虽然地面上的通信越来越完善，基本可以满足人们日常生活中的通信需求，但是传统的地面通信却有很多的局限性，例如地面通信的通信距离较短，对通信环境要求较高，需要有固定的地面站来中转信号等^[1]。为了弥补地面通信的不足，同时也为了满足某些特殊情况下的通信需求，在移动载体上进行的通信技术——“动中通”(Communication on the Move)技术应运而生^[2]。动中通技术中关键技术之一就是天线指向的问题，天线指向精确度是动中通技术的一项重要指标^[3−4]。天通一号卫星是我国自主研制的卫星通信系统的首颗发射卫星，不仅在民用市场具有广阔的前景，更会成为军事通信的中坚力量^[5−6]。本文以天通卫星作为指向卫星，主要研究天线的指向精确度，改进了地理坐标系中天线对星角度的数学模型，同时引入天线的高度，使对星角度更加精确。

文献[7]利用船载平台罗经信号特点分析了地理坐标系的对星角度，但是缺乏不稳定因素影响即有姿态角情况下载体坐标系的推导。文献[8]设大地坐标系为单位矢量，给出了地理坐标系下天线指向计算公式，但地球模型不够准确，有较大的误差。文献[9]引入了地球卯酉圈曲率半径，修正了地球模型，但是忽略了天线的海拔高度，也会有一定的误差。本文在分析这些文献的基础上，引入了天线高度，在不增加算法复杂度的基础上提高了对星精确度，并对3种算法模型进行了分析比较，同时也分析了姿态信息对天线指向角度的影响。

1 地理坐标系下天线对星角度模型

如图 1所示，${o_i}{x_i}{y_i}{z_i}$为地心惯性坐标系，o_gx_gy_gz_g为地理坐标系。天通一号卫星位于地心惯性坐标系中，其经度、纬度和高度分别为${\lambda _i}$, ${\theta _i}$和$H$。天通一号运行在地球同步轨道，运行轨道与地球赤道重合，所以其纬度${\theta _i} = 0$，其星下点为${p'}$。车载天线A点位于地理坐标系中，其经度和纬度分别为λ, θ。

地球半径为$R$，天通一号卫星在地心惯性坐标系中的投影为：

$\left\{ \begin{array}{l} {x_i} = \left( {R + H} \right)\sin {\theta _i} \\ {y_i} = \left( {R + H} \right)\cos {\theta _i}\cos {\lambda _i} \\ {z_i} = \left( {R + H} \right)\cos {\theta _i}\sin {\lambda _i} \\ \end{array} \right.$

(1)

其中${\theta _i} = 0$，所以：

$\left\{ \begin{array}{l} {x_i} = {\rm{0}} \\ {y_i} = \left( {R + H} \right)\cos {\lambda _i} \\ {z_i} = \left( {R + H} \right)\sin {\lambda _i} \\ \end{array} \right.$

(2)

获取天通一号在地理坐标系${o_\text{g}}{x_\text{g}}{y_\text{g}}{z_\text{g}}$的位置，需要经过以下步骤：第一步，绕${o_i}{x_i}$逆时针旋转$\lambda $，得到坐标系${o_i}'{x_i}'{y_i}'{z_i}^{}$，旋转后${o_i}'{z_i}'$在A点所在经线圈的垂线方向上，与${o_\text{g}}{z_\text{g}}$重合，同时得到旋转因子${M_1}$；第二步，绕${o_i}'{z_i}'$逆时针旋转$\theta $，得到坐标系${o_i}{''}{x_i}{''}{y_i}'{z_i}'$，得到旋转因子${M_2}$；第三步，在${o_i}{''}{y_i}{''}$方向上平移地球半径$R$。

所以，天通一号卫星在地理坐标系中的位置矢量可以表示为：

$\left[ \begin{gathered} {x_\text{g}} \\ {y_\text{g}} \\ {z_\text{g}} \\ \end{gathered} \right] = {M_2} \times {M_1} \times \left[ \begin{gathered} {x_i} \\ {y_i} \\ {z_i} \\ \end{gathered} \right] - \left[ \begin{gathered} 0 \\ R \\ 0 \\ \end{gathered} \right]$

(3)

式中旋转因子${M_1}$, ${M_2}$分别为：

${M_1} = \left[ \begin{array}{l} 1{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} 0{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} 0 \\ 0{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \;\, {\kern 1pt} \cos \lambda {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \;\sin \lambda \\ 0{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \;\; - \sin \lambda {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \;{\kern 1pt} \cos \lambda \\ \end{array} \right]$

(4)

${M_2} = \left[ \begin{gathered} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \cos \theta {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \sin \theta {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \;\;{\kern 1pt} {\kern 1pt} 0 \\ {\rm{ - }}\sin \theta {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \cos \theta {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \;\;{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} 0 \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} 0{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} 0{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} 1 \\ \end{gathered} \right]$

(5)

将${M_1}$, ${M_2}$代入式(3)中，得到

$\left[ \begin{gathered} {x_\text{g}} \\ {y_\text{g}} \\ {z_\text{g}} \\ \end{gathered} \right] = \left( {R + H} \right)\left[ \begin{gathered} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \sin \theta \cos \left( {{\lambda _i} - \lambda } \right) \\ \cos \theta \cos \left( {{\lambda _i} - \lambda } \right) - \frac{R}{{R + H}} \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \sin \left( {{\lambda _i} - \lambda } \right) \\ \end{gathered} \right]$

(6)

移动卫星通信系统的地面接收天线是利用指向卫星的直视波束来跟踪卫星的，表示地面接收天线指向的基本参数有两个：一个是方位角度(表示方位方向位置)；另一个是俯仰角度(表示俯仰方向位置)^[10−11]。这也是系统初始对准天通卫星和跟踪天通卫星的两个基本的参数^[12]。天通一号卫星在地理坐标系中的位置矢量如图 2所示，$\theta a1$为俯仰角，向上为正；$\phi 1$为方位角，与正北方向的夹角，且顺时针旋转为正。根据图 2，可以计算出天通一号卫星在地理坐标系下的投影为：

$\left\{ \begin{array}{l} {x_\text{g}} = H\cos \left( {\theta 1} \right)\cos \left( {\phi 1} \right) \\ {y_\text{g}} = H\sin \left( {\theta 1} \right) \\ {z_\text{g}} = H\cos \left( {\theta 1} \right)\sin \left( {\phi 1} \right) \\ \end{array} \right.$

(7)

地理坐标系中，天通一号卫星的俯仰角为：

$\tan (\theta 1) = \frac{{{y_\text{g}}}}{{\sqrt {{x_\text{g}}^2{\rm{ + }}{{\rm{z}}_\text{g}}^2} }}$

(8)

方位角为：

$\tan (\phi 1) = \frac{{{z_\text{g}}}}{{{x_\text{g}}}}$

(9)

将式(7)带代入式(8)、式(9)中，得到式(10)、式(11)，可以计算出天线在地理坐标系中的方位角和俯仰角。

$\theta 1 = {\rm{arctan}}\left( {\frac{{\cos \theta \cos \left( {{\lambda _i} - \lambda } \right) - \frac{R}{{R + H}}}}{{1 - {{\left( {\cos \theta \cos ({\lambda _i} - \lambda )} \right)}^2}}}} \right)$

(10)

$\phi 1 = {\rm{arctan}}\left( {\frac{{\tan ({\lambda _i} - \lambda )}}{{\sin \theta }}} \right){\rm{ + 180}}$

(11)

2 地理坐标系下天线对星角度修正 2.1 引入地球卯酉圈曲率半径

根据公式(1)可以看出，在计算天通一号卫星在地心惯性坐标系下的投影时，是将地球看成一个半径为R的标准球体，这样虽然能简化计算，但是对于精确度要求比较高的天线来说，会带来很大的误差，所以需要进一步考虑地球是椭球体时的情况。如图 3所示，在建立地心惯性坐标系时，考虑了椭球体的长半轴和短半轴不一的情况，椭圆的长半轴(赤道半径)为$a = 6\;378\;137{\rm{ m}}$，短半轴(极半径)为$a = 6\;356\;752{\rm{ m}}$。制定了更接近真实情况的地球模型^[13]，此时地球上某一点到地心的距离为该点的卯酉圈曲率半径$N$。星下点${P'}$在赤道上，所以其半径长度为赤道半径，可以将式(1)修正为：

$\left\{ \begin{array}{l} {x_i} = \left( {a + H} \right)\sin {\theta _i} \\ {y_i} = \left( {a + H} \right)\cos {\theta _i}\cos {\lambda _i} \\ {z_i} = \left( {a + H} \right)\cos {\theta _i}\sin {\lambda _i} \\ \end{array} \right.$

(12)

公式(3)修正为：

$\left[ \begin{gathered} {x_\text{g}} \\ {y_\text{g}} \\ {z_\text{g}} \\ \end{gathered} \right] = {M_2} \times {M_1} \times \left[ \begin{gathered} {x_i} \\ {y_i} \\ {z_i} \\ \end{gathered} \right] - \left[ \begin{gathered} {\kern 1pt} 0 \\ N \\ {\kern 1pt} 0 \\ \end{gathered} \right]$

(13)

将公式(12)代入公式(13)中，得到天通卫星在椭球体地理坐标系下的位置矢量为：

$\left[ \begin{gathered} {x_\text{g}} \\ {y_\text{g}} \\ {z_\text{g}} \\ \end{gathered} \right] = \left( {a + H} \right)\left[ \begin{gathered} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \sin \theta \cos \left( {{\lambda _i} - \lambda } \right) \\ \cos \theta \cos \left( {{\lambda _i} - \lambda } \right) - \frac{N}{{a + H}} \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \sin \left( {{\lambda _i} - \lambda } \right) \\ \end{gathered} \right]$

(14)

式中$N$为地球卯酉圈曲率半径，其表达式为：

$\left\{ \begin{array}{l} N = \frac{a}{{\sqrt {1 - {e^2}{{\sin }^2}{\theta _i}} }} \\ e = \frac{{\sqrt {{a^2} - {b^2}} }}{a} \\ \end{array} \right.$

(15)

式中$e$为椭球体的第一曲率半径^[14]。根据公式(14)，将地球的长半轴和短半轴考虑到对星角度数学模型中，能够求出更加精确的地理坐标系下天线指向角度。

2.2 引入天线海拔高度

2.1中的数学模型将地球看成了一个椭球体，相比将地球看成一个规则的球体，对星角度的精确度进一步提高。但是，在实际工程实践中，天线是有一定高度的。比如到海拔较高的山脉或者装载在飞机上，其高度是不可以忽略的。所以，在引入地球卯酉圈曲率半径的基础上，又引入了天线高度，假设为$h$。

在地心惯性坐标系转换到地理坐标系的第三步中，在${o_i}{''}{y_i}{''}$方向上平移地球半径$R$，此时是没有考虑天线高度的。所以将公式(13)修正为：

$\left[ \begin{gathered} {x_\text{g}} \\ {y_\text{g}} \\ {z_\text{g}} \\ \end{gathered} \right] = {M_2} \times {M_1} \times \left[ \begin{gathered} {x_i} \\ {y_i} \\ {z_i} \\ \end{gathered} \right] - \left[ \begin{gathered} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} 0 \\ N + h \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} 0 \\ \end{gathered} \right]$

(16)

所以，得到天通卫星在椭球体地理坐标系且考虑天线高度的情况下的位置矢量为：

$\left[ \begin{gathered} {x_\text{g}} \\ {y_\text{g}} \\ {z_\text{g}} \\ \end{gathered} \right] = \left( {N + H} \right)\left[ \begin{gathered} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \sin \theta \cos \left( {{\lambda _i} - \lambda } \right) \\ \cos \theta \cos \left( {{\lambda _i} - \lambda } \right) - \frac{{N{\rm{ + }}h}}{{N + H}} \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \sin \left( {{\lambda _i} - \lambda } \right) \\ \end{gathered} \right]$

(17)

此时，得到的方位角和俯仰角为：

$\left\{ \begin{array}{l} \phi 1 = {\rm{arctan}}\left( {\frac{{\tan \left( {{\lambda _{\rm{i}}} - \lambda } \right)}}{{\sin \theta }}} \right) + 180^\circ \\ \theta 1 = \left( {\frac{{\cos \theta \cos \left( {{\lambda _{\rm{i}}} - \lambda } \right) - \frac{{N{\rm{ + }}h}}{{N + H}}}}{{1 - {{\left( {\cos \theta \cos \left( {{\lambda _{\rm{i}}} - \lambda } \right)} \right)}^2}}}} \right) \\ \end{array} \right.$

(18)

3 载体坐标系下天线对星角度

上面描述的情况是在载体静止的情况下考虑的，但是实际中载体是一直变化的。天通一号卫星在载体坐标系中的方位角、俯仰角如图 4所示。此时就需要将地理坐标系下的天线指向转换到载体坐标系下。地理坐标系到载体坐标系，实际是将姿态角度进行补偿，即补偿航向角、纵摇角和偏航角。要将地理坐标系下的位置矢量转换到载体坐标系，需要经过三次旋转。

第一转，$o{x_\text{g}}{y_\text{g}}{z_\text{g}}$坐标系绕$o{z_\text{g}}$轴逆时针旋转偏航角$Y$，得到坐标系$o{x_1}{y_1}{z_1}$，表示为：

$\left[ \begin{gathered} {x_{\rm{1}}} \\ {y_1} \\ {z_1} \\ \end{gathered} \right] = \left[ {\begin{array}{*{20}{c}} {\kern 1pt} {\kern 1pt} {\kern 1pt} \cos Y{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} - {\kern 1pt} {\kern 1pt} {\kern 1pt} \sin Y{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt}0 \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} \sin Y{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \cos Y{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt}{\kern 1pt} {\kern 1pt} {\kern 1pt}{\kern 1pt} {\kern 1pt} {\kern 1pt} 0 \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} 0{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} 0{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt}{\kern 1pt} {\kern 1pt} {\kern 1pt}{\kern 1pt} {\kern 1pt} {\kern 1pt}{\kern 1pt} {\kern 1pt} {\kern 1pt}{\kern 1pt} {\kern 1pt} {\kern 1pt}1 \\ \end{array}} \right ]\left[ \begin{gathered} {x_\text{g}} \\ {y_\text{g}} \\ {z_\text{g}} \\ \end{gathered} \right]$

(19)

第二转，$o{x_1}{y_1}{z_1}$坐标系绕$o{x_1}$轴顺时针旋转纵摇角$P$，得到$o{x_2}{y_2}{z_2}$坐标系。可以表示为

$\left[ \begin{gathered} {x_2} \\ {y_2} \\ {z_2} \\ \end{gathered} \right] = \left[\begin{array}{l} 1{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} 0{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} 0 \\ 0{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \cos P{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} - \sin P \\ {\kern 1pt} 0{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \sin P{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \cos P{\kern 1pt} {\kern 1pt} \\ \end{array} \right ]\left[ \begin{gathered} {x_1} \\ {y_1} \\ {z_1} \\ \end{gathered} \right]$

(20)

第三转，$o{x_2}{y_2}{z_{\rm{2}}}$坐标系绕$o{y_{\rm{2}}}$轴顺时针旋转横摇角$R$，得到$o{x_b}{y_b}{z_b}$。可以表示为：

$\left[ \begin{gathered} {x_b} \\ {y_b} \\ {z_b} \\ \end{gathered} \right] = \left[ \begin{gathered} \cos R{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\rm{ }}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} 0{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \sin R \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} 0{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\rm{ }}{\kern 1pt} {\kern 1pt} 1{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} 0{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \\ - \sin R{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} 0{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \cos R \\ \end{gathered} \right]\left[ \begin{gathered} {x_{\rm{2}}} \\ {y_2} \\ {z_2} \\ \end{gathered} \right]$

(21)

计算得出：

$\left[ \begin{gathered} {x_b} \\ {y_b} \\ {z_b} \\ \end{gathered} \right] = \left[ \begin{gathered} \cos R\cos Y{\rm{ + }}\sin R\sin P\sin Y{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} - \cos R\sin Y{\rm{ + }}\sin R\sin P\cos Y{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \sin R\cos P \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \cos P\sin Y{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \cos P\cos Y{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} - \sin P \\ - \sin R\cos Y{\rm{ + }}\cos R\sin P\sin Y{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \sin Y\sin R{\rm{ + }}\cos R\sin P\cos Y{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \cos R\cos P{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \\ \end{gathered} \right]\left[ \begin{gathered} {x_\text{g}} \\ {y_\text{g}} \\ {z_\text{g}} \\ \end{gathered} \right]$

(22)

根据图 4，可以得到公式(23)：

$\left[ \begin{gathered} {x_b} \\ {y_b} \\ {z_b} \\ \end{gathered} \right] = \left[ \begin{gathered} \cos \theta \sin (\phi ) \\ \cos \theta \cos (\phi ) \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \sin \theta \\ \end{gathered} \right]$

(23)

将公式(23)代入公式(22)中，得到公式(24):

$\left[ \begin{gathered} {x_b} \\ {y_b} \\ {z_b} \\ \end{gathered} \right] = \left[ \begin{gathered} \cos R\cos (\theta 1)\sin (\phi 1 - Y){\rm{ + }}\sin R\sin P\cos (\theta 1)\cos (\phi 1 - Y){\rm{ + }}\sin R\cos P\sin (\theta 1) \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \cos P\cos (\theta 1)\cos (\phi 1 - Y){\rm{ - }}\sin P\sin (\theta 1) \\ \sin R\cos (\theta 1)\sin (\phi 1 - Y){\rm{ + }}\cos R\sin P\cos (\theta 1)\cos (\phi 1 - Y) + \cos R\cos P\sin (\theta 1) \\ \end{gathered} \right]$

(24)

根据公式(24)，可以计算出载体坐标系中的俯仰角($\theta $)和方位角($\phi $)。

$\left\{ \begin{array}{l} \theta = {\rm{arcsin}}\left( {{z_b}} \right) \\ \phi = {\rm{arctan}}\left( {\frac{{{x_b}}}{{{y_b}}}} \right) \\ \end{array} \right.$

(25)

将公式(24)带入公式(25)到如下公式：

$\left\{ \begin{array}{l} \theta = {\rm{arcsin}}\left( {\sin R\cos (\theta 1)\sin (\phi 1 - Y){\rm{ + }}\cos R\sin P\cos (\theta 1)\cos (\phi 1 - Y) + \cos R\cos P\sin (\theta 1)} \right) \\ \phi = {\rm{arctan}}\left( {\frac{{\left( {\cos R\cos (\theta 1)\sin (\phi 1 - Y){\rm{ + }}\sin R\sin P\cos (\theta 1)\cos (\phi 1 - Y){\rm{ + }}\sin R\cos P\sin (\theta 1)} \right)}}{{\cos P\cos (\theta 1)\cos (\phi 1 - Y){\rm{ - }}\sin P\sin (\theta 1)}}} \right) \\ \end{array} \right.$

(26)

在工程应用实践中，偏航角、纵摇角、横滚角可以通过惯性单元测量，俯仰角的定义区间范围一般为[−90°, 90°]，这和反正弦函数主值的区间范围是一致的，所以$\theta $就是俯仰角度的真值。方位角定义区间范围一般为[0°, 360°]，这个区间范围是不同于反正切函数的主值区域[−90°, 90°]的，所以需要跟据${x_b}$和${y_b}$的符号来确定方位角度的真值，其判断方法如表 1所示。

4 仿真实验

文献[15]提出天线的指向精确度需达到0.002°左右时，才能做到精确跟踪，所以对于天线来说，即使有0.001°的误差也可能导致对星角度有偏差。本文以天通一号卫星为选星对象，其经度、纬度及高度分别为101.4°、0°和35 786 000 m。对推导出来的公式进行仿真分析。首先分析地球为标准圆球模型和标准椭球模型时，其俯仰误差随经度、纬度的变化，如图 5、图 6所示。根据图 5、图 6可以看出，标准椭球球体模型与标准圆球模型的误差是逐渐增大的，且基本上会超过0.002°。所以改进的标准的椭球体模型能够提高对星角度的精确度。图 5、图 6验证了椭球体模型能使对星角度更加精确。接着在椭球体模型的基础上，增加天线的海拔高度，来验证修改后公式的正确性。根据图 7，可以看出在高度为1 km时，误差已经达到了0.001°，并且随着高度的增加，误差会逐渐增大，所以将高度加入数学模型中是非常有必要的。

上述验证了在地理坐标系下数学模型中加入地球卯酉圈曲率半径和天线的海拔高度能进一步增加天线指向的对星精确度。但是，并没有考虑载体坐标系。然而在实际中，车辆是有姿态角的，所以需要分析偏航角、纵摇角以及横滚角对方位角和俯仰角的影响。可以通过GPS获取车辆的经纬度信息，通过惯性测量单元获取车辆的姿态角信息，然后计算出车辆在载体坐标系下的方位角和俯仰角。通过具体的数据分析和对比，得出载体坐标系下的方位角、俯仰角与姿态角度之间存在的关系。表 2以河北石家庄为对星地理位置进行实际分析。

根据表 2可以看出偏航角对俯仰角基本没有影响，对方位角有影响。随着偏航角的增大，方位角会减小；纵摇角变化不仅会造成仰角变化，同时也会造成方位角变化。仰角会随着纵摇角的增大而减小，而对方位角变化造成的影响不大；横滚角对俯仰角和方位角都会有影响。其中仰角会随着横滚角的增大而减小，方位角变化的影响会随着横滚角的增大而增大。

5 结论

本文以天通卫星作为指向对星，研究了天线对星角度的问题。改进了原有的地理坐标系下天线指向数学模型，即引入地球卯酉圈曲率半径和天线高度，给出了更加准确的天线对星角度公式，至少能提高0.002°。同时也考虑到实际中不稳定因素的影响，即引入姿态角，给出了载体坐标系下天线对星数学模型，并分析偏航角、纵摇角和横滚角对方位角和俯仰角的影响。经过仿真实验可以看出，改进的数学模型能够提高天线对星性能。