兹力对系统的离轨作用。
引入两种参数［23］：
式中　 与表示摆角的方向；为感生电势，其中为系统相对于地球的轨道速度；与表示广义力相对于摆角的方向。
电流控制律为：
当且仅当系绳摆角均小于固定值或两种广义力对系绳摆动的瞬时总功率为负时，打开电流。
根据轨道特性给出常值电流输入，其电流大小与电流方向均保持恒定，给出电流：
记该常值电流输入方案为控制方案1。
电动力绳系卫星系统离轨机动过程中由于电动力产生的耗散功率［24］为：
式中　 表示电动力。当感生电势与电流同向时，所产生的耗散功率为负，即电动力对系统做负功。据此，给出方向可变的电流输入：
记该可变电流输入方案为控制方案2。
2.2　最优控制设计方案
在本节中，基于两体电动力绳系卫星系统的简单哑铃模型，设计离轨过程中的最优控制设计方案。在最优控制设计方案中，将系统的离轨问题建立为带非线性约束的优化问题，通过优化给定的性能指标函数，规划系统状态变量和设计控制输入使系统最终达到期望的目标状态。最优控制设计方案通过控制设计调节系统特性，属于动力学反问题的范畴［25］。
为简化计算并保持磁场精度，在最优控制模型中采用非倾斜偶极子模型描述地磁场，地磁场强度在轨道系中表示为：
式中　 。为地磁场轴的赤经；表示格林尼治恒星时，为地球磁偶极子强度。初始参数选定为：
引入状态向量：
引入无量纲变换，系统的无量纲姿态动力学方程可以写成状态空间形式：
与为电动力产生的广义力矩：
上述方程可以记为：
定义控制输入变量：
将非线性最优控制问题按照数学形式表述为：
式中　控制周期从到时刻结束，为最优控制中的最大系绳电流输入；为目标函数；为拉格朗日形式的性能指标函数。考虑离轨效率与系统的稳定性，设计性能指标函数为：
式中　 ，； 表示系统的洛伦兹力所做的离轨功；为系统沿飞行方向的速度；为权系数。另外，性能指标函数中的第一和第二项在离轨过程中对系绳摆角起抑制作用，第三项用于系统的高效离轨。各项的增益系数通过反复计算确定，旨在保证姿态稳定性的同时，获得较高的离轨效率。在求解过程中，利用勒让德-高斯伪谱法将连续时间的非线性最优控制问题离散化为非线性规划的离散参数优化问题。得到的离散参数优化问题基于MATLAB的软件GPOPS进行求解。
为简化计算与控制流程，采用时间尺度分离的概念进行最优控制方案设计，如图3所示。在较短的时间域内，相对于系统的姿态运动，系统的轨道参数变化非常缓慢从而可以忽略不计。为提高计算效率，在计算最优控制轨迹时，忽略系统的轨道参数变化，将系统的轨道参数设定为常数，仅考虑系统的姿态变化。而在计算整个离轨过程时，考虑系统完整的轨道动力学和姿态动力学。在进行离轨控制时，基于最优控制计算得到参考轨迹，引入跟踪控制算法以修正轨道动力学产生的误差，实现稳定的离轨机动。
将基于最优控制设计方案得到的参考电流控制输入应用在系统的完整动力学模型上时，需考虑轨道参数变化对系统姿态动力学的影响。当轨道参数变化时，为保持系统姿态稳定，对参考的电流输入进行跟踪修正，所设计的电流的跟踪控制律为：
且有：
式中　 与为控制增益，与分别表示系绳初始摆动角和摆动角速度，与表示系绳参考摆动角和摆动角速度，表示最优控制方法得到的参考控制电流，和表示用于对系绳俯仰角和滚转角方向运动误差修正的控制电流，为修正的控制电流输入，可表示为：
记最优控制设计方案为控制方案3。
3　算例研究
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本节通过对几种不同轨道工况进行数值仿真，进而分析三种控制设计的离轨效果。所有算例均在MATLAB（R2023a）上进行，数值模拟通过ode45求解。
表1给出了系统的物理参数，表2给出了两种不同工况下系统的初始轨道参数。
在直接控制设计方案中，设置电流数值均为0.1 A。
另外，在最优控制设计方案中，控制参数设置为：
三种控制方案的系统初始摆角与摆角速度，即初始状态参数均设置为0。
在仿真过程中，将轨道半长轴下降至450 km视为任务目标。在44°倾角轨道和不同摆角限制情况下，表3给出了系统在控制方案1与控制方案2作用下的离轨时间。可以发现，两种直接控制设计方案效果基本相同。图4给出了不同控制方案的离轨时间。需要注意的是，最优控制设计方案中在利用跟踪控制律控制电流时，未设置摆角限制，因此系统在控制方案3作用下的离轨时间保持定值。关注控制方案1，表3仅给出了摆角限制从12°增长至18°时系统的离轨时间，这是由于继续增大摆角限制，系统会由于电动力作用而失稳。将12°摆角限制下控制设计方案1所消耗的离轨时间作为基准进行对比，如表3所示，效率栏中的数值表示相应工况的效率变化，其中“+”代表效率的提升。
图5为不同控制方案作用下系统的轨道半长轴变化曲线，其中控制方案1和控制方案2的摆角限制为18°。可以看出，此时三种控制方案的效率接近一致。图6给出了摆角限制增大到21°时，系统在控制方案1作用下的俯仰角变化曲线。可以发现，系绳摆角迅速增大，导致系统失稳。从图7中可以看出，当摆角限制为30°时，控制方案2作用下的系统发生翻滚。结果表明，对于在相同轨道倾角下的系统，控制方案2较控制方案1有更大的摆角限制阈值。另外可以发现，在保证系统稳定的情况下，增大摆角限制阈值，可以有效提高直接控制设计方案的离轨效率。
图8给出了44°轨道倾角时，不同控制方案的电流变化对比，其中控制方案1和2的摆角限制为18°。图9和10描述了摆角限制分别为18°和25°时，控制方案2和控制方案3的摆角对比。在44°倾角的轨道中，控制方案3的离轨时间为125.92天，相对于基准离轨时间，效率提升13.10%。当控制方案2摆角限制为18°时，控制方案3的系绳摆角相对于控制方案2更小，且离轨效率更高。当控制方案2摆角限制增大至25°时，系统离轨效率提高，但系绳最大摆角增大至50°左右时，系统稳定性较差。基于上述结果可以发现，当摆角限制大于18°以后，控制方案1的离轨效率优于最优控制设计方案。这是由于最优控制设计方案综合考虑了系统稳定性和高效离轨。尽管在最优控制设计方案中未设置摆角约束，但由于所设计的目标函数，得到的电流控制输入也被用于稳定系绳姿态，不是完全用于系统离轨。从图9和10中可以发现，最优控制设计方案下的系绳摆角幅值较小，系统稳定性较好。因此，最优控制设计方案可以有效稳定系统姿态并使系统快速离轨，基于该方案得到的离轨天数是综合考虑系统稳定性和高效离轨得到的离轨时长。
考虑位于65°倾角轨道上的系统，表4给出了不同摆角限制下控制方案1与2系统离轨时间的对比结果。另外，图11中给出了不同控制方案下的离轨时间曲线。可以发现，在摆角限制为18°时，控制方案2有较高的离轨效率。与轨道倾角44°的情况类似，当摆角限制大于12°时，在控制方案1作用下系统会发生失稳，因此表4仅给出了摆角限制12°时控制方案1的离轨时间。值得注意的是，控制方案2在摆角限制为18°时有最高的离轨效率，这是由于系绳姿态、电流与磁场的非线性耦合关系，使控制方案2在65°轨道倾角和18°摆角限制情况下有最高的电流利用效率，此时系绳电流的作用时间更久，可以更快地使系统离轨。
图12给出了不同控制方案下系统的轨道半长轴变化曲线，其中直接控制设计方案的摆角限制为12°。可以看出，控制方案2与控制方案3的效率接近一致，相对控制方案1有明显的效率提升。图13给出了控制方案2与控制方案3的轨道半长轴高度变化。此时摆角限制为18°，可以发现控制方案2较控制方案3的离轨效率更高。
如图14所示，当摆角限制增大到15°时，控制方案1作用下系绳的俯仰角会急剧增大，表明系绳在俯仰方向形成了旋转运动，系统姿态无法稳定。对比图4和11可以发现，降低轨道倾角可以提高不同控制方案下系统的离轨效率。
图15为不同控制方案下的电流变化对比，其中直接控制设计方案的摆角限制为12°。图16给出了摆角限制为12°时，系统在控制方案1和控制方案2下的俯仰角与滚转角变化曲线。结合表4，控制方案2相对于控制方案1表现出更好的稳定性、更高的离轨效率以及更大的摆角限制阈值。
图17给出了摆角限制为12°时，系统在控制方案2和控制方案3下的俯仰角与滚转角变化曲线。在65°倾角的轨道中，控制方案3的离轨时间为172.74天，相对于基准离轨时间，效率提升25.18%。可以发现，控制方案3在保持系统稳定的同时，有较短的离轨时间。图18表明，通过增大摆角限制阈值，控制方案2可以获得较高的离轨效率，但系绳摆角会因此增大。
4　结论
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本文针对电动力绳系卫星系统的长期离轨动力学与控制问题，建立了考虑多种摄动力的精确动力学模型，提出了多种电流控制设计方法用于高效离轨。通过仿真对比分析，方向可变的电流控制方法相对于常值电流控制方法，表现出更好的稳定性与更高的离轨效率。在直接控制设计方案中，为防止系绳摆角过大，提出了根据摆角是否超过限制阈值判断电流开关的控制策略，以维持系统稳定性。结果表明，通过增大摆角限制阈值，可以在保证系统稳定性的同时，有效提高离轨效率。需要注意的是，在增大系绳摆角限制时，需考虑系统稳定性。当摆角限制阈值过高时，系统可能会发生失稳，从而导致离轨效率降低。在所提出的控制设计方法中，最优控制设计方案可以有效限制系绳摆角，具有较好的系统稳定性，并且具有较高的离轨效率。
本文在进行离轨效率分析时，仅从控制参数设计进行考虑，后续研究可分析主子星质量比等物理参数对系统离轨效率的影响，针对系统物理参数进行优化设计，进一步提高离轨效率。
附录
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考虑到系绳长度远小于地球半径，故近似系统附近各处的磁场强度是相等的。本文考虑绝缘系绳，假设电流沿电动力系绳均匀分布。系绳所受的洛伦兹力为：
可表示为：
采用IGRF13阶模型来表述地球地磁场，系统周围的各矢量方向的磁场强度表示为：
式中　 和为IGRF模型的高斯系数；为地球余纬；为地球经度；为地球半径；为所求磁场点距地球质心的距离；为施密特准归一化伴随勒让德函数。经过转换可以求得轨道坐标系下描述的磁场强度［26］。
基于NRLMSISE-00大气模型得到系统所处位置的大气密度，并利用经验阻力系数、系统相对于地球的飞行速度、迎风面积（系绳截面面积+端部体截面面积）得到系统所受到的大气阻力：
另外，考虑地球引力场不均匀性导致的摄动，其引起的摂动加速度在轨道系下的分量为：
基金
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  • 国防科技工业局空间碎片专项资助项目(KJSP2020010301)
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2024年第37卷第10期
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doi: 10.16385/j.cnki.issn.1004-4523.2024.10.018
  • 接收时间：2024-05-19
  • 首发时间：2026-02-12
  • 出版时间：2024-10-28
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  • 收稿日期：2024-05-19
  • 修回日期：2024-07-15
基金
国防科技工业局空间碎片专项资助项目(KJSP2020010301)
作者信息
    1南京航空航天大学航空航天结构力学及控制全国重点实验室,江苏　南京　210016
    2上海宇航系统工程研究所,上海　201108

通讯作者:

罗操群（1993―），男，博士，助理教授。E-mail： 。
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https://castjournals.cast.org.cn/joweb/zdgcxb/CN/10.16385/j.cnki.issn.1004-4523.2024.10.018
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2种不同金属材料的力学参数
科
Family		 属数
Number of
genus		 种数
Number of
species		 占总种数比例
Percentage of
total species (%)		 属
Genus		 种数
Number of
species		 占总种数比例
Percentage of total
species (%)
鹅膏菌科Amanitaceae 2 11 5.26 鹅膏菌属 Amanita 10 4.78
小菇科 Mycenaceae 2 12 5.74 丝盖伞属 Inocybe 5 2.39
多孔菌科 Polyporaceae 8 14 6.70 蜡蘑属 Laccaria 5 2.39
红菇科 Russulaceae 3 23 11.00 小皮伞属 Marasmius 6 2.87
小菇属 Mycena 11 5.26
光柄菇属 Pluteus 5 2.39
红菇属 Russula 17 8.13
栓菌属 Trametes 5 2.39
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