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A new method for double IRST cooperative passive detection and positioning for maneuvering target
ZHANG Feng
Luoyang Institute of Electro-Optical Equipment, AVIC, Luoyang 471009, China
Abstract: When Infrared Search and Track (IRST) System works in double station cooperative detection and positioning in operational use, it is difficult to match a maneuvering target model with the actual target motion, which worses position estimation error. Therefore, an adaptive filtering algorithm based on curvilinear model is proposed. The method here shifts the thinking that turning angular velocity is estimated by direction angle and tangential acceleration is estimated by inserted acceleration in traditional method. In this paper, turning angular velocity and target acceleration is both treated as augmented state variable and its corresponding process noise covariance expression is derived, which greatly decreases the calculation burden of traditional two-layer filtering structure, as well as improving the estimation accuracy to tangential acceleration. In the meanwhile, the paper optimizes the heading angle calculation method based on library function "atan2()" in embedded software and its transfer relationship between four quadrants. It demonstrates that the proposed method has the better adaptation to maneuvering target and has better positioning accuracy through double IRST cooperative simulation instance.
Keywords: Infrared Search and Track (IRST) System    double IRST cooperative positioning    curvilinear model    maneuvering target positioning

1 系统建模

 ${\mathit{\boldsymbol{X}}_k} = f\left( {{\mathit{\boldsymbol{X}}_{k - 1}}} \right) + {\mathit{\boldsymbol{W}}_k}$ （1）
 ${\mathit{\boldsymbol{Z}}_k} = h\left( {{\mathit{\boldsymbol{X}}_k}} \right) + {\mathit{\boldsymbol{V}}_k}$ （2）

 ${\alpha _i} = \arctan \frac{{{y_{\rm{T}}} - y_z^i}}{{{x_{\rm{T}}} - x_z^i}}$ （3）
 ${\beta _i} = \arctan \frac{{{z_{\rm{T}}} - z_z^i}}{{\sqrt {{{\left( {{x_{\rm{T}}} - x_z^i} \right)}^2} + {{\left( {{y_{\rm{T}}} - y_z^i} \right)}^2}} }}$ （4）
2 曲线模型状态方程的建立

 图 1 定位模糊区示意图(仅考虑俯仰平面) Fig. 1 Schematic diagram of positioning fuzzy area (pitch only)

 图 2 转弯运动示意图 Fig. 2 Schematic diagram of turning motion
 $v(t) = r\omega (t)$ （5）
 ${\alpha _{\rm{t}}} = \frac{{{\rm{d}}v(t)}}{{{\rm{d}}t}} = r{\alpha _\omega }(t)$ （6）
 ${\alpha _{\rm{n}}} = r\omega {(t)^2} = v(t)\omega (t)$ （7）
 $\dot x(t) = v(t)\sin \phi (t)$ （8）
 $\dot y(t) = v(t)\cos \phi (t)$ （9）

 $\begin{array}{l} \left[ {\begin{array}{*{20}{l}} {x(k + 1)}\\ {\dot x(k + 1)}\\ {y(k + 1)}\\ {\dot y(k + 1)} \end{array}} \right] = \left[ {\begin{array}{*{20}{l}} 1&{\frac{{\sin \left( {T{\omega _k}} \right)}}{{{\omega _k}}}}&0&{\cos \left( {T{\omega _k}} \right)}\\ 0&{\frac{{\cos \left( {T{\omega _k}} \right)}}{{{\omega _k}}}}&0&{ - \sin \left( {T{\omega _k}} \right)}\\ 0&{\frac{{1 - \cos \left( {T{\omega _k}} \right)}}{{{\omega _k}}}}&0&{\sin \left( {T{\omega _k}} \right)}\\ 0&{\frac{{\sin \left( {T{\omega _k}} \right)}}{{{\omega _k}}}}&0&{\cos \left( {T{\omega _k}} \right)} \end{array}} \right]\left[ {\begin{array}{*{20}{l}} {x(k)}\\ {x(k)}\\ {y(k)}\\ {\dot y(k)} \end{array}} \right] + \left[ {\begin{array}{*{20}{c}} {\frac{{T\cos {\phi _k}}}{{{\omega _k}}} + \frac{{\sin {\phi _k}}}{{\omega _k^2}} - \frac{{\sin \left( {{\phi _k} + {T_{{\omega _k}}}} \right)}}{{\omega _k^2}}}\\ {\frac{{\cos {\phi _k}}}{{{\omega _k}}} - \frac{{\cos \left( {{\phi _k} + T{\omega _k}} \right)}}{{{\omega _k}}}}\\ {\frac{{\cos {\phi _k}}}{{\omega _k^2}} - \frac{{T\sin {\phi _k}}}{{{\omega _k}}} - \frac{{\cos \left( {{\phi _k} + T{\omega _k}} \right)}}{{\omega _k^2}}}\\ {\frac{{\sin \left( {{\phi _k} + T{\omega _k}} \right)}}{{{\omega _k}}} - \frac{{\sin {\phi _k}}}{{{\omega _k}}}} \end{array}} \right] \cdot \\ {\alpha _{\rm{t}}}\left( k \right) + \mathit{\boldsymbol{ \boldsymbol{\varGamma} W}}\left( k \right) \end{array}$ （10）

 $\theta \left( {k + 1} \right) = \theta \left( k \right) + T\omega \left( k \right) + \frac{1}{2}{\alpha _\omega }{T^2}$ （11）
 $\omega \left( {k + 1} \right) = \omega \left( k \right) + {\alpha _\omega }T \cong \omega \left( k \right) + \frac{{\Delta \theta }}{{v\left( k \right)}}{\alpha _{\rm{t}}}\left( k \right)$ （12）

 $\begin{array}{l} \omega \left( {k + 1} \right) = \omega \left( k \right) + \\ \;\;\;\;\;\;\frac{{\arctan \frac{{\dot x\left( k \right)}}{{\dot y\left( k \right)}} - \arctan \frac{{\dot x\left( {k - 1} \right)}}{{\dot y\left( {k - 1} \right)}}}}{{\sqrt {{{\dot x}^2}\left( k \right) + \dot y2\left( k \right)} }}{\alpha _{\rm{t}}}\left( k \right) \end{array}$ （13）

 $\begin{array}{l} \left[ {\begin{array}{*{20}{l}} {x(k + 1)}\\ {\dot x(k + 1)}\\ {y(k + 1)}\\ {\dot y(k + 1)}\\ {\omega (k + 1)} \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} 1&{\frac{{\sin \left( {T{\omega _k}} \right)}}{{{\omega _k}}}}&0&{\frac{{\cos \left( {T{\omega _k}} \right) - 1}}{{{\omega _k}}}}&0\\ 0&{\cos \left( {T{\omega _k}} \right)}&0&{ - \sin \left( {T{\omega _k}} \right)}&0\\ 0&{\frac{{1 - \cos \left( {T{\omega _k}} \right)}}{{{\omega _k}}}}&1&{\frac{{\sin \left( {T{\omega _k}} \right)}}{{{\omega _k}}}}&0\\ 0&{\sin \left( {T{\omega _k}} \right)}&0&{\cos \left( {T{\omega _k}} \right)}&0\\ 0&0&0&0&1 \end{array}} \right]\left[ {\begin{array}{*{20}{l}} {x(k)}\\ {\dot x(k)}\\ {y(k)}\\ {\dot y(k)}\\ {\omega (k)} \end{array}} \right] + \\ \left[ {\begin{array}{*{20}{c}} {\frac{{T\cos {\phi _k}}}{{{\omega _k}}} + \frac{{\sin {\phi _k}}}{{\omega _k^2}} - \frac{{\sin \left( {{\phi _k} + T{\omega _k}} \right)}}{{\omega _k^2}}}\\ {\frac{{\cos {\phi _k}}}{{{\omega _k}}} - \frac{{\cos \left( {{\phi _k} + T{\omega _k}} \right)}}{{{\omega _k}}}}\\ {\frac{{\cos {\phi _k}}}{{\omega _k^2}} - \frac{{T\sin {\phi _k}}}{{{\omega _k}}} - \frac{{\cos \left( {{\phi _k} + T{\omega _k}} \right)}}{{\omega _k^2}}}\\ {\frac{{\sin \left( {{\phi _k} + T{\omega _k}} \right)}}{{{\omega _k}}} - \frac{{\sin {\phi _k}}}{{{\omega _k}}}}\\ \eta \end{array}} \right]{\alpha _{\rm{t}}}\left( k \right) + \mathit{\boldsymbol{ \boldsymbol{\varGamma} W}}\left( k \right) \end{array}$ （14）

 $\eta = \frac{{\arctan \frac{{\dot x(k)}}{{\dot y(k)}} - \arctan \frac{{\dot x(k - 1)}}{{\dot y(k - 1)}}}}{{\sqrt {{{\dot x}^2}(k) + \dot y2(k)} }}$
 $\mathit{\boldsymbol{ \boldsymbol{\varGamma} }} = \left[ {\begin{array}{*{20}{c}} {\frac{{{T^2}}}{2}}&0\\ T&0\\ 0&{\frac{{{T^2}}}{2}}\\ 0&T\\ 0&0 \end{array}} \right]$

3 一种新的自适应曲线模型滤波方法 3.1 自适应曲线模型建模

 ${\mathit{\boldsymbol{X}}_k} = {\left[ {\begin{array}{*{20}{l}} {{x_{\rm{T}}}}&{{{\dot x}_{\rm{T}}}}&{{{\ddot x}_{\rm{T}}}}&{{y_{\rm{T}}}}&{{{\dot y}_{\rm{T}}}}&{{{\ddot y}_{\rm{T}}}}&{{z_{\rm{T}}}}&{{{\dot z}_{\rm{T}}}}&{{{\ddot z}_{\rm{T}}}}&\omega \end{array}} \right]^{\rm{T}}}$

 $\mathit{\boldsymbol{F}}(k) = \left[ {\begin{array}{*{20}{c}} {{f_{11}}}& \cdots &{{f_{110}}}\\ \vdots & \vdots & \vdots \\ {{f_{101}}}& \cdots &{{f_{1010}}} \end{array}} \right]$ （15）

 ${f_{11}} = 1,{f_{12}} = \frac{{\sin \left( {T{\omega _k}} \right)}}{{{\omega _k}}}$
 ${f_{15}} = \frac{{\cos \left( {T{\omega _k}} \right) - 1}}{{{\omega _k}}},{f_{22}} = \cos \left( {T{\omega _k}} \right)$
 ${f_{25}} = - \sin \left( {T{\omega _k}} \right),{f_{32}} = - {\omega _k} \cdot \sin \left( {T{\omega _k}} \right)$
 ${f_{33}} = 1,{f_{35}} = - {\omega _k}\left( {\cos \left( {T{\omega _k}} \right) - 1} \right)$
 ${f_{42}} = - \frac{{\cos \left( {T{\omega _k}} \right) - 1}}{{{\omega _k}}},{f_{44}} = 1$
 ${f_{45}} = \frac{{\sin \left( {T{\omega _k}} \right)}}{{{\omega _k}}},{f_{52}} = \sin \left( {T{\omega _k}} \right)$
 ${f_{55}} = \cos \left( {T{\omega _k}} \right),{f_{62}} = {\omega _k}\left( {\cos \left( {T{\omega _k}} \right) - 1} \right)$
 ${f_{65}} = - {\omega _k}\sin \left( {T{\omega _k}} \right),{f_{66}} = 1$
 ${f_{77}} = 1,{f_{77}} = 1,{f_{78}} = T$
 ${f_{88}} = 1,{f_{1010}} = 1$

 $\mathit{\boldsymbol{ \boldsymbol{\varGamma} }} = \left[ {\begin{array}{*{20}{c}} {0.167{T^3}}&0&0\\ {0.5{T^2}}&0&0\\ T&0&0\\ 0&{0.167{T^3}}&0\\ 0&{0.5{T^2}}&0\\ 0&0&{0.167{T^3}}\\ 0&0&{0.5{T^2}}\\ 0&0&T\\ 0&0&0 \end{array}} \right]$ （16）

 $\mathit{\boldsymbol{B}}\left( k \right) = \left[ {\begin{array}{*{20}{c}} {\frac{{T\cos {\phi _k}}}{{{\omega _k}}} + \frac{{\sin {\phi _k}}}{{\omega _k^2}} - \frac{{\sin \left( {{\phi _k} + T{\omega _k}} \right)}}{{\omega _k^2}}}\\ {\frac{{\cos {\phi _k}}}{{{\omega _k}}} - \frac{{\cos \left( {{\phi _k} + T{\omega _k}} \right)}}{{{\omega _k}}}}\\ 0\\ {\frac{{\cos {\phi _k}}}{{\omega _k^2}} - \frac{{T\sin {\phi _k}}}{{{\omega _k}}} - \frac{{\cos \left( {{\phi _k} + T{\omega _k}} \right)}}{{\omega _k^2}}}\\ {\frac{{\sin \left( {{\phi _k} + T{\omega _k}} \right)}}{{{\omega _k}}} - \frac{{\sin {\phi _k}}}{{{\omega _k}}}}\\ 0\\ 0\\ 0\\ 0\\ \eta \end{array}} \right]$ （17）

 $\begin{array}{l} \left[ {\begin{array}{*{20}{l}} {x\left( {k + 1} \right)}\\ {\dot x\left( {k + 1} \right)}\\ {\ddot x\left( {k + 1} \right)}\\ {y\left( {k + 1} \right)}\\ {\dot y\left( {k + 1} \right)}\\ {\ddot y\left( {k + 1} \right)}\\ {z\left( {k + 1} \right)}\\ {\dot z\left( {k + 1} \right)}\\ {\ddot z\left( {k + 1} \right)}\\ {\omega \left( {k + 1} \right)} \end{array}} \right] = \mathit{\boldsymbol{F}}\left( k \right)\left[ {\begin{array}{*{20}{l}} {x\left( k \right)}\\ {\dot x\left( k \right)}\\ {\ddot x\left( k \right)}\\ {y\left( k \right)}\\ {\dot y\left( k \right)}\\ {\ddot y\left( k \right)}\\ {z\left( k \right)}\\ {\dot z\left( k \right)}\\ {\ddot z\left( k \right)}\\ {\omega \left( k \right)} \end{array}} \right] + \mathit{\boldsymbol{B}}\left( k \right){\alpha _{\rm{t}}}\left( k \right) + \\ \;\;\;\;\;\;\;\;\mathit{\boldsymbol{ \boldsymbol{\varGamma} W}}\left( t \right) \end{array}$ （18）

 $\begin{array}{l} \mathit{\boldsymbol{Q}} = \sigma _\omega ^2\int_0^T {\mathit{\boldsymbol{F}}\left[ {T,\omega } \right]{\mathit{\boldsymbol{F}}^{\rm{T}}}\left[ {T,\omega } \right]{\rm{d}}t} = \\ \;\;\;\;\;\;\;\sigma _\omega ^2\left[ {\begin{array}{*{20}{c}} {{q_{00}}}& \cdots &{{q_{09}}}\\ \vdots & \cdots & \vdots \\ {{q_{90}}}&{}&{{q_{99}}} \end{array}} \right] \end{array}$ （19）

 ${q_{00}} = \frac{{2{\omega ^5}T + {\omega ^3}T - {\omega ^2}\sin \left( {\omega T} \right)\cos \left( {\omega T} \right)}}{{2{\omega ^5}}} + \frac{{3\omega T - 4\sin \left( {\omega T} \right) + \sin \left( {\omega T} \right)\cos \left( {\omega T} \right)}}{{2{\omega ^5}}}$
 ${q_{01}} = - \frac{{{\omega ^2}\cos {{(\omega T)}^2} + 2\cos (\omega T) - \cos {{(\omega T)}^2} - {\omega ^2} - 1}}{{2{\omega ^4}}}$
 ${q_{02}} = \frac{{{\omega ^3}T - {\omega ^2}\sin (\omega T)\cos (\omega T) - 2\sin (\omega T) + \sin (\omega T)\cos (\omega T) + \omega T}}{{ - 2{\omega ^3}}}$
 ${q_{10}} = {q_{01}}$
 ${q_{11}} = \frac{{{\omega ^2}\sin (\omega T)\cos (\omega T) + {\omega ^3}T + \omega T - \sin (\omega T)\cos (\omega T)}}{{2{\omega ^3}}}$
 ${q_{12}} = \frac{{{\omega ^2}\cos {{(\omega T)}^2} - \cos {{(\omega T)}^2} - {\omega ^2} + 1}}{{2{\omega ^2}}},{q_{20}} = {q_{02}},{q_{21}} = {q_{12}}$
 ${q_{22}} = \frac{{{\omega ^3}T - {\omega ^2}\sin (\omega T)\cos (\omega T) + \sin (\omega T)\cos (\omega T) + \omega T}}{{2\omega }}$
 ${q_{33}} = {q_{00}},{q_{34}} = {q_{01}},{q_{35}} = {q_{02}},{q_{43}} = {q_{10}}$
 ${q_{44}} = {q_{11}},{q_{45}} = {q_{12}},{q_{53}} = {q_{20}},{q_{54}} = {q_{21}}$
 ${q_{55}} = {q_{22}},{q_{66}} = T + \frac{{{T^3}}}{3},{q_{67}} = \frac{{{T^2}}}{2}$
 ${q_{76}} = {q_{67}},{q_{77}} = T,{q_{99}} = \delta$

Q阵的其他元素为0。

 ${\alpha _{\rm{t}}}(k) = {\left[ {\alpha {{(k)}^2} - {\alpha _n}{{(k)}^2}} \right]^{1/2}}$ （20）

 ${\alpha _{\rm{n}}}(k) = v(k)\omega (k),v(k) = \sqrt {\dot x_{\rm{T}}^2 + \dot y_{\rm{T}}^2}$

αt(k)代入式(18)即可确定k时刻曲线模型的状态方程表达式。

3.2 自适应滤波方法

1) 因式分解

k-1时刻协方差矩阵Pk-1|k-1正定，对其进行因式分解得到Sk-1|k-1

 ${\mathit{\boldsymbol{P}}_{k - 1|k - 1}} = {\mathit{\boldsymbol{S}}_{k - 1|k - 1}}{\left( {{\mathit{\boldsymbol{S}}_{k - 1|k - 1}}} \right)^{\rm{T}}}$ （21）

2) 容积点估计

 $\mathit{\boldsymbol{x}}_{k - 1|k - 1}^i = {\mathit{\boldsymbol{S}}_{k - 1|k - 1}}{\mathit{\boldsymbol{\xi }}_i} + {{\mathit{\boldsymbol{\hat x}}}_{k - 1|k - 1}}$ （22）

3) 容积点传播

 $\mathit{\boldsymbol{x}}_{k|k - 1}^{ * ,i} = f\left( {\mathit{\boldsymbol{x}}_{k - 1|k - 1}^i} \right)$ （23）

4) 一步预测

 ${\mathit{\boldsymbol{x}}_{k|k - 1}} = \sum\limits_{i = 1}^L {\mathit{\boldsymbol{x}}_{k|k - 1}^{ * ,i}} /L$ （24）
 $\begin{array}{*{20}{c}} {{\mathit{\boldsymbol{P}}_{k|k - 1}} = \sum\limits_{i = 1}^L {\mathit{\boldsymbol{x}}_{k|k - 1}^{ * ,i}} {{\left( {\mathit{\boldsymbol{x}}_{k|k - 1}^{ * ,i}} \right)}^{\rm{T}}}/L - }\\ {{{\mathit{\boldsymbol{\hat x}}}_{k|k - 1}}{{\left( {{{\mathit{\boldsymbol{\hat x}}}_{k|k - 1}}} \right)}^{\rm{T}}} + {\mathit{\boldsymbol{Q}}_k}} \end{array}$ （25）

1) 因式分解

 ${\mathit{\boldsymbol{P}}_{k|k - 1}} = {\mathit{\boldsymbol{S}}_{k|k - 1}}{\left( {{\mathit{\boldsymbol{S}}_{k|k - 1}}} \right)^{\rm{T}}}$ （26）

2) 容积点估计

 $\mathit{\boldsymbol{x}}_{k|k - 1}^i = {\mathit{\boldsymbol{S}}_{k|k - 1}}{\mathit{\boldsymbol{\xi }}_i} + {{\mathit{\boldsymbol{\hat x}}}_{k|k - 1}}$ （27）

3) 容积点传播

 $\mathit{\boldsymbol{z}}_{k|k - 1}^i = \mathit{\boldsymbol{h}}\left( {\mathit{\boldsymbol{x}}_{k|k - 1}^i} \right)$ （28）

4) 一步预测

 ${{\mathit{\boldsymbol{\hat z}}}_{k|k - 1}} = \sum\limits_{i = 1}^L {\mathit{\boldsymbol{z}}_{k|k - 1}^i} /L$ （29）

5) 协方差矩阵估计

 $\mathit{\boldsymbol{P}}_{k|k - 1}^{zz} = \sum\limits_{i = 1}^L {\frac{{\mathit{\boldsymbol{z}}_{k|k - 1}^i{{\left( {\mathit{\boldsymbol{z}}_{k|k - 1}^i} \right)}^{\rm{T}}}}}{L}} - {{\mathit{\boldsymbol{\hat z}}}_{k|k - 1}}{\left( {{{\mathit{\boldsymbol{\hat z}}}_{k|k - 1}}} \right)^{\rm{T}}}$ （30）

 $\mathit{\boldsymbol{P}}_{k|k - 1}^{xz} = \sum\limits_{i = 1}^L {\frac{{\mathit{\boldsymbol{x}}_{k|k - 1}^i{{\left( {\mathit{\boldsymbol{x}}_{k|k - 1}^i} \right)}^{\rm{T}}}}}{L}} - {{\mathit{\boldsymbol{\hat x}}}_{k|k - 1}}{\left( {{{\mathit{\boldsymbol{\hat x}}}_{k|k - 1}}} \right)^{\rm{T}}}$ （31）

6) 滤波增益计算

 ${\mathit{\boldsymbol{K}}_k} = \mathit{\boldsymbol{P}}_{k|k - 1}^{xz}{\left( {\mathit{\boldsymbol{P}}_{k|k - 1}^{zz}} \right)^{\rm{T}}}$ （32）

7) 状态估计

 ${{\mathit{\boldsymbol{\hat x}}}_{k|k}} = {{\mathit{\boldsymbol{\hat x}}}_{k|k - 1}} + {\mathit{\boldsymbol{K}}_k}\left( {{\mathit{\boldsymbol{z}}_k} - {{\mathit{\boldsymbol{\hat z}}}_{k|k - 1}}} \right)$ （33）

8) 估计误差协方差计算

 ${\mathit{\boldsymbol{P}}_{k|k}} = {\mathit{\boldsymbol{P}}_{k|k - 1}} - {\mathit{\boldsymbol{K}}_k}\mathit{\boldsymbol{P}}_{k|k - 1}^{xz}{\left( {{\mathit{\boldsymbol{K}}_k}} \right)^{\rm{T}}}$ （34）
3.3 方向角模型建立

 图 3 方向角转移示意图 Fig. 3 Schematic diagram of heading angle transfer

$\phi_{k}=-\arctan \frac{\dot{x}(k)}{\dot{y}(k)}+\Delta \phi$，其中Δϕ表示方向角跨象限时的计算补偿量，记i时刻的补偿量记作

 $\Delta {\phi _i} = \left\{ \begin{array}{l} - 2{\rm{ \mathsf{ π} }}{n_1}\;\;{Q_4}\;转\;{Q_3},{n_1} + 1\\ 2{\rm{ \mathsf{ π} }}{n_2}\;\;\;\;{Q_3}\;转\;{Q_4},{n_2} + 1\\ - {\rm{ \mathsf{ π} }}{n_3}\;\;\;\;{Q_1}\;转\;{Q_3}\;或\;{Q_4}\;转\;{Q_2},{n_3} + 1\\ {\rm{ \mathsf{ π} }}{n_4}\;\;\;\;\;\;{Q_3}\;转\;{Q_1}\;或\;{Q_2}\;转\;{Q_4},{n_4} + 1\\ 0\;\;\;\;\;\;\;\;\;\;其他 \end{array} \right.$ （35）

4 仿真实验与分析

 图 4 双机组网定位态势图 Fig. 4 Situation map of location by two planes

 图 5 方向角估计结果 Fig. 5 Results of heading angle estimation
 图 6 转弯率估计精度比较 Fig. 6 Comparison of accuracy of turning rate estimation
 图 7 滤波性能对比 Fig. 7 Comparison of filtering performance

5 结论

1) 将转弯率和目标线运动加速度作为状态变量进行了状态扩维，提高了切向加速度的估计精度，从而增强了曲线模型的自适应性。

2) 针对方向角在4个象限间转移时存在波动的问题，基于反正切函数arctan ()的值域，结合四象限空间关系，优化了方向角的设计方法，从而提高了对转弯角速率的估计精度。

 $\ddot x(t) = v\omega \cos \phi (t) = \omega \dot y$ （A1）
 $\ddot y(t) = - v\omega \sin \phi (t) = - \omega \dot x$ （A2）

 $\dddot x(t) = - v{\omega ^2}\sin \phi (t) = - {\omega ^2}\dot x$ （A3）
 $\ddot y(t) = - v{\omega ^2}\cos \phi (t) = - {\omega ^2}\dot y$ （A4）

 $\begin{gathered} \left[ {\begin{array}{*{20}{l}} {\dot x(t)} \\ {\ddot x(t)} \\ {\dddot x(t)} \\ {\dot y(t)} \\ {\ddot y(t)} \\ {\dddot y(t)} \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} 0&1&0&0&0&0 \\ 0&0&0&0&{ - \omega }&0 \\ 0&{ - {\omega ^2}}&0&0&0&0 \\ 0&0&0&0&1&0 \\ 0&\omega &0&0&0&0 \\ 0&0&0&0&{ - {\omega ^2}}&0 \end{array}} \right]\left[ {\begin{array}{*{20}{l}} {x(t)} \\ {\dot x(t)} \\ {\ddot y(t)} \\ {y(t)} \\ {\dot y(t)} \\ {\ddot y(t)} \end{array}} \right] + \hfill \\ \;\;\;\;\;\;\mathit{\boldsymbol{W}}\left( t \right) \hfill \\ \end{gathered}$ （A5）

 $\mathit{\boldsymbol{A}} = \left[ {\begin{array}{*{20}{c}} 0&1&0&0&0&0\\ 0&0&0&0&{ - \omega }&0\\ 0&{ - {\omega ^2}}&0&0&0&0\\ 0&0&0&0&1&0\\ 0&\omega &0&0&0&0\\ 0&0&0&0&{ - {\omega ^2}}&0 \end{array}} \right]$ （A6）

 $s\mathit{\boldsymbol{I}} - \mathit{\boldsymbol{A}} = \left[ {\begin{array}{*{20}{c}} s&{ - 1}&0&0&0&0\\ 0&s&0&0&\omega &0\\ 0&{{\omega ^2}}&s&0&0&0\\ 0&0&0&s&{ - 1}&0\\ 0&{ - \omega }&0&0&s&0\\ 0&0&0&0&{{\omega ^2}}&s \end{array}} \right]$ （A7）
 ${\left( {s\mathit{\boldsymbol{I}} - \mathit{\boldsymbol{A}}} \right)^{ - 1}} = \left[ {\begin{array}{*{20}{c}} {\frac{1}{s}}&{\frac{1}{{{s^2} + {\omega ^2}}}}&0&0&{ - \frac{\omega }{{s\left( {{s^2} + {\omega ^2}} \right)}}}&0\\ 0&{\frac{s}{{{s^2} + {\omega ^2}}}}&0&0&{\frac{{ - \omega }}{{{s^2} + {\omega ^2}}}}&0\\ 0&{\frac{{ - {\omega ^2}}}{{{s^2} + {\omega ^2}}}}&{\frac{1}{s}}&0&{\frac{{{\omega ^3}}}{{s\left( {{s^2} + {\omega ^2}} \right)}}}&0\\ 0&{\frac{\omega }{{s\left( {{s^2} + {\omega ^2}} \right)}}}&0&{\frac{1}{s}}&{\frac{1}{{{s^2} + {\omega ^2}}}}&0\\ 0&{\frac{\omega }{{{s^2} + {\omega ^2}}}}&0&0&{\frac{s}{{{s^2} + {\omega ^2}}}}&0\\ 0&{\frac{{ - {\omega ^3}}}{{s\left( {{s^2} + {\omega ^2}} \right)}}}&0&0&{\frac{{ - {\omega ^2}}}{{{s^2} + {\omega ^2}}}}&{\frac{1}{s}} \end{array}} \right]$ （A8）
 ${\mathit{\boldsymbol{F}}_k} = {\mathit{\boldsymbol{L}}^{ - 1}}{\left( {s\mathit{\boldsymbol{I}} - \mathit{\boldsymbol{A}}} \right)^{ - 1}} = \left[ {\begin{array}{*{20}{c}} 1&{\frac{{\sin (\omega T)}}{\omega }}&0&0&{ - \frac{{1 - \cos (\omega T)}}{\omega }}&0\\ 0&{\cos (\omega T)}&0&0&{ - \sin (\omega T)}&0\\ 0&{ - \omega \sin (\omega T)}&1&0&{ - \omega (\cos (\omega T) - 1)}&0\\ 0&{\frac{{1 - \cos (\omega T)}}{\omega }}&0&1&{\frac{{\sin (\omega T)}}{\omega }}&0\\ 0&{\sin (\omega T)}&0&0&{\cos (\omega T)}&0\\ 0&{\omega (\cos (\omega T) - 1)}&0&0&{ - \omega \sin (\omega T)}&1 \end{array}} \right]$ （A9）

 ${\mathit{\boldsymbol{X}}_k} = {\left[ {\begin{array}{*{20}{c}} {{x_{\rm{T}}}}&{{{\dot x}_{\rm{T}}}}&{{{\ddot x}_{\rm{T}}}}&{{y_{\rm{T}}}}&{{{\dot y}_{\rm{T}}}}&{{{\ddot y}_{\rm{T}}}}&{{z_{\rm{T}}}}&{{{\dot z}_{\rm{T}}}}&{{{\ddot z}_{\rm{T}}}}&\omega \end{array}} \right]^{\rm{T}}}$

 $\mathit{\boldsymbol{F}} = \left[ {\begin{array}{*{20}{c}} {}&{{\mathit{\boldsymbol{F}}_k}}&{{{\bf{0}}_{6 \times 4}}}&{}&{}\\ {}&1&T&0&0\\ {{{\bf{0}}_{4 \times 6}}}&0&1&0&0\\ {}&0&0&0&1 \end{array}} \right]$ （A10）

 $\mathit{\boldsymbol{FF}} = \mathit{\boldsymbol{F}}\left[ {T,\omega } \right] \cdot {\mathit{\boldsymbol{F}}^{\rm{T}}}\left[ {T,\omega } \right] = \left[ {\begin{array}{*{20}{c}} {{q_{00}}}& \cdots &{{q_{09}}}\\ \vdots & \cdots & \vdots \\ {{q_{90}}}& \cdots &{{q_{99}}} \end{array}} \right]$ （B1）
 ${q_{00}} = \frac{{{\omega ^2} + 2 - 2\cos \left( {\omega T} \right)}}{{{\omega ^2}}}$ （B2）
 ${q_{01}} = \frac{{\sin \left( {\omega T} \right)}}{\omega }$ （B3）
 ${q_{02}} = - 2 + 2\cos \left( {\omega T} \right)$ （B4）
 ${q_{04}} = \frac{{1 - \cos \left( {\omega T} \right)}}{\omega }$ （B5）
 ${q_{10}} = \frac{{\sin \left( {\omega T} \right)}}{\omega }$ （B6）
 ${q_{11}} = 1$ （B7）
 ${q_{12}} = - \omega \sin \left( {\omega T} \right)$ （B8）
 ${q_{13}} = - \frac{{1 - \cos \left( {\omega T} \right)}}{\omega }$ （B9）
 ${q_{15}} = - \omega \left( {\cos \left( {\omega T} \right) - 1} \right)$ （B10）
 ${q_{20}} = - 2 + 2\cos \left( {\omega T} \right)$ （B11）
 ${q_{21}} = - \omega \sin \left( {\omega T} \right)$ （B12）
 ${q_{22}} = 1 + 2{\omega ^2} - 2{\omega ^2}\cos \left( {\omega T} \right)$ （B13）
 ${q_{24}} = \omega \left( {\cos \left( {\omega T} \right) - 1} \right)$ （B14）
 ${q_{31}} = - \frac{{1 - \cos \left( {\omega T} \right)}}{\omega }$ （B15）
 ${q_{33}} = \frac{{{\omega ^2} + 2 - 2\cos \left( {\omega T} \right)}}{{{\omega ^2}}}$ （B16）
 ${q_{34}} = \frac{{\sin (\omega T)}}{\omega }$ （B17）
 ${q_{35}} = - 2 + 2\cos (\omega T)$ （B18）
 ${q_{40}} = \frac{{1 - \cos (\omega T)}}{\omega }$ （B19）
 ${q_{42}} = \omega (\cos (\omega T) - 1)$ （B20）
 ${q_{43}} = \frac{{\sin (\omega T)}}{\omega }$ （B21）
 ${q_{44}} = 1$ （B22）
 ${q_{45}} = - \omega \sin (\omega T)$ （B23）
 ${q_{51}} = - \omega (\cos (\omega T) - 1)$ （B24）
 ${q_{53}} = - 2 + 2\cos (\omega T)$ （B25）
 ${q_{54}} = - \omega \sin (\omega \cdot T)$ （B26）
 ${q_{55}} = 1 + 2{\omega ^2} - 2{\omega ^2}\cos (\omega T)$ （B27）
 ${q_{54}} = - \omega \sin (\omega \cdot T)$ （B28）
 ${q_{66}} = 1 + {T^2}$ （B29）
 ${q_{66}} = 1 + {T^2}$ （B30）
 ${q_{76}} = T$ （B31）
 ${q_{77}} = 1$ （B32）
 ${q_{99}} = 1$ （B33）

 $\mathit{\boldsymbol{Q}} = \sigma _\omega ^2\int_0^T {\mathit{\boldsymbol{FF}}{\rm{d}}t}$ （B34）
 $= \sigma _\omega ^2\left[ {\begin{array}{*{20}{c}} {{q_{00}}}& \cdots &{{q_{09}}}\\ \vdots & \cdots & \vdots \\ {{q_{90}}}& \cdots &{{q_{99}}} \end{array}} \right]$ （B35）

 ${q_{00}} = \frac{{{\omega ^3}T + 2\omega T - 2\omega \sin \left( {\omega T} \right)}}{{{\omega ^3}}}$ （B36）
 ${q_{01}} = \frac{{\cos \left( {\omega T} \right)}}{{{\omega ^2}}}$ （B37）
 ${q_{02}} = \frac{{ - 2\omega T + 2\sin \left( {\omega T} \right)}}{\omega }$ （B38）
 ${q_{04}} = \frac{{\sin \left( {\omega T} \right) - \omega T}}{{{\omega ^2}}}$ （B39）
 ${q_{20}} = \frac{{ - \cos \left( {\omega T} \right)}}{{{\omega ^2}}}$ （B40）
 ${q_{21}} = \cos \left( {\omega T} \right)$ （B41）
 ${q_{22}} = T + 2{\omega ^2}T - 2\omega \sin \left( {\omega T} \right)$ （B42）
 ${q_{24}} = \sin \left( {\omega T} \right) - \omega T$ （B43）
 ${q_{31}} = \frac{{\sin \left( {\omega T} \right) - \omega T}}{{{\omega ^2}}}$ （B44）
 ${q_{33}} = \frac{{{\omega ^3}T + 2\omega T - 2\omega \sin \left( {\omega T} \right)}}{{{\omega ^3}}}$ （B45）
 ${q_{34}} = \frac{{ - \cos \left( {\omega T} \right)}}{{{\omega ^2}}}$ （B46）
 ${q_{35}} = \frac{{ - 2\omega T + 2\sin \left( {\omega T} \right)}}{\omega }$ （B47）
 ${q_{40}} = \frac{{\sin \left( {\omega T} \right) - \omega T}}{{{\omega ^2}}}$ （B48）
 ${q_{42}} = \sin \left( {\omega T} \right) - \omega T$ （B49）
 ${q_{43}} = \frac{{ - \cos \left( {\omega T} \right)}}{{{\omega ^2}}}$ （B50）
 ${q_{44}} = T$ （B51）
 ${q_{45}} = \cos \left( {\omega T} \right)$ （B52）
 ${q_{51}} = - \sin \left( {\omega T} \right) + \omega T$ （B53）
 ${q_{53}} = \frac{{ - 2\omega T + 2\sin \left( {\omega T} \right)}}{\omega }$ （B54）
 ${q_{54}} = \cos \left( {\omega T} \right)$ （B55）
 ${q_{55}} = T + 2{\omega ^2}T - 2\omega \sin \left( {\omega T} \right)$ （B56）
 ${q_{66}} = \frac{{{T^3} + 3T}}{3}$ （B57）
 ${q_{67}} = \frac{{{T^2}}}{2}$ （B58）
 ${q_{76}} = \frac{{{T^2}}}{2}$ （B59）
 ${q_{77}} = T$ （B60）
 ${q_{99}} = T$ （B61）

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http://dx.doi.org/10.7527/S1000-6893.2019.22988

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#### 文章信息

ZHANG Feng

A new method for double IRST cooperative passive detection and positioning for maneuvering target

Acta Aeronautica et Astronautica Sinica, 2020, 41(2): 322988.
http://dx.doi.org/10.7527/S1000-6893.2019.22988