CN101907638B

CN101907638B - Method for calibrating accelerometer under unsupported state

Info

Publication number: CN101907638B
Application number: CN2010102050331A
Authority: CN
Inventors: 张海; 毛友泽; 沈晓蓉; 任章
Original assignee: Beihang University
Current assignee: Beihang University
Priority date: 2010-06-11
Filing date: 2010-06-11
Publication date: 2011-09-28
Anticipated expiration: 2030-06-11
Also published as: CN101907638A

Abstract

The invention discloses a method for calibrating an accelerometer in an unsupported state, and the application object is an inertial navigation system with the ability to initially align the attitude of a static base. Firstly, simplify the error model, change the position of the carrier several times, use the rough alignment and fine alignment of the static base to obtain the attitude information for each position, and calculate the three-axis component of the gravitational acceleration g projected in the carrier coordinate system according to the attitude information, and Accelerometer measurements are recorded for each position. The present invention provides a 10-position calibration scheme. Based on the calculated and measured values of the above projection components, the non-linear least squares method is used to identify the zero position deviation. After obtaining the zero position deviation, the error model can be degenerated into a linear model, and then the linear The least squares method is used to identify the coupling matrix of the installation error and the scale factor, so as to realize the calibration of the key parameters of the accelerometer. The invention can realize the non-support calibration of the accelerometer, has simple process and high speed, and can effectively estimate the key parameters of the accelerometer with time-varying property.

Description

A Calibration Method of Accelerometer in Unsupported State

技术领域technical field

本发明属于惯性导航技术领域，具体涉及一种无依托状态下加速度计的标定方法。The invention belongs to the technical field of inertial navigation, and in particular relates to a method for calibrating an accelerometer in an unsupported state.

背景技术Background technique

对加速度计进行有效的标定可解决其性能参数时变性的问题，对提高测量精度非常必要。传统的标定方法需要借助转台、离心机等标定设备，对未知参数进行完整的系统级标定，标定精度高，但操作复杂，标定周期长，实时性较差；所以针对上述问题提出了无依托状态下加速度计标定的概念，即要求脱离实验室条件，只借助惯性器件自身，在使用现场即可实现标定，由于现场标定条件上的限制，通常只针对加速度计性能影响最严重的参数进行辨识，目前公开发表的加速度计无依托现场标定文献还比较少。Effective calibration of accelerometers can solve the problem of time-varying performance parameters, which is very necessary to improve measurement accuracy. The traditional calibration method requires complete system-level calibration of unknown parameters with the help of calibration equipment such as turntables and centrifuges. The calibration accuracy is high, but the operation is complicated, the calibration cycle is long, and the real-time performance is poor; Under the concept of accelerometer calibration, it is required to break away from the laboratory conditions, and only use the inertial device itself to achieve calibration at the use site. Due to the limitations of on-site calibration conditions, usually only the parameters that have the most serious impact on the performance of the accelerometer are identified. Currently, there are relatively few publications on unsupported on-site calibration of accelerometers.

清华大学精密仪器与机械学院的尚捷、顾启泰提出基于虚拟噪声的现场最优标定方法，即两步估计法，该方法同多位置方法比较，具有结构简单、省时、易于实现的优点，但不能标定出安装误差和刻度因数误差，并且只适用于短时间、低中精度导航系统。Shang Jie and Gu Qitai from the School of Precision Instruments and Mechanics of Tsinghua University proposed an on-site optimal calibration method based on virtual noise, that is, a two-step estimation method. Compared with the multi-position method, this method has the advantages of simple structure, time saving, and easy implementation. The installation error and scale factor error cannot be calibrated, and it is only suitable for short-term, low-medium precision navigation systems.

北京航空航天大学的刘百奇、房建成针对光纤陀螺IMU提出六位置旋转现场标定方法，借助水平面在六个位置上进行十二次旋转，建立42个非线性输入输出方程，通过旋转积分和对称位置误差相消，进行惯性器件参数标定，该方法操作较复杂，且对旋转平面是否水平具有较高要求，并且不能辨识出加速度计的安装误差和刻度因数误差。Liu Baiqi and Fang Jiancheng of Beijing University of Aeronautics and Astronautics proposed a six-position rotating on-site calibration method for the fiber optic gyroscope IMU, using the horizontal plane to perform twelve rotations at six positions, and establishing 42 nonlinear input and output equations, through rotation integral and symmetrical position error Elimination, calibration of inertial device parameters, this method is more complicated to operate, and has higher requirements on whether the rotation plane is horizontal, and cannot identify the installation error and scale factor error of the accelerometer.

新加坡国立大学的W T Fong、S K Ong和A Y C Nee提出一种基于downhill simple最优化的辨识方案，操作简单，且可以辨识出完整的误差参数，该文献主要针对微机械IMU设计，受器件测量精度的限制，误差模型简化严重，辨识结果精度较低。W T Fong, S K Ong and A Y C Nee from the National University of Singapore proposed an identification scheme based on downhill simple optimization, which is easy to operate and can identify complete error parameters. This document is mainly aimed at the design of micromechanical IMUs. Due to the limitation of device measurement accuracy, the error model is seriously simplified, and the accuracy of identification results is low.

发明内容Contents of the invention

为了解决上述不能标出或者识别加速度计的安装误差和刻度因数误差，辨识结果精度低的问题，本发明提出一种无依托状态下加速度计的标定方法，在进行标定过程前首先需要考虑对加速度计性能影响最严重的误差参数，简化三轴加速度计的误差模型为In order to solve the above-mentioned problems that the installation error and scale factor error of the accelerometer cannot be marked or identified, and the accuracy of the identification result is low, the present invention proposes a calibration method for the accelerometer in the unsupported state. Before performing the calibration process, it is first necessary to consider the acceleration The most serious error parameter affecting the performance of the accelerometer, the simplified error model of the three-axis accelerometer is

$(\begin{matrix} {g g}_{xa xa} \\ {g g}_{ya the ya} \\ {g g}_{za za} \end{matrix}) = = (\begin{matrix} {K K}_{1111} {k k}_{x x} & {K K}_{1212} {k k}_{y the y} & {K K}_{1313} {k k}_{z z} \\ {K K}_{21 twenty one} {k k}_{x x} & {K K}_{22 twenty two} {k k}_{y the y} & {K K}_{23 twenty three} {k k}_{z z} \\ {K K}_{3131} {k k}_{x x} & {K K}_{3232} {k k}_{y the y} & {K K}_{3333} {K K}_{z z} \end{matrix}) (\begin{matrix} (({g g}_{xm xm} - - {g g}_{x x 00})) \\ (({g g}_{ym ym} - - {g g}_{y the y 00})) \\ (({g g}_{zm zm} - - {g g}_{z z 00})) \end{matrix}) - - - - - - ((11))$

式中，k_x、k_y、k_z为加速度计x、y、z轴刻度因数；K₁₁，K₁₂，K₁₃，K₂₁，K₂₂，K₂₃，K₃₁，K₃₂，K₃₃为安装误差矩阵元素；g_x0、g_y0、g_z0为加速度计x、y、z轴零位偏差；g_xm、g_ym、g_zm为加速度计x、y、z轴测量值；g_xa、g_ya、g_za为重力加速度g在载体坐标系下投影分量；

为安装误差和刻度因数的耦合矩阵。In the formula, k _x , _ky , k _z are scale factors of accelerometer x, y, z axis; K ₁₁ , K ₁₂ , K ₁₃ , K ₂₁ , K ₂₂ , K ₂₃ , K ₃₁ , K ₃₂ , K ₃₃ are Install error matrix elements; g _x0 , g _y0 , g _z0 are accelerometer x, y, z axis zero position deviation; g _xm , g _ym , g _zm are accelerometer x, y, z axis measurement values; g _xa , g _ya and g _za are the projected components of the gravitational acceleration g in the carrier coordinate system;

is the coupling matrix of installation errors and scale factors.

进而通过5个步骤来完成标定：And then through 5 steps to complete the calibration:

步骤1：静基座粗对准，得到载体横滚角γ、俯仰角θ；Step 1: Roughly align the static base to obtain the carrier roll angle γ and pitch angle θ;

步骤2：利用Kalman滤波进行静基座精对准，提高横滚角γ、俯仰角θ的对准精度，并进一步得到重力加速度g在载体坐标系下的投影的三轴分量的计算值(g_xa，g_ya，g_za)；Step 2: Use the Kalman filter for fine alignment of the static base, improve the alignment accuracy of the roll angle γ and the pitch angle θ, and further obtain the calculated value of the three-axis component of the projection of the gravitational acceleration g in the carrier coordinate system (g _xa , _gya , _gza );

步骤3：将载体在10个不同的位置上进行步骤1、步骤2获得10组加速度计x、y、z轴测量值(g_xm，g_ym，g_zm)以及10组重力加速度g在载体坐标系下的投影的三轴分量的计算值(g_xa，g_ya，g_za)。本步骤中根据载体是否在10个位置上放置完成来判断进行，若没有放置完成，则转步骤1进行，若已经完成，则得到所需数据，继续步骤4。Step 3: Perform steps 1 and 2 on the carrier at 10 different positions to obtain 10 sets of accelerometer x, y, and z-axis measurement values (g _xm , g _ym , g _zm ) and 10 sets of gravitational acceleration g at the carrier coordinates Calculated values (g _xa , g _ya , g _za ) of the three-axis components of the projection under the system. In this step, it is judged according to whether the carrier has been placed in the 10 positions. If the placement is not completed, go to step 1. If it has been completed, obtain the required data and continue to step 4.

步骤4：通过非线性最小二乘法得到加速度计的零位偏差g_x0，g_y0，g_z0；Step 4: Obtain the zero position deviation g _x0 , g _y0 , g _z0 of the accelerometer by nonlinear least square method;

步骤5：通过线性最小二乘法得到耦合矩阵

Step 5: Obtain the coupling matrix by linear least squares

将步骤4得到零位偏差g_x0，g_y0，g_z0以及步骤5中耦合矩阵的值代入式(1)中，构建式(1)误差模型。从而可对加速度计的测量值g_xm，g_ym，g_zm，进行有效的修正，得到修正后的重力加速度g在载体坐标系下投影分量g_xa，g_ya，g_za，从而提高了测量的精度。Substitute the zero position deviation g _x0 , g _y0 , g _z0 obtained in step 4 and the value of the coupling matrix in step 5 into formula (1) to construct the error model of formula (1). Therefore, the measured values g _xm , g _ym , g _zm of the accelerometer can be effectively corrected, and the projected components g _xa , g _ya , g _za of the corrected gravitational acceleration g in the carrier coordinate system can be obtained, thereby improving the measurement accuracy precision.

本发明的优点在于：The advantages of the present invention are:

(1)可实现外场条件下加速度计的无依托标定，脱离实验室条件，不借助于传统的标定设备；(1) It can realize the unsupported calibration of the accelerometer under the external field conditions, which is separated from the laboratory conditions and does not rely on traditional calibration equipment;

(2)标定过程简单，标定速度快，可实现加速度计每次使用前均进行标定，有效的解决了性能参数时变性的问题。(2) The calibration process is simple, the calibration speed is fast, and the accelerometer can be calibrated before each use, effectively solving the problem of time-varying performance parameters.

(3)非线性最小二乘方法的应用可以解决误差模型非线性的问题，避免了传统标定对转台等标定设备的依赖，实现了加速度计无依托状态下的标定。(3) The application of the nonlinear least squares method can solve the problem of nonlinearity of the error model, avoid the traditional calibration's dependence on the calibration equipment such as the turntable, and realize the calibration of the accelerometer without support.

附图说明Description of drawings

图1为本发明的标定方法的步骤流程图。Fig. 1 is a flowchart of steps of the calibration method of the present invention.

具体实施方式Detailed ways

下面结合附图对本发明进行进一步的详细说明。The present invention will be further described in detail below in conjunction with the accompanying drawings.

本发明进行标定前需要首先简化三轴加速度计的误差模型：Before the present invention is calibrated, it is necessary to simplify the error model of the three-axis accelerometer at first:

在现场标定的前提下，只考虑对加速度计性能影响最严重的误差参数，可将三轴加速度计的误差模型简化为：On the premise of on-site calibration, only the error parameters that have the most serious impact on the performance of the accelerometer are considered, and the error model of the three-axis accelerometer can be simplified as:

$(\begin{matrix} {g g}_{xa xa} \\ {g g}_{ya the ya} \\ {g g}_{za za} \end{matrix}) = = (\begin{matrix} {K K}_{1111} & {K K}_{1212} & {K K}_{1313} \\ {K K}_{21 twenty one} & {K K}_{22 twenty two} & {K K}_{23 twenty three} \\ {K K}_{3131} & {K K}_{3232} & {K K}_{3333} \end{matrix}) (\begin{matrix} {k k}_{x x} (({g g}_{xm xm} - - {g g}_{x x 00})) \\ {k k}_{y the y} (({g g}_{ym ym} - - {g g}_{y the y 00})) \\ {k k}_{z z} (({g g}_{zm zm} - - {g g}_{z z 00})) \end{matrix}) - - - - - - ((11))$

式中，

为安装误差矩阵，K₁₁、K₁₂、K₁₃、K₂₁、K₂₂、K₂₃、K₃₁、K₃₂及K₃₃为安装误差矩阵元素；k_x、k_y、k_z为加速度计x、y、z轴刻度因数；g_x0、g_y0、g_z0为加速度计x、y、z轴零位偏差；g_xm、g_ym、g_zm为加速度计x、y、z轴测量值；g_xa、g_ya、g_za为重力加速度g在载体坐标系下投影分量的计算值。In the formula,

is the installation error matrix, K ₁₁ , K ₁₂ , K ₁₃ , K ₂₁ , K ₂₂ , K ₂₃ , K ₃₁ , K ₃₂ and K ₃₃ are the installation error matrix elements; k _x , _ky , k _z are the accelerometer x, y, z-axis scale factor; g _x0 , g _y0 , g _z0 are accelerometer x, y, z-axis zero position deviation; g _xm , g _ym , g _zm are accelerometer x, y, z-axis measurement values; g _xa , g _ya , g _za are the calculated values of the projected components of the gravitational acceleration g in the carrier coordinate system.

将式(1)进一步化简合并可得：Further simplification and combination of formula (1) can be obtained:

$(\begin{matrix} {g g}_{xa xa} \\ {g g}_{ya the ya} \\ {g g}_{za za} \end{matrix}) = = (\begin{matrix} {K K}_{1111} {k k}_{x x} & {K K}_{1212} {k k}_{y the y} & {K K}_{1313} {k k}_{z z} \\ {K K}_{21 twenty one} {k k}_{x x} & {K K}_{22 twenty two} {k k}_{y the y} & {K K}_{23 twenty three} {k k}_{z z} \\ {K K}_{3131} {k k}_{x x} & {K K}_{3232} {k k}_{y the y} & {K K}_{3333} {K K}_{z z} \end{matrix}) (\begin{matrix} (({g g}_{xm xm} - - {g g}_{x x 00})) \\ (({g g}_{ym ym} - - {g g}_{y the y 00})) \\ (({g g}_{zm zm} - - {g g}_{z z 00})) \end{matrix}) - - - - - - ((22))$

其中，

为安装误差和刻度因数的耦合矩阵S。in,

is the coupling matrix S of installation error and scale factor.

将安装误差和刻度因数的耦合矩阵S表示为如下的简化方式：The coupling matrix S of installation error and scale factor is expressed as the following simplified way:

$S S = = (\begin{matrix} {S S}_{1111} & {S S}_{1212} & {S S}_{1313} \\ {S S}_{21 twenty one} & {S S}_{22 twenty two} & {S S}_{23 twenty three} \\ {S S}_{3131} & {S S}_{3232} & {S S}_{3333} \end{matrix}) - - - - - - ((33))$

其中，式(3)所示矩阵中元素与式(2)中安装误差和刻度因数的耦合矩阵中元素一一对应；Among them, the elements in the matrix shown in formula (3) correspond to the elements in the coupling matrix of installation error and scale factor in formula (2) one by one;

本发明一种无依托状态下加速度计的标定方法，通过以下5步来对加速度计进行标定，具体步骤如下：The calibration method of accelerometer under a kind of unsupported state of the present invention, carries out calibration to accelerometer by following 5 steps, concrete steps are as follows:

步骤1：静基座粗对准，得到载体横滚角γ、俯仰角θ。本发明实施例中将载体横滚角γ、俯仰角θ的对准误差控制在1°以内，具体得到载体横滚角γ、俯仰角θ通过下述过程来实现。Step 1: Roughly align the static base to obtain the carrier roll angle γ and pitch angle θ. In the embodiment of the present invention, the alignment errors of the carrier roll angle γ and pitch angle θ are controlled within 1°, and the carrier roll angle γ and pitch angle θ are specifically obtained through the following process.

通过：pass:

$(\begin{matrix} {g g}_{x x}^{b b} \\ {g g}_{y the y}^{b b} \\ {g g}_{z z}^{b b} \end{matrix}) = = {C C}_{n no}^{b b} (\begin{matrix} {g g}_{x x}^{n no} \\ {g g}_{y the y}^{n no} \\ {g g}_{z z}^{n no} \end{matrix}) - - - - - - ((44))$

$(\begin{matrix} {ω ω}_{iex iex}^{b b} \\ {ω ω}_{iey iey}^{b b} \\ {ω ω}_{iez iez}^{b b} \end{matrix}) = = {C C}_{n no}^{b b} (\begin{matrix} {ω ω}_{iex iex}^{n no} \\ {ω ω}_{iey iey}^{n no} \\ {ω ω}_{iez iez}^{n no} \end{matrix}) - - - - - - ((55))$

得到载体的载体坐标系和地理坐标系之间的方向余弦矩阵：Get the direction cosine matrix between the vector's vector frame and the geographic frame:

${C C}_{n no}^{b b} = = [\begin{matrix} {g g}_{x x}^{b b} & {ω ω}_{iex iex}^{b b} & {ω ω}_{iez iez}^{b b} {g g}_{y the y}^{b b} - - {ω ω}_{iey iey}^{b b} {g g}_{z z}^{b b} \\ {g g}_{y the y}^{b b} & {ω ω}_{iey iey}^{b b} & {ω ω}_{iex iex}^{b b} {g g}_{z z}^{b b} - - {ω ω}_{iez iez}^{b b} {g g}_{x x}^{b b} \\ {g g}_{z z}^{b b} & {ω ω}_{iez iez}^{b b} & {ω ω}_{iez iez}^{b b} {g g}_{x x}^{b b} - - {ω ω}_{iex iex}^{b b} {g g}_{y the y}^{b b} \end{matrix}] \cdot &Center Dot; [\begin{matrix} 00 & \frac{11}{g g} tan the tan L L & - - \frac{11}{g g} \\ 00 & \frac{11}{{ω ω}_{ie ie}} sec sec L L & 00 \\ \frac{11}{{gω gω}_{ie ie}} sec sec L L & 00 & 00 \end{matrix}]$

$= = [\begin{matrix} (({ω ω}_{iez iez}^{b b} {g g}_{y the y}^{b b} - - {ω ω}_{iey iey}^{b b} {g g}_{z z}^{b b})) \cdot \cdot \frac{11}{g g {ω ω}_{ie ie}} sec sec L L & {g g}_{x x}^{b b} \cdot &Center Dot; \frac{11}{g g} tan the tan L L + + {ω ω}_{iex iex}^{b b} \cdot \cdot \frac{11}{{ω ω}_{ie ie}} sec sec L L & - - {g g}_{x x}^{b b} \cdot &Center Dot; \frac{11}{g g} \\ (({ω ω}_{iex iex}^{b b} {g g}_{z z}^{b b} - - {ω ω}_{iez iez}^{b b} {g g}_{x x}^{b b})) \cdot &Center Dot; \frac{11}{g g {ω ω}_{ie ie}} sec sec l l & {g g}_{y the y}^{b b} \cdot \cdot \frac{11}{g g} tan the tan L L + + {ω ω}_{iey iey}^{b b} \cdot \cdot \frac{11}{{ω ω}_{ie ie}} sec sec L L & - - {g g}_{y the y}^{b b} \cdot \cdot \frac{11}{g g} \\ (({ω ω}_{iey iey}^{b b} {g g}_{x x}^{b b} - - {ω ω}_{iex iex}^{b b} {g g}_{y the y}^{b b})) \cdot &Center Dot; \frac{11}{{gω gω}_{ie ie}} sec sec L L & {g g}_{z z}^{b b} \cdot &Center Dot; \frac{11}{g g} tan the tan L L + + {ω ω}_{iez iez}^{b b} \cdot &Center Dot; \frac{11}{{ω ω}_{ie ie}} sec sec L L & - - {g g}_{z z}^{b b} \cdot &Center Dot; \frac{11}{g g} \end{matrix}] - - - - - - ((66))$

其中，

为重力加速度g在载体坐标系下投影的三轴分量测量值；

为重力加速度g在地理坐标系下投影的三轴分量；ω_ie为地球自转角速度，为一常量，数值为7.292115147e-5弧度每秒；

为地球自转角速度在载体坐标系投影的三轴分量；

为地球自转角速度在地理系下投影的三轴分量；L为当地地理纬度；g为当地重力加速度。in,

is the measured value of the three-axis component projected in the carrier coordinate system of the gravitational acceleration g;

is the three-axis component of the gravitational acceleration g projected in the geographic coordinate system; ω _ie is the angular velocity of the earth's rotation, which is a constant, and the value is 7.292115147e-5 radians per second;

are the three-axis components projected on the carrier coordinate system of the earth's rotation angular velocity;

is the three-axis component of the earth's rotation angular velocity projected under the geographic system; L is the local geographic latitude; g is the local gravitational acceleration.

将式(6)中方向余弦矩阵表示为如下的简化方式：Express the direction cosine matrix in formula (6) as the following simplified way:

${C C}_{n no}^{b b} = = [\begin{matrix} {T T}_{1111} & {T T}_{1212} & {T T}_{1313} \\ {T T}_{21 twenty one} & {T T}_{22 twenty two} & {T T}_{23 twenty three} \\ {T T}_{3131} & {T T}_{3232} & {T T}_{3333} \end{matrix}] - - - - - - ((77))$

其中，式(7)所示矩阵中元素与式(6)矩阵中元素一一对应；Wherein, the elements in the matrix shown in formula (7) correspond to the elements in the matrix of formula (6) one by one;

由此，根据式(7)可得到载体误差在1°以内的俯仰角θ、横滚角γ：Thus, according to formula (7), the pitch angle θ and roll angle γ of the carrier error within 1° can be obtained:

θ＝sin^-1T₂₃ (8)θ＝sin ^-1 T ₂₃ (8)

γ＝tg^-1(-T₁₃/T₃₃) (9)γ＝tg ^-1 (-T ₁₃ /T ₃₃ ) (9)

步骤2：利用Kalman滤波进行静基座精对准，提高横滚角γ、俯仰角θ的对准精度，得到俯仰角θ′、横滚角γ′。Step 2: Use the Kalman filter for fine alignment of the static base, improve the alignment accuracy of the roll angle γ and the pitch angle θ, and obtain the pitch angle θ′ and roll angle γ′.

本发明实施例中将载体对准误差在50角秒以内，具体得到俯仰角θ′、横滚角γ′的过程如下。In the embodiment of the present invention, the alignment error of the carrier is within 50 arc seconds, and the specific process of obtaining the pitch angle θ' and the roll angle γ' is as follows.

将Kalman滤波运用到静基座精对准中，Kalman滤波状态方程为：The Kalman filter is applied to the fine alignment of the static base, and the state equation of the Kalman filter is:

$(\begin{matrix} {\overset{\cdot &Center Dot;}{φ φ}}_{x x} \\ {\overset{\cdot \cdot}{φ φ}}_{y the y} \\ {\overset{\cdot &Center Dot;}{φ φ}}_{z z} \\ {\overset{\cdot &Center Dot;}{ϵ ϵ}}_{x x} \\ {\overset{\cdot \cdot}{ϵ ϵ}}_{y the y} \\ {\overset{\cdot &Center Dot;}{ϵ ϵ}}_{z z} \\ {\overset{\cdot \cdot}{&dtri; &dtri;}}_{x x} \\ {\overset{\cdot \cdot}{&dtri; &dtri;}}_{y the y} \\ \overset{\cdot \cdot}{δ δ} {V V}_{x x} \\ \overset{\cdot &Center Dot;}{δ δ} {V V}_{y the y} \end{matrix}) = = (\begin{matrix} 00 & {ω ω}_{ie ie} sin sin L L & - - {ω ω}_{ie ie} cos cos L L & 11 & 00 & 00 & 00 & 00 & 00 & - - 11 / / R R \\ - - {ω ω}_{ie ie} sin sin L L & 00 & 00 & 00 & 11 & 00 & 00 & 00 & 11 / / R R & 00 \\ {ω ω}_{ie ie} cos cos L L & 00 & 00 & 00 & 00 & 11 & 00 & 00 & tgL wxya / / R R & 00 \\ 00 & 00 & 00 & 00 & 00 & 00 & 00 & 00 & 00 & 00 \\ 00 & 00 & 00 & 00 & 00 & 00 & 00 & 00 & 00 & 00 \\ 00 & 00 & 00 & 00 & 00 & 00 & 00 & 00 & 00 & 00 \\ 00 & 00 & 00 & 00 & 00 & 00 & 00 & 00 & 00 & 00 \\ 00 & 00 & 00 & 00 & 00 & 00 & 00 & 00 & 00 & 00 \\ 00 & - - g g & 00 & 00 & 00 & 00 & 00 & 00 & 00 & 00 \\ g g & 00 & 00 & 00 & 00 & 00 & 00 & 00 & 00 & 00 \end{matrix}) (\begin{matrix} {φ φ}_{x x} \\ {φ φ}_{y the y} \\ {φ φ}_{z z} \\ {ϵ ϵ}_{x x} \\ {ϵ ϵ}_{y the y} \\ {ϵ ϵ}_{z z} \\ {&dtri; &dtri;}_{x x} \\ {&dtri; &dtri;}_{y the y} \\ δ δ {V V}_{x x} \\ δ δ {V V}_{y the y} \end{matrix}) - - - - - - ((1010))$

Kalman滤波测量方程为：The Kalman filter measurement equation is:

$(\begin{matrix} δ δ {V V}_{x x} \\ δ δ {V V}_{y the y} \end{matrix}) = = (\begin{matrix} 00 & 00 & 00 & 00 & 00 & 00 & 00 & 00 & 11 & 00 \\ 00 & 00 & 00 & 00 & 00 & 00 & 00 & 00 & 00 & 11 \end{matrix}) (\begin{matrix} {φ φ}_{x x} \\ {φ φ}_{y the y} \\ {φ φ}_{z z} \\ {ϵ ϵ}_{x x} \\ {ϵ ϵ}_{y the y} \\ {ϵ ϵ}_{z z} \\ {&dtri; &dtri;}_{x x} \\ {&dtri; &dtri;}_{y the y} \\ δ δ {V V}_{x x} \\ δ δ {V V}_{y the y} \end{matrix}) + + (\begin{matrix} {η η}_{x x} \\ {η η}_{y the y} \end{matrix}) - - - - - - ((1111))$

其中，φ_x、φ_y、φ_z分别为载体俯仰误差角、横滚误差角、航向误差角；ε_x、ε_y、ε_z分别为载体安装坐标系x、y、z轴向陀螺误差；

分别为载体安装坐标系x、y轴向加速度计误差；δV_x、δV_y分别为载体安装坐标系x、y轴向载体速度误差；R为地球半径；η_x、η_y分别为载体安装坐标系x、y轴速度噪声；L为当地地理纬度。Among them, φ _x , φ _y , φ _z are the pitch error angle, roll error angle, and heading error angle of the carrier respectively; ε _x , ε _y , ε _z are the gyro errors in the x, y, and z axis of the carrier installation coordinate system, respectively;

are the accelerometer errors in the x and y axis of the carrier installation coordinate system; δV _x and δV _y are the carrier velocity errors in the x and y axis of the carrier installation coordinate system respectively; R is the radius of the earth; η _x and η _y are the carrier installation coordinates is the speed noise of the x and y axes; L is the local geographic latitude.

通过式(10)、式(11)可得到φ_x、φ_y、φ_z、ε_x、ε_y、ε_z、

δV_x、δV_y。Through formula (10) and formula (11), we can get φ _x , φ _y , φ _z , ε _x , ε _y , ε _z ,

δV _x , δV _y .

由φ_x、φ_y、φ_z组成误差矩阵，令其为C，表示如下：The error matrix is composed of φ _x , φ _y , φ _z , let it be C, expressed as follows:

$C C = = [\begin{matrix} 11 & - - {φ φ}_{z z} & - - {φ φ}_{y the y} \\ {φ φ}_{z z} & 11 & {φ φ}_{x x} \\ {φ φ}_{y the y} & - - {φ φ}_{x x} & 11 \end{matrix}] - - - - - - ((1212))$

则可得精对准后的方向余弦矩阵Then the direction cosine matrix after fine alignment can be obtained

${C C}_{n no}^{' ' b b} = = {C C}_{n no}^{b b} * * C C = = [\begin{matrix} {T T}_{1111} + + {T T}_{1212} * * {φ φ}_{z z} + + {T T}_{1313} * * {φ φ}_{y the y} & - - {T T}_{1111} * * {φ φ}_{z z} + + {T T}_{1212} - - {T T}_{1313} * * {φ φ}_{x x} & - - {T T}_{1111} * * {φ φ}_{y the y} + + {T T}_{1212} * * {φ φ}_{x x} + + {T T}_{1313} \\ {T T}_{21 twenty one} + + {T T}_{22 twenty two} * * {φ φ}_{z z} + + {T T}_{23 twenty three} * * {φ φ}_{y the y} & - - {T T}_{21 twenty one} * * {φ φ}_{z z} + + {T T}_{22 twenty two} - - {T T}_{23 twenty three} * * {φ φ}_{x x} & - - {T T}_{21 twenty one} * * {φ φ}_{y the y} + + {T T}_{22 twenty two} * * {φ φ}_{x x} + + {T T}_{23 twenty three} \\ {T T}_{3131} + + {T T}_{3232} * * {φ φ}_{z z} + + {T T}_{3333} * * {φ φ}_{y the y} & - - {T T}_{3131} * * {φ φ}_{z z} + + {T T}_{3232} - - {T T}_{3333} * * {φ φ}_{x x} & - - {T T}_{3131} * * {φ φ}_{y the y} + + {T T}_{3232} * * {φ φ}_{x x} + + {T T}_{3333} \end{matrix}] - - - - - - ((1313))$

将精对准后的方向余弦矩阵

表示为如下的简化方式：Direction cosine matrix after fine alignment

Expressed in a simplified form as follows:

${C C}_{n no}^{' ' b b} = = [\begin{matrix} {T T}_{1111}^{' '} & {T T}_{1212}^{' '} & {T T}_{1313}^{' '} \\ {T T}_{21 twenty one}^{' '} & {T T}_{22 twenty two}^{' '} & {T T}_{23 twenty three}^{' '} \\ {T T}_{3131}^{' '} & {T T}_{3232}^{' '} & {T T}_{3333}^{' '} \end{matrix}] - - - - - - ((1414))$

其中，式(14)所示矩阵中元素与式(13)精对准后的方向余弦矩阵

中元素一一对应。通过式(14)可得：Among them, the direction cosine matrix after the elements in the matrix shown in formula (14) are precisely aligned with formula (13)

One-to-one correspondence of elements. Through formula (14) can get:

θ′＝sin^-1T′₂₃ (15)θ'=sin ^-1 T' ₂₃ (15)

γ′＝tg^-1(-T′₁₃/T′₃₃) (16)γ'=tg ^-1 (-T' ₁₃ /T' ₃₃ ) (16)

从而获得载体对准误差在50角秒以内的俯仰角θ′、横滚角γ′。将重力加速度g投影到载体坐标系下，可得到重力加速度g在载体坐标系下的投影的三轴分量的计算值：In this way, the pitch angle θ' and the roll angle γ' of the carrier alignment error within 50 arc seconds are obtained. Projecting the gravitational acceleration g into the carrier coordinate system, the calculated value of the three-axis component of the projection of the gravitational acceleration g in the carrier coordinate system can be obtained:

g_xa＝-sin(γ′)·cos(θ′)·g (17)g _xa = -sin(γ′) cos(θ′) g (17)

g_ya＝sin(θ′)·g (18)g _ya = sin(θ′) g (18)

g_za＝cos(γ′)·cos(θ′)·g (19)g _za =cos(γ′)·cos(θ′)·g (19)

步骤3：将载体在10个不同的位置放置，在每个位置上都需要进行步骤1和步骤2中的静基座粗对准和静基座精对准，判断10个位置是否放置完成，若没有完成则转步骤1进行，若完成，则获得10组加速度计x、y、z轴测量值(g_xm，g_ym，g_zm)以及10组重力加速度g在载体坐标系下的投影的三轴分量的计算值(g_xa，g_ya，g_za)。Step 3: Place the carrier in 10 different positions. In each position, the coarse alignment of the static base and the fine alignment of the static base in Step 1 and Step 2 are required to determine whether the placement of the 10 positions is complete. If it is not completed, go to step 1. If it is completed, obtain 10 sets of accelerometer x, y, z-axis measurement values (g _xm , g _ym , g _zm ) and 10 sets of gravitational acceleration g in the carrier coordinate system. Calculated values of the triaxial components (g _xa , g _ya , g _za ).

步骤4：通过非线性最小二乘法得到加速度计的零位偏差g_x0，g_y0，g_z0。Step 4: Obtain the zero position deviation g _x0 , g _y0 , g _z0 of the accelerometer by nonlinear least square method.

a、构建目标函数；a. Construct the objective function;

根据式(3)可将式(2)变形，令目标函数According to formula (3), formula (2) can be deformed, so that the objective function

f(p)＝[S₁₁(g_xm-g_x0)+S₁₂(g_ym-g_y0)+S₁₃(g_zm-g_z0)]²+[S₂₁(g_xm-g_x0)+S₂₂(g_ym-g_y0)+S₂₃(g_zm-g_z0)]² (20)f(p)＝[S ₁₁ (g _xm -g _x0 )+S ₁₂ (g _ym -g _y0 )+S ₁₃ (g _zm -g _z0 )] ² +[S ₂₁ (g _xm -g _x0 )+S ₂₂ (g _ym -g _y0 )+S ₂₃ (g _zm -g _z0 )] ² (20)

+[S₃₁(g_xm-g_x0)+S₃₂(g_ym-g_y0)+S₃₃(g_zm-g_z0)]²-(g_xp ²+g_yp ²+g_zp ²)+[S ₃₁ (g _xm -g _x0 )+S ₃₂ (g _ym -g _y0 )+S ₃₃ (g _zm -g _z0 )] ² -(g _xp ² +g _yp ² +g _zp ² )

其中，p＝[g_x0，g_y0，g_z0，S₁₁，S₁₂，S₁₃，S₂₁，S₂₂，S₂₃，S₃₁，S₃₂，S₃₃]；Wherein, p=[g _x0 , g _y0 , g _z0 , S ₁₁ , S ₁₂ , S ₁₃ , S ₂₁ , S ₂₂ , S ₂₃ , S ₃₁ , S ₃₂ , S ₃₃ ];

b、求目标函数f(p)的雅可比矩阵；b. Find the Jacobian matrix of the objective function f(p);

目标函数f(p)的雅可比矩阵J为：The Jacobian matrix J of the objective function f(p) is:

式中，f₁...f₁₀为第1到第10位置下的目标函数f(p)；p₁...p₁₂为向量p的第1到第12个元素。In the formula, f ₁ ... f ₁₀ is the objective function f(p) at the 1st to 10th positions; p ₁ ... p ₁₂ is the 1st to 12th elements of the vector p.

c、迭代逼近未知向量p；c. Iteratively approximate the unknown vector p;

解方程：Solving equations:

$(({J J}_{k k}^{T T} {J J}_{k k} + + {v v}_{k k} I I)) {δ δ}^{((k k))} = = - - {J J}_{k k}^{T T} {f f}^{((k k))} - - - - - - ((22 twenty two))$

得到向量p第k步修正量δ^(k)。Get the kth step correction amount δ ^(k) of the vector p.

其中，J_k为雅可比矩阵J第k步更新值，

为J_k的转置；v_k为可调迭代步长；I为12阶单位阵；

为目标函数f(p)在第1到第10位置时第k步的更新结果；Among them, J _k is the update value of the Jacobian matrix J at the kth step,

is the transposition of J _k ; v _k is the adjustable iteration step; I is the 12th order unit matrix;

is the update result of step k when the objective function f(p) is in the 1st to 10th positions;

通过pass

p^(k+1)＝p^(k)+δ^(k) (23)p ^(k+1) = p ^(k) + δ ^(k) (23)

式中，p^(k+1)为p向量第k+1步更新值；p^(k)为p向量第k步更新值；对式(20)，(21)，(22)，(23)反复求解迭代，直到向量p收敛到稳定值。向量p的前三个元素g_x0，g_y0，g_z0即为加速度计的零位偏差。In the formula, p ^(k+1) is the update value of p vector at step k+1; p ^(k) is the update value of p vector at step k; for formula (20), (21), (22), (23) The solution iterations are repeated until the vector p converges to a stable value. The first three elements g _x0 , g _y0 , g _z0 of the vector p are the zero position deviation of the accelerometer.

步骤5：通过线性最小二乘法得到耦合矩阵S；Step 5: Obtain the coupling matrix S by linear least square method;

将步骤4中获得的零位偏差g_x0，g_y0，g_z0带入式(2)，可将式(2)误差模型蜕化为线性，利用线性最小二乘法可求得耦合矩阵S为：Bringing the zero position deviations g _x0 , g _y0 , and g _z0 obtained in step 4 into formula (2), the error model of formula (2) can be reduced to linear, and the coupling matrix S can be obtained by using the linear least square method as:

S＝(R*(G^T*(G*G^T)^-1))^-1 (25)S＝(R*(G ^T *(G*G ^T ) ^-1 )) ^-1 (25)

其中：in:

$G = (\begin{matrix} g_{xa 1} & . . . & g_{xa 10} \\ g_{ya 1} & . . . & g_{ya 10} \\ g_{za 1} & . . . & g_{za 10} \end{matrix});$ G^T为G的转置； $R = (\begin{matrix} g_{xm 1} - g_{x 0} & . . . & g_{xm 10} - g_{x 0} \\ g_{ym 1} - g_{y 0} & . . . & g_{ym 10} - g_{y 0} \\ g_{zm 1} - g_{z 0} & . . . & g_{zm 10} - g_{z 0} \end{matrix})$ $G = (\begin{matrix} g_{xa 1} & . . . & g_{xa 10} \\ g_{the ya 1} & . . . & g_{the ya 10} \\ g_{za 1} & . . . & g_{za 10} \end{matrix});$ G ^T is the transpose of G; $R = (\begin{matrix} g_{xm 1} - g_{x 0} & . . . & g_{xm 10} - g_{x 0} \\ g_{ym 1} - g_{the y 0} & . . . & g_{ym 10} - g_{the y 0} \\ g_{zm 1} - g_{z 0} & . . . & g_{zm 10} - g_{z 0} \end{matrix})$

式中，g_xa1...g_xa10，g_ya1...g_ya10，g_za1...g_za10为重力加速度g在第1到第10个位置下在载体坐标系下的投影；g_xm1...g_xm10，g_ym1...g_ym10，g_zm1...g_zm10为加速度计在第1到第10个位置下x、y、z轴测量值。In the formula, g _xa1 ...g _xa10 , g _ya1 ...g _ya10 , g _za1 ...g _za10 are projections of gravitational acceleration g in the carrier coordinate system at the 1st to 10th positions; g _xm1 . ..g _xm10 , g _ym1 ... g _ym10 , g _zm1 ... g _zm10 are the x, y, z axis measurement values of the accelerometer at the 1st to 10th positions.

将步骤4得到零位偏差g_x0，g_y0，g_z0以及步骤5中安装误差和刻度因数的耦合矩阵S的值代入式(2)中，便可构建式(2)误差模型，从而可对加速度计的测量值g_xm，g_ym，g_zm，进行有效的修正，得到修正后的重力加速度g在载体坐标系下投影分量g_xa，g_ya，g_za，从而提高了测量的精度。Substituting the zero position deviation g _x0 , g _y0 , g _z0 obtained in step 4 and the coupling matrix S of installation error and scale factor in step 5 into formula (2), the error model of formula (2) can be constructed, so that the The measured values g _xm , g _ym , g _zm of the accelerometer are effectively corrected to obtain the projected components g _xa , g _ya , g _za of the corrected gravitational acceleration g in the carrier coordinate system, thereby improving the measurement accuracy.

上述步骤4与步骤5中进行的非线性最小二乘和线性最小二乘均需要步骤1、步骤2中载体所在的10个位置间的相关性要小，即10组(g_xa，g_ya，g_za)之间、10组(g_xm，g_ym，g_zm)之间相关性要小。本发明采用如下位置选取方法(以俯仰角、横滚角的不同代表不同的位置)，如表1所示。The nonlinear least squares and linear least squares carried out in the above step 4 and step 5 all require that the correlation between the 10 positions of the carrier in step 1 and step 2 is small, that is, 10 groups (g _xa , g _ya , g _za ) and between 10 groups (g _xm , g _ym , g _zm ) are less correlated. The present invention adopts the following position selection method (different positions are represented by different pitch angles and roll angles), as shown in Table 1.

表1 10位置选择方案一Table 1 10 position selection scheme 1

姿态角attitude angle 1 1 2 2 33 44 55 66 77 8 8 9 9 1010 俯仰角/(°)Pitch angle/(°) 00 -20-20 4040 -60-60 8080 -100-100 120120 -140-140 160160 -180-180 横滚角/(°)Roll angle/(°) -10-10 3030 -50-50 7070 -90-90 110110 -130-130 150150 -170-170 1010

上述方案满足相关性要求，且在这种选取方法下标定精度高。对该位置方案的仿真验证如下：经过数学软件MATLAB编程验证，根据位置方案进行标定实验，如果可以精确的辨识出零位偏差等参数，则说明10个标定位置不相关，否则反之。验证过程中共进行3000次实验，记录其中可以准确辨识出待求参数的实验次数作为方案的评价指标，并选取不同的10位置方案进行对比，结果如表5。The above scheme meets the correlation requirements, and the calibration accuracy is high under this selection method. The simulation verification of the position scheme is as follows: After the mathematical software MATLAB programming verification, the calibration experiment is carried out according to the position scheme. If the parameters such as zero position deviation can be accurately identified, it means that the 10 calibration positions are irrelevant, otherwise, the opposite is true. During the verification process, a total of 3,000 experiments were carried out, and the number of experiments that could accurately identify the parameters to be sought was recorded as the evaluation index of the scheme, and different 10-position schemes were selected for comparison. The results are shown in Table 5.

10位置选择如表2的方案二如下：The 10 position selection is as shown in the second scheme of Table 2 as follows:

表2 10位置选取方案二Table 2 10 position selection scheme two

姿态角attitude angle 1 1 2 2 33 44 55 66 77 8 8 9 9 1010 俯仰角/(°)Pitch angle/(°) 00 -20-20 4040 -60-60 8080 -100-100 120120 -140-140 160160 -180-180 横滚角/(°)Roll angle/(°) 00 3030 -50-50 7070 -90-90 110110 -130-130 150150 -170-170 1010

10位置选择如表3的方案三如下：10 Locations are selected as shown in Table 3, Scheme 3 is as follows:

表3 10位置选取方案三Table 3 10 position selection scheme three

姿态角attitude angle 1 1 2 2 33 44 55 66 77 8 8 9 9 1010 俯仰角/(°)Pitch angle/(°) 00 -20-20 4040 -60-60 8080 -100-100 120120 -140-140 160160 -180-180 横滚角/(°)Roll angle/(°) -10-10 1010 -30-30 5050 -70-70 9090 -110-110 130130 -150-150 170170

10位置选择如表4的方案四如下：The 10 position selection is as shown in Table 4, plan four is as follows:

表4 10位置选取方案四Table 4 10 position selection scheme four

姿态角attitude angle 1 1 2 2 33 44 55 66 77 8 8 9 9 1010 俯仰角/(°)Pitch angle/(°) 00 -20-20 4040 -60-60 8080 -100-100 120120 -140-140 160160 -180-180 横滚角/(°)Roll angle/(°) 00 -30-30 5050 -70-70 9090 -110-110 130130 -150-150 170170 -10-10

10位置选择在四种方案下，各进行3000次实验，各方案下可以准确辨识出待求参数的实验次数的统计结果如表5所示：10 Location selection Under the four schemes, 3000 experiments were carried out each, and the statistical results of the number of experiments that can accurately identify the parameters to be sought under each scheme are shown in Table 5:

表5 对比结果Table 5 Comparison results

从表5中可以看出，10位置选择在方案一的情况下，准确辨识出待求参数的实验次数最高，因此选择方案一为优选方案。It can be seen from Table 5 that the number of experiments to accurately identify the parameters to be sought is the highest in the case of 10 position selection in the case of scheme one, so scheme one is the preferred scheme.

Claims

1. A method for calibrating an accelerometer in an unsupported state is characterized in that: before the calibration process, firstly, the error parameter which has the most serious influence on the performance of the accelerometer needs to be considered, and the error model of the simplified triaxial accelerometer is as follows:

(\begin{matrix} g_{xa} \\ g_{ya} \\ g_{za} \end{matrix}) = (\begin{matrix} K_{11} k_{x} & K_{12} k_{y} & K_{13} k_{z} \\ K_{21} k_{x} & K_{22} k_{y} & K_{23} k_{z} \\ K_{31} k_{x} & K_{32} k_{y} & K_{33} k_{z} \end{matrix}) (\begin{matrix} (g_{xm} - g_{x 0}) \\ (g_{ym} - g_{y 0}) \\ (g_{zm} - g_{z 0}) \end{matrix}) - - - (1)

wherein k is_x、k_y、k_zScale factors for x, y, z axes of the accelerometer; k₁₁、K₁₂、K₁₃、K₂₁、K₂₂、K₂₃、K₃₁、K₃₂And K₃₃Is an installation error matrix element; g_x0、g_y0、g_z0Zero offset of x, y and z axes of the accelerometer; g_xm、g_ym、g_zmAre accelerometer x, y, z axis measurements; g_xa、g_ya、g_zaThe calculated value of the projection component of the gravity acceleration g under the carrier coordinate system is obtained;

(\begin{matrix} K_{11} k_{x} & K_{12} k_{y} & K_{13} k_{z} \\ K_{21} k_{x} & K_{22} k_{y} & K_{23} k_{z} \\ K_{31} k_{x} & K_{32} k_{y} & K_{33} k_{z} \end{matrix})

a coupling matrix of mounting errors and scale factors;

further calibration is accomplished by the following 5 steps:

step 1: roughly aligning the static base to obtain a roll angle gamma and a pitch angle theta of the carrier;

step 2: the Kalman filtering is utilized to carry out precise alignment on the static base, the alignment precision of the roll angle gamma and the pitch angle theta is improved, and the calculated value (g) of the three-axis component of the projection of the gravity acceleration g under the carrier coordinate system is obtained_xa，g_ya，g_za)；

And step 3: placing the carrier at 10 positions with different pitch angles and roll angles, performing the steps 1 and 2, judging whether the placement at 10 positions is finished, if the placement at the 10 positions is not finished, executing the step 1, and if the placement at the 10 positions is finished, obtaining the x-axis, y-axis and z-axis measured values (g) of 10 groups of accelerometers_xm，g_ym，g_zm) And 10 sets of calculation values (g) of three-axis components of the projection of the gravity acceleration g in the carrier coordinate system_xa，g_ya，g_za) Continuing to step 4;

and 4, step 4: obtaining triaxial zero offset g of accelerometer by nonlinear least square method_x0，g_y0，g_z0；

And 5: obtaining an' by a linear least squares methodCoupling matrix with error and scale factor

(\begin{matrix} K_{11} k_{x} & K_{12} k_{y} & K_{13} k_{z} \\ K_{21} k_{x} & K_{22} k_{y} & K_{23} k_{z} \\ K_{31} k_{x} & K_{32} k_{y} & K_{33} k_{z} \end{matrix});

The zero deviation g obtained in the step 4 is compared_x0、g_y0、g_z0And substituting the values of the coupling matrix of the installation error and the scale factors into the formula (1), constructing an error model of the formula (1), and realizing the calibration of the accelerometer in an unsupported state.

2. The method for calibrating the accelerometer in the unsupported state as claimed in claim 1, wherein: and 4, obtaining the three-axis zero offset through a nonlinear least square method, and specifically comprising the following steps:

step a, constructing a target function;

transforming equation (1) into the order of the objective function f (p)

f(p)＝[K₁₁k_x(g_xm-g_x0)+K₁₂k_y(g_ym-g_y0)+K₁₃k_z(g_zm-g_z0)]²+[K₂₁k_x(g_xm-g_x0)+K₂₂k_y(g_ym-g_y0)+K₂₃k_z(g_zm-g_z0)]² (2)

+[K₃₁k_x(g_xm-g_x0)+K₃₂k_y(g_ym-g_y0)+K₃₃k_z(g_zm-g_z0)]²-(g_xa ²+g_ya ²+g_za ²)

Wherein the vector

p＝[g_x0，g_y0，g_z0，K₁₁k_x，K₁₂k_y，K₁₃k_z，K₂₁k_x，K₂₂k_y，K₂₃k_z，K₃₁k_x，K₃₂k_y，K₃₃k_z]；

B, solving a Jacobian matrix of the objective function f (p);

the jacobian matrix J of the objective function f (p) is:

wherein f is₁...f₁₀Is an objective function f (p) at the 1 st to 10 th positions; p is a radical of₁…p₁₂Is given as vector p ═ g_x0，g_y0，g_z0，S₁₁，S₁₂，S₁₃，S₂₁，S₂₂，S₂₃，S₃₁，S₃₂，S₃₃]1 st to 12 th element of (1);

step c, iterative approximation is carried out on an unknown vector p;

solving the equation:

obtaining the k step correction quantity delta of the vector p^(k)；

Wherein, J_kUpdating the k-th step of the Jacobian matrix J, wherein k is 1, 2, …;

is J_kTransposing; v. of_kIs an adjustable iteration step length; i is a 12-order unit array;

is the updated result of the k step of the objective function f (p) at the 1 st to 10 th positions;

by passing

p^(k+1)＝p^(k)+δ^(k) (5)

Wherein p is^(k+1)Updating the k +1 step of the vector p; p is a radical of^(k)Updating the value of the k step for the vector p; solving iterations for equations (2), (3), (4), (5) until vector p converges to a stable value; the first three elements g of the vector p_x0，g_y0，g_z0Is the null of the accelerometer.

3. The method for calibrating the accelerometer in the unsupported state as claimed in claim 1, wherein: and 5, obtaining the coupling matrix of the installation error and the scale factor by a linear least square method, and specifically comprising the following steps:

deviation of zero position g_x0，g_y0，g_z0Taking formula (1), degrading the error model of formula (1) into linearity, and obtaining the coupling matrix by using a linear least square method

(\begin{matrix} K_{11} k_{x} & K_{12} k_{y} & K_{13} k_{z} \\ K_{21} k_{x} & K_{22} k_{y} & K_{23} k_{z} \\ K_{31} k_{x} & K_{32} k_{y} & K_{33} k_{z} \end{matrix})

Comprises the following steps:

(\begin{matrix} K_{11} k_{x} & K_{12} k_{y} & K_{13} k_{z} \\ K_{21} k_{x} & K_{22} k_{y} & K_{23} k_{z} \\ K_{31} k_{x} & K_{32} k_{y} & K_{33} k_{z} \end{matrix}) = {(R * (G^{T} * {(G * G^{T})}^{- 1}))}^{- 1} - - - (6)

wherein:

G = (\begin{matrix} g_{xa 1} & . . . & g_{xa 10} \\ g_{ya 1} & . . . & g_{ya 10} \\ g_{za 1} & . . . & g_{za 10} \end{matrix});

G^Tis the transpose of G;

R = (\begin{matrix} g_{xm 1} - g_{x 0} & . . . & g_{xm 10} - g_{x 0} \\ g_{ym 1} - g_{y 0} & . . . & g_{ym 10} - g_{y 0} \\ g_{zm 1} - g_{z 0} & . . . & g_{zm 10} - g_{z 0} \end{matrix})

in the formula, g_xa1…g_xa10，g_ya1…g_ya10，g_za1…g_za10The projection of the gravity acceleration g under the carrier coordinate system under the 1 st to the 10 th positions; g_xm1…g_xm10，g_ym1…g_ym10，g_zm1…g_zm10The x, y and z axis measurement values of the accelerometers of the carrier in the 1 st to 10 th positions are obtained.

4. The method for calibrating the accelerometer in the unsupported state as claimed in claim 1, wherein: the pitch angle and the roll angle were (0 °, -10 °), (-20 °, 30 °), (40 °, -50 °) (60 °, 70 °), (80 °, -90 °), (-100 °, 110 °), (120 °, -130 °), (-140 °, 150 °), (160 °, (170 °), (-180 °, 10 °) respectively, and 10 positions of the carrier were selected.