Relativistic Navigation Method for Deep Space Probes
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摘要:在相对论导航系统中，通过毫角秒恒星角距测量装置获取反映恒星光行差和光线引力偏折变化的恒星角距观测量，利用导航滤波器处理观测量，估计深空探测器在惯性空间的位置和速度矢量，以及敏感器的测量基准偏差。建立了面向导航滤波器设计和系统性能分析的状态方程和观测方程，根据导航系统的克拉美劳下界（Cramer-Rao Lower Bound，CRLB）考察了相对论导航方法用于深空探测器的可行性，设计了通过导航滤波器自学习提升相对论导航系统性能的策略。仿真研究表明，对于环绕火星运行的深空探测器，在恒星角距测量精度为1 mas的情况下，相对论导航方法能达到百米量级的定位精度水平。为相对论导航方法在深空中的应用提供了支持。
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关键词:
- 深空探测/
- 天文导航/
- 相对论效应/
- Q学习扩展卡尔曼滤波器
Abstract:An autonomous navigation method based on the observations of the relativistic perturbations for deep space probes is presented in this paper. The relativistic perturbations including the stellar aberration and the starlight gravitational deflection are new type of celestial navigation measurement, which can provide the kinematic state information of the deep space probes in the inertial frame. In the relativistic navigation system, the position and velocity vectors of the deep space probes, and the measurement bias of the optical sensor can be estimated through measuring the inter-star angle perturbed by the stellar aberration and the gravitational deflection of light with an optical sensor for LOS (line-of-sight) direction with extremely high accuracy. In this paper, the state equation and measurement equation for the design of the navigation filter and the navigation performance evaluation are established. The feasibility of the relativistic navigation method for deep space probes is investigated via the calculation of the Cramer-Rao Lower Bound (CRLB). In addition, the self-learning strategy of the navigation filter is designed to enhance the relativistic navigation performance. It is illustrated through the numerical simulation that, for a Mars-circling probe, the position error of the relativistic navigation method is on the order of 100 m with the inter-star angle measurement accuracy of 1 mas.Highlights● An autonomous celestial navigation method based on the observations of the relativistic perturbations of starlight for deep space probes is presented. ● The feasibility of the presented method is investigated via the calculation of the Cramer-Rao Lower Bound (CRLB) of the relativistic navigation system. ● A novel Q-learning extended Kalman filter (QLEKF) is designed to optimize the navigation filtering parameters adaptively. -
表 1Q学习扩展卡尔曼滤波器
Table 1Q-learning extended Kalman filter
算法 1：Q学习扩展卡尔曼滤波器 输入：状态估计初始值 $ {\hat{\boldsymbol{x}}}_{0} $ 及其误差方差阵 $ {\boldsymbol{P}}_{0} $，事先确定的方差阵集合 $ \left\{{\hat{\boldsymbol{Q}}}_{k}^{(s,a)}\right\} $，观测量$ {\boldsymbol{y}}_{k} $ 1：$ {\hat{\boldsymbol{x}}}_{0}^{(s,a)}\leftarrow {\hat{\boldsymbol{x}}}_{0} $，$ {\boldsymbol{P}}_{0}^{(s,a)}\leftarrow {\boldsymbol{P}}_{0} $，$ Q(\mathrm{s},\mathrm{a})\leftarrow 0 $，$ k\leftarrow 0 $ 2：for特定导航任务周期，do 3： for任意 $ a\in \mathbf{A} $，do 4： $ R(s,a)\leftarrow 0 $ 5： for$t=\mathrm{1,2},\cdots ,T$，do 6： $ k\leftarrow k+1 $ 7： $ [{\hat{\boldsymbol{x}}}_{k}^{\left(s,a\right)},{\boldsymbol{P}}_{k}^{\left(s,a\right)},{\tilde{\boldsymbol{y}}}_{k}^{\left(s,a\right)}]\leftarrow $ $\mathrm{E}\mathrm{K}\mathrm{F}({\hat{\boldsymbol{x} } }_{k-1}^{\left(s,a\right)},{\boldsymbol{P} }_{k-1}^{\left(s,a\right)},{\boldsymbol{y} }_{k},{\hat{\boldsymbol{Q} } }_{k}^{\left(s,a\right)},{\mathit{R} }_{k})$ $\rhd $ 探索滤波器 8： $ R\left(s,a\right)\leftarrow R\left(s,a\right)+ $ $\dfrac{1}{{T} }\left\{ {\left[{\left({\tilde{\boldsymbol{y} } }_{k}^{\left(s,a\right)}\right)}^{-1}{\mathit{R} }_{k}^{-1}{\tilde{\boldsymbol{y} } }_{k}^{\left(s,a\right)}\right]}^{-1}-R\left(s,a\right)\right\}$ 9： $ [{\hat{\boldsymbol{x}}}_{k},{\boldsymbol{P}}_{k},{\tilde{\boldsymbol{y}}}_{k}]\leftarrow \mathrm{E}\mathrm{K}\mathrm{F}({\hat{\boldsymbol{x}}}_{k-1},{\boldsymbol{P}}_{k-1},{\boldsymbol{y}}_{k},{\hat{\boldsymbol{Q}}}_{k},{\mathit{R}}_{k}) $ $\rhd$ 导航滤波器 10： end for 11： $ Q\left(s,a\right)=\left(1-\alpha \right)Q\left(s,a\right)+ $$\mathrm{\alpha }[R\left(s,a\right)+\mathrm{\gamma }\underset{a' }{\mathrm{max} }Q\left(s',a'\right)]$ 12：end for 13： ${a}_{\max}\leftarrow \mathrm{arg}\underset{a}{\mathrm{max} }Q\left(s,a\right)$ 14： ${\hat{\boldsymbol{Q} } }_{k}\leftarrow {\hat{\boldsymbol{Q} } }_{k}^{\left(s,{a}_{\max}\right)}$ 15：end for 16：输出：$ \left\{{\hat{\boldsymbol{x}}}_{k}\right\} $ and $ \left\{{\boldsymbol{P}}_{k}\right\} $ 
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表 2扩展卡尔曼滤波器的计算公式
Table 2Equations of extended Kalman filter
算法 2：扩展卡尔曼滤波器 1：function$ \mathrm{E}\mathrm{K}\mathrm{F}({\hat{\boldsymbol{x}}}_{k-1},{\boldsymbol{P}}_{k-1},{\boldsymbol{y}}_{k},{\boldsymbol{Q}}_{k},{\boldsymbol{R}}_{k}) $ 2： ${\hat{\boldsymbol{x} } }_{k|k-1}\leftarrow{f}\left({\hat{\boldsymbol{x} } }_{k-1}\right)$ $ \rhd $ 预测 3： $ {\boldsymbol{P}}_{k|k-1}\leftarrow {\boldsymbol{F}}_{k}{\boldsymbol{P}}_{k-1}{\boldsymbol{F}}_{k}^{\mathrm{T}}+{\boldsymbol{Q}}_{k} $ 4： $ {\boldsymbol{K}}_{k}\leftarrow {\boldsymbol{P}}_{k|k-1}{\boldsymbol{H}}_{k}^{\mathrm{T}}{({\boldsymbol{H}}_{k}{\boldsymbol{P}}_{k|k-1}{\boldsymbol{H}}_{k}^{\mathrm{T}}+{\boldsymbol{R}}_{k})}^{-1} $ 5： ${\tilde{\boldsymbol{y} } }_{k}\leftarrow {\boldsymbol{y} }_{k}-{h}\left({\hat{\boldsymbol{x} } }_{k|k-1}\right)$ 6： $ {\hat{\boldsymbol{x}}}_{k}\leftarrow {\hat{\boldsymbol{x}}}_{k|k-1}+{\boldsymbol{K}}_{k}{\tilde{\boldsymbol{y}}}_{k} $ $ \rhd $ 更新 7： ${\boldsymbol{P} }_{k}\leftarrow \left({\boldsymbol I}-{\boldsymbol{K} }_{k}{\boldsymbol{H} }_{k}\right){\boldsymbol{P} }_{k|k-1}{({\boldsymbol I}-{\boldsymbol{K} }_{k}{\boldsymbol{H} }_{k})}^{\mathrm{T} }+{\boldsymbol{K} }_{k}{\boldsymbol{R} }_{k}{\boldsymbol{K} }_{k}^{\mathrm{T} }$ 8： return$ {\hat{\boldsymbol{x}}}_{k} $，$ {\boldsymbol{P}}_{k} $，$ {\tilde{\boldsymbol{y}}}_{k} $ 9：end function 下载:
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表 3火星探测器初始轨道参数
Table 3Initial orbital elements of Mars probe
半长轴/km 偏心率 轨道倾角/(°) 升交点赤经/(°) 近地点幅角/(°) 平近点角/(°) 3 697 0.011 90.1549 1.647×10–5 0.3490 0.5235 下载:
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表 4导航滤波器参数
Table 4Parameters of navigation filter
滤波器参数 仿真中参数设置 初始系统噪声方差阵 $ {\hat{\boldsymbol{Q}}}_{0}=\left[\begin{array}{ccc}{\sigma }_{r}^{2}{\boldsymbol{I}}_{3\times 3}& & \\ & {\sigma }_{v}^{2}{\boldsymbol{I}}_{3\times 3}& \\ & & {\sigma }_{\kappa }^{2}{\boldsymbol{I}}_{3\times 3}\end{array}\right] $，其中，$ {\sigma }_{r}=5\times {10}^{-5}\;\mathrm{m} $，$ {\sigma }_{v}=5\times {10}^{-5}\;\mathrm{m}/\mathrm{s} $，$ {\sigma }_{\kappa }=0.1\;\mathrm{m}\mathrm{a}\mathrm{s} $ 测量噪声方差阵 ${\boldsymbol{R} }_{k}={\sigma }_{a}^{2}{\boldsymbol{I} }_{3\times 3}$，其中${\sigma }_{a}=1\;\mathrm{m}\mathrm{a}\mathrm{s}$ 初始估计误差方差阵 $ {\boldsymbol{P}}_{0}=\left[\begin{array}{ccc}{p}_{r}^{2}{\boldsymbol{I}}_{3\times 3}& & \\ & {p}_{v}^{2}{\boldsymbol{I}}_{3\times 3}& \\ & & {p}_{\kappa }^{2}{\boldsymbol{I}}_{3\times 3}\end{array}\right] $，其中，${p}_{r}=1\;\mathrm{k}\mathrm{m}$，${p}_{v}=0.1\;\mathrm{m}/\mathrm{s}$，${p}_{\kappa }=0.2''$ 学习速率 $ \mathrm{\alpha }=0.1 $ 折扣因子 $ \gamma =0.9 $ 计算奖赏的周期/s $ T=500 $ 集合中的元素 对$ {\hat{\boldsymbol{Q}}}_{0} $进行放大或缩小得到$ {\hat{\boldsymbol{Q}}}_{k}^{(s,a)} $ 下载:
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表 5相对论导航系统的定位测速精度
Table 5Position and velocity accuracy of relativistic navigation system
导航滤波器 EKF AEKF REKF QLEKF 平均位置误差RMS/m 348.4 198.1 553.3 163.8 平均速度误差RMS/（m·s–1） 0.29 0.17 0.47 0.11 下载:
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