﻿ 低速水面目标航速精度分析及精确解算
 舰船科学技术  2018, Vol. 40 Issue (7): 116-120 PDF

Accuracy analysis and accurate calculation of low-speed surface target speed
MIAO Gao-jie
Jiangsu Automation Research Institute, Lianyungang 222061, China
Abstract: The movement distance of low-speed surface target is small during the radar scan cycle, subject to the measurement accuracy of the radar, the speed error output through the track filter is often large relative to the target speed. For the problem, analyzes the influence of distance、bearing and elevation measurement error on speed accuracy, and proposes a speed accurate calculation method based on two-level strategy. At last, the algorithm is validated by simulation experiment, the results show that the algorithm can effectively improve the speed accuracy.
Key words: low-speed surface target     speed accuracy     mean shift algorithm     speed accurate calculation
0 引　言

1 航速精度分析

 $\left\{ \begin{array}{l}{x_i} = {r_i}\cos {\eta _i}\sin {\theta _i}{\text{，}}\\{y_i} = {r_i}\cos {\eta _i}\cos {\theta _i}{\text{，}}\\{z_i} = {r_i}\sin {\eta _i}{\text{。}}\end{array} \right.$ (1)

 $\left\{ \begin{array}{l}{v_{{x_i}}} = {{\left( {{x_i} - {x_{i - 1}}} \right)} / {{T_i}}}{\text{，}}\\{v_{{y_i}}} = {{\left( {{y_i} - {y_{i - 1}}} \right)} / {{T_i}}}{\text{，}}\\{v_{{z_i}}} = {{\left( {{z_i} - {z_{i - 1}}} \right)} / {{T_i}}}{\text{。}}\end{array} \right.$ (2)

 ${V_i} = \sqrt {v_{{x_i}}^2 + v_{{y_i}}^2 + v_{{z_i}}^2} {\text{。}}$ (3)

 $\left\{ \begin{array}{l}{{\bar r}_i} = {r_i} + \Delta {r_i}{\text{，}}\\{{\bar \theta }_i} = {\theta _i} + \Delta {\theta _i}{\text{，}}\\{{\bar \eta }_i} = {\eta _i} + \Delta {\eta _i}{\text{。}}\end{array} \right.$ (4)

 ${\bar V_i} = \frac{{\sqrt {{{\left( {{{\bar r}_i}\cos {{\bar \eta }_i}\sin {{\bar \theta }_i} - {{\bar r}_{i - 1}}\cos {{\bar \eta }_{i - 1}}\sin {{\bar \theta }_{i - 1}}} \right)}^2} + {{\left( {{{\bar r}_i}\cos {{\bar \eta }_i}\cos {{\bar \theta }_i} - {{\bar r}_{i - 1}}\cos {{\bar \eta }_{i - 1}}\cos {{\bar \theta }_{i - 1}}} \right)}^2} + {{\left( {{{\bar r}_i}\sin {{\bar \eta }_i} - {{\bar r}_{i - 1}}\sin {{\bar \eta }_{i - 1}}} \right)}^2}} }}{{{T_i}}}{\text{，}}$ (5)

 $\Delta {V_i} = {\bar V_i} - {V_i}{\text{。}}$ (6)

${\bar V_i}$ 在目标真实位置参数 ${\left( {{r_i},{\theta _i},{\eta _i},{r_{i - 1}},{\theta _{i - 1}},{\eta _{i - 1}}} \right)^{\rm T}}$ 处一阶泰勒展开，可得

 $\begin{split}{\bar V_i} = {V_i} + \frac{{\partial \bar V}}{{\partial \Delta {r_i}}}\Delta {r_i} + \frac{{\partial \bar V}}{{\partial \Delta {r_{i - 1}}}}\Delta {r_{i - 1}} + \frac{{\partial \bar V}}{{\partial \Delta {\theta _i}}}\Delta {\theta _i} +\\ \frac{{\partial \bar V}}{{\partial \Delta {\theta _{i - 1}}}}\Delta {\theta _{i - 1}} + \frac{{\partial \bar V}}{{\partial \Delta {\eta _i}}}\Delta {\eta _i} + \frac{{\partial \bar V}}{{\partial \Delta {\eta _{i - 1}}}}\Delta {\eta _{i - 1}}\end{split}{\text{，}}$ (7)

 $\frac{{\partial \bar V}}{{\partial \Delta {r_i}}} = \frac{{{v_{{x_i}}}}}{{{V_i}{T_i}}}\cos {\eta _i}\sin {\theta _i} + \frac{{{v_{{y_i}}}}}{{{V_i}{T_i}}}\cos {\eta _i}\cos {\theta _i} + \frac{{{v_{{z_i}}}}}{{{V_i}{T_i}}}\sin {\eta _i}{\text{，}}$
 $\begin{split}\frac{{\partial \bar V}}{{\partial \Delta {r_{i - 1}}}} = - \frac{{{v_{{x_i}}}}}{{{V_i}{T_i}}}\cos {\eta _{i - 1}}\sin {\theta _{i - 1}}-\\\frac{{{v_{{y_i}}}}}{{{V_i}{T_i}}}\cos {\eta _{i - 1}}\cos {\theta _{i - 1}} - \frac{{{v_{{z_i}}}}}{{{V_i}{T_i}}}\sin {\eta _{i - 1}}\end{split}{\text{，}}$
 $\frac{{\partial \bar V}}{{\partial \Delta {\theta _i}}} = \frac{{{v_{{x_i}}}}}{{{V_i}{T_i}}}{y_i} - \frac{{{v_{{y_i}}}}}{{{V_i}{T_i}}}{x_i}{\text{，}}$
 $\frac{{\partial \bar V}}{{\partial \Delta {\theta _{i - 1}}}} = \frac{{{v_{{x_i}}}}}{{{V_i}{T_i}}}{y_{i - 1}} - \frac{{{v_{{y_i}}}}}{{{V_i}{T_i}}}{x_{i - 1}}{\text{，}}$
 $\begin{split}\frac{{\partial \bar V}}{{\partial \Delta {\eta _i}}} = - \frac{{{v_{{x_i}}}}}{{{V_i}{T_i}}}{r_i}\sin {\eta _i}\sin {\theta _i} -\\\frac{{{v_{{y_i}}}}}{{{V_i}{T_i}}}{r_i}\sin {\eta _i}\cos {\theta _i} + \frac{{{v_{{z_i}}}}}{{{V_i}{T_i}}}{r_i}\cos {\eta _i}\end{split}{\text{，}}$
 $\begin{split}\frac{{\partial \bar V}}{{\partial \Delta {\eta _{i - 1}}}} = \frac{{{v_{{x_i}}}}}{{{V_i}{T_i}}}{r_{i - 1}}\sin {\eta _{i - 1}}\sin {\theta _i} +\\\frac{{{v_{{y_i}}}}}{{{V_i}{T_i}}}{r_{i - 1}}\sin {\eta _{i - 1}}\cos {\theta _{i - 1}} - \frac{{{v_{{z_i}}}}}{{{V_i}{T_i}}}{r_{i - 1}}\cos {\eta _{i - 1}}{\text{，}}\end{split}$

 $\begin{split}\Delta {V_i} = \frac{{\partial \bar V}}{{\partial \Delta {r_i}}}\Delta {r_i} + \frac{{\partial \bar V}}{{\partial \Delta {r_{i - 1}}}}\Delta {r_{i - 1}} + \frac{{\partial \bar V}}{{\partial \Delta {\theta _i}}}\Delta {\theta _i} + \\\frac{{\partial \bar V}}{{\partial \Delta {\theta _{i - 1}}}}\Delta {\theta _{i - 1}} + \frac{{\partial \bar V}}{{\partial \Delta {\eta _i}}}\Delta {\eta _i} + \frac{{\partial \bar V}}{{\partial \Delta {\eta _{i - 1}}}}\Delta {\eta _{i - 1}}\end{split}{\text{。}}$ (8)

1.1 距离误差影响

 $d\left( {\Delta {r_i}} \right) = \Delta {r_i} - \Delta {r_{i - 1}}{\text{，}}$ (9)

 $\begin{split}\!\!\Delta {V_{ri}} =& \left[ {\frac{{{v_{{x_i}}}}}{{{V_i}{T_i}}}\left( {\cos {\eta _i}\sin {\theta _i} - \cos {\eta _{i - 1}}\sin {\theta _{i - 1}}} \right) + } \right.\\&\left[ {\frac{{{v_{{y_i}}}}}{{{V_i}{T_i}}}\left( {\cos {\eta _i}\cos {\theta _i} - \cos {\eta _{i - 1}}\cos {\theta _{i - 1}}} \right) + } \right.\\&\left. {\frac{{{v_{{z_i}}}}}{{{V_i}{T_i}}}\left( {\sin {\eta _i} - \sin {\eta _{i - 1}}} \right)} \right]\Delta {r_i} + \left( {\frac{{{v_{{x_i}}}}}{{{V_i}{T_i}}}\cos {\eta _{i - 1}}\sin {\theta _{i - 1}} + } \right.\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\\&\left. {\frac{{{v_{{y_i}}}}}{{{V_i}{T_i}}}\cos {\eta _{i - 1}}\cos {\theta _{i - 1}} + \frac{{{v_{{z_i}}}}}{{{V_i}{T_i}}}\sin {\eta _{i - 1}}} \right)d\left( {\Delta {r_i}} \right){\text{。}}\!\!\!\!\!\!\end{split}$ (10)

 $\begin{split}\Delta {V_{ri}} \approx & \left( {\frac{{{v_{{x_i}}}}}{{{V_i}}}\cos {\eta _{i - 1}}\sin {\theta _{i - 1}} + \frac{{{v_{{y_i}}}}}{{{V_i}}}\cos {\eta _{i - 1}}\cos {\theta _{i - 1}} + } \right.\\&\left. {\frac{{{v_{{z_i}}}}}{{{V_i}}}\sin {\eta _{i - 1}}} \right)\frac{{{\rm d}\left( {\Delta {r_i}} \right)}}{{{T_i}}}{\text{。}}\end{split}$ (11)

 $\Delta {V_{ri}} = \cos \phi \frac{{{\rm ds}\left( {\Delta {r_i}} \right)}}{{{T_i}}}{\text{，}}$ (12)

1.2 方位误差影响

 $d\left( {\Delta {\theta _i}} \right) = \Delta {\theta _i} - \Delta {\theta _{i - 1}},$ (13)

 $\begin{split}\Delta {V_{\theta i}} = &\left( {\frac{{{v_{{x_i}}}{v_{{y_i}}}}}{{{V_i}}} - \frac{{{v_{{y_i}}}{v_{{x_i}}}}}{{{V_i}}}} \right)\Delta {\theta _i} + \left( {\frac{{{v_{{x_i}}}{y_{i - 1}}}}{{{V_i}{T_i}}} - \frac{{{v_{{y_i}}}{x_{i - 1}}}}{{{V_i}{T_i}}}} \right)d\left( {\Delta {\theta _i}} \right) =\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \\&\left( {\frac{{{v_{{x_i}}}{y_{i - 1}}}}{{{V_i}}} - \frac{{{v_{{y_i}}}{x_{i - 1}}}}{{{V_i}}}} \right)\frac{{d\left( {\Delta {\theta _i}} \right)}}{{{T_i}}}{\text{。}}\end{split}$ (14)

 $\Delta {V_{\theta i}} = {r_{i - 1}}\left( {\cos \phi ' - \frac{{{v_{zi}}{z_{i - 1}}}}{{{r_{i - 1}}}}} \right)\frac{{d\left( {\Delta {\theta _i}} \right)}}{{{T_i}}}{\text{，}}$ (15)

 $\Delta {V_{\theta i}} = {r_{i - 1}}\cos \phi '\frac{{d\left( {\Delta {\theta _i}} \right)}}{{{T_i}}}{\text{。}}$ (16)

1.3 俯仰误差影响

 $d\left( {\Delta {\eta _i}} \right) = \Delta {\eta _i} - \Delta {\eta _{i - 1}}{\text{，}}$ (17)

 $\!\!\!\!\!\!\!\!\begin{split}& \Delta {V_{\eta i}} = \left[ { - \frac{{{v_{{x_i}}}}}{{{V_i}{T_i}}}\left( {{r_i}\sin {\eta _i}\sin {\theta _i} - {r_{i - 1}}\sin {\eta _{i - 1}}\sin {\theta _{i - 1}}} \right) - } \right.\\& \frac{{{v_{{y_i}}}}}{{{V_i}{T_i}}}\left( {{r_i}\sin {\eta _i}\cos {\theta _i} - {r_{i - 1}}\sin {\eta _{i - 1}}\cos {\theta _{i - 1}}} \right) + \\& \frac{{{v_{{z_i}}}}}{{{V_i}{T_i}}}\left. {\left( {{r_i}\cos {\eta _i} \!-\! {r_{i - 1}}\cos {\eta _{i - 1}}} \right)} \right]\Delta {\eta _i} \!+\! \left( { \!-\! \frac{{{v_{{x_i}}}}}{{{V_i}{T_i}}}{r_{i - 1}}\sin {\eta _{i - 1}}\sin {\theta _i} - } \right. \\& \left. {\frac{{{v_{{y_i}}}}}{{{V_i}{T_i}}}{r_{i - 1}}\sin {\eta _{i - 1}}\cos {\theta _{i - 1}} + \frac{{{v_{{z_i}}}}}{{{V_i}{T_i}}}{r_{i - 1}}\cos {\eta _{i - 1}}} \right)d\left( {\Delta {\eta _i}} \right){\text{。}}(18)\end{split}\!\!\!\!\!\!\!\!\!\!\!\!\!$

 $\begin{split}\Delta {V_{\eta i}} \approx & {r_{i - 1}}\left( { - \frac{{{v_{{x_i}}}}}{{{V_i}}}\sin {\eta _{i - 1}}\sin {\theta _i} - \frac{{{v_{{y_i}}}}}{{{V_i}}}\sin {\eta _{i - 1}}\cos {\theta _{i - 1}} + } \right.\\&\left. {\frac{{{v_{{z_i}}}}}{{{V_i}_i}}\cos {\eta _{i - 1}}} \right)\frac{{d\left( {\Delta {\eta _i}} \right)}}{{{T_i}}}{\text{。}}\end{split}$ (19)

 $\Delta {V_{\eta i}} \approx {r_{i - 1}}\cos \phi ''\frac{{d\left( {\Delta {\eta _i}} \right)}}{{{T_i}}}{\text{，}}$ (20)

2 航速精确解算

 图 1 周期大小对直线近似的影响 Fig. 1 The effect of the periodic size on the linear approximation

 图 2 基于两级策略的航速精确解算示意图 Fig. 2 Accurate calculation of speed based on two-level strategy

① 初始化 $k = 1$ $\varepsilon = 0.01$ ${\tilde v_k} = {v_m}$

②计算核函数

 ${G_{hi}} = \sum\limits_{i = 1}^m {\exp \left[ { - {{\left( {\frac{{{v_i} - {{\tilde v}_k}}}{h}} \right)}^2}} \right]} \begin{array}{*{20}{c}} ,&{} \end{array}i = 1, \cdots ,m{\text{。}}$ (21)

③计算mean shift值

 ${\tilde v_{k + 1}} = \frac{{\sum\limits_{i = 1}^m {{G_{hi}}{w_i}} {v_i}}}{{\sum\limits_{i = 1}^m {{G_{hi}}{w_i}} }}{\text{，}}$ (22)

④判断 $\left| {{{\tilde v}_{k + 1}} - {{\tilde v}_k}} \right| < \varepsilon$ 是否成立，如果不成立，置 $k = k + 1$ 后转②；如果成立，退出。

${\tilde v_{k + 1}}$ 即目标精确解算航速。

3 仿真验证

 图 3 目标1精算前后航速对比结果 Fig. 3 Speed comparison result before and after the accurate calculation of target 1

 图 4 目标2精算前后航速对比结果 Fig. 4 Speed comparison result before and after the accurate calculation of target 2

 图 5 目标3精算前后航速对比结果 Fig. 5 Speed comparison result before and after the accurate calculation of target 3

4 结　语

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