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SCIENTIA SINICA Informationis, Volume 51 , Issue 6 : 971(2021) https://doi.org/10.1360/SSI-2020-0085

TOA and TDTOA-based augmented state pulsar integrated navigation error suppression method

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  • ReceivedApr 8, 2020
  • AcceptedJun 9, 2020
  • PublishedMay 18, 2021

Abstract


Funded by

国家自然科学基金(61722301)


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References

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  • Figure 1

    (Color online) Results of traditional pulsar navigation with TOA as observation. (a) Position error; (b) velocity error

  • Figure 1

    (Color online) Results of traditional pulsar navigation with TOA as observation. (a) Position error; (b) velocity error

  • Figure 2

    Time-difference pulsar navigation results using TDTOA as observation. (a) Position error; (b) velocity error

  • Figure 2

    Time-difference pulsar navigation results using TDTOA as observation. (a) Position error; (b) velocity error

  • Figure 3

    Pulsar integrated navigation results using TOA and TDTOA as observations. (a) Position error; (b) velocity error

  • Figure 3

    Pulsar integrated navigation results using TOA and TDTOA as observations. (a) Position error; (b) velocity error

  • Figure 4

    Results of augmented state pulsar navigation using TOA as observation. (a) Position error; (b) velocity error

  • Figure 4

    Results of augmented state pulsar navigation using TOA as observation. (a) Position error; (b) velocity error

  • Figure 5

    Results of augmented state pulsar integrated navigation with TOA and TDTOA as observations. (a) Position error; (b) velocity error

  • Figure 5

    Results of augmented state pulsar integrated navigation with TOA and TDTOA as observations. (a) Position error; (b) velocity error

  • Figure 6

    Navigation results when observing a pulsar. (a) Position error; (b) velocity error

  • Figure 6

    Navigation results when observing a pulsar. (a) Position error; (b) velocity error

  • Figure 7

    Navigation results when observing two pulsars. (a) Position error; (b) velocity error

  • Figure 7

    Navigation results when observing two pulsars. (a) Position error; (b) velocity error

  • Table 1   Orbital parameters of the Mars Reconnaissance Orbiter
    Parameter Value
    Semimajor axis 3684.5 km
    Eccentricity 0.010
    Track inclination 93.0$^{\circ}$
    Right ascension 278.0$^{\circ}$
    Near rising angle 270.0$^{\circ}$
  • Table 2   Pulsar parameters
    Parameter B0531+21 B1821$-$24 B0540$-$69
    Right ascension $\varphi$ $(^{\circ})$ 83.63 276.13 85.046
    Right ascension uncertainty $\delta\varphi$ (mas) 75 0.90 4.5
    Declination $\theta$ $(^{\circ})$ 22.01 $-$24.87 $-$69.332
    Declination uncertainty $\delta\theta$ (mas) 60 12 4.99
    $D_0$ (kpc) 2.0 5.5 47.3
    $P$ (s) 0.0334 0.00305 0.0504
    $W$ (s) 1.7$\times$$10^{-3}$ 5.5$\times$$10^{-5}$ 2.5$\times$$10^{-3}$
    $F_x$ (ph/cm$^2$/s) 1.54 1.93$\times$$10^{-4}$ 5.15$\times$$10^{-3}$
    $P_f$ (%) 70 98 67
  • Table 3   Results of various pulsar navigation methods
    Method Average position error (km) Average velocity error (m/s)
    Traditional pulsar navigation
    with TOA as observation (without systematic error)
    1.20 1.29
    Traditional pulsar navigation
    with TOA as observation (with systematic error)
    3.58 2.89
    Time-difference pulsar navigation
    using TDTOA as observation (with systematic error)
    1.79 1.58
    Pulsar integrated navigation using
    TOA and TDTOA as observations (with systematic error)
    1.63 1.48
    Augmented state pulsar navigation
    using TOA as observation (with systematic error)
    2.26 2.04
    Augmented state pulsar integrated navigation
    with TOA and TDTOA as observations (with systematic error)
    1.28 1.31
  • Table 41  
    SSISCIENTIA SINICA Informationis中国科学: 信息科学Sci Sin-Inf1674-72672095-948611-5846/TPScience China PressSCP 10.1360/SSI-2020-0085论文基于TOA和TDTOA的增广状态脉冲星组合导航误差抑制方法TOA and TDTOA-based augmented state pulsar integrated navigation error suppression method晓琳1*明臻2*月清2*建成1*伟仁3*

    通讯作者, E-mail: Xiaolin NINGningxiaolin@buaa.edu.cn, Mingzhen GUI, Yueqing HUANGsy1917224@buaa.edu.cn, Jiancheng FANG, Weiren WU

    Corresponding author ()

    20216516971080420200906202018052021

    脉冲星导航是一种极具潜力的深空自主导航技术, 通常采用脉冲到达时间(time of arrival, TOA)作为量测信息. 但脉冲星星历误差和星载原子钟误差等系统误差对导航性能有显著影响. 为了解决上述问题, 提出了一种基于TOA和时间差分TOA (TDTOA)的增广状态脉冲星组合导航误差抑制方法, 通过将每个脉冲星的星历误差和时钟误差增加到状态向量, 并利用TOA和TDTOA量测值对其进行估计和校正. 仿真结果表明, 该方法提高了脉冲星星历误差和时钟误差的可观测性, 有效地消除了这些系统误差的影响, 导航精度相比传统脉冲星导航提高了29%.

    Pulsar navigation is a promising deep space autonomous navigation technology, which usually uses the time of arrival (time of arrival, TOA) as the measurement information. However, system errors such as pulsar ephemeris errors and space-borne atomic clock errors have a significant impact on navigation performance. To solve the above problems, an error suppression method for augmented state pulsar integrated navigation based on TOA and time difference TOA (TDTOA) is proposed. By adding the ephemeris error and clock error of each pulsar to the state vector, the TOA and TDTOA measurements are used to estimate and correct them. Simulation results show that this method improves the observability of the pulsar ephemeris and clock errors, eliminates the effect of these systematic errors, and improves navigation accuracy by 29% compared to traditional pulsar navigation.

    自主导航脉冲星导航系统误差时间差分可观测性分析autonomous navigationpulsar navigationsystem errortime differenceobservability analysis国家自然科学基金61722301国家自然科学基金(批准号: 61722301)资助项目Citation宁晓琳, 桂明臻, 黄月清, 等. 基于TOA和TDTOA的增广状态脉冲星组合导航误差抑制方法. 中国科学: 信息科学, 2021, 51: 971-, doi: 10.1360/SSI-2020-0085Crossmark2021-05-31T09:50:24AuthorMark宁晓琳等AuthorMarkCite宁晓琳, 桂明臻, 黄月清, 等AuthorMarkCiteEnNing X L, Gui M Z, Huang Y Q, et alTitleCite基于TOA和TDTOA的增广状态脉冲星组合导航误差抑制方法TitleCiteEnTOA and TDTOA-based augmented state pulsar integrated navigation error suppression method

    《中国科学》杂志社

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    <x>1</x><x> </x><x>引言</x>

    有效的导航和控制是确保深空探测任务成功的一个重要前提. 因为深空任务地面测控的局限性[1-3], 提高深空探测器的自主导航能力势在必行. 脉冲星导航是一种极具吸引力的自主导航方法[4,5]. 通过对脉冲到达时间(time of arrival, TOA)信号的观测, 经过相应的信号和数据处理, 深空探测器可以自主确定轨道、时间和姿态等导航参数[6-8]. 2006 年, Sheikh等[7]将卡尔曼(Kalman)滤波引入脉冲星导航, 在不考虑系统误差影响的情况下, 实现了精度高于200 m的脉冲星导航. 然而, 在目前的脉冲星观测技术中, 脉冲星的星历参数, 如脉冲星的方向, 存在着不可避免的误差. 研究表明, 脉冲星方向1毫弧秒(mas)的误差将引起约800 m的系统误差[9]. 此外, 星载原子钟的误差也是影响长期行星际飞行任务导航性能的重要因素[10,11]. 1 $\mu$s的时钟误差将导致约300 m的系统误差.

    目前消除系统误差的方法, 主要可以分为两类: 一类是把系统误差增广到状态向量中并在滤波器中进行估计和校正[12], 另一种是对量测进行位置差分或时间差分, 从而消除共同的系统误差. Liu等[13,14]分析了由脉冲星星历误差和时钟误差引起的系统误差, 得出了系统误差是慢变的结论. 在此基础上, 将系统误差增广到状态量中, 采用无迹卡尔曼滤波(unscented Kalman filter, UKF)对系统误差进行估计和校正, 提高了导航精度. 然而, 由于TOA量测中无法区分星历误差和时钟误差的影响, 因此该方法可观测性差, 对系统误差的估计精度是有限的.

    在差分方面, Xiong等[15]提出了一种基于卫星星座的位置差分技术, 用两个航天器间的TOA之差来代替TOA量测. 差分量测对系统误差不敏感, 可以提高导航精度. 然而, 这项技术要求多颗卫星同步观测一颗脉冲星, 增加了任务的复杂度. 时间差分是一种新的消除系统误差的方法, 最早是在GPS中提出的[16]. 载波相位时间差分可用于消除对流层延迟、电离层延迟、卫星和接收机时钟误差等常见误差的影响[17]. 由于时间差分量测只能提供相对位置信息, 所以时间差分GPS通常借助惯性导航提供绝对信息[18,19]. Wang等[9]提出了一种采用时间差分TOA (TDTOA)量测的脉冲星导航方法, 即用相邻时刻TOA量测值的差值代替不同航天器TOA量测值的. 该方法可以消除系统误差的影响, 适用于单航天器. 然而, 差分TOA量测只包含相对位置信息, 缺乏绝对位置信息, 位置误差可能会随时间而发散. 本文作者的科研团队提出了一种同时使用TOA量测和TDTOA量测的脉冲星导航方法, 其中使用TOA量测提供绝对位置信息, 使用TDTOA量测提供相对位置信息[20]. 该方法的性能优于单独使用TDTOA的脉冲星导航方法, 但是TOA量测中存在系统误差, 仍然会影响导航精度.

    针对上述问题, 本文提出了一种新的基于TOA和TDTOA的增广状态脉冲星组合导航误差抑制方法, 每个脉冲星的星历误差和时钟误差都被增加到状态向量, 由TOA和TDTOA量测值联合进行估计和校正. 基于奇异值分解的可观测性分析表明, 该方法提高了各脉冲星星历误差和时钟误差的可观测性. 仿真结果表明, 该方法可以有效地抑制系统误差的影响, 导航精度相比传统脉冲星导航方法精度提高29%.

    <x>2</x><x> </x><x>基于TOA和TDTOA的增广状态脉冲星组合导航误差抑制</x>

    为了减小脉冲星星历误差和星载时钟误差对系统误差的影响, 将脉冲星星历误差和时钟误差增广到状态向量, 并以TOA和TDTOA组合作为量测对其进行估计和校正.

    <x>2.1</x><x> </x><x>增广状态模型</x>

    受目前观测技术的限制, 不可避免存在脉冲星星历误差. 假设$\varphi^i$和$\theta^i$分别为第$i$个脉冲星的实际赤经和赤纬. $\widetilde{\varphi}^i$和$\widetilde{\theta}^i$分别为存在误差的第$i$个脉冲星的赤经和赤纬. 则第$i$个脉冲星的星历误差可表示为 除了上述脉冲星星历误差, 还存在星载时钟误差$(\delta~t_c)$. 因为脉冲星的星历误差和时钟误差是慢变的[21], 所以在状态模型中可被建模为常量[13]. 本文以火星探测器为例, 将这些误差增加到状态向量中, 同时考虑太阳和地球的引力, 并将其他扰动视为过程噪声, 则火星中心惯性系(J2000.0)中的增广状态模型可表示为 其中${\boldsymbol~r}_{pm}=[r_{pmx},~r_{pmy},r_{pmz}]^{\rm~T}$ 和 ${\boldsymbol~v}_{pm}=[v_{pmx},~v_{pmy},v_{pmz}]^{\rm~T}$ 分别是探测器相对于火星的位置矢量和速度矢量. $\mu_m$, $\mu_e$ 和 $\mu_s$ 分别为火星、地球和太阳的引力常数. ${\boldsymbol~r}_{ps}={\boldsymbol~r}_{pm}-{\boldsymbol~r}_{sm}$ 是探测器相对于太阳的位置矢量, ${\boldsymbol~r}_{sm}$ 是太阳相对于火星的位置矢量. ${\boldsymbol~r}_{pe}={\boldsymbol~r}_{pm}-{\boldsymbol~r}_{em}$ 是探测器相对于地球的位置矢量, ${\boldsymbol~r}_{em}$ 是地球相对于火星的位置矢量. ${\boldsymbol~w}_{v_{pm}}$, $w_{\delta\varphi^i}$, $w_{\delta\theta^i}$ 和 $w_{\delta~t_c}$ 是过程噪声.

    假设 ${\boldsymbol~X}=[{\boldsymbol~r}_{pm},~{\boldsymbol~v}_{pm},~\delta\varphi^i,~\delta\theta^i,~\delta~t_c]^{\rm~T}$ 是增广状态向量, ${\boldsymbol~W}=[~\bf{0},~{\boldsymbol~w}_{v_{pm}},~w_{\delta\varphi^i},~w_{\delta\theta^i},~w_{\delta~t_c}]^{\rm~T}$ 是过程噪声向量, 则$k$时刻的状态模型可以用如下一般形式表示:

    <x>2.2</x><x> </x><x>TOA和TDTOA的量测模型</x> <x>2.2.1</x><x> </x><x>TOA的量测模型</x>

    采用探测器上第$i$颗脉冲星的脉冲到达时间$(t^i_{sc})$与其到太阳系质心(solar system barycenter, SSB) 的到达时间$(t^i_b)$的差值作为量测量. 在考虑相对论效应和几何效应的情况下, 量测方程可表示为[7] 其中 ${\boldsymbol~b}$ 是SSB相对于太阳的位置矢量. c 是光速. $D^i_0$ 是从脉冲星到SSB的距离. $v^i$ 是第$i$个脉冲星的量测噪声. $\widetilde{{\boldsymbol~n}}^i$ 是修正后的脉冲星方向矢量, 可以用以下形式表示:

    假设 TOA量测 ${\boldsymbol~Z}_p=[t^1_b-t^1_{sc},~\ldots,~t^i_b-t^i_{sc},~\ldots,~t^{n_p}_b-t^{n_p}_{sc}]^{\rm~T}$, 量测噪声 ${\boldsymbol~V}_p=[v^1,~\ldots,~v^i,~\ldots,~v^{n_p}]^{\rm~T}$, 其中 $1<i<n_p$, $n_p$是观测到的脉冲星个数. 则$k$时刻的TOA量测模型可表示如下:

    <x>2.2.2</x><x> </x><x>TDTOA的量测模型</x>

    两个相邻历元之间的脉冲星星历误差和时钟误差的变化很小[9,10]. 因此, 可以通过将相邻历元的脉冲星TOA量测${\boldsymbol~Z}_{p,~k}$和${\boldsymbol~Z}_{p,~k-1}$作差, 以消除其中的共同系统误差. TDTOA的量测模型可表示如下: 其中 $f&apos;(\cdot)$ 是状态模型 $f(\cdot)$ 的逆过程, 即从 ${\boldsymbol~X}_k$ 获得 ${\boldsymbol~X}_{k-1}$ 的过程.

    <x>2.2.3</x><x> </x><x>增广状态时间差分量测模型</x>

    本文方法以${\boldsymbol~Z}_{p,~k}$和$\Delta{\boldsymbol~Z}_{p,~k}$作为量测, 即${\boldsymbol~Z}_k=[{\boldsymbol~Z}_{p,~k},~\Delta{\boldsymbol~Z}_{p,~k}~]^{\rm~T}$, 量测噪声 ${\boldsymbol~V}_k=[{\boldsymbol~V}_{p,~k},~{\boldsymbol~V}_{dp,~k}]^{\rm~T}$, 基于增广状态和时间差分的量测模型为 其中 ${\boldsymbol~h}({\boldsymbol~X}_k)=[{\boldsymbol~h}_p({\boldsymbol~X}_k),~{\boldsymbol~h}_{dp}({\boldsymbol~X}_k)]^{\rm~T}$.

    <x>2.3</x><x> </x><x>滤波过程</x>

    由于脉冲星TOA量测量观测周期较长, 在没有脉冲星量测的情况时状态量和预测误差协方差仅进行时间更新. 在获得脉冲星TOA量测量时, 采用UKF滤波技术同时进行时间更新和量测更新.

    <x>3</x><x> </x><x>基于UKF的可观测性分析</x> <x>3.1</x><x> </x><x>基于UKF的可观测性矩阵的计算</x>

    从状态模型和量测模型可以看出, 所提出的脉冲星导航系统是非线性的、时变的, 且时间差分的量测模型非常复杂, 如果使用传统的线性化状态模型和量测模型的方法计算可观测性矩阵是非常复杂且耗时的[22]. 文献[23]中给出了利用UKF计算转移矩阵$~\bf{\Phi}_k~\in~i^{n~\times~n}$和量测矩阵${\boldsymbol~H}_k~\in~i^{m~\times~n}$的方法, $n$和$m$分别是状态向量和量测向量的维数.

    <x>3.1.1</x><x> </x><x>通过sigma点获得等效状态转移矩阵</x>

    在$k-1$时刻获得的后验状态估计${\hat{\boldsymbol~X}}^+_{k-1}$附近选取$2n+1$个采样点, 其中$n$表示状态变量的维数, 这些样本点的均值等于后验状态估计${\hat{\boldsymbol~X}}^+_{k-1}$, 协方差等于$k-1$时刻获得的后验误差协方差${\boldsymbol~P}^+_{k-1}$; 那么选取的采样点$x^{(0)}_{k-1}$, $x^{(1)}_{k-1}$, $\ldots$, $x^{(2n)}_{k-1}$及其权重$w_0$, $w_1$, $\ldots$, $w_{2n}$分别如下: 其中$\tau$表示缩放参数, $(\sqrt{{\boldsymbol~P}^+_{k-1}})_i$表示取平方根矩阵的第$i$行或列.

    传递sigma采样点, 得到每个采样点的一步预测$x^{(i)}_k$为 其中$F(g)$为系统非线性连续状态转移函数. 由$2n+1$个sigma采样点$x^{(i)}_{k-1}$构成矩阵 再由$2n+1$个sigma采样点的一步预测$x^{(i)}_k$构成矩阵 可通过矩阵${\boldsymbol~x}_{k-1}$的广义逆矩阵求得等效状态转移矩阵$\tilde{~\bf{\Phi}}_k$:

    <x>3.1.2</x><x> </x><x>获得等效量测矩阵</x>

    通过卡尔曼滤波的统计学推导可知 其中${\boldsymbol~P}_{xy,~k}$是互协方差矩阵, ${\boldsymbol~P}^-_k$是先验估计协方差矩阵, $\tilde{{\boldsymbol~H}}^{\rm~T}_k$表示等效量测矩阵; 由式(16)可计算$\tilde{{\boldsymbol~H}}^{\rm~T}_k$.

    <x>3.1.3</x><x> </x><x>构建可观测矩阵</x>

    基于3.1.1小节及3.1.2小节得到的$\tilde{~\bf{\Phi}}_k$及$\tilde{{\boldsymbol~H}}^{\rm~T}_k$, 构造每时段的可观测矩阵[24] 其中$j=1,2,~\ldots,~l$; 构造系统条带化可观测矩阵${\boldsymbol~Q}_s$:

    <x>3.2</x><x> </x><x>基于奇异值分解的可观测度分析</x>

    利用奇异值分解来分析每个状态的可观测度[25,26]. 可观测矩阵${\boldsymbol~Q}^{\rm~T}_j$ 可以分解为 其中${\boldsymbol~U}^{\rm~T}=[{\boldsymbol~u}_1,~{\boldsymbol~u}_2,~\ldots,~{\boldsymbol~u}_n]$, ${\boldsymbol~V}^{\rm~T}=[{\boldsymbol~v}_1,~{\boldsymbol~v}_2,~\ldots,~{\boldsymbol~v}_n]$ 是正交矩阵. $~\bf{\Sigma}={\rm~diag}(\sigma_1,~\sigma_2,~\ldots,~\sigma_n)$是对角矩阵, $\sigma_1~\geq~\sigma_2~\geq~\cdots~\geq~\sigma_n~\geq~0$ 是可观测性矩阵的奇异值. 奇异值可以表示出每维状态的可观测程度.

    <x>4</x><x> </x><x>仿真分析</x>

    仿真中首先对比了以TOA作为观测量的传统脉冲星导航、以TDTOA作为观测量的时间差分脉冲星导航、 以TOA和TDTOA作为观测量的脉冲星组合导航、以TOA作为观测量的增广状态脉冲星导航、以TOA和TDTOA作 为观测量的增广状态脉冲星组合导航5种不同方法的导航结果, 其次对提出的以TOA和TDTOA作为观测量的增广状态脉冲星组合导航系统误差抑制方法与仅以TOA作为观测量的增广状态脉冲星导航方法的可观测度进行了比较, 证明了通过增加TDTOA可有效地增强系统的可观测度.

    <x>4.1</x><x> </x><x>仿真条件</x>

    探测器的标称轨迹依据美国“火星勘测轨道器"的轨道参数, 由STK软件产生, 如表1所示, 仿真时间从2021年3月5日0时至3月6日0时.

    火星的理想轨道是用JPLDE421从STK得到[27]. 仿真中使用的脉冲星是B0531+21, B1821$-$24和B0540$-$69, 其参数如表2所示[28]. 假设初始时钟误差为$1\times~10^{-6}$ s, 时钟误差漂移和漂移变化率分别为$3.637979~\times~10^{-11}$, 以及 $6.66~\times~10^{-18}$/s. 航天器的初始位置误差和速度误差分别为 $[1000,~1000,$$1000]^{\rm~T}$ m, $[1,~1,~1]^{\rm~T}$ m/s.

    <x>4.2</x><x> </x><x>5种导航方法的仿真结果比较</x> <x>4.2.1</x><x> </x><x>以TOA作为观测量的传统脉冲星导航结果</x>

    1给出了不存在系统误差和存在系统误差两种情况下以TOA作为观测量的传统脉冲星导航方法的导航结果. 仿真结果表明, 在没有系统误差时, 位置和速度估计误差能够快速收敛并达到精确的稳态. 由于脉冲星量测的观测周期较长, 位置和速度估计误差曲线呈波浪形. 在没有脉冲星量测的情况下, 仅进行时间更新, 导致了位置和速度估计误差的快速增大. 当有脉冲星量测时, 同时进行时间更新和量测更新, 位置和速度估计误差急剧减小. 无系统误差时以TOA作为观测量的传统脉冲星导航方法的平均位置误差约为1.20 km, 平均速度误差约为1.29 m/s. 然而, 当存在系统误差时, 以TOA作为观测量的传统脉冲星导航方法的位置和速度估计误差会显著增大, 且其中存在恒定的误差. 有系统误差的以TOA作为观测量的传统脉冲星导航系统的平均位置误差和速度误差分别是无系统误差时的3倍和2.2倍.

    <xref rid="FIG1" xml:base="fig">图 1</xref>

    (网络版彩图) 以TOA作为观测量的传统脉冲星导航结果

    <xref rid="FIG1" xml:base="fig">Figure 1</xref>

    (Color online) Results of traditional pulsar navigation with TOA as observation. (a) Position error; (b) velocity error

    <x>4.2.2</x><x> </x><x>以TDTOA作为观测量的时间差分脉冲星导航结果</x>

    2给出了以TDTOA作为观测量的时间差分脉冲星导航在有系统误差时的结果. 可以看出, 在有系统误差时, 以TDTOA作为观测量的时间差分脉冲星导航的位置误差相比以TOA作为观测量的传统脉冲星导航的位置误差减少了约50%, 速度误差减少了约45%, 这是由于时间差分方法有效地消除了系统误差的影响. 但是, 由于缺少绝对位置信息, 其位置误差仍是无系统误差时以TOA作为观测量的传统脉冲星导航的 位置误差的1.5 倍.

    <xref rid="FIG2" xml:base="fig">图 2</xref>

    以TDTOA作为观测量的时间差分脉冲星导航结果

    <xref rid="FIG2" xml:base="fig">Figure 2</xref>

    Time-difference pulsar navigation results using TDTOA as observation. (a) Position error; (b) velocity error

    <x>4.2.3</x><x> </x><x>以TOA和TDTOA作为观测量的脉冲星组合导航结果</x>

    将TOA量测量与TDTOA量测量组合, 可利用TOA提供绝对位置信息, 利用TDTOA提供相对位置信息, 提高导航性能. 图3给出了以TOA和TDTOA作为观测量的脉冲星组合导航结果. 可以看出, 通过将TOA与TDTOA组合, 位置误差相比单独以TDTOA作为观测量的时间差分脉冲星导航位置误差减小了8$%$. 以TOA和TDTOA作为观测量的脉冲星组合导航的位置和速度估计误差曲线与单独以TDTOA作为观测量的时间差分脉冲星导航的位置和速度估计误差曲线较为相似. 这是由于TOA仍然受系统误差影响, 并且其量测噪声方差阵大于TDTOA的量测噪声方差阵, 因此在量测更新中TDTOA发挥主要作用.

    <xref rid="FIG3" xml:base="fig">图 3</xref>

    以TOA和TDTOA作为观测量的脉冲星组合导航结果

    <xref rid="FIG3" xml:base="fig">Figure 3</xref>

    Pulsar integrated navigation results using TOA and TDTOA as observations. (a) Position error; (b) velocity error

    <x>4.2.4</x><x> </x><x>以TOA作为观测量的增广状态脉冲星导航结果</x>

    以TOA作为观测量的增广状态脉冲星导航的状态模型为式(4), 量测模型为式(8). 图4给出了以TOA 作为观测量的增广状态脉冲星导航结果. 可以看出, 在有系统误差时, 以TOA作为观测量的增广状态脉冲星导航的结果显著优 于以TOA作为观测量的传统脉冲星导航的结果, 位置误差减少了约 48%, 速度误差减少了约 38%. 但是, 由位置和速度估计误差曲线可看出, 位置和速度估计误差仍存在常值误差, 这是由于仅利用TOA量测量无法准确估计脉冲星星历误差及时钟误差.

    <xref rid="FIG4" xml:base="fig">图 4</xref>

    以TOA作为观测量的增广状态脉冲星导航结果

    <xref rid="FIG4" xml:base="fig">Figure 4</xref>

    Results of augmented state pulsar navigation using TOA as observation. (a) Position error; (b) velocity error

    <x>4.2.5</x><x> </x><x>以TOA和TDTOA作为观测量的增广状态脉冲星组合导航结果</x>

    本文提出的以TOA和TDTOA作为观测量的增广状态脉冲星组合导航的状态模型为式(4), 量测模型为式(10). 图5给出了以TOA和TDTOA作为观测量的增广状态脉冲星组合导航结果. 可以看出, 本文的方法可以获得更高的导航精度, 平均位置误差相比以TOA作为观测量的增广状态脉冲星导航减少了约 29%, 平均速度误差减少了约27%. 上述5种导航方法的结果比较如表3所示. 可以看到, 本文的方法可以获得与无系统误差的理想状态相似的精度, 验证了以TOA和TDTOA作为观测量的增广状态脉冲星组合导航方法抑制系统误差影响的有效性.

    <xref rid="FIG5" xml:base="fig">图 5</xref>

    以TOA和TDTOA作为观测量的增广状态脉冲星组合导航结果

    <xref rid="FIG5" xml:base="fig">Figure 5</xref>

    Results of augmented state pulsar integrated navigation with TOA and TDTOA as observations. (a) Position error; (b) velocity error

    <x>4.3</x><x> </x><x>脉冲星星历误差和时钟误差的可观测度</x>

    本小节比较了本文提出的以TOA和TDTOA作为观测量的增广状态脉冲星组合导航方法与 以TOA作为观测量的增广状态脉冲星导航方法的可观测度. 表4分别给出了两种方法中脉冲星星历误差及时钟误差的估计误差和对应的基于奇异值的可观测度. 可以看到, 本文方法的各脉冲星星历误差及时钟误差的奇异值相比以TOA作为观测量的增广状态脉冲星导航方法均明显增大, 验证了本文提出的方法可提高星历误差和时钟误差的可观测度, 另外星历误差和时钟误差估计结果也验证了可观测分析结果的正确性.

    <x>4.4</x><x> </x><x>脉冲星个数对导航精度的影响</x>

    67分别给出了本文的方法在只观测脉冲星B0531+21 及同时观测B0531+21和B1821$-$24两颗脉冲星时的导航结果. 从图5$\sim$7可以看到, 本文提出的方法在仅观测1颗脉冲星时同样可以获得收敛的导航结果. 表5给出了本文提出的方法在脉冲星个数不同时的导航结果. 随着观测脉冲星个数的减少, 位置和速度估计误差增大. 观测1颗脉冲星时的平均位置误差及平均速度误差分别是观测3颗脉冲星时的1.43倍及1.35倍.

    <xref rid="FIG6" xml:base="fig">图 6</xref>

    观测一颗脉冲星时导航结果

    <xref rid="FIG6" xml:base="fig">Figure 6</xref>

    Navigation results when observing a pulsar. (a) Position error; (b) velocity error

    <xref rid="FIG7" xml:base="fig">图 7</xref>

    观测两颗脉冲星时导航结果

    <xref rid="FIG7" xml:base="fig">Figure 7</xref>

    Navigation results when observing two pulsars. (a) Position error; (b) velocity error

    <x>4.5</x><x> </x><x>时间复杂度</x>

    时间复杂度也是一个重要的指标, 尤其在实时性较高的场景中. 运行环境如表6所示. 表7给出了5种方法的时间复杂度.

    可以看出, 以TOA作为观测量的传统脉冲星导航和以TDTOA作为观测量的时间差分脉冲星导航的运行耗时相近; 同时以TOA和TDTOA作为观测量时, 由于量测量的维数增加, 导致运行耗时略有增加; 当在状态向量中加入脉冲星的星历误差及星载时钟误差进行在线估计时, 由于状态量的维数由6维增加到13维, 导致其运行耗时增加较多, 相比传统脉冲星导航运行耗时增加0.942 s; 以TOA和TDTOA作为观测量的增广状态脉冲星组合导航的运行耗时与以TOA作为观测量的增广状态脉冲星导航相近. 它们的运行耗时比为1: 1.06: 1.15: 1.24: 1.25.

    <x>5</x><x> </x><x>结论</x>

    本文将增广状态法与时间差分法相结合, 提出了基于TOA和TDTOA的增广状态脉冲星组合导航误差抑制方法, 将各脉冲星星历误差和时钟误差均作为状态量增广到状态向量中, 并同时利用TOA及TDTOA 共同估计、补偿各系统误差. 仿真结果表明, 本文提出的方法相比以TDTOA作为观测量的时间差分脉冲星导航、以TOA作为观测量的增广状态脉冲星导航等其他方法可以获得更高的导航精度, 可以有效地消除系统误差的影响. 利用奇异值分解对以TOA和TDTOA作为观测量的增广状态脉冲星组合导航和以TOA作为观测量的增广状态脉冲星导航的可观测度进行了分析. 可观测性分析结果表明, 本文提出的方法提高了各脉冲星星历误差和时钟误差的可观测性. 本文所提出的方法也可应用于地球卫星等航天器.

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    表 1“火星勘测轨道器"轨道参数

    Table 1Orbital parameters of the Mars Reconnaissance Orbiter

    Parameter Value
    Semimajor axis 3684.5 km
    Eccentricity 0.010
    Track inclination 93.0$^{\circ}$
    Right ascension 278.0$^{\circ}$
    Near rising angle 270.0$^{\circ}$

    表 2脉冲星参数

    Table 2Pulsar parameters

    Parameter B0531+21 B1821$-$24 B0540$-$69
    Right ascension $\varphi$ $(^{\circ})$ 83.63 276.13 85.046
    Right ascension uncertainty $\delta\varphi$ (mas) 75 0.90 4.5
    Declination $\theta$ $(^{\circ})$ 22.01 $-$24.87 $-$69.332
    Declination uncertainty $\delta\theta$ (mas) 60 12 4.99
    $D_0$ (kpc) 2.0 5.5 47.3
    $P$ (s) 0.0334 0.00305 0.0504
    $W$ (s) 1.7$\times$$10^{-3}$ 5.5$\times$$10^{-5}$ 2.5$\times$$10^{-3}$
    $F_x$ (ph/cm$^2$/s) 1.54 1.93$\times$$10^{-4}$ 5.15$\times$$10^{-3}$
    $P_f$ (%) 70 98 67
    <xref rid="FIG1" xml:base="fig">图 1</xref>

    (网络版彩图) 以TOA作为观测量的传统脉冲星导航结果

    <xref rid="FIG1" xml:base="fig">Figure 1</xref>

    (Color online) Results of traditional pulsar navigation with TOA as observation. (a) Position error; (b) velocity error

    <xref rid="FIG2" xml:base="fig">图 2</xref>

    以TDTOA作为观测量的时间差分脉冲星导航结果

    <xref rid="FIG2" xml:base="fig">Figure 2</xref>

    Time-difference pulsar navigation results using TDTOA as observation. (a) Position error; (b) velocity error

    <xref rid="FIG3" xml:base="fig">图 3</xref>

    以TOA和TDTOA作为观测量的脉冲星组合导航结果

    <xref rid="FIG3" xml:base="fig">Figure 3</xref>

    Pulsar integrated navigation results using TOA and TDTOA as observations. (a) Position error; (b) velocity error

    <xref rid="FIG4" xml:base="fig">图 4</xref>

    以TOA作为观测量的增广状态脉冲星导航结果

    <xref rid="FIG4" xml:base="fig">Figure 4</xref>

    Results of augmented state pulsar navigation using TOA as observation. (a) Position error; (b) velocity error

    <xref rid="FIG5" xml:base="fig">图 5</xref>

    以TOA和TDTOA作为观测量的增广状态脉冲星组合导航结果

    <xref rid="FIG5" xml:base="fig">Figure 5</xref>

    Results of augmented state pulsar integrated navigation with TOA and TDTOA as observations. (a) Position error; (b) velocity error

    表 3各种脉冲星导航方法的结果

    Table 3Results of various pulsar navigation methods

    Method Average position error (km) Average velocity error (m/s)
    Traditional pulsar navigation
    with TOA as observation (without systematic error)
    1.20 1.29
    Traditional pulsar navigation
    with TOA as observation (with systematic error)
    3.58 2.89
    Time-difference pulsar navigation
    using TDTOA as observation (with systematic error)
    1.79 1.58
    Pulsar integrated navigation using
    TOA and TDTOA as observations (with systematic error)
    1.63 1.48
    Augmented state pulsar navigation
    using TOA as observation (with systematic error)
    2.26 2.04
    Augmented state pulsar integrated navigation
    with TOA and TDTOA as observations (with systematic error)
    1.28 1.31

    表 4表 1

    脉冲星星历误差和时钟误差的结果

    Table 4Table 1

    Pulsar ephemeris and clock error using augmented state pulsar navigation

    pt

    Method

    Precision

    and

    $\delta\varphi^1$

    $\delta\theta^1$

    $\delta\varphi^2$

    $\delta\theta^2$

    $\delta\varphi^3$

    $\delta\theta^3$

    $\delta~t_c$

    observability

    Using

    TOA

    as

    Estimation

    error

    0.52

    mas

    6.00

    mas

    0.65

    mas

    8.39

    mas

    4.24

    mas

    4.96

    mas

    2.52

    $\times~10^{-7}$

    observation

    Singular

    value

    2305.29

    6.56

    $\times$$10^{-2}$

    2203.41

    3.03

    $\times$$10^{-2}$

    879.43

    1.42

    4.47

    $\times~10^{-4}$

    Using

    TOA

    and

    TDTOA

    Estimation

    error

    0.17

    mas

    4.35

    mas

    0.41

    mas

    6.17

    mas

    0.99

    mas

    3.48

    mas

    2.51

    $\times~10^{-7}$

    as

    observations

    Singular

    value

    2305.31

    7.44

    $\times$$10^{-2}$

    2203.43

    4.36

    $\times$$10^{-2}$

    879.45

    1.47

    5.05

    $\times~10^{-4}$

    <xref rid="FIG6" xml:base="fig">图 6</xref>

    观测一颗脉冲星时导航结果

    <xref rid="FIG6" xml:base="fig">Figure 6</xref>

    Navigation results when observing a pulsar. (a) Position error; (b) velocity error

    <xref rid="FIG7" xml:base="fig">图 7</xref>

    观测两颗脉冲星时导航结果

    <xref rid="FIG7" xml:base="fig">Figure 7</xref>

    Navigation results when observing two pulsars. (a) Position error; (b) velocity error

    表 5观测不同个数脉冲星时提出方法的导航结果

    Table 5Navigation results of the proposed method when observing different numbers of pulsars

    Number of pulsars Pulsars Average position error (km) Average velocity error (m/s)
    1 B0531+21 2.18 1.97
    2 B0531+21, B1821$-$24 1.80 1.76
    3 B0531+21, B1821$-$24, B0540$-$69 1.28 1.31

    表 6仿真运行环境

    Table 6Operating environment

    Item Description
    CPU Intel Core i7-8565U 1.80 GHz
    RAM 16 GB
    Operating system Windows 10
    Simulation software Matlab R2018b

    表 7时间复杂度的仿真结果

    Table 7The results of time consuming

    Method Time-consuming Single cycle Time-consuming
    operation (s)time (ms)ratio
    Traditional pulsar
    navigation with TOA as observation
    3.976 2.76 1
    Time-difference pulsar navigation
    using TDTOA as observation
    4.222 2.93 1.06
    Pulsar integrated navigation
    using TOA and TDTOA as observations
    4.553 3.16 1.15
    Augmented state pulsar
    navigation using TOA as observation
    4.918 3.42 1.24
    Augmented state pulsar integrated
    navigation with TOA and TDTOA as observations
    4.981 3.46 1.25

    pt

  • Table 5   Navigation results of the proposed method when observing different numbers of pulsars
    Number of pulsars Pulsars Average position error (km) Average velocity error (m/s)
    1 B0531+21 2.18 1.97
    2 B0531+21, B1821$-$24 1.80 1.76
    3 B0531+21, B1821$-$24, B0540$-$69 1.28 1.31
  • Table 6   Operating environment
    Item Description
    CPU Intel Core i7-8565U 1.80 GHz
    RAM 16 GB
    Operating system Windows 10
    Simulation software Matlab R2018b
  • Table 7   The results of time consuming
    Method Time-consuming Single cycle Time-consuming
    operation (s)time (ms)ratio
    Traditional pulsar
    navigation with TOA as observation
    3.976 2.76 1
    Time-difference pulsar navigation
    using TDTOA as observation
    4.222 2.93 1.06
    Pulsar integrated navigation
    using TOA and TDTOA as observations
    4.553 3.16 1.15
    Augmented state pulsar
    navigation using TOA as observation
    4.918 3.42 1.24
    Augmented state pulsar integrated
    navigation with TOA and TDTOA as observations
    4.981 3.46 1.25
qqqq

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