﻿ 基于船舶航行定位参数识别的数学模型设计
 舰船科学技术  2023, Vol. 45 Issue (13): 142-145    DOI: 10.3404/j.issn.1672-7649.2023.13.028 PDF

Research on mathematical model design based on ship navigation positioning parameter identification
LI Zi-ling
College of Information Engineering, Wuhan Huaxia Institute of Technology, Wuhan 430223, China
Abstract: In order to improve the performance of ship navigation positioning system by using the optimal parameters, a mathematical model based on ship navigation positioning parameters identification is designed.The geodetic coordinate system and shipboard coordinate system are constructed, and the nonlinear motion equation of ship navigation positioning system is constructed by the conversion of the two coordinate systems. Based on the motion equation and the untracked Kalman filter method, the mathematical model of ship navigation positioning system parameter identification is designed. The particle swarm optimization algorithm was used to optimize the designed model, and the covariance matrix of the initial state errors of positioning parameters was selected as the optimization target. The covariance matrix of Kalman filter algorithm was determined by updating the particle position and velocity, and the state vector of parameters to be identified was updated to output the identification results of ship navigation positioning parameters. The experimental results show that the model can realize the identification of navigation positioning parameters of ships, and ensure that the actual navigation trajectory of ships closely follow the target navigation trajectory according to the identification results of parameters.
Key words: navigation of ships     positioning parameter identification     mathematical model     Kalman filter     covariance matrix     particle swarm optimization
0 引　言

1 船舶航行定位参数识别 1.1 船舶航行动力定位系统的非线性运动方程

 $\dot \eta = {\boldsymbol{R}}\left( \phi \right)\upsilon，$ (1)

 $R\left( \phi \right) = \left[ {\begin{array}{*{20}{c}} {\cos \phi }&{ - \sin \phi }&0 \\ {\sin \phi }&{ - \cos \phi }&0 \\ 0&0&1 \end{array}} \right] 。$ (2)

 $\begin{gathered} {\boldsymbol{M}}\dot \upsilon + {{\boldsymbol{C}}_R}\upsilon + {{\boldsymbol{C}}_A}\upsilon + {{\boldsymbol{D}}_L}\upsilon + {V_R}\left( {{\upsilon _r},\gamma } \right)= \\ {\tau _1} + {\tau _2} + {\tau _3}。\\ \end{gathered}$ (3)

1.2 基于无迹卡尔曼滤波的参数识别数学模型

1）初始化

 $\left\{ {\begin{array}{*{20}{c}} {{{\hat w}_0} = E\left( {{w_0}} \right)} ，\\ {{P_0} = E \geqslant \left( {\left( {{w_0} - {{\hat w}_0}} \right){{\left( {{w_0} - {{\hat w}_0}} \right)}^{\rm{T}}}} \right)} 。\end{array}} \right.$ (4)

2）计算船舶航行定位采样点及其权值

 ${\chi _{k - 1}} = \left[ {{{\hat w}_{k - 1}},{{\hat w}_{k - 1}} + \sqrt {n\beta {P_{k - 1}}} ,{{\hat w}_{k - 1}} - \sqrt {n\beta {P_{k - 1}}} } \right]，$ (5)

 ${\varphi _{0m}} = \frac{\beta }{{n + \beta }} ，$ (6)
 ${\varphi _{0c}} = \frac{\beta }{{n + \beta }} + \left( {1 - {\varepsilon ^2} + \xi } \right) ，$ (7)
 ${\varphi _{im}} = {\varphi _{ic}} = \frac{\beta }{{2\left( {n + \beta } \right)}}。$ (8)

3）船舶航行定位采样点更新

 ${\hat w_{k\left| {k - 1} \right.}} = \sum\limits_{i = 0}^{2n} {{\varphi _{im}}{\chi _{k - 1}}}，$ (9)

 ${P_{k\left| {k - 1} \right.}} = \sum\limits_{i = 0}^{2n} {{\varphi _{ic}}\left( {{\chi _{k - 1}} - {{\hat w}_{k\left| {k - 1} \right.}}} \right)} {\left( {{\chi _{k - 1}} - {{\hat w}_{k\left| {k - 1} \right.}}} \right)^{\rm{T}}}，$ (10)

 $\begin{split} {\chi _{k\left| {k - 1} \right.}} = &\left[ {{{\hat w}_{k\left| {k - 1} \right.}},} \right.{{\hat w}_{k\left| {k - 1} \right.}} + \\ & \left. {\sqrt {\left( {n + \beta } \right){P_{k\left| {k - 1} \right.}}} ,{{\hat w}_{k\left| {k - 1} \right.}} - \sqrt {\left( {n + \beta } \right){P_{k\left| {k - 1} \right.}}} } \right] 。\end{split}$ (11)

4）确定协方差矩阵

 ${\hat y_k} = \sum\limits_{i = 0}^{2n} {{\varphi _{im}}\left[ {\chi _{i,k\left| {k - 1} \right.}^{\rm{T}}X\left( k \right) - \left( {\chi _{i,k\left| {k - 1} \right.}^{\rm{T}}X\left( {k - 1} \right)} \right)} \right]}$ (12)

 $\begin{split} {P_y} = &\sum\limits_{i = 0}^{2n} {{\varphi _{ic}}} \left( {\chi _{i,k\left| {k - 1} \right.}^{\rm{T}}X\left( k \right) - \left( {\chi _{i,k\left| {k - 1} \right.}^{\rm{T}}X\left( {k - 1} \right)} \right)} \right)\times \\ &{\left( {\chi _{i,k\left| {k - 1} \right.}^{\rm{T}}X\left( k \right) - \left( {\chi _{i,k\left| {k - 1} \right.}^{\rm{T}}X\left( {k - 1} \right)} \right) - {{\hat y}_k}} \right)^{\rm{T}}} ，\end{split}$ (13)

 $\begin{split} {P_w} = &\sum\limits_{i = 0}^{2n} {{\varphi _{ic}}} \left( {\chi _{i,k\left| {k - 1} \right.}^{} - {{\hat w}_{k\left| {k - 1} \right.}}} \right) - \\ & {{\hat y}_k}{\left( {\chi _{i,k\left| {k - 1} \right.}^{\rm{T}}X\left( k \right) - f\left( {\chi _{i,k\left| {k - 1} \right.}^{\rm{T}}X\left( {k - 1} \right)} \right)} \right)^{\rm{T}}} 。\end{split}$ (14)

5）更新参数状态向量及协方差矩阵

 ${K_k} = {P_w}{\left( {{P_y}} \right)^{ - 1}}，$ (15)

 ${\hat w_k} = {\hat w_{k\left| {k - 1} \right.}} + {K_k}\left( {{{\hat y}_k} - {{\hat y}_{k\left| {k - 1} \right.}}} \right)，$ (16)

 ${P_k} = {P_{k\left| {k - 1} \right.}} - {K_k}{P_y}K_k^{\rm{T}}。$ (17)

1.3 基于粒子群算法的定位参数识别优化

 ${v_{i + 1}} = \omega {v_i} + {c_1}{r_1}\left( {{q_i} - {x_i}} \right) + {c_2}{r_2}\left( {{g_i} - {x_i}} \right)，$ (18)
 ${x_{i + 1}} = {x_i} + {v_{i + 1}}。$ (19)

 $\omega = {\omega _{\max }} - k\left( {{\omega _{\max }} - {\omega _{\min }}} \right)/{k_{\max }} 。$ (20)

2 实例分析

 图 1 Z型操作实验 Fig. 1 Z-type operation experiment

 图 2 船舶航行定位参数识别结果 Fig. 2 Identification results of ship navigation positioning parameters

 图 3 船舶速度统计结果 Fig. 3 Statistical results of ship speed

 图 4 船舶航迹图 Fig. 4 Ship track chart
3 结　语

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