﻿ 一种应变片估计挠曲变形的快速传递对准方法
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 哈尔滨工程大学学报  2019, Vol. 40 Issue (6): 1142-1148  DOI: 10.11990/jheu.201811081 0

### 引用本文

ZHANG Lining, XU Xu, HE Kunpeng. Fast transfer alignment based on deflection angle estimated by strain gauge[J]. Journal of Harbin Engineering University, 2019, 40(6), 1142-1148. DOI: 10.11990/jheu.201811081.

### 文章历史

1. 北京机电工程总体设计部, 北京 100854;
2. 哈尔滨工程大学 自动化学院, 黑龙江 哈尔滨 150001

Fast transfer alignment based on deflection angle estimated by strain gauge
ZHANG Lining 1, XU Xu 2, HE Kunpeng 2
1. Beijing General Design Department of Electromechanical Engineering, Beijing 100854, China;
2. College of Automation, Harbin Engineering University, Harbin 150001, China
Abstract: In the traditional transfer alignment, using the second-order Markov process to empirically model the dynamic deformation angle can result in inaccuracy. To solve this problem, an inverse modeling method based on measured data is proposed. It involves recording the actual deflection angle in the motion process by strain gauges, and the second-order differential equation and the state space correlation coefficient are deduced. This can effectively solve the problem of the traditional scheme that requires related parameters to be repeatedly modified and tried. Experiments prove that the transfer alignment scheme adopting this method can not only realize rapid convergence of misalignment angle to a reliable accuracy within 10 s after the maneuvering is finished, but also achieve a final accuracy that is within 3'.
Keywords: transfer alignment    strain gauges    dynamic deformation angle    inertial navigation    Kalman filter    Markov process    misalignment angle    matching with computation parameter

1 传递对准基础模型 1.1 传统的传递对准模型

 Download: 图 1 传递对准的2种变形角示意 Fig. 1 Two types deformation angles of transfer alignment

 $\begin{gathered} \mathit{\boldsymbol{\dot X}} = \left[ {\begin{array}{*{20}{c}} { - \left( {\omega _{in}^n \times } \right)}&{{{\bf{0}}_{3 \times 3}}}&{ - \mathit{\boldsymbol{C}}_{bs}^n}&{{{\bf{0}}_{3 \times 3}}}&{{{\bf{0}}_{3 \times 3}}} \\ {{\mathit{\boldsymbol{M}}_1}}&{{\mathit{\boldsymbol{M}}_2}}&{{{\bf{0}}_{3 \times 3}}}&{{\mathit{\boldsymbol{M}}_3}}&{{{\bf{0}}_{3 \times 3}}} \\ {{{\bf{0}}_{3 \times 3}}}&{{{\bf{0}}_{3 \times 3}}}&{{{\bf{0}}_{3 \times 3}}}&{{{\bf{0}}_{3 \times 3}}}&{{{\bf{0}}_{3 \times 3}}} \\ {{{\bf{0}}_{3 \times 3}}}&{{{\bf{0}}_{3 \times 3}}}&{{{\bf{0}}_{3 \times 3}}}&{{{\bf{0}}_{3 \times 3}}}&{{{\bf{0}}_{3 \times 3}}} \\ {{{\bf{0}}_{3 \times 3}}}&{{{\bf{0}}_{3 \times 3}}}&{{{\bf{0}}_{3 \times 3}}}&{{{\bf{0}}_{3 \times 3}}}&{{{\bf{0}}_{3 \times 3}}} \end{array}} \right]\mathit{\boldsymbol{X}} + \hfill \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\left[ {\begin{array}{*{20}{c}} { - \mathit{\boldsymbol{C}}_{bs}^n\mathit{\boldsymbol{\varepsilon }}} \\ {{\mathit{\boldsymbol{M}}_3}\nabla } \\ {{{\bf{0}}_{3 \times 1}}} \\ {{{\bf{0}}_{3 \times 1}}} \\ {{{\bf{0}}_{3 \times 1}}} \end{array}} \right] = {\mathit{\boldsymbol{F}}_1}\mathit{\boldsymbol{X}} + {\mathit{\boldsymbol{W}}_1} \hfill \\ \end{gathered}$ (1)

 $\begin{array}{*{20}{c}} {{\mathit{\boldsymbol{M}}_1} = \left[ {\begin{array}{*{20}{c}} 0&{{f_U}}&{{f_N}} \\ {{f_U}}&0&{ - {f_E}} \end{array}} \right];} \\ {{\mathit{\boldsymbol{M}}_2} = \left[ {\begin{array}{*{20}{c}} {\frac{{{V_N}\tan L - {V_U}}}{{{R_N} + h}}}&{2{\omega _{ie}}\sin L + \frac{{{V_E}}}{{{R_N} + h}}\tan L} \\ { - 2{\omega _{ie}}\sin L - \frac{{{V_E}}}{{{R_N} + h}}\tan L}&{\frac{{{V_U}}}{{{R_M} + h}}} \end{array}} \right];} \\ {{\mathit{\boldsymbol{M}}_3} = \left[ {\begin{array}{*{20}{l}} {{C_{11}}}&{{C_{12}}}&{{C_{13}}} \\ {{C_{21}}}&{{C_{22}}}&{{C_{23}}} \end{array}} \right]。} \end{array}$

 $\mathit{\boldsymbol{Z}} = \left[ {\begin{array}{*{20}{l}} {{Z_\theta }} \\ {{Z_V}} \end{array}} \right]$

 $\mathit{\boldsymbol{Z}} = \left[ {\begin{array}{*{20}{l}} {{\mathit{\boldsymbol{I}}_{3 \times 3}}}&{{{\bf{0}}_{3 \times 2}}}&{{{\bf{0}}_{3 \times 3}}}&{{{\bf{0}}_{3 \times 3}}}&{ - C_{bs}^n} \\ {{{\bf{0}}_{2 \times 3}}}&{{{\bf{I}}_{2 \times 2}}}&{{{\bf{0}}_{2 \times 3}}}&{{{\bf{0}}_{2 \times 3}}}&{{{\bf{0}}_{2 \times 3}}} \end{array}} \right]\mathit{\boldsymbol{X}} + \left[ {\begin{array}{*{20}{c}} {{V_\theta }} \\ {{V_v}} \end{array}} \right]$ (2)

1.2 基于马尔可夫过程的动态变形角建模

 $\left\{ {\begin{array}{*{20}{l}} {{w_{{fx}}}(t) = {{\dot \lambda }_x}(t)} \\ {{w_{fy}}(t) = {{\dot \lambda }_y}(t)} \\ {{w_{fz}}(t) = {\dot \lambda _z}(t)} \end{array}} \right.$ (3)

 $\left\{ {\begin{array}{*{20}{l}} {{{\dot w}_{{fx}}}(t) = - \beta _x^2{\lambda _x}(t) - 2{\beta _x}{w_{fx}}(t) + {W_x}(t)} \\ {{{\dot w}_{{fy}}}(t) = - \beta _y^2{\lambda _y}(t) - 2{\beta _y}{w_{fy}}(t) + {W_y}(t)} \\ {{{\dot w}_{{fz}}}(t) = - \beta _z^2{\lambda _z}(t) - 2{\beta _z}{w_{fy}}(t) + {W_z}(t)} \end{array}} \right.$ (4)

 $\begin{array}{*{20}{c}} {\mathit{\boldsymbol{\dot X}} = \mathit{\boldsymbol{FX}} + \mathit{\boldsymbol{W}} = \left[ {\begin{array}{*{20}{c}} {{\mathit{\boldsymbol{F}}_1}}&{{{\bf{0}}_{3 \times 3}}}&{{{\bf{0}}_{3 \times 3}}} \\ {{{\bf{0}}_{3 \times 3}}}&{{{\bf{0}}_{3 \times 3}}}&{{\mathit{\boldsymbol{I}}_{3 \times 3}}} \\ {{{\bf{0}}_{3 \times 3}}}&{{\mathit{\boldsymbol{B}}_1}}&{{\mathit{\boldsymbol{B}}_2}} \end{array}} \right]\mathit{\boldsymbol{X}} + \left[ {\begin{array}{*{20}{c}} {{\mathit{\boldsymbol{W}}_1}} \\ {{{\bf{0}}_{3 \times 1}}} \\ {{\sigma _{3 \times 1}}} \end{array}} \right]} \\ {{\mathit{\boldsymbol{B}}_1} = \left[ {\begin{array}{*{20}{c}} { - \beta _x^2}&0&0 \\ 0&{ - \beta _y^2}&0 \\ 0&0&{ - \beta _z^2} \end{array}} \right]} \\ {{\mathit{\boldsymbol{B}}_2} = \left[ {\begin{array}{*{20}{c}} { - 2{\beta _x}}&0&0 \\ 0&{ - 2{\beta _y}}&0 \\ 0&0&{ - 2{\beta _s}} \end{array}} \right]} \end{array}$ (5)

 $\begin{array}{*{20}{c}} {\mathit{\boldsymbol{Z}} = \left[ {\begin{array}{*{20}{c}} {{\mathit{\boldsymbol{I}}_{3 \times 3}}}&{{{\bf{0}}_{3 \times 2}}}&{{{\bf{0}}_{3 \times 2}}}&{{{\bf{0}}_{3 \times 3}}}&{ - \mathit{\boldsymbol{C}}_{bs}^n}&{ - \mathit{\boldsymbol{C}}_{bs}^n}&{{{\bf{0}}_{3 \times 3}}} \\ {{{\bf{0}}_{2 \times 3}}}&{{\mathit{\boldsymbol{I}}_{2 \times 2}}}&{{{\bf{0}}_{2 \times 3}}}&{{{\bf{0}}_{2 \times 3}}}&{{{\bf{0}}_{2 \times 3}}}&{{{\bf{0}}_{2 \times 3}}}&{{{\bf{0}}_{2 \times 3}}} \end{array}} \right] + } \\ {\left[ {\begin{array}{*{20}{l}} {{V_\theta }} \\ {{V_V}} \end{array}} \right]} \end{array}$ (6)

2 应变片量测改进后模型 2.1 采用应变片测量动态变形角及建模

 Download: 图 2 使用应变片测量挠曲变形角 Fig. 2 Measuring deflection angle with strain gauge

 $\begin{array}{*{20}{c}} {J\left( {\ddot \theta - {{\ddot \theta }_t}} \right) = c{{\dot \theta }_t} - mgl\sin \left( {\theta - {\theta _t}} \right) - } \\ {ml\left( {\ddot y\sin \left( {\theta - {\theta _t}} \right) + \ddot x\cos \left( {\theta - {\theta _t}} \right)} \right)} \end{array}$ (7)

 $J\left( {\ddot \theta - {{\ddot \theta }_t}} \right) = c{\dot \theta _t} - mgl\left( {\theta - {\theta _t}} \right) - ml\ddot x$ (8)

 ${\mathit{\Theta} _t}(s) = \frac{{{\beta _0}{s^2} + {\beta _1}s + {\beta _2}}}{{{s^2} + {\alpha _1}s + {\alpha _2}}}\mathit{\Theta} (s) + \frac{{{\gamma _1}}}{{{s^2} + {\alpha _1}s + {\alpha _2}}}X(s)$ (9)

 $\begin{array}{*{20}{c}} {\left[ {\begin{array}{*{20}{c}} {{{\dot x}_1}} \\ {{{\dot x}_2}} \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} { - {\alpha _1}}&1 \\ { - {\alpha _2}}&0 \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {{x_1}} \\ {{x_2}} \end{array}} \right] + \left[ {\begin{array}{*{20}{c}} {{\beta _1} - {\beta _0}{\alpha _1}} \\ {{\beta _2} - {\beta _0}{\alpha _2}} \end{array}} \right]\theta + \left[ {\begin{array}{*{20}{c}} 0 \\ {{\gamma _1}} \end{array}} \right]\ddot x} \\ {{{\dot \theta }_t} = \left[ {\begin{array}{*{20}{l}} 1&0 \end{array}} \right]\left[ {\begin{array}{*{20}{l}} {{x_1}} \\ {{x_2}} \end{array}} \right] + {\beta _0}\theta + {w_t}} \end{array}$ (10)

 $\left\{ {\begin{array}{*{20}{l}} {{{\dot x}_1} = - {\alpha _1}{x_1} + {x_2} - {\alpha _1}\theta = {x_2} - {\alpha _1}\left( {{x_1} + \theta } \right)} \\ {{{\dot x}_2} = - {\alpha _2}{x_1} + {\gamma _1}\ddot x} \\ {{{\dot \theta }_t} = \left( {{x_1} + \theta } \right) + {w_t}} \end{array}} \right.$ (11)

2.2 构建新的传递对准状态方程

 $\begin{array}{*{20}{c}} {{\mathit{\boldsymbol{B}}_1} = \left[ {\begin{array}{*{20}{c}} {{A_{1x}}}&0&0 \\ 0&{{A_{1y}}}&0 \\ 0&0&{{A_{1z}}} \end{array}} \right]} \\ {{\mathit{\boldsymbol{B}}_2} = \left[ {\begin{array}{*{20}{c}} {{A_{2x}} - {\alpha _{1x}}}&0&0 \\ 0&{{A_{2y}} - {\alpha _{1y}}}&0 \\ 0&0&{{A_{2z}} - {\alpha _{1z}}} \end{array}} \right]} \end{array}$ (12)

3 实验结果及分析 3.1 实验方法

 Download: 图 3 车载试验验证流程 Fig. 3 The verification process of vehicle test
 Download: 图 4 传递对准综合信息处理系统 Fig. 4 The CPU of transfer alignment test

 Download: 图 5 车载试验安装示意 Fig. 5 The installation diagram of vehicle test

 Download: 图 6 高精度寻北仪作为精度检验基准设备 Fig. 6 The high precision gyro north finder
3.2 传统传递对准模型下计算结果

 Download: 图 8 方位、俯仰和横滚动态变形角估计结果 Fig. 8 The estimation result of dynamic deformation angle

 Download: 图 9 方位、俯仰和横滚失准角估计误差 Fig. 9 The estimation error of misalignment angle
3.3 采用应变片量测辅助建模结果

 Download: 图 10 方位、俯仰和横滚失准角估计结果 Fig. 10 The estimation result of misalignment angle

4 结论

1) 使用了应变片估计动态变形角的传递对准模型有效解决了传统传递对准模型中因经验建模参数不匹配导致的失准角估计无法收敛问题，可在机动结束10 s内将失准角估计收敛至一个较高精度；

2) 在机动结束后，传递对准系统估计的方位失准角误差小于3′，俯仰和横滚失准角误差小于2′，其精度等级基本达到了该精度子惯导系统的理想结果。

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