﻿ 舰载雷达两轴转台稳定方程推导
 舰船科学技术  2017, Vol. 39 Issue (10): 127-131 PDF

Steady-state equation derivation for the two-shaft -stabilized platform of the ship-borne radar
LEI Jie, WU Yan, FENG Yuan-yuan
The 724 Research Institute of CSIC, Nanjing 211153, China
Abstract: The mechanical structure for the two-shaft-stabilized platform of the ship-borne radar is simple, which keep the direction of radar beam by controlling the attitude angles of the antenna normal, but the radar beam rotates around the antenna normal when ship swung. The relationship between geodetic coordinate system and ship coordinate system is analyzed, the two -shaft-stabilized system’s attitude angles of the ship-borne radar in ship coordinate system are defined, when the value of platform compass and the ship-borne radar antenna normal’s attitude angles are given, this paper provides a way to the two-shaft -stabilized system’s steady-state equations and the radar beam’s rotation angle equation by use of space analytic geometry method, which provides a theoretical basis for the ship-borne radar serve control and correlative compensation method.
Key words: two- shaft -stabilized system     space analytic geometry method     attitude angle     steady-state equation     rotation angle equation
0 引 言

 图 1 舰载雷达两轴转台系统示意图 Fig. 1 The sketch figure of the two-shaft-stabilized platform of the ship-borne radar
1 坐标系及姿态角定义

 图 2 大地直角坐标系舰船姿态角 Fig. 2 The attitude angles of ship in the geodetic coordinate system

H为航向角，正北线OX顺时针到线OXc （舰首线OM在水平面投影线）的角度，水平面内测量。

P为纵摇角，舰首线OM与水平面的夹角，在过舰首线OM的铅垂面内测量，舰首线在水平面上方（舰首抬高）为正。

R为横摇角，甲板面（面MON）绕舰首线OM旋转的角度，即舰船左舷线ON和线OYc（过左舷线的舰船横剖面与水平面交线）的夹角；其测量平面（横摇面）垂直于甲板面且垂直于舰首线（即OM⊥横摇面NOYc），左舷线在水平面上方（左舷抬高）为正。

Z轴在过舰首线OM的铅垂面内，OMOYcOZOYc，可知OYc⊥铅垂面MOZOYcOXcO-XcYcZ为直角坐标系，即舰船直角坐标系（见图3）。

 图 3 舰船直角坐标系雷达两轴转台系统姿态角 Fig. 3 The attitude angles of the two-shaft-stabilized platform in ship coordinate system

Ac 为天线甲板面舷角，天线转台绕舷角轴顺时针运动的角度，即舰首线OM顺时针到舷角线OJ（天线法线OG在甲板面MON内投影线）的角度，甲板面内测量。

Ec 为天线甲板面俯仰角，天线绕俯仰轴运动的角度，即天线法线OG与舷角线OJ的夹角，在平面GOJ（过天线法线且垂直于甲板面的平面）内测量，夹角在甲板面上方为正（因雷达机械结构限制，该角度一般不超过±30°）。

A为天线方位角，线OXc顺时针到方位线OK（天线法线OG在水平面投影线）的角度，水平面内测量。

E 为天线俯仰角，天线法线OG与方位线OK的夹角，在平面GOK（过天线法线的铅垂面）内测量，夹角在水平面上方为正。

φ 为波束旋转角，平面GOK与平面GOJ的夹角（锐角）。沿天线法线方向看（点OG），面GOK绕天线法线OG逆时针到面GOJ的角度φ为正，顺时针为负。

A 为波束旋转方位角，线OXc顺时针到旋转方位线OS（平面GOJ与水平面的交线，取该交线与天线法线OG夹角为锐角的部分）的角度，水平面内测量；At 为雷达目标真方位角，由大地直角坐标系和舰船直角坐标系的关系可知，雷达目标真方位值即为航向角H与天线方位角A值之和。

2 稳定方程推导

2.1 甲板面方程

 ${\rm{sin}}P{{x}} + {\rm{tan}}R{{y}} - {\rm{cos}}P{{z}} = 0 \text{。}$ (1)
 图 4 N点空间坐标 Fig. 4 The space coordinates of point N

 图 5 G点空间坐标 Fig. 5 The space coordinates of point G
2.2 过天线法线垂直于甲板面的平面方程

 ${{{m}}_1}{{x}} + {{{n}}_1}{{y}} + {{{k}}_1}{{z}} = 0\text{，}$ (2)

 $\left\{ {\begin{array}{*{20}{l}}\!\!\!\! {{{{m}}_1} = \left( {\displaystyle\frac{{\cos {{P}}}}{{\sin {{P}}}} - \displaystyle\frac{{\tan {{R}}}}{{\sin {{P}}}}\displaystyle\frac{{\cos {{A}}\cos {{E}}\cos {{P}} + \sin {{E}}\sin {{P}}}}{{\cos {{A}}\cos {{E}}\tan {{R}} + \sin {{A}}\cos {{E}}\sin {{P}}}}} \right){{{k}}_1}}\text{，}\\\!\!\!\! {{{{n}}_1} = \left( {\displaystyle\frac{{\cos {{A}}\cos {{E}}\cos {{P}} + \sin {{E}}\sin {{P}}}}{{\cos {{A}}\cos {{E}}\tan {{R}} + \sin {{A}}\cos {{E}}\sin {{P}}}}} \right){{{k}}_1}}\text{，}\\\!\!\!\! {{{{k}}_1} = 1}\text{。}\end{array}} \right.$
2.3 舷角线向量

 \left\{ {\begin{aligned}& {\sin {{P}}{{{m}}_2} + \tan {{R}}{{{n}}_2} - \cos {{P}}{{{k}}_2} = 0}\text{，}\\& {{{{m}}_1}{{{m}}_2} + {{{n}}_1}{{{n}}_2} + {{{k}}_1}{{{k}}_2} = 0}\text{。}\end{aligned}} \right.

 \left\{ {\begin{aligned}& {{{{m}}_2} = \left( { - \displaystyle\frac{{{{{k}}_1}}}{{{{{m}}_1}}} - \displaystyle\frac{{{{{n}}_1}}}{{{{{m}}_1}}}\displaystyle\frac{{\cos {{P}} + \displaystyle\frac{{{{{k}}_1}}}{{{{{m}}_1}}}\sin {{P}}}}{{\tan {{R}} - \displaystyle\frac{{{{{n}}_1}}}{{{{{m}}_1}}}\sin {{P}}}}} \right){{{k}}_2}}\text{，}\\& {{{{n}}_2} = \displaystyle\frac{{\cos {{P}} + \displaystyle\frac{{{{{k}}_1}}}{{{{{m}}_1}}}\sin {{P}}}}{{\tan {{R}} - \displaystyle\frac{{{{{n}}_1}}}{{{{{m}}_1}}}\sin {{P}}}}{{{k}}_2}}\text{。}\end{aligned}} \right.

 $\cos {{\alpha }} = \frac{{{{{m}}_2}\cos {{A}}\cos {{E}} - {{{n}}_2}\sin {{A}}\cos {{E}} + {{{k}}_2}\sin {{E}}}}{{\sqrt {{{m}}_2^2 + {{n}}_2^2 + {{k}}_2^2} }}\text{。}$

2.4 天线甲板面舷角Ac和俯仰角Ec

 $\cos {{\beta }} = {{{m}}_4}\cos {{P}} + {{{k}}_4}\sin {{P}}\text{，}$ (3)

n4 $\leqslant$ 0时， ${{Ac}} = {{\beta }} = {\cos ^{ - 1}}\left( {{{{m}}_4}\cos {{P}} + {{{k}}_4}\sin {{P}}} \right)$ ，当n4>0时， ${{Ac}} = 2{{π }} - {{\beta }} = 2{{π }} - {\cos ^{ - 1}}\left( {{{{m}}_4}\cos {{P}} + {{{k}}_4}\sin {{P}}} \right)$

 $\cos {{\gamma }} = {{{m}}_4}\cos {{A}}\cos {{E}} - {{{n}}_4}\sin {{A}}\cos {{E}} + {{{k}}_4}\sin {{E}}\text{。}$ (4)

k4 $\leqslant$ sinE时， ${{Ec}} = \gamma = {\cos ^{ - 1}}({{{m}}_4}\cos {{A}}\cos {{E}} -$ ${{{n}}_4}\sin {{A}}\cos {{E}} + {{{k}}_4}\sin {{E}})$ ，当k4>sinE时， ${{Ec}} = {{\gamma }} =$ $- {\cos ^{ - 1}}\left( {{{{m}}_4}\cos {{A}}\cos {{E}} - {{{n}}_4}\sin {{A}}\cos {{E}} + {{{k}}_4}\sin {{E}}} \right)$

2.5 波束旋转方位角A

 $\left\{ {\begin{array}{*{20}{l}}\!\!\! {z = 0}\text{，}\\\!\!\! {{{{m}}_1}x + {{{n}}_1}y + {{{k}}_1}z = 0}\text{。}\end{array}} \right.$

 $\left\{ {\begin{array}{*{20}{l}}\!\!\! {\displaystyle\frac{{{x}}}{{{{{n}}_1}}} = \displaystyle\frac{{{y}}}{{ - {{{m}}_1}}}}\text{，}\\\!\!\! {z = 0}\text{。}\end{array}} \right.$

 $\cos {{\omega }} = \frac{{{{{n}}_1}\cos {{A}}\cos {{E}} + {{{m}}_1}\sin {{A}}\cos {{E}}}}{{\sqrt {{{n}}_1^2 + {{m}}_1^2} }}\text{。}$

$\left( {{{{m}}_5}{\rm{,}}{{{n}}_5}{\rm{,}}0} \right)$ 单位化后得旋转方位线OS方向向量，记为 $\left( {{{{m}}_6},{{{n}}_6},0} \right)$

 $\cos {{\alpha }} = \frac{{{{{m}}_6}}}{{\sqrt {{{m}}_6^2 + {{n}}_6^2} }}\text{。}$

m6 $\leqslant$ 0时， ${{{A}}'} = {{\alpha }} = {\cos ^{ - 1}}\left( {\displaystyle\frac{{{{{m}}_6}}}{{\sqrt {{{m}}_6^2 + {{n}}_6^2} }}} \right)$ ，当m6>0时， ${{{A}}^{\rm{'}}} = 2{{π }} - {{\alpha }} = 2{{π }} - {\cos ^{ - 1}}\left( {\displaystyle\frac{{{{{m}}_6}}}{{\sqrt {{{m}}_6^2 + {{n}}_6^2} }}} \right)$

2.6 波束旋转角φ

$\sin {{A}}\cos {{Ex}} + \cos {{A}}\cos {{Ey}} = 0$ （过z轴）。

 ${{{m}}_1}{{x}} + {{{n}}_1}{{y}} + {{{k}}_1}{{z}} = 0\text{，}$

 $\cos {{\varphi }} = \frac{{\left| {{{{m}}_1}\sin {{A}}\cos {{E}} + {{{n}}_1}\cos {{A}}\cos {{E}}} \right|}}{{\sqrt {{{\left( {\sin {{A}}\cos {{E}}} \right)}^2} + {{\left( {\cos {{A}}\cos {{E}}} \right)}^2}} \sqrt {{{m}}_1^2 + {{n}}_1^2 + {{k}}_1^2} }}\text{。}$

$\sin {{E}} > 0,\left| {{{A'}} - {{A}}} \right|\left\langle {180,{{A'}}} \right\rangle A$ 时：

 ${{\varphi }} \!=\! {\cos ^{ - 1}}\left( {\displaystyle\frac{{\left| {{{{m}}_1}\sin {{A}}\cos {{E}} + {{{n}}_1}\cos {{A}}\cos {{E}}} \right|}}{{\sqrt {{{\left( {\sin {{A}}\cos {{E}}} \right)}^2} + {{\left( {\cos {{A}}\cos {{E}}} \right)}^2}} \sqrt {{{m}}_1^2 + {{n}}_1^2 + {{k}}_1^2} }}} \right)\text{，}$

$\sin {{E}} > 0,\left| {{{A'}} - {{A}}} \right| < 180,{{A'}} < A$ 时：

 ${{\varphi }} \!=\!- {\cos ^{ - 1}}\left( {\displaystyle\frac{{\left| {{{{m}}_1}\sin {{A}}\cos {{E}} + {{{n}}_1}\cos {{A}}\cos {{E}}} \right|}}{{\sqrt {{{\left( {\sin {{A}}\cos {{E}}} \right)}^2} + {{\left( {\cos {{A}}\cos {{E}}} \right)}^2}} \sqrt {{{m}}_1^2 + {{n}}_1^2 + {{k}}_1^2} }}} \right)\text{，}$

$\sin {{E}} > 0,\left| {{{A'}} - {{A}}} \right| > 180,{{A'}} > A$ 时：

 ${{\varphi }} = - {\cos ^{ - 1}}\left( {\displaystyle\frac{{\left| {{{{m}}_1}\sin {{A}}\cos {{E}} + {{{n}}_1}\cos {{A}}\cos {{E}}} \right|}}{{\sqrt {{{\left( {\sin {{A}}\cos {{E}}} \right)}^2} + {{\left( {\cos {{A}}\cos {{E}}} \right)}^2}} \sqrt {{{m}}_1^2 + {{n}}_1^2 + {{k}}_1^2} }}} \right)\text{，}$

$\sin {{E}} > 0,\left| {{{A'}} - {{A}}} \right| > 180,{{A'}} < A$ 时：

 ${{\varphi }} = {\cos ^{ - 1}}\left( {\displaystyle\frac{{\left| {{{{m}}_1}\sin {{A}}\cos {{E}} + {{{n}}_1}\cos {{A}}\cos {{E}}} \right|}}{{\sqrt {{{\left( {\sin {{A}}\cos {{E}}} \right)}^2} + {{\left( {\cos {{A}}\cos {{E}}} \right)}^2}} \sqrt {{{m}}_1^2 + {{n}}_1^2 + {{k}}_1^2} }}} \right);$

$\sin {{E}} < 0,\left| {{{A'}} - {{A}}} \right|\left\langle {180,{{A'}}} \right\rangle A$ 时：

 ${{\varphi }} = - {\cos ^{ - 1}}\left( {\displaystyle\frac{{\left| {{{{m}}_1}\sin {{A}}\cos {{E}} + {{{n}}_1}\cos {{A}}\cos {{E}}} \right|}}{{\sqrt {{{\left( {\sin {{A}}\cos {{E}}} \right)}^2} + {{\left( {\cos {{A}}\cos {{E}}} \right)}^2}} \sqrt {{{m}}_1^2 + {{n}}_1^2 + {{k}}_1^2} }}} \right)\text{，}$

$\sin {{E}} < 0,\left| {{{A'}} - {{A}}} \right| < 180,{{A'}} < A$ 时：

 ${{\varphi }} = {\cos ^{ - 1}}\left( {\displaystyle\frac{{\left| {{{{m}}_1}\sin {{A}}\cos {{E}} + {{{n}}_1}\cos {{A}}\cos {{E}}} \right|}}{{\sqrt {{{\left( {\sin {{A}}\cos {{E}}} \right)}^2} + {{\left( {\cos {{A}}\cos {{E}}} \right)}^2}} \sqrt {{{m}}_1^2 + {{n}}_1^2 + {{k}}_1^2} }}} \right)\text{，}$

$\sin {{E}}\left\langle {0,\left| {{{A'}} - {{A}}} \right|} \right\rangle 180,{{A'}} > A$ 时：

 ${{\varphi }} = {\cos ^{ - 1}}\left( {\displaystyle\frac{{\left| {{{{m}}_1}\sin {{A}}\cos {{E}} + {{{n}}_1}\cos {{A}}\cos {{E}}} \right|}}{{\sqrt {{{\left( {\sin {{A}}\cos {{E}}} \right)}^2} + {{\left( {\cos {{A}}\cos {{E}}} \right)}^2}} \sqrt {{{m}}_1^2 + {{n}}_1^2 + {{k}}_1^2} }}} \right)\text{，}$

$\sin {{E}}\left\langle {0,\left| {{{A'}} - {{A}}} \right|} \right\rangle 180,{{A'}} < A$ 时：

 ${{\varphi }} = - {\cos ^{ - 1}}\left( {\displaystyle\frac{{\left| {{{{m}}_1}\sin {{A}}\cos {{E}} + {{{n}}_1}\cos {{A}}\cos {{E}}} \right|}}{{\sqrt {{{\left( {\sin {{A}}\cos {{E}}} \right)}^2} + {{\left( {\cos {{A}}\cos {{E}}} \right)}^2}} \sqrt {{{m}}_1^2 + {{n}}_1^2 + {{k}}_1^2} }}} \right);$

${{A'}} = A$ 时， ${{\varphi }} = 0$

3 数据验证

 图 6 两轴转台系统数据 Fig. 6 Data of the two-shaft-stabilized system
4 结 语