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目标、观测站、坐标系共同构成一个观测系统。观察站采集目标数据,经过计算将数据映射到坐标系内形成位置或轨迹,实现定位。若采用两个观测站,且使用的探测器是CCD成像设备,则构成基于CCD的双站定位系统。
三维空间坐标定位模型适合于对运动目标的定位特性进行动态分析,比较复杂。平面直角坐标系定位模型相对简单,便于分析静态目标的定位特性。另外,由于动态目标在每一特定时刻的位置点是确定的,因此对其进行定位分析时,通常是选取某一时刻,确定目标位置,然后采用平面直角坐标模型进行分析和评价[10]。文中的测量对象是静止目标,为便于分析,建立平面直角坐标模型。
如图1所示,两个观测站点的间距为D,站点之间的连线为观测基线,在站点采用可见光成像设备CCD1、CCD2同时对静止目标进行观测,观测角度分别为
$ \alpha $ 、$ \;\beta $ 。假定CCD1位置坐标为(0,0),CCD2坐标为(D,0),目标出现位置为(x,y),有如下关系:
$$ \left\{\begin{array}{c}\tan \alpha =\dfrac{y}{x}\\ \tan \beta =\dfrac{y}{x-D}\end{array}\right. $$ (1) 转化为矩阵形式表示:
$$ \left[\begin{array}{cc}\tan \alpha & -1\\ \tan \beta & -1\end{array}\right]\left[\begin{array}{c}x\\ y\end{array}\right]=\left[\begin{array}{c}0\\ D \tan \beta \end{array}\right] $$ (2) 其中,角度测量过程为:目标通过镜头在CCD靶面成像,经信号处理后在监视器上显示,人眼从监视器上观察目标图像,并操作转台实现目标瞄准,测角设备自动进行角度测量。由于设备精度及人为因素,每一个角度观测值不可避免存在误差,最终导致目标坐标误差。为提高测算精度,采用最小二乘原理求公式(2)的最小均方误差解,以实现对目标的准确定位。该线性方程的最小均方误差解为:
$$ \left[\begin{array}{c}x\\ y\end{array} \right]={\left({\left[\begin{array}{cc}\tan\alpha -1\\ \tan\beta -1\end{array}\right]}^{{\rm{T}}}\left[\begin{array}{cc}\tan\alpha -1\\ \tan\beta -1\end{array}\right]\right)}^{-1}\cdot \\ {\left[\begin{array}{cc}\tan\alpha -1\\ \tan\beta -1\end{array}\right]}^{{\rm{T}}}\left[\begin{array}{c}0\\ D\tan\beta \end{array} \right] $$ (3) -
图2为人眼借助CCD成像设备观察目标的示意图,设目标尺寸为
$ L $ ,目标距离为$ R $ ,镜头焦距为$ f $ ,当$ R\gg f $ 时,像距接近于$ f $ 。CCD传感器靶面尺寸(对角线长度)为$ {S}_{p} $ ,监视器的对角线长度为$ {S}_{c} $ ,设$ \theta $ 为目标图像相对人眼的张角,则有:$$ \theta =2 \arctan\frac{gLf{S}_{c}}{2R{{S}_{o}S}_{p}} $$ (4) 式中:
$ g $ 为弧分换算系数;${S}_{o}$ 为人眼的明视距离,一般取250 mm。目标处于特定的背景中,由于反射特性、空间位置、表面材料等物理因素,导致目标与背景存在色彩和亮度对比[11]。其中色彩对比在近距离观察时影响较大,但在战场侦察中一般距离较远,亮度对比度具有决定性作用。
目标和背景的亮度对比为[4]:
$$ {K}_{l}=\frac{\left|{L}_{b}-{L}_{o}\right|}{\mathrm{m}\mathrm{a}\mathrm{x}({L}_{b},{L}_{o})} $$ (5) 式中:
$ {L}_{b} $ 为背景亮度;$ {L}_{o} $ 为目标亮度。当目标距离较远时,目标与背景的空间位置近似一致,$ {K}_{l} $ 可直接以目标和背景的亮度系数$ {r}_{b} $ 、$ {r}_{o} $ 表示,即:$$ {K}_{l}=\frac{\left|{r}_{b}-{r}_{o}\right|}{\mathrm{m}\mathrm{a}\mathrm{x}({r}_{b},{r}_{o})} $$ (6) 由于大气衰减作用,目标与背景亮度将被降低,由朗伯-比耳定律,影响传输的主要是消光系数,大气透过率可表示为:
$$ \tau ={{\rm{e}}}^{-aR} $$ (7) 式中:R为水平方向大气层厚度(近似为观察距离);
$ a $ 为水平方向单位大气层消光系数。此外,大气中由于太阳辐射所产生的散射光形成了一定的气幕亮度将增大目标与背景的亮度,气幕亮度可表示为:
$$ {L}_{a}={L}_{h}(1-{{\rm{e}}}^{-aR}) $$ (8) 式中:
$ {L}_{h} $ 为空气层无穷厚时的气幕亮度,一般取天边靠近地平线处的天空亮度。故目标与背景的亮度经过一段距离R传输到CCD成像设备时,实际镜头处的视觉亮度分别为:
$$ \left\{\begin{array}{c}{L}_{o}^{\text{'}}={L}_{o}{{\rm{e}}}^{-aR}+{L}_{h}(1-{{\rm{{e}}}}^{-aR})\\ {L}_{b}^{\text{'}}={L}_{b}{{\rm{e}}}^{-aR}+{L}_{h}(1-{{\rm{e}}}^{-aR})\end{array}\right. $$ (9) 由公式(5)可知,目标与背景的视觉对比度为:
$$ {K}_{ls}=\dfrac{\left|{L}_{b}^{\text{'}}-\right.\left.{L}_{o}^{\text{'}}\right|}{\mathrm{m}\mathrm{a}\mathrm{x}({L}_{b}^{\text{'}},{L}_{o}^{\text{'}})}=\dfrac{{K}_{l}}{1+\dfrac{{L}_{h}({{\rm{e}}}^{aR}-1)}{\mathrm{m}\mathrm{a}\mathrm{x}({L}_{b},{L}_{o})}}=\dfrac{{K}_{l}}{1+\dfrac{{r}_{h}({{\rm{e}}}^{aR}-1)}{\mathrm{m}\mathrm{a}\mathrm{x}({r}_{b},{r}_{o})}} $$ (10) 式中:
$ {r}_{h} $ 为水平方向的天空亮度系数。CCD成像设备性能也会导致视觉对比度的变化,但在正常情况下影响较小,故不考虑。
当人眼通过CCD设备侦察时,目标最终能否被探测到还与人眼恰好能发现目标的最小亮度对比值有关,称为亮度对比阈值
$ \varepsilon $ 。实验表明:$ \varepsilon $ 是一个呈正态$ N(\mu ,{\sigma }^{2}) $ 分布的随机变量,其均值$ \;\mu $ 与视角$ \theta $ 的关系为:$$ \mu =\left\{\begin{array}{l}0.05,\;\;\;\;\;\theta \geqslant {30}^{\mathrm{\text{'}}}\\ 0.812{\theta }^{-0.819},\;\theta <{30}^{\text{'}}\end{array}\right. $$ (11) 由
$ {K}_{ls} $ 及$ N(\mu ,{\sigma }^{2}) $ 可求得某一观察距离$ R $ 上目标的探测概率为:$$ \begin{split} P\left(D\right)=&P\left({K}_{ls}\ge \varepsilon \right)={\int }_{-\infty }^{{K}_{ls}}\frac{1}{\sqrt{2\pi }\sigma }{{\rm{e}}}^{-\frac{{\left(\varepsilon -\mu \right)}^{2}}{2{\sigma }^{2}}}{\rm{d}}\varepsilon =\\ &{\int }_{-\infty }^{\frac{{K}_{ls}-\mu }{\sigma }}\frac{1}{\sqrt{2\pi }}{{\rm{e}}}^{-\frac{{t}^{2}}{2}}{\rm{d}}t=\varnothing \left(\frac{{K}_{ls}-\mu }{\sigma }\right) \end{split} $$ (12) 其中
$$ \varnothing \left(\frac{{K}_{ls}-\mu }{\sigma }\right)={\int }_{-\infty }^{x}\frac{1}{\sqrt{2\pi }}{{\rm{e}}}^{-\frac{{t}^{2}}{2}}{\rm{d}}t $$ (13) 根据约翰逊准则,标准差
$ \sigma =0.387\;6 $ μ。 -
由前节分析可知,双站间距数值D决定了定位范围。在该范围内,随机抽取n个目标点进行定位精度分析。
在实际应用中,角度测量是借助CCD成像系统及测角设备实现,由人工完成对目标的侦察与瞄准,由测角设备自动完成目标的角度测量和数据传输。影响角度测量误差的因素主要有镜头光学分辨率、CCD成像角度分辨率、测角设备的角度测量精度[13-14]。其中镜头光学分辨率一般较高,可不考虑。CCD角度分辨率为:
$$ \mathrm{\omega }=2\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{t}\mathrm{a}\mathrm{n}(∆S/2f) $$ (14) 式中:
$ ∆S $ 为CCD像元尺寸;$ f $ 为镜头焦距。根据成像设备的实际参数,可计算出CCD1、CCD2的角度分辨率分别为$ {6.2} $ ″、$ {3.6} $ ″。测角设备的角度误差为00-01密位(≈$ {216} $ ″)。因此,测角设备的精度是角度测量误差的主要因素。根据弧秒换算关系,设定角度观测噪声方差Q=1.097×10−6。在对公式(2)进行仿真计算时,对每个目标的角度观测值$ \rm{\alpha }\rm{、}\rm{\beta } $ 叠加一个方差为Q的高斯白噪声。$$ \left\{\begin{array}{c}\alpha ={{\rm{arctan}}}\left(\dfrac{y}{x}\right)+\sqrt{Q}\cdot{\rm{rand}}n\\ \beta ={{\rm{arctan}}}\left(\dfrac{y}{x-D}\right)+\sqrt{Q}\cdot{\rm{rand}}n\end{array}\right. $$ (15) 式中:
${\rm{rand}}n$ 表示介于0与1之间的随机数。设定第i个目标的真实位置为
$ {T}_{r}\left(i\right) $ ,根据目标的观测角度,用最小二乘法计算目标的估计位置为$ {T}_{e}\left(i\right) $ ,计算$ {T}_{r}\left(i\right) $ 与$ {T}_{e}\left(i\right) $ 的直线距离,得到目标的定位误差$ {T}_{l}\left(i\right) $ 。$ {T}_{l}\left(i\right) $ 的均值及标准差分别为:$$ \stackrel-{{T}_{l}}=\frac{1}{n}\sum _{i=1}^{n}{T}_{l}\left(i\right) $$ (16) $$ \mathrm{\sigma }=\sqrt{\frac{\displaystyle\sum _{i=1}^{n}{({T}_{l}\left(i\right)-\stackrel-{{T}_{l}})}^{2}}{n}} $$ (17) 注意到当目标点与观测站基线距离太近时,实际测量意义不大,结合仿真软件需求,剔除了与观测基线距离小于500 m的目标点。选取n=100个目标点,由公式(16)、公式(17)计算出定位误差的均值
$ \stackrel{-}{{T}_{l}} $ 及母体标准差$ \mathrm{\sigma } $ 。经过计算,当双站间距D由0逐步增大时,定位误差的均值及标准差的变化曲线如图6所示。
可以看出,当两个CCD间距小于5 km时,在有效定位范围内,平均定位误差在50 m以上,标准差大于50,此时目标定位数据精度较低,误差离散程度较大;当CCD间距继续增加时,定位误差逐步减小,并在约14.5 km处达到最小值13.8384 m,标准差也减小至13以下;当CCD间距继续增加时,定位误差均值及标准差逐步增大。这是因为,当间距增加到一定值后,最大定位距离快速减小,目标的观测角度
$ \alpha $ 趋近于0°,$\; \beta $ 趋近于180°,导致计算误差增大;当间距D超过30 km以上,误差急剧增大。综合以上分析,当观测站间距变化时,定位范围(最大定位宽度、最大定位距离)及定位误差情况如表1所示。
从表1可以得出以下结论:对于确定的大气条件和CCD设备参数,双站定位范围取决于双站间距大小,当间距超过一定数值后,最大定位宽度、最大定位距离将迅速减小;双站间距直接影响目标定位精度,从统计分析数据看,定位误差及误差值离散程度随着间距的逐步增大先减小后增大。要更好地发挥双站定位的效能,需结合气候条件和大气衰减影响,根据每个观测站传感器的特性参数,综合考虑目标位置、定位范围、定位精度的要求,对观测站位置进行优化配置。
表 1 定位范围、定位误差与双站间距的关系
Table 1. Location range and deviation vs. the distance of double station
Distance between CCD’s/km Maximum location width/km Maximum
location range/kmMean of
location deviation/mStd. of
location deviation/m2.5 30.5 15.8 90.9 88.6 5.0 28.0 15.8 48.2 58.2 7.5 25.5 15.7 24.6 21.5 10.0 23.0 15.6 23.9 18.4 12.5 20.5 15.2 18.8 15.4 15.0 18.0 14.6 14.2 10.5 17.5 15.5 13.9 14.3 10.1 20.0 13.0 13.1 15.6 10.3 22.5 10.5 12.0 22.7 11.2 25.0 8.0 10.8 27.5 14.1 27.5 5.5 9.1 35.4 24.9 30.0 3.0 6.9 48.3 36.2 32.5 0.5 2.9 322.3 273.5
Effectiveness evaluation for double-station passive location based on CCD
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摘要: 光电被动定位系统隐蔽性好,有利于提高复杂电磁环境中对抗作战的效能。建立基于CCD的双站被动定位系统模型,为提高测算精度,依据最小二乘法原理推导出静止目标坐标的求解方法。基于大气消光系数、目标背景亮度系数、CCD成像设备参数建立CCD侦察模型。以目标探测概率大于10%为标准,确定单个CCD的作用距离,并在此基础上分析CCD双站定位范围随双站间距的变化关系。在有效定位范围内抽取批量目标样本,计算双站定位误差,分析误差的均值和标准差随双站间距的变化规律。计算结果表明:对于确定的天气条件及CCD成像设备,当双站间距超过一定距离后,定位范围将逐步减小;定位误差及误差离散程度随着双站间距的增加先变小后变大。分析结果对于优化CCD设备参数、合理配置观测站位置具有一定的借鉴意义。Abstract: The system of photoelectric passive location with good concealment could improve the combat effectiveness of confrontation in complex electromagnetic environment. The system model of double-station passive location based on CCD was established. In order to improve the accuracy of calculation, the calculation method of target coordinate was provided according to the principle of least square method. The CCD reconnaissance model was established based on atmospheric extinction coefficient, the luminance coefficient of target and background, parameters of CCD imaging equipment. Based on the rule that the detection probability must be more than 10%, the operating distance of single CCD was calculated. Then, the change relations of the double-station location range and the distance of the stations was studied. By the target samples within the effective location range, the double-station location error was calculated. Then, the variation of mean value and standard deviation with the distance of the stations was analyzed. The calculation show that to certain weather and CCD, when the distance of the stations exceed a certain value, the location range would decrease rapidly. With the increase of distance, the location error and its dispersion first increase and then decrease. The analysis results have certain reference significance for optimizing CCD equipment parameters and reasonably configuring the position of observation stations observation stations. The analysis conclusion can be use for reference to optimize the CCD parameters and choose the reasonable location of observation station.
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Key words:
- passive location /
- CCD sensor /
- photo-electric countermeasure /
- operating distance
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表 1 定位范围、定位误差与双站间距的关系
Table 1. Location range and deviation vs. the distance of double station
Distance between CCD’s/km Maximum location width/km Maximum
location range/kmMean of
location deviation/mStd. of
location deviation/m2.5 30.5 15.8 90.9 88.6 5.0 28.0 15.8 48.2 58.2 7.5 25.5 15.7 24.6 21.5 10.0 23.0 15.6 23.9 18.4 12.5 20.5 15.2 18.8 15.4 15.0 18.0 14.6 14.2 10.5 17.5 15.5 13.9 14.3 10.1 20.0 13.0 13.1 15.6 10.3 22.5 10.5 12.0 22.7 11.2 25.0 8.0 10.8 27.5 14.1 27.5 5.5 9.1 35.4 24.9 30.0 3.0 6.9 48.3 36.2 32.5 0.5 2.9 322.3 273.5 -
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