一種基于Herm ite函數(shù)約束的光學(xué)經(jīng)緯儀多站定位方法研究

宮志華,徐旭,段鵬偉,雷紅

(中國(guó)白城兵器試驗(yàn)中心,吉林白城 137001)

一種基于Herm ite函數(shù)約束的光學(xué)經(jīng)緯儀多站定位方法研究

宮志華,徐旭,段鵬偉,雷紅

(中國(guó)白城兵器試驗(yàn)中心,吉林白城 137001)

傳統(tǒng)光學(xué)幾何交會(huì)方法中采用的方向余弦定位法因其算法簡(jiǎn)單清晰一直廣泛應(yīng)用于工程實(shí)踐,但該方法受空間幾何關(guān)系影響較大,對(duì)測(cè)元誤差抑制能力差,因此定位精度并不理想,而且不能多站同時(shí)定位計(jì)算。為此提出一種基于Hermite函數(shù)約束的光學(xué)經(jīng)緯儀多站定位解算方法,稱為函數(shù)約束誤差模型彈道最佳估計(jì)(EMBET)方法。該方法首先在采用Hermite函數(shù)表征目標(biāo)定位參數(shù)以壓縮待估參數(shù)數(shù)量的基礎(chǔ)上進(jìn)而實(shí)現(xiàn)對(duì)光學(xué)經(jīng)緯儀測(cè)元的數(shù)學(xué)表征,然后將多站多測(cè)元觀測(cè)數(shù)據(jù)組成聯(lián)合測(cè)量方程,最后依據(jù)最小二乘準(zhǔn)則優(yōu)化解算方程獲取目標(biāo)定位參數(shù),并具有對(duì)測(cè)元系統(tǒng)誤差校準(zhǔn)的能力。結(jié)合仿真實(shí)例計(jì)算,對(duì)方向余弦定位方法和函數(shù)約束EMBET方法進(jìn)行了比較分析,驗(yàn)證了函數(shù)約束EMBET方法在光學(xué)經(jīng)緯儀多站定位中具有更高的定位精度和更強(qiáng)的實(shí)用性,為工程應(yīng)用提供了參考依據(jù)。

光學(xué)；光學(xué)經(jīng)緯儀；時(shí)序關(guān)聯(lián)；函數(shù)約束；Hermite函數(shù)；EMBET

0 引言

光學(xué)經(jīng)緯儀是我國(guó)靶場(chǎng)獲取外彈道跟蹤數(shù)據(jù)和飛行狀態(tài)的最基本測(cè)量手段之一。由于單臺(tái)經(jīng)緯儀只能得到空間目標(biāo)的二維坐標(biāo)信息(方位角和高低角),所以為了獲得目標(biāo)在空間中的三維坐標(biāo),至少需要兩臺(tái)光學(xué)經(jīng)緯儀以交會(huì)方式計(jì)算被測(cè)目標(biāo)空中位置。為保證跟蹤測(cè)試可靠性,采用更多臺(tái)經(jīng)緯儀對(duì)目標(biāo)進(jìn)行跟蹤測(cè)量是常選的測(cè)量方案?，F(xiàn)階段光學(xué)經(jīng)緯儀多采用空間幾何交會(huì)方法確定目標(biāo)位置,如目前工程上廣泛應(yīng)用方向余弦定位法[1],亦稱為公垂線法、最短距離法[2],或異面交會(huì)方法[3],但這種方法對(duì)于測(cè)元中含有的誤差抑制能力差,且受布站幾何關(guān)系的影響較大,因此定位精度并不很理想。關(guān)于兩臺(tái)光學(xué)經(jīng)緯儀的空間幾何交會(huì)測(cè)量問(wèn)題已得到解決[1-4],但對(duì)于3臺(tái)甚至3臺(tái)以上光學(xué)經(jīng)緯儀的空間幾何交會(huì)計(jì)算問(wèn)題尚未有統(tǒng)一的處理方法,如果只選用其中兩臺(tái)經(jīng)緯儀測(cè)量數(shù)據(jù)進(jìn)行處理,這造成了一定數(shù)據(jù)資源的浪費(fèi),而且精度上也不能保證最優(yōu);文獻(xiàn)[5]中闡述的一種對(duì)4臺(tái)光學(xué)經(jīng)緯儀分兩側(cè)布站的交會(huì)結(jié)果進(jìn)行數(shù)據(jù)融合的方法,其實(shí)質(zhì)還是在多臺(tái)經(jīng)緯儀測(cè)量條件下兩兩組合交會(huì)計(jì)算,再按一定權(quán)值分配方法給出唯一一組數(shù)據(jù),雖然交會(huì)結(jié)果精度會(huì)有提高,但這種方法的計(jì)算效率非常低。

針對(duì)這些問(wèn)題,本文提出采用基于函數(shù)約束的光學(xué)經(jīng)緯儀多站定位方法,稱為函數(shù)約束誤差模型最佳彈道估計(jì)(EMBET)方法[6-7],并利用仿真算例,對(duì)光學(xué)經(jīng)緯儀傳統(tǒng)幾何交會(huì)和函數(shù)約束EMBET定位方法進(jìn)行了比較分析,結(jié)果顯示,新方法具有4個(gè)方面的優(yōu)點(diǎn):一是抑制和探查測(cè)元誤差能力強(qiáng);二是降低布站幾何關(guān)系影響;三是多站能夠同時(shí)參與定位計(jì)算;四是定位精度高。

1 計(jì)算模型

以m臺(tái)光學(xué)經(jīng)緯儀聯(lián)合測(cè)量目標(biāo)軌跡為例,則其中第k臺(tái)經(jīng)緯儀在ti(i=1,2,…,n)時(shí)刻所得測(cè)元Aki、Eki與空間坐標(biāo)(xi,zi,yi)之間的關(guān)系式為αT為判象限角?？紤]測(cè)量方程的非線性,需按泰勒公式展開(kāi)成線性方程,又因?yàn)闇y(cè)量誤差(包含系統(tǒng)誤差S和隨機(jī)誤差ε)的存在,則由m臺(tái)經(jīng)緯儀測(cè)得的2m個(gè)測(cè)元組成的聯(lián)合測(cè)量方程如(2)式所示:

假設(shè)系統(tǒng)誤差參數(shù)有M個(gè),則未知待估參數(shù)數(shù)量為2(mx+mz+my+6)+M個(gè),方程總數(shù)量2mn.可見(jiàn),當(dāng)采樣數(shù)量n一定大時(shí),待估參數(shù)被極大壓縮,不僅增加測(cè)量方程的冗余度,提高方程計(jì)算穩(wěn)定度,而且對(duì)各測(cè)元系統(tǒng)誤差探查和估計(jì)能力大大增強(qiáng)。(3)式以矩陣形式給出為

式中:V=(εA1,εE1,…εAm,εEm)T為隨機(jī)誤差向量; X=(βx,βz,βy)T為擬合函數(shù)系數(shù)向量,即待估參數(shù);A為與X有關(guān)的系數(shù)矩陣;B為系統(tǒng)誤差模型系數(shù)矩陣;C為系統(tǒng)誤差待估參數(shù)向量;ΔL=(A1i-A1i0,E1i-E1i0,…,Ami-Ami0,Emi-Emi0)T為常數(shù)向量。

在參數(shù)估計(jì)時(shí),在大數(shù)據(jù)量處理的情況下,相關(guān)性和非高斯性對(duì)估計(jì)結(jié)果影響不大,因此采用白噪聲條件下的參數(shù)估計(jì)方法。利用最小二乘估計(jì)[9-13]可以得到待估參數(shù)解X(包括擬合函數(shù)待估系數(shù)和系統(tǒng)誤差模型系數(shù)C)和待估參數(shù)的誤差協(xié)方差陣,如(7)式所示:

式中:P為權(quán)值矩陣,由各測(cè)元統(tǒng)計(jì)方差相互標(biāo)定確定。在實(shí)際求解中,由于非線性函數(shù)級(jí)數(shù)展開(kāi)帶來(lái)的截?cái)嗾`差,需要進(jìn)行疊代計(jì)算。由函數(shù)擬合系數(shù)代入(4)式,最終得到目標(biāo)空間位置參數(shù)。

2 仿真分析

仿真數(shù)據(jù)設(shè)計(jì):依據(jù)質(zhì)點(diǎn)彈道方程生成某型火箭彈理論彈道,并以其為基準(zhǔn),設(shè)計(jì)3臺(tái)光學(xué)經(jīng)緯儀站址(1號(hào)站、2號(hào)站和3號(hào)站)均勻布設(shè)在彈道同側(cè),反推產(chǎn)生3臺(tái)經(jīng)緯儀6個(gè)真值角度測(cè)元,數(shù)據(jù)率10 Hz,并給每個(gè)測(cè)元加入相應(yīng)的隨機(jī)誤差和系統(tǒng)誤差值。初值以理論彈道為基礎(chǔ),在x、z、y 3個(gè)方向上分別加上300 m、500 m、200 m的固定誤差和6 m、6m、8m的隨機(jī)誤差。

2.1 兩站定位分析

對(duì)加入誤差的仿真測(cè)元數(shù)據(jù)分別采用方向余弦定位方法和函數(shù)約束EMBET方法進(jìn)行兩站定位計(jì)算,每種解算方法得到3組彈道坐標(biāo)數(shù)據(jù),取平均值與理論彈道真值進(jìn)行比對(duì),誤差曲線如圖1所示,誤差統(tǒng)計(jì)值見(jiàn)表1.仿真加入的隨機(jī)誤差和系統(tǒng)誤差真值以及探查出的隨機(jī)誤差和系統(tǒng)誤差值見(jiàn)表2.

圖1 兩種定位方法兩站定位誤差曲線Fig.1 The positioning error curves of twomethods for two optical theodolites

從圖1和表1中看到,在兩站定位條件下,方向余弦定位方法的定位總誤差計(jì)算值達(dá)到6.3 m,而函數(shù)約束EMBET方法的定位總誤差計(jì)算值2.4m,定位精度比方向余弦定位方法提高1倍以上;從表2中看到,所有測(cè)元隨機(jī)誤差探查的最大誤差計(jì)算值是0.7″,系統(tǒng)誤差探查的最大誤差計(jì)算值達(dá)到12.1″,可見(jiàn),系統(tǒng)誤差探查的準(zhǔn)確度不高。

表1 兩種定位方法兩站定位誤差統(tǒng)計(jì)Tab.1 The positioning error statistics of twomethods for two optical theodolites

表2 函數(shù)約束EMBET方法兩站解算測(cè)元隨機(jī)誤差和系統(tǒng)誤差統(tǒng)計(jì)Tab.2 The random error and system error statistics ofmeasuring elements got by function restriced EMBET method with two optical theodolites

2.2 三站定位分析

對(duì)加入誤差的仿真測(cè)元數(shù)據(jù)采用函數(shù)約束EMBET方法進(jìn)行三站定位計(jì)算,與理論彈道真值進(jìn)行比對(duì),誤差曲線如圖2所示,誤差統(tǒng)計(jì)值見(jiàn)表3,探查出的隨機(jī)誤差和系統(tǒng)誤差值見(jiàn)表4。

表3 函數(shù)約束EMBET方法三站定位誤差統(tǒng)計(jì)Tab.3 The positioning error statistics of function restriced EMBETmethod for three optical theodolites

從圖2和表3中看到,應(yīng)用函數(shù)約束EMBET方法,三站定位總誤差計(jì)算值只有0.98 m,定位精度比兩站定位提高1倍以上;從表4中看到,所有測(cè)元隨機(jī)誤差探查的最大誤差計(jì)算值還是0.7″,但系統(tǒng)誤差探查的最大誤差計(jì)算值只有1.3″,可見(jiàn),系統(tǒng)誤差探查的準(zhǔn)確度得到極大提高。

圖2 函數(shù)約束EMBET方法三站定位誤差曲線Fig.2 The positioning error curves of function restriced EMBETmethod for three optical theodolites

采用函數(shù)約束EMBET方法對(duì)四站經(jīng)緯儀進(jìn)行仿真計(jì)算,定位總誤差為0.87m,所有測(cè)元隨機(jī)誤差探查的最大誤差0.7″,系統(tǒng)誤差探查的最大誤差1.1″,可見(jiàn),定位精度和誤差探查準(zhǔn)確度進(jìn)一步提高,但提高幅度已經(jīng)不大了。

表4 函數(shù)約束EMBET方法三站定位測(cè)元隨機(jī)誤差和系統(tǒng)誤差統(tǒng)計(jì)Tab.4 The random errors and system error statics ofmeasuring elements got by function restriced EMBET method for three optical theodolites

3 結(jié)論

通過(guò)以上仿真計(jì)算分析,可以得到如下結(jié)論:

1)僅就兩站定位來(lái)說(shuō),方向余弦定位方法受隨機(jī)誤差和系統(tǒng)誤差影響大,而函數(shù)約束EMBET方法由于對(duì)待估參數(shù)進(jìn)行了極大壓縮,受隨機(jī)誤差影響降低,因此,定位精度得到提高;函數(shù)約束EMBET方法在兩站定位時(shí),由于參與計(jì)算的測(cè)元少,冗余程度不高,系統(tǒng)誤差探查的準(zhǔn)確度不高;從計(jì)算效率來(lái)看,方向余弦定位方法和函數(shù)約束EMBET方法計(jì)算效率都很高。

2)多站(大于兩站)同時(shí)定位時(shí),方向余弦定位方法不能實(shí)現(xiàn)同時(shí)定位計(jì)算,函數(shù)約束EMBET方法定位精度更高,且優(yōu)于兩站定位精度;函數(shù)約束EMBET方法在多站定位時(shí),由于參與計(jì)算的測(cè)元增多,冗余程度增強(qiáng),具有了較強(qiáng)的系統(tǒng)誤差探查和校準(zhǔn)能力,系統(tǒng)誤差探查的準(zhǔn)確度較高。

綜合以上結(jié)論,在靶場(chǎng)工程應(yīng)用時(shí),方向余弦定位方法和函數(shù)約束EMBET方法在兩站定位時(shí),都具有較強(qiáng)的實(shí)用性,精度也可以,但函數(shù)約束EMBET方法對(duì)系統(tǒng)誤差還具有了一定的探查和校準(zhǔn)能力;在多站定位時(shí),函數(shù)約束EMBET方法具有更強(qiáng)的實(shí)用性,系統(tǒng)誤差探查能力強(qiáng),且對(duì)初值要求不高,因此,在高精度定位和設(shè)備校準(zhǔn)等方面具有很大的工程應(yīng)用價(jià)值。

References)

[1] 劉利生,李杰.外測(cè)數(shù)據(jù)事后處理[M].北京:國(guó)防工業(yè)出版社,2000.LIU Li-sheng,LI jie.Post-flight data processing of trajectorymeasurement[M].Beijing:National Defense Industry Press,2000.(in Chinese)

[2] 賈濤,趙長(zhǎng)壽.光測(cè)設(shè)備交會(huì)計(jì)算最優(yōu)方法研究[J].兵器試驗(yàn),1999,183(3):1-14.

JIA Tao,ZHAO Chang-shou.A study on the optimized intersection method of optical theodolite[J].Ordance Test,1999,183(3):1-14.(in Chinese)

[3] 吳能偉,陳濤.異面交會(huì)法在光電經(jīng)緯儀中的應(yīng)用[J].儀器儀表學(xué)報(bào),2006,27(6):15-21.

WU Neng-wei,CHEN Tao.Application of non-coplane intersection method to photoelectric theodolite[J].Chinese Journal of Scientific Instrurment,2006,27(6):15-21.(in Chinese)

[4] 侯宏錄,李宏.光電經(jīng)緯儀測(cè)量飛行器三維坐標(biāo)方法及誤差分析[J].光電工程,2002,29(3):4-8.

HOU Hong-lu,LIHong.Measurement of vehicle three dimension coordinate with photoelectrical theodolite and error analysis[J]. Opto-Electronic Engineering,2002,29(3):4-8.(in Chinese)

[5] 張玲霞,馬彩文,陳明,等.靶場(chǎng)光電經(jīng)緯儀多臺(tái)交會(huì)測(cè)量數(shù)據(jù)的一種處理方法[J].測(cè)繪學(xué)報(bào),2003,32(2):139-142.

ZHANG Ling-xia,MA Cai-wen,CHEN Ming,et al.A data processingmethod on trajectorymeasurement by intersection in shooting range[J].Acta Geodaetica et Cartographica Sinica,2003,32 (2):139-142.(in Chinese)

[6] 王正明,易東云,周海銀.彈道跟蹤數(shù)據(jù)的校準(zhǔn)與評(píng)估[M].長(zhǎng)沙:國(guó)防科技大學(xué)出版社,1999.

WANG Zheng-ming,YIDong-yun,ZHOU Hai-yin.Tracking trajectory data calibrating and evaluating[M].Changsha:Publishing House of National University of Defense Technology,1999.(in Chinese)

[7] 郭軍海.彈道測(cè)量數(shù)據(jù)融合技術(shù)[M].北京:國(guó)防工業(yè)出版社社,2012.

GUO Jun-hai.Trajectory data fusion technology[M].Beijing:National Defense Industry Press,2012.(in Chinese)

[8] 詹武平,諶廷政,劉成,等.組網(wǎng)雷達(dá)測(cè)量色噪聲數(shù)據(jù)的Hermite插值融合方法[J].現(xiàn)代雷達(dá),2012,34(6):41-44.

ZHANWu-ping,SHEN Ting-zheng,LIU Cheng,et al.Hermite interpolation fusion method for netted radar measuring colored noise data[J].Modern Radar,2012,34(6):41-44.(in Chinese)

[9] 吳澤民,任姝婕.雷達(dá)時(shí)差和系統(tǒng)誤差的聯(lián)合估計(jì)方法[J].兵工學(xué)報(bào),2011,32(7):848-852.

WU Ze-min,REN Shu-jie.Jointestimation for time offsetand radar system error[J].Acta Armamentarii,2011,32(7):848-852.(in Chinese)

[10] 柴敏,余慧,宋衛(wèi)紅,等.光學(xué)無(wú)線電測(cè)量信息融合定位方法[J].光學(xué)學(xué)報(bào),2012,32(12):158-164.

CHAIMin,YU Hui,SONG Wei-hong,et al.Joint positioning method for data fusion of photo-electric theodolite and radar[J]. Acta Optica Sinica,2012,32(12):158-164.(in Chinese)

[11] 崔書(shū)華,胡紹林,宋衛(wèi)紅,等.多測(cè)速系統(tǒng)最優(yōu)彈道估計(jì)方法及應(yīng)用[J].彈箭與制導(dǎo)學(xué)報(bào),2012,32(4):215-218.

CUIShu-hua,HU Shao-lin,SONGWei-hong,et al.Optimal estimation and application of velocitymeasurement system[J].Journal of Projectiles,Rockets,Missiles and Guidance,2012,32(4): 215-218.(in Chinese)

[12] 崔書(shū)華,許愛(ài)華,趙樹(shù)強(qiáng),等.高精度跟蹤測(cè)量系統(tǒng)測(cè)元數(shù)據(jù)融合處理方法[J].彈箭與制導(dǎo)學(xué)報(bào),2013,33(3):167-170.

CUIShu-hua,XU Ai-hua,ZHAO Shu-qiang,et al.Fusion processingmethod ofmeasurement elements in high-precision trackingmeasuring system[J].Journal of Projectiles,Rockets,Missiles and Guidance,2013,33(3):167-170.(in Chinese)

[13] 宮志華,周海銀,郭文勝,等.基于樣條函數(shù)表征目標(biāo)運(yùn)動(dòng)軌跡事后數(shù)據(jù)融合方法研究[J].兵工學(xué)報(bào),2014,35(1):120-127.

GONG Zhi-hua,ZHOU Hai-yin,GUOWen-sheng,et al.Data fusion algorithm for target trajectory determination based on spline function representation[J].Acta Armamentarii,2014,35(1): 120-127.(in Chinese)

Research on the Positioning M ethod of M ulti-optical Theodolites Based on Herm ite Function Restriction

GONG Zhi-hua,XU Xu,DUAN Peng-wei,LEIHong
(Baicheng Ordnance Test Center of China,Baicheng 137001,Jilin,China)

The direction cosine locationmethod which is used in traditional optical geometric intersection is simple,so it iswidely used in engineering practice.But thismethod is greatly influenced by the geometric relationship,and cannot inhibit themeasuring error,so its positioning accuracy is unsatisfactory. In addition,it cannot be used to compute the locationswithmore than three stations at a time.A new positioningmethod with multi-optical theodolites is proposed based on Hermite function restriction.This method can be called function restricted EMBETmethod.Thismethod uses Hermite function to express the positioning parameters to compress the number of parameters to be estimated,and meanwhile the measuring elements of optical theodolite can be expressed.The measuring equation can be formed by measuring data ofmultiplemeasuring elements.Based on least square criterion,the positioning parameters are obtained by computing the combined equation,and the system errors ofmeasuring elements are calibrated.Through the calculation of simulated example,the function restricted EMBETmethod is compared with the direction cosine location method.The result shows that the function restricted EMBET method ismore accurate and more practical.

optics;optical theodolite;time sequence relevance;function restriction;Hermite function;errormodel best estimate of trajectory

V557+.4

A

1000-1093(2014)12-2092-06

10.3969/j.issn.1000-1093.2014.12.023

2014-01-07

宮志華(1975—),男,高級(jí)工程師。E-mail:gzh63298@126.com