﻿ 基于带约束最小二乘的超短基线安装偏差校准
 舰船科学技术  2021, Vol. 43 Issue (7): 131-136    DOI: 10.3404/j.issn.1672-7649.2021.07.027 PDF

USBL calibration for installation bias based on constrained least square
ZHAO Jun-bo, GE Xi-yun, LI Jin, CHENG Yue
China Ship Scientific Research Center, Wuxi 214000, China
Abstract: Ultra short baseline(USBL) systems are widely used in channel and port construction, ocean exploration in recent years.In order to realize its hign precision positioning, USBL needs to be calibrated accurately beforehand. Aiming at the problem of USBL calibration for installation bias, this paper proposed an calibration method based on constrained least square, and also put forward the handing method on condition of sound speed measurement error. Simulation result shows that the constrained least square method is suitable for USBL calibration, and has ability of gross error resistance; when there exists sound speed measurement error, the proposed handing method is efficient. The purpose of this paper is to provide a reference for USBL accurate calibration for installation bias.
Key words: USBL     calibration     constrained least square     sound speed measurement error
0 引　言

USBL定位系统，一般由应答器、声基阵、母船航姿及定位系统组成。其中，应答器固定安装在水下目标上，其余设备固定安装在母船上。超短基线定位系统通过实时测量水下目标相对于母船的相对位置矢量，再结合母船的实时位置信息，实现对水下目标的精确定位。在实际应用时，声基阵相对于母船的安装偏差是影响该类系统定位精度的一个重要因素[5-6]。一般来说，1°的首向安装偏角将会产生1.7%斜距的定位误差[7]，因此，在使用该类系统进行定位之前，必须先对声基阵安装偏差进行校准。

1 超短基线的基本测量原理

$p_{AB}^u = {[{x_a},\; \; \; {y_a},\; \; \; {z_a}]^{\rm{T}}}$ 为声基阵中心A点至应答器中心B点的相对位置矢量（见图1），根据超短基线测量原理可知[4]

 图 1 相对位置矢量示意图 Fig. 1 Schematic diagram of relative position vector
 $\left\{ {\begin{array}{*{20}{c}} {{x_a} = \dfrac{c}{{2{\text{π}} df}}l{\phi _x},}\\ {{y_a} = \dfrac{c}{{2{\text{π}} df}}l{\phi _y},}\\ {{z_a} = \sqrt {{l^2} - {{\left(\dfrac{{cl{\phi _x}}}{{2{\text{π}} df}}\right)}^2} - {{\left(\dfrac{{cl{\phi _y}}}{{2{\text{π}} df}}\right)}^2}} } {\text。}\end{array}} \right.$ (1)

2 超短基线安装偏差的校准方法 2.1 校准问题及其一般求解方法

 $p_{OB}^b = p_{OA}^b + {{C}}_u^bp_{AB}^u{\text{。}}$ (2)

 $p_{OB}^b = C_n^b(p_{NB}^n - p_{NO}^n){\text{。}}$ (3)

 $\hat p_{OB}^b = \hat p_{OA}^b + \hat {{C}}_u^bp_{AB}^u {\text。}$ (4)

2.2 基于带约束最小二乘的校准方法

1）带旋转矩阵约束的最小二乘估计法

 $\left\{ {\begin{array}{*{20}{l}} {{\mu _x} = \dfrac{1}{N}\displaystyle\sum\limits_{i = 1}^N {{x_i}} } {\text，}\\ {{\mu _y} = \dfrac{1}{N}\displaystyle\sum\limits_{i = 1}^N {{y_i}} } {\text，}\\ {\sigma _x^2 = \dfrac{1}{N}\displaystyle\sum\limits_{i = 1}^N {{{\left\| {{x_i} - {\mu _x}} \right\|}^2}} } {\text，}\\ {\sigma _y^2 = \dfrac{1}{N}\displaystyle\sum\limits_{i = 1}^N {{{\left\| {{y_i} - {\mu _y}} \right\|}^2}} }{\text，}\\ {{\displaystyle\sum _{xy}} = \dfrac{1}{N}\displaystyle\sum\limits_{i = 1}^N {({y_i} - {\mu _y})} {{({x_i} - {\mu _x})}^{\rm{T}}}} {\text，} \end{array}} \right.$ (5)

 $\left\{ {\begin{array}{*{20}{l}} {{\displaystyle\sum _{xy}} = UD{V^{\rm{T}}}{\text，}\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;} \\ {D\; = {\rm{diag}}({d_i}),\;\;\;{d_1} \geqslant \cdots \geqslant {d_m} \geqslant 0} {\text，} \end{array}} \right.$ (6)

 ${e^2}({{R}},t,c) = \dfrac{1}{N}\sum\limits_{i = 1}^N {{{\left\| {{y_i} - (c{{R}}{x_i} + t)} \right\|}^2}}{\text，}$ (7)

 ${\varepsilon ^2} = \sigma _y^2 - {{tr{{(DS)}^2}} / {\sigma _x^2}}{\text，}$ (8)

 $\left\{ {\begin{array}{*{20}{c}} {{{R}} = US{V^{\rm{T}}}{\text，}\;\;\;\;\;\;\;\;\;} \\ {t = {\mu _y} - c{{R}}{\mu _x}{\text，}\;\;} \\ {c = {{tr{{(DS)}^2}} / {\sigma _x^2}}} {\text{。}} \end{array}} \right.$ (9)

 $S = \left\{ {\begin{array}{*{20}{c}} {I\;,\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\det \left({\displaystyle\sum _{xy}}\right) \geqslant 0} {\text，}\\ {{\rm{diag}}(1,1, \cdots ,1, - 1)\;,\;\;\det \left({\displaystyle\sum _{xy}}\right) < 0}{\text。} \end{array}} \right.$ (10)

2）安装偏差的校准思路

2.3 声速测量误差条件下的处理方法

1）固定声速误差的影响分析

${p^*}= {[{x^*},\;\; {y^*},\;\; {z^*}]^{\rm{T}}}$ $\tilde p = {[\tilde x,\;\; \tilde y,\;\; \tilde z]^{\rm{T}}}$ $\Delta p = [\Delta x,\;\; \Delta y,\;\;$ $\Delta z]^{\rm{T}}$ 分别为相对位置矢量 $p_{AB}^u$ 的真值、测量值和测量误差值， ${c^*}$ $\tilde c$ $\Delta c$ 分别为声速 $c$ 的真值、测量值和测量误差值。因此， $p_{AB}^u$ 各分量的真值与测量值的比值（以下简称真测比）为：

 $\left\{ {\begin{array}{*{20}{l}} {\dfrac{{{x^*}}}{{\tilde x}} = \dfrac{{\tilde x - \left( {{{\left. {\dfrac{{\partial p_{AB}^u}}{{\partial c}}(1)} \right|}_{c = \tilde c}}} \right)\Delta c}}{{\tilde x}} = 1 - \dfrac{{2 \cdot \Delta c}}{{\tilde c}}} {\text，}\\ {\dfrac{{{y^*}}}{{\tilde y}} = \dfrac{{\tilde y - \left( {{{\left. {\dfrac{{\partial p_{AB}^u}}{{\partial c}}(2)} \right|}_{c = \tilde c}}} \right)\Delta c}}{{\tilde y}} = 1 - \dfrac{{2 \cdot \Delta c}}{{\tilde c}}} {\text，}\\ {\dfrac{{{z^*}}}{{\tilde z}} = \dfrac{{\tilde z - \left( {{{\left. {\dfrac{{\partial p_{AB}^u}}{{\partial c}}(3)} \right|}_{c = \tilde c}}} \right)\Delta c}}{{\tilde z}} = 1 - \left( {1 - \dfrac{1}{{2{{\cos }^2}{\theta _z}}}} \right)\dfrac{{2 \cdot \Delta c}}{{\tilde c}}} {\text{。}} \end{array} } \right.$ (11)

2）处理方法及校准思路

$\dfrac{{{z^*}}}{{m \cdot \tilde z}} = \dfrac{{{x^*}}}{{\tilde x}} = \dfrac{{{y^*}}}{{\tilde y}}$ ，则比例系数为：

 $m = \dfrac{{1 - \left( {1 - \dfrac{1}{{2{{\cos }^2}{\theta _z}}}} \right)\dfrac{{2 \cdot \Delta c}}{{\tilde c}}}}{{1 - \dfrac{{2 \cdot \Delta c}}{{\tilde c}}}}{\text，}$ (12)

 $\hat m = \dfrac{{1 - \left( {1 - \dfrac{1}{{2{{\cos }^2}{\theta _z}}}} \right)\dfrac{{2 \cdot \Delta \hat c}}{{\tilde c}}}}{{1 - \dfrac{{2 \cdot \Delta \hat c}}{{\tilde c}}}}{\text。}$ (13)

1）获取2组观测值 $\bigg\{ \tilde p_{AB,i}^u\bigg\} (i = 1,\cdots,N)$ $\bigg\{ \tilde p_{OB,i}^b\bigg\} (i =$ $1,\cdots,N)$ ，建立模型 $\tilde p_{OB,i}^b = \hat p_{OA}^b + {k_1}\hat C_u^b\tilde p_{AB,i}^u\;(i = 1, \cdots , N)$ ，应用带约束最小二乘法求解尺度因子 ${k_1}$

2）根据尺度因子 ${k_1}$ 和声速测量值 $\tilde c$ ，利用公式 $\Delta \hat c = \dfrac{{\tilde c}}{2}\left( {1 - {k_1}} \right)$ 估算声速误差 $\Delta \hat c$

3）将 $\Delta \hat c$ 代入式（7），估算比例系数 $\hat m$

4）根据 $\hat m$ 计算 $\bigg\{ p_i^u\bigg\} (i = 1,\cdots,N)$

5）针对矢量集 $\bigg\{ p_i^u\bigg\} (i = 1,\cdots,N)$ 和矢量集 $\bigg\{ \tilde p_{OB,i}^b\bigg\} (i =$ $1,\cdots,N)$ ，建立模型 $\tilde p_{OB,i}^b = \hat p_{OA}^b + {k_2}\hat C_u^bp_i^u\;(i = 1, \cdots ,N)$ ，应用带约束最小二乘法求解 $\hat C_u^b$ $\hat p_{OA}^b$

3 仿真分析

 图 2 航行轨迹图 Fig. 2 Navigation trajectory

1)仿真实验1

2)仿真实验2

 图 3 校准误差随声速误差的变化曲线 Fig. 3 Curve of calibration error versus sound velocity error

4 结　语

1)针对超短基线安装偏差的校准问题，提出应用带约束的最小二乘法来进行校准。仿真结果表明，该校准方法适用于求解超短基线安装偏差校准问题，并且具有一定的抗粗差能力。

2)在上述校准方法的基础上。仿真结果表明，该方法具有一定的抗粗差能力和抗固定声速测量误差能力，因此，该方法适用于存在固定声速测量误差的情况。

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