﻿ 一种目标运动要素纯方位解算方法
 舰船科学技术  2020, Vol. 42 Issue (12): 129-132    DOI: 10.3404/j.issn.1672-7649.2020.12.025 PDF

A movement parameters calculating method base on bearings-only
CHENG Shan-zheng, CHEN Shuang, HE Xin-yi
No. 92578 Unit of the PLA, Beijing 100161, China
Abstract: Movement parameters calculating method only needs to measure object bearings. This method requires less parameter and is propitious to measure for submarine. The traditional calculating method base on bearings-only has the problems of long solution time and low convergence rate. This paper provides a new method based on object bearings parameters only, which can advance convergence rate and shorten solution time. This method can be used for underwater object and weapon firing data.
Key words: underwater object     movement parameters     bearings only     calculating method
0 引　言

1 纯方位解算的传统算法 1.1 符号说明

${{{D}}_0}$ 为目标的初始距离； ${{{V}}_{{m}}}$ 为目标速度； ${{{Q}}_{{m}}}$ 为目标初始舷角； ${y_i}$ 为测量方位变化量（瞬时方位 ${f_i}$ 与初始方位 ${f_0}$ 的方位差）； $\overline {{{{y}}_i}}$ 为拟合推算方位变化量（拟合推算方位 ${f_i}$ 与初始方位 ${f_0}$ 的方位差）。

1.2 数学模型

 图 1 纯方位解算方法示意图 Fig. 1 Schematic diagram of azimuth-only solution method
 ${\rm{tan}}{y_i} = \frac{{{t_i}{V_m}\sin {Q_m} - {J_{si}}}}{{{D_0} - {t_i}{V_m}\cos {Q_m} - {J_{ci}}}}\text{，}$ (1)
 ${J_{si}} = \int_0^{{t_i}} {{V_w}\sin ({C_w} - {f_0}){\rm{d}}t} \text{，}$ (2)
 ${J_{ci}} = \int_0^{{t_i}} {{V_w}\cos ({C_w} - {f_0}){\rm{d}}t}\text{，}$ (3)

 $A = {D_0}\text{，}$ (4)
 $B = {V_m}\sin ({Q_m})\text{，}$ (5)
 $C = {V_m}\cos ({Q_m})\text{，}$ (6)

 $\left\{ {\begin{array}{*{20}{c}} {{V_m} = \sqrt {{B^2} + {C^2}} } \text{，}\\ {{Q_m} = t{g^{ - 1}}\dfrac{B}{C}} \text{，} \end{array}} \right.$ (7)

 $tg{y_i} = \frac{{B{t_i} - {J_{si}}}}{{A - C{t_i} - {J_{ci}}}}\;\;\;{\rm{i}}=1,2\text{，}$ (8)

 ${J_{ci}}\sin {y_i} - {J_{si}}\cos {y_i} = A\sin {y_i} - B{t_i}\cos {y_i} - C{t_i}\sin {y_i} \text{。}$ (9)

 $X = \left( {\begin{array}{*{20}{c}} {\sin {y_1}}&{ - {t_1}\cos {y_1}}&{ - {t_1}\sin {y_1}} \\ {\sin {y_2}}&{ - {t_2}\cos {y_2}}&{ - {t_2}\sin {y_2}} \\ {\cdots}&{\cdots}&{\cdots} \\ {\sin {y_n}}&{ - {t_n}\cos {y_n}}&{ - {t_n}\sin {y_n}} \end{array}} \right)\text{，}$ (10)
 $Y = \left( {\begin{array}{*{20}{c}} {{J_{c1}}\sin {y_1} - {J_{s1}}\cos {y_1}} \\ {{J_{C2}}\sin {y_2} - {J_{s2}}\cos {y_2}} \\ {\cdots} \\ {{J_{cn}}\sin {y_n} - {J_{sn}}\cos {y_n}} \end{array}} \right)\text{，}$ (11)
 $\beta = {(A,B,C)'}\text{，}$ (12)
 $\varepsilon = ({\varepsilon _1},{\varepsilon _2},\cdots,{\varepsilon _n})'\text{，}$ (13)

 ${{Y}} = {{X}}\beta + \varepsilon \text{，}$ (14)

 ${{X'X}}\beta = {{X'Y}}\text{。}$ (15)

2 一种新的纯方位解算方法

${ D}_{0}, { V}_{m}, { Q}_{m}$ 分别划分为 ${n}_{1}, {n}_{2}, {n}_{3}$ 个水平，水平间隔在解算误差允许的范围之内，离散化后可能的解算值应该有 ${n_1} \times {n_2} \times {n_3}$ 个，在 ${n_1} \times {n_2} \times {n_3}$ 个可能的解算值中，找出一个解 ${ D}_{0i}, { V}_{mi}, { Q}_{mi}$ 使得拟合值 $\overline {{y_i}}$ 与实际测量值 ${y_i}$ 之间的偏差平方和达到最小。

 ${D}_{0}={f}_{1}({V}_{m}){\text{，}}$ (16)
 ${Q}_{m}={f}_{2}({V}_{m}){\text{，}}$ (17)

 $B = {V_m}\sin ({Q_m})\text{，}$ (18)
 $C = {V_m}\cos ({Q_m})\text{，}$ (19)

 ${\rm{tan}}{y_i} = \frac{{B{t_i} - {J_{si}}}}{{{D_{0j}} - C{t_i} - {J_{ci}}}},\;\;{{i}}=1,2, \cdots {{n}}\text{，}$ (20)

 ${J_{ci}}\sin {y_i} - {J_{si}}\cos {y_i} = {D_{0j}}\sin {y_i} - B{t_i}\cos {y_i} - C{t_i}\sin {y_i} \text{。}$ (21)

 ${{X}} = \left( {\begin{array}{*{20}{c}} { - {t_1}\cos {y_1}}&{ - {t_1}\sin {y_1}} \\ { - {t_2}\cos {y_2}}&{ - {t_2}\sin {y_2}} \\ {\cdots}&{\cdots} \\ { - {t_n}\cos {y_n}}&{ - {t_n}\sin {y_n}} \end{array}} \right)\text{，}$ (22)
 ${{Y}} = \left( {\begin{array}{*{20}{c}} {{J_{c1}}\sin {y_1} - {J_{s1}}\cos {y_1} - {D_{0j}}\sin {y_1}} \\ {{J_{C2}}\sin {y_2} - {J_{s2}}\cos {y_2} - {D_{0j}}\sin {y_2}} \\ {\cdots} \\ {{J_{cn}}\sin {y_n} - {J_{sn}}\cos {y_n} - {D_{0j}}\sin {y_n}} \end{array}} \right)\text{，}$ (23)
 $\beta = {(B,C)'}\text{，}$ (24)
 $\varepsilon = ({\varepsilon _1},{\varepsilon _2},\cdots,{\varepsilon _n})'\text{，}$ (25)

 ${{Y}} = {{X}}\beta + \varepsilon \text{，}$ (26)

 ${{X'X}}\beta = {{X'Y}}\text{，}$ (27)

 ${J_{fsi}} = \int_0^{{t_i}} {{V_w}\sin ({C_w}){\rm{d}}t} \text{，}$ (28)
 ${J_{fci}} = \int_0^{{t_i}} {{V_w}\cos ({C_w}){\rm{d}}t} \text{，}$ (29)
 $\begin{split} {D_{ji}} =& {\rm{(}}{({t_i}{V_{mj}}\sin ({C_m}) + {d_{0j}}\sin {f_0} - {J_{fsi}})^2} + \\ & {({t_i}{V_{mj}}\cos ({C_m}) + {d_{0j}}\cos {f_0} - {J_{fci}})^2}{{\rm{)}}^{1/2}} \text{，} \end{split}$ (30)
 $\sin x = \frac{{{t_i}{V_{mj}}\sin ({C_m}) + {d_{0j}}\sin {f_0} - {J_{fsi}}}}{{{D_{ji}}}}\text{，}$ (31)
 $\cos x = \frac{{{t_i}{V_m}\cos ({C_m}) + {d_{0j}}\cos {f_0} - {J_{fci}}}}{{{D_{ji}}}}\text{，}$ (32)
 ${\overline f _i} = t{g^{ - 1}}\dfrac{{\sin x}}{{\cos x}}\text{，}$ (33)
 ${\overline y _i} = {\overline f _i} - {f_0}\text{。}$ (34)

3 实　例

 图 2 初距解算误差随时间变化比较图 Fig. 2 Comparison diagram of initial distance solution error with time

 图 3 敌速解算误差随时间变化比较图 Fig. 3 Comparison diagram of initial speed solution error with time

 图 4 解算敌向误差随时间变化比较图 Fig. 4 Comparison diagram of initial course solution error with time

4 结　语

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