Innovative self-calibration method for accelerometer scale factor of the missile-borne RINS with fiber optic gyro

Qian Zhang; Lei Wang; Zengjun Liu; Yiming Zhang

doi:10.1364/OE.24.021228

1. Introduction

With the requirements of missile systems becoming more and more significant for modern war, it should not only be capable to aim the targets accurately, but should also be able to respond quickly and flexibly. The inertial navigation system (INS) is entirely self-contained and can provide velocity, position and attitude information in all terrain and in all time [1–5], therefore, INS holds a dominant position in missile navigation [6]. An inertial measurement unit (IMU) is the core component of INS which consists of three orthogonal accelerometers and three orthogonal gyroscopes and is used to provide specific forces and angular rates for INS [7,8]. As the speed range of missile is larger than aircraft and ships, it puts higher requirements on the scale factor errors of accelerometer.

Calibration is very important in the operation of INS [9–13], when it comes to missile, it is especially important for the scale factor errors of accelerometer. The calibration parameters will change as working time increasing. Generally, the INS of missile would be removed for calibration every few months, which is a repetitive and boring job. The traditional calibration method is usually performed by comparing the IMU outputs with known references provided by precise orientation control which requires specialized high-precision equipment such as a turntable [14,15]. For example, multi-position calibration methods are used in [1,3,5], by keeping the sensors in several known positions with respect to the local gravity and the angular rate of the earth, the sensor errors are calculated by several groups of sensor output errors. However, this calibration method commonly used is realized by a high accuracy turntable in the laboratory and the IMU is required to be mounted on a precise turntable with the input axes of each accelerometers pointing in the directions which are designed before in calibration process. Generally speaking, the traditional calibration method is usually expensive due to the cost of the essential high accuracy turntable. The other calibration method is based on the state estimation in which several sets of rotation sequences will be designed to estimate the parameters of IMU [6,10]. This systematic calibration method is based on the observation of the navigaiton errors for the navigation, after establishing the relationship between the calibrated parameters and the navigation information, the parameters can be calculated by the filtering method. However, for missile application, these calibration methods have a fundamental disadvantage that the IMU must be detached down from missile which leads to a situation that the calibration results are not calculated under the actual operating environment. Therefore, the results of traditional calibration method may not be accurate when the INS is reinstalled on the missile.

With the development of optical inertial sensors and photoelectric encoder, a new type of INS named rotational INS (RINS) is proposed and has been used in several missiles [16–19]. For example, in the Minuteman III ICBM, the INS which has three mechanical gimbals with limited rotational range ue to the use of limited cable bundles [20]. In the INS of Peacekeeper, the mechanical gimbals are replaced by hydraulic gimbal system, in which the gimbals are suspended in hydraulic fluid to provide unlimited rotational range in any direction [21,22]. In RINS, IMU is mounted on the multi-axis gimbals, the gimbals are controlled precisely by photoelectric encoder, the drifts and bias errors of inertial sensors are mitigated by rotating the IMU periodically [18, 23–25]. Since the multi-axis RINS has the main functions of gimbals and photoelectric encoders, it is natural to have the idea of calibrating its IMU on the system itself after the system is deployed [16]. In this way, the missile could complete timely self-calibration of IMU in wartime without detaching the IMU from missile. Due to some reasons such as mechanical processing, the gimbals cannot be ideally perpendicular to each other and the non-orthogonality errors still exist. What’s more, the mobility of the missile launch vehicle lead to the state of the missile is random before launching. Therefore, the initial state of calibration is unknow and the INS is non-horizontal. However, the unknown local horizontal of RINS and non-orthogonality errors of mechanical processing will give rise to the inaccuracy of calibration results and the complexity of calibration analysis. In this paper, an innovative accelerometer scale factor calibration procedure using the gimbals of RINS is proposed, so that the orthogonality requirement of gimbals and exact information of the local horizontal could be relaxed. This method is especially applicable to the missile for the characteristics of quick response and mobility.

2. Theoretical analysis

2.1 Configuration of RINS and frame definition

The proposed RINS structure is shown in Fig. 1. The RINS mainly includes three rotation gimbals, IMU, three groups of photoelectric encoder and torque motor. The IMU could rotate continuously around the axis of inner gimbal.

Fig. 1 The structure of RINS.

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In RINS, the three rotating axes are presumed to be orthogonal to each other when photoelectric encoders of three photoelectric encoders are all zero. However, due to the processing accuracy and the mounting errors, the adjacent two gimbals cannot be completely orthogonal to each other. Therefore, the non- orthogonality angles are coupled into projecting matrices and will bring new errors for the calculation solutions.

To explain the mechanism of calibration error in this paper clearly, the frames used in this paper are defined as follows. The local geographical coordinate frame is called navigation frame (N-frame), the $X_{n}$ -axis points east in the local horizontal, $Y_{n}$ -axis points to north in the local horizontal and $Z_{n}$ -axis is defined according to the right-hand rule. The body coordinate frame (B-frame) is fixed to the outer gimbal, $Y_{b}$ -axis points to the outer gimbal axis, $X_{b}$ -axis points to the right of RINS and $Z_{b}$ -axis is defined according to the right-hand rule. The B-frame also be called as outer gimbal frame (O-frame). The middle coordinate frame (M-frame) is fixed to the middle gimbal, $X_{m}$ -axis points to the middle gimbal axis and $Z_{m}$ -axis points to the vertical direction of middle frame plane, $Y_{m}$ -axis is defined according to the right-hand rule. The inner coordinate frame (I-frame) is fixed to the inner gimbal, $Z_{i}$ -axis points the inner axis and $X_{i}$ -axis points the $x$ accelerometer, $Y_{i}$ -axis is defined according to the right-hand rule. The I-frame can also be called as measurement coordinate frame. Ignoring the non-orthogonality angles of gimbals, the I-frame should be aligned with the M-frame and O-frame when three photoelectric encoders are all zero.

2.2 Analysis of calibration error

Multi-position method and continuous rotating method are usually used in traditional calibration. In multi-position method, rotation angular rate of earth and local gravity are taken as the input of sensors, based on the fact that the norms of the measured outputs of the accelerometer and gyroscope are equal to the magnitudes of specific force and rotational velocity inputs, respectively [12]. This paper mainly focuses on the calibration of $x$ accelerometer scale factor and the detail of calibration method is described as follows, the scale factor of other accelerometers could be calibrated in the same way.

As shown in Fig. 2, the non-orthogonality angles of gimbals and non-horizontal are ignored and three photoelectric encoders are all zero. Then, rotating the inner gimbal to the position of 90° and middle gimbal to the position of 90°,the $x$ accelerometer axis points up and the outputs of three accelerometers can be expressed by Eq. (1).

(\begin{matrix} 0 \\ 0 \\ g \end{matrix}) = R_{X} (90) R_{Z} (90) (\begin{matrix} a_{x u} \\ a_{y u} \\ a_{z u} \end{matrix}) = (\begin{matrix} 1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & - 1 & 0 \end{matrix}) (\begin{matrix} 0 & 1 & 0 \\ - 1 & 0 & 0 \\ 0 & 0 & 1 \end{matrix}) (\begin{matrix} a_{x u} \\ a_{y u} \\ a_{z u} \end{matrix}) = (\begin{matrix} a_{y u} \\ a_{z u} \\ a_{x u} \end{matrix})

where

a_{x u}

,

a_{y u}

and

a_{z u}

are the input values of three accelerometers, respectively.

R_{X} (90)

represents a 90° rotation around the

X

axis and

R_{Z} (90)

represents a 90° rotation around the

Z

axis.

Fig. 2 The $x$ accelerometer axis points up.

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Then, rotating the inner gimbal to the position of 90° and middle gimbal to the position of −90°,the $x$ accelerometer axis points down (Fig. 3) and the outputs of three accelerometers can be expressed by Eq. (2).

Fig. 3 The $x$ accelerometer axis points down.

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(\begin{matrix} 0 \\ 0 \\ g \end{matrix}) = R_{X} (- 90) R_{Z} (90) (\begin{matrix} a_{x u} \\ a_{y u} \\ a_{z u} \end{matrix}) = (\begin{matrix} 1 & 0 & 0 \\ 0 & 0 & - 1 \\ 0 & 1 & 0 \end{matrix}) (\begin{matrix} 0 & 1 & 0 \\ - 1 & 0 & 0 \\ 0 & 0 & 1 \end{matrix}) (\begin{matrix} a_{x u} \\ a_{y u} \\ a_{z u} \end{matrix}) = (\begin{matrix} a_{y u} \\ - a_{z u} \\ - a_{x u} \end{matrix})

Based on the Eq. (1) and Eq. (2), the relationship between $x$ accelerometer scale factor and raw sampling data can be described as

{\begin{matrix} K_{a x} (g + \nabla_{x}) = {\bar{N}}_{a x u} \\ K_{a x} (- g + \nabla_{x}) = {\bar{N}}_{a x d} \end{matrix}

where

K_{a x}

is the scale factor of

x

accelerometer,

\nabla_{x}

is the accelerometer bias error.

{\bar{N}}_{a x u}

and

{\bar{N}}_{a x d}

are averaged sampling data when

x

accelerometer axis points to up and down, respectively.

Then the scale factor of $x$ accelerometer can be calculated using the Eq. (3) and the result is

K_{a x} = \frac{{\bar{N}}_{a x u} - {\bar{N}}_{a x d}}{2 g} .

Different from the precision turntable, non-orthogonality angles are universally existed in the RINS and the operating environment cannot guarantee the absolute local horizontal of the system. Then the method mentioned above cannot be suitable for this condition. The actual situation is shown in Fig. 4 and the outputs of three accelerometers are described in Eq. (5) and Eq. (6).

(\begin{matrix} a_{x u} \\ a_{y u} \\ a_{z u} \end{matrix}) = R_{Y} (- η) R_{Y} (- γ) R_{Z} (- 90) R_{X} (- 90) (\begin{matrix} 0 \\ 0 \\ g \end{matrix}) = (\begin{matrix} - g (\sin η \sin γ - \cos η \cos γ) \\ 0 \\ - g (\sin η \cos γ + \cos η \sin γ) \end{matrix})

(\begin{matrix} a_{x d} \\ a_{y d} \\ a_{z d} \end{matrix}) = R_{Y} (- η) R_{Y} (- γ) R_{Z} (- 90) R_{X} (90) (\begin{matrix} 0 \\ 0 \\ g \end{matrix}) = (\begin{matrix} g (\sin η \sin γ - \cos η \cos γ) \\ 0 \\ g (\sin η \cos γ + \cos η \sin γ) \end{matrix})

where

η

is the non-horizontal angle and

γ

is the non- orthogonality angle.

Fig. 4 The actual situation of the gimbals.

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Then the scale factor of $x$ accelerometer can be calculated using the Eq. (5) and Eq. (6), the result is

K_{a x} = \frac{{\bar{N}}_{a x u} - {\bar{N}}_{a x d}}{2 g \cos η \cos γ}

considering the Eq. (4) and Eq. (7), the $x$ accelerometer scale factor error (ppm) could be expressed

Δ K_{a x} = 10^{6} \frac{\frac{{\bar{N}}_{a x u} - {\bar{N}}_{a x d}}{2 g \cos η \cos γ} - \frac{{\bar{N}}_{a x u} - {\bar{N}}_{a x d}}{2 g}}{\frac{{\bar{N}}_{a x u} - {\bar{N}}_{a x d}}{2 g \cos η \cos γ}} .

The Eq. (8) indicates the accelerometer scale factor error is tightly influenced by non-orthogonality angle of inner axis and middle axis and the non-horizontal angle of middle axis. Figure 5 indicates the error of accelerometer scale factor when non-horizontal angle changes from 0° to 10° and non-orthogonality angle changes from 0° to 1°. For example, assuming that the non-horizontal angle $η$ is 10° and non-orthogonality angle $γ$ is 1°, the accelerometer scale factor error will be 15342ppm, which is definitely apparent and unacceptable in INS.

Fig. 5 The error of accelerometer scale factor.

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3. Calibration algorithm

3.1 Calibration of non-orthogonality angle

In multi-axis RINS, the constant errors of sensors are mitigated by rotating the IMU periodically [9]. If the adjacent two gimbals are presumed to be orthogonal and the I-frame should be aligned with the M-frame and O-frame when three photoelectric encoders are all zero. Therefore, each fiber optic gyro could only capture the vertical rotation angular velocity of frame axis while cross-axis sensitivities are inexistent. On the contrary, the non-orthogonality angle between gimbals could be calculated by the gyroscope signal with a designed rotation strategy.

Figure 6 indicates the relationship between non-orthogonality angle $γ$ and other error angles. Where $G_{x}$ and $G_{z}$ point to the direction of $x$ and $z$ gyroscope, respectively. $*_{- V}$ means the vertical direction of $*$ . $θ_{G z G x}^{X}$ is the non-orthogonality angle between $G_{x}$ and $G_{z}$ , $β_{G x}^{Y}$ is the non-orthogonality angle between $G_{x}$ and $Z_{i}$ , $θ_{G z}^{Y}$ is the non-orthogonality angle between $G_{z}$ and $X_{m}$ . $θ_{G z G x}^{X}, θ_{G z}^{Y}, β_{G x}^{Y}$ are all orthogonal decomposition values in the vertical plane of $Y_{b}$ . $γ$ is the non-orthogonality angle between inner axis and middle axis.

Fig. 6 Schematic of non- orthogonality angle.

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As shown in Fig. 6, the non-orthogonality angle cannot be calibrated directly and could be calculated by the Eq. (9).

γ = θ_{G z G x}^{X} - θ_{G z}^{Y} - β_{G x}^{Y}

$β_{G x}^{Y}$ is the non-orthogonality angle between $G_{x}$ and $Z_{i}$ , it means that if the IMU rotating around $Z_{i}$ -axis, there will be an output signal in $x$ gyroscope because of cross-axis sensitivities. As shown in Fig. 7, rotating the middle gimbal to the position of 0° and outer gimbal to the position of 0°,then IMU rotates bi-directionally along with the $Z_{i}$ -axis at the rotation angular speed of $Ω$ continuously. The output of $x$ gyroscope can be expressed as follows.

{\begin{cases} \bar{ω_{x}^{+}} = - (Ω + ω_{e x}) β_{G x}^{Y} + Δ ε_{x} \\ \bar{ω_{x}^{-}} = - (- Ω + ω_{e x}) β_{G x}^{Y} + Δ ε_{x} \end{cases}

where

\bar{ω_{x}^{+}}

and

\bar{ω_{x}^{-}}

are averaged sampling data of

x

gyroscope when the IMU rotates bi-directionally.

Ω

is the rotation angular speed around

Z_{i}

-axis,

ω_{e x}

is the component of earth rotation angular speed and

Δ ε_{x}

is

x

gyroscope drift. Then the non-orthogonality angle between

G_{x}

and

Z_{i}

can be calculated as

Fig. 7 Rotation along with the $Z_{i}$ -axis.

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β_{G x}^{Y} = - \frac{\bar{ω_{x}^{+}} - \bar{ω_{x}^{-}}}{2 Ω}

$θ_{G z}^{Y}$ is the non-orthogonality angle between $G_{z}$ and $X_{m}$ . As shown in Fig. 8, rotating the inner gimbal to the position of 0° and outer gimbal to the position of 0°,then IMU rotates bi-directionally along with the $X_{m}$ -axis at the rotation angular speed of $Ω$ continuously. The output of $z$ gyroscope can be expressed as follows.

{\begin{cases} \bar{ω_{z}^{+}} = - (Ω + ω_{e z}) θ_{G z}^{Y} + Δ ε_{z} \\ \bar{ω_{z}^{-}} = - (- Ω + ω_{e z}) θ_{G z}^{Y} + Δ ε_{z} \end{cases}

where

\bar{ω_{z}^{+}}

and

\bar{ω_{z}^{-}}

are averaged sampling data of

z

gyroscope when the middle gimbal rotates bi-directionally, respectively.

Ω

is the rotation angular speed around

X_{m}

-axis,

ω_{e z}

is the component of earth rotation angular speed and

Δ ε_{z}

is

z

gyroscope drift. Then the non-orthogonality angle between

G_{z}

and

X_{m}

can be calculated as

Fig. 8 Rotation along with the $X_{m}$ -axis.

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θ_{G z}^{Y} = - \frac{\bar{ω_{z}^{+}} - \bar{ω_{z}^{-}}}{2 Ω} .

As the $θ_{G z G x}^{X}$ is the non-orthogonality angle between $G_{x}$ and $G_{z}$ , it cannot be directly calibrated by rotating any axis. So a transitional axis should be used as an intermediate between $G_{x}$ and $G_{z}$ . As shown in Fig. 9, the outer gimbal axis $Y_{b}$ is selected as the transitional axis. Rotating the inner gimbal to the position of 90° and middle gimbal to the position of 0°,then IMU rotates bi-directionally along with the $Y_{b}$ -axis at the rotation angular speed of $Ω$ continuously. The output of $z$ gyroscope can be expressed as follows.

{\begin{cases} \bar{{ω^{'}}_{z}^{+}} = - (Ω + {ω^{'}}_{e z}) θ_{G z}^{X} + Δ ε_{z} \\ \bar{{ω^{'}}_{z}^{-}} = - (- Ω + {ω^{'}}_{e z}) θ_{G z}^{X} + Δ ε_{z} \end{cases}

θ_{G z}^{X} = - \frac{\bar{{ω^{'}}_{z}^{+}} - \bar{{ω^{'}}_{z}^{-}}}{2 Ω}

where

\bar{{ω^{'}}_{z}^{+}}

and

\bar{{ω^{'}}_{z}^{-}}

are averaged sampling data of

z

gyroscope when the outer gimbal rotates bi-directionally, respectively.

Ω

is the rotation angular speed around

Y_{b}

-axis,

{ω^{'}}_{e z}

is the component of earth rotation angular speed.

θ_{G z}^{X}

represents the non-orthogonality angle between

G_{z}

and

Y_{b}

-axis.

Fig. 9 Rotation along with the $Y_{b}$ -axis.

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Then rotates the inner gimbal to the position of 90° and middle gimbal to the position of 90°, rotates bi-directionally along with the $Y_{b}$ -axis at the rotation angular speed of $Ω$ continuously. As shown in Fig. 10, the output of $x$ gyroscope can be expressed as follows.

{\begin{cases} \bar{{ω^{'}}_{x}^{+}} = - (Ω + {ω^{'}}_{e x}) θ_{G x}^{X} + Δ ε_{x} \\ \bar{{ω^{'}}_{x}^{-}} = - (- Ω + {ω^{'}}_{e x}) θ_{G x}^{X} + Δ ε_{x} \end{cases}

θ_{G x}^{X} = - \frac{\bar{{ω^{'}}_{x}^{+}} - \bar{{ω^{'}}_{x}^{-}}}{2 Ω}

where

\bar{{ω^{'}}_{x}^{+}}

and

\bar{{ω^{'}}_{x}^{-}}

are averaged sampling data of

x

gyroscope when the outer gimbal rotates bi-directionally, respectively.

{ω^{'}}_{e x}

is the component of earth rotation angular speed.

θ_{G x}^{X}

represents the non-orthogonality angle between

G_{x}

and

Y_{b}

-axis. Then the non-orthogonality angle

θ_{G z G x}^{X}

can be described as

Fig. 10 Rotation along with the $Y_{b}$ -axis.

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θ_{G z G x}^{X} = θ_{G x}^{X} - θ_{G z}^{X}

Now the non-orthogonality angle $γ$ could be calculated according the Eq. (9).

3.2 Calibration of non-horizontal angle

The non-horizontal angle of $Y_{m}$ -axis is caused by the local horizontal of the system and it is variable based on the operating environment of the RINS. Therefore, a universally applicable method should be proposed to adapt to any kinds of situation. Here, before the calibration, a spinning coarse alignment method is introduced to eliminate the initial platform angles, then the horizontal two gimbals need to be rotated to local horizontal for calculating the dictate deviation angle.

The rotation strategy is described as follows. Firstly, rotates the middle and outer gimbals to the position of 0° and locks the horizontal two gimbals in the current position, then rotates the IMU bi-directionally around the $Z_{i}$ -axis once as a single-axis RINS. Because of the rotation around the $Z_{i}$ -axis, the constant errors of sensors are modulated into periodical form, whose average values are zero in a rotation period $T$ . It means that the gyroscope drifts and accelerometer bias errors can be ignored during coarse alignment.

As the middle and outer gimbals are all at the position of 0°, the transformation matrix from I-frame to B-frame can be described as follows:

C_{i}^{b} = [\begin{matrix} \cos ϕ_{Z R} & - \sin ϕ_{Z R} & 0 \\ \sin ϕ_{Z R} & \cos ϕ_{Z R} & 0 \\ 0 & 0 & 1 \end{matrix}]

where

ϕ_{Z R}

is the rotation angle of the

Z_{i}

-axis which can be read from the photoelectric encoder on inner gimbal. The angular speed and acceleration in B-frame can be described as

[\begin{matrix} Δ θ_{x}^{b} \\ Δ θ_{y}^{b} \\ Δ θ_{z}^{b} \end{matrix}] = [\begin{matrix} \cos ϕ_{Z R} & - \sin ϕ_{Z R} & 0 \\ \sin ϕ_{Z R} & \cos ϕ_{Z R} & 0 \\ 0 & 0 & 1 \end{matrix}] [\begin{matrix} Δ θ_{x}^{i} \\ Δ θ_{y}^{i} \\ Δ θ_{z}^{i} \end{matrix}]

[\begin{matrix} Δ V_{x}^{b} \\ Δ V_{y}^{b} \\ Δ V_{z}^{b} \end{matrix}] = [\begin{matrix} \cos ϕ_{Z R} & - \sin ϕ_{Z R} & 0 \\ \sin ϕ_{Z R} & \cos ϕ_{Z R} & 0 \\ 0 & 0 & 1 \end{matrix}] [\begin{matrix} Δ V_{x}^{i} \\ Δ V_{y}^{i} \\ Δ V_{z}^{i} \end{matrix}]

where

Δ θ_{x}^{i}, Δ θ_{y}^{i}, Δ θ_{z}^{i}

are the angle increment in I-frame and

Δ θ_{x}^{b}, Δ θ_{y}^{b}, Δ θ_{z}^{b}

are the angle increment in B-frame,

Δ V_{x}^{i}, Δ V_{y}^{i}, Δ V_{z}^{i}

are the velocity increment in I-frame and

Δ V_{x}^{b}, Δ V_{y}^{b}, Δ V_{z}^{b}

are the velocity increment in B-frame. After rotation, the averaged values of angular speed and acceleration can be calculated as

{\bar{ω}}^{b} = [\begin{matrix} ω_{x}^{b} \\ ω_{y}^{b} \\ ω_{z}^{b} \end{matrix}] = \frac{1}{2 T} [\begin{matrix} \sum Δ θ_{x}^{b} \\ \sum Δ θ_{y}^{b} \\ \sum Δ θ_{z}^{b} \end{matrix}]

{\bar{f}}^{b} = [\begin{matrix} f_{x}^{b} \\ f_{y}^{b} \\ f_{z}^{b} \end{matrix}] = \frac{1}{2 T} [\begin{matrix} \sum Δ V_{x}^{b} \\ \sum Δ V_{y}^{b} \\ \sum Δ V_{z}^{b} \end{matrix}]

Then the attitude transformation matrix can be calculated based on the Eq. (24).

[\begin{matrix} ω_{x}^{b} & ω_{y}^{b} & ω_{z}^{b} \\ f_{x}^{b} & f_{y}^{b} & f_{z}^{b} \end{matrix}] = [\begin{matrix} 0 & ω_{i e} \cos φ & ω_{i e} \sin φ \\ 0 & 0 & g \end{matrix}] C_{b}^{n}

where

C_{b}^{n}

is the attitude transformation matrix between B-frame and N-frame,

ω_{i e}

is the earth rotation angular speed and

φ

is the local latitude. So the components of

C_{b}^{n}

are expressed as

{\begin{array}{l} T_{31} = \frac{f_{x}^{b}}{g} \\ T_{32} = \frac{f_{y}^{b}}{g} \\ T_{33} = \frac{f_{z}^{b}}{g} \\ T_{21} = \frac{ω_{x}^{b} - T_{31} ω_{i e} \sin φ}{ω_{i e} \cos φ} \\ T_{22} = \frac{ω_{x}^{b} - T_{32} ω_{i e} \sin φ}{ω_{i e} \cos φ} \\ T_{23} = \frac{ω_{x}^{b} - T_{33} ω_{i e} \sin φ}{ω_{i e} \cos φ} \\ T_{11} = T_{22} T_{33} - T_{23} T_{32} \\ T_{12} = T_{23} T_{31} - T_{21} T_{33} \\ T_{13} = T_{21} T_{32} - T_{22} T_{31} \end{array}

Then the attitude of RINS can be computed as

{\begin{matrix} θ = \sin^{- 1} (T_{23}) \\ ϕ = \tan^{- 1} (- \frac{T_{13}}{T_{33}}) \\ ψ = \tan^{- 1} (\frac{T_{21}}{T_{22}}) \end{matrix}

where

θ, ϕ, ψ

are pitch angle, roll angle and yaw angle. As the three gimbals are all rotated to the position of 0° before spinning coarse alignment, the calculated results

θ, ϕ, ψ

are non-horizontal angles of three gimbals, namely middle gimbal, outer gimbal and inner gimbal. Then, the three gimbals could rotate to keep the IMU horizontal according to the results of spinning coarse alignment.

4. Experimental results

In this section, several experimental tests have been implemented to confirm the validity of the proposed method.

4.1 Laboratory static experiment

A calibration experiment with multi-axis RINS is carried out to validate the preceding analysis. The IMU used in the RINS consists of three fiber optic gyroscopes and three quartz flexure accelerometers. The specifications of the IMU and photoelectric encoder are shown in Table 1.

Table 1. Specification of the IMU.

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Figure 11 is the calibration experiment platform. The RINS is mounted rigidly on the turntable, and its middle axis is vertical with the outer axis of the turntable. Firstly, keep the each axis of RINS at the photoelectric encoder position of 0°, and then set the turntable outer frame angle to 0°, ± 5°, ± 10°, ± 15°, ± 20°. Then non-orthogonality angle $γ$ is calibrated at each position. Table 2 shows the calibration results, it represents that the non-orthogonality angle $γ$ is stable with the change of the turntable outer frame angle.

Fig. 11 The laboratory experimental equipment.

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Table 2. The calibration results.

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Then, rotates the IMU bi-directionally around the $Z_{i}$ -axis once at each turntable outer frame angle, the initial platform angles can be calculated according to the spinning coarse alignment method. Before calibration, rotates the horizontal two gimbals to local horizontal according to the dictate deviation angle. What’s more, as a comparative experiment, the calibration method with no coarse alignment is also used to calibrate the scale factor of $x$ accelerometer according to the Eq. (4). The comparison results are shown in Fig. 12, with standard deviations of 2390 ppm and 30 ppm for traditional method and improved method, respectively.

Fig. 12 Comparison of traditional and improved method.

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From the laboratory experiments results shown above, it can be seen that the traditional calibration method, limited by the accuracy of the gimbal, cannot be directly applied in RINS because of some non-orthogonality errors of mechanical processing and the installation state of the system. The calibration method described in this paper overcomes these disadvantages and the orthogonality requirements of gimbals and exact information of the local horizontal could be relaxed.

4.2 Dynamic vehicle experiment

To verify the dynamic performance of the system, a vehicle experiment is done. The calibration method with no coarse alignment is also used as a comparative experiment. As shown in Fig. 13, the RINS and generator are placed in the trunk of the vechile, and the GPS antenna is fixed on the roof of the car. The source data of IMU and GPS are calculated by a notebook computer synchronously. The inetial navigation calculation are carried out with the same IMU data but with different parameters which are calculated by two calibration strategies. To avoid the effect of other parameters, the accelerometer scale factor is the only variable. The GPS data is used as a comparative navigation result and Fig. 14 shows the trajectory of GPS data.

Fig. 13 The vehicle experiment device.

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Fig. 14 Trajectory of the navigation result in vehicle experiment.

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The velocity error and position error are shown in Fig. 15 and Fig. 16, respectively. It is obvious that the oscillation of velocity error is suppressed and accuracy of position is increased. What’s more, some performance indicators are described in Table 3. The vehicle experiment results show that the proposed calibration method could accurately calibrate the accelerometer scale factor, so that the navigation performance could be significantly improved.

Fig. 15 The camparition of velocity error.

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Fig. 16 The camparition of position error.

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Table 3. The navigation performance indicators^a

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5. Conclusion

In this paper, an innovative calibration procedure for accelerometer scale factor using the gimbals of RINS is proposed. After analyzing the mechanism of accelerometer scale factor calibration error considering non-orthogonality angles and the non-horizontal angles, an innovative calibration method is suggested. First of all, a detailed analysis of the calibration errors is built and the relationship between the calibration error of accelerometer scale factor and non-orthogonality and non-horizontal angle is presented. Then a special calibration procedure is designed with the calibration of the non-orthogonality angle and coarse alignment of the system. All in all, this method overcomes the challenge that traditional calibration method must be conducted under strict conditions, such as absolute orthogonal of gimbals and absolute horizontal of the system, which is especially suitable for missile in wartime. In addition, the proposed calibration procedure is a general method which is not only suitable for the RINS in this paper, but also in some calibration conditions that the turntable has machining errors or application errors.

Funding

BUAA for Graduate Innovation and Practice (YCSJ-01-2015-08).

References and links

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Sensor performace	Gyroscope	Accelerometer	Photoelectric encoder
Full scale	$- 300 ~ + 300 (\circ / s)$	$- 20 ~ + 20 (g)$	$0 ~ 360 (\circ)$
Bias stability	$\leq 0.01 (\circ / h)$	$\leq 20 (μ g)$	$\pm 2.97 (a r c - s e c o n d)$
Scale factor stability	$\leq 20 (p p m)$	$\leq 50 (p p m)$	$17 (p p m /^{\circ} C)$
Sensitivity	$\leq 0.005 (\circ / h)$	$\leq 5 (μ g)$	$\leq 0.82 (a r c - s e c o n d)$

Tilt Angle(°)	$β_{G x}^{Y} (^{″})$	$θ_{G z}^{Y} (^{″})$	$θ_{G z}^{X} (^{″})$	$θ_{G x}^{X} (^{″})$	$γ (^{″})$
-20	185	35.3	433.7	388.9	−265.1
-15	181	35.9	433.8	388.8	−261.9
-10	182	35.2	433.8	388.8	−262.2
-5	185	39.6	434.7	388.9	−270.4
0	179	42.5	434.7	388.9	−267.3
+ 5	182	42.6	434.2	389.0	−269.8
+ 10	181	45.3	434.2	388.9	−271.6
+ 15	183	46.4	433.9	388.9	−274.4
+ 20	183	40.8	434.7	388.8	−269.7

Calibration method	East velocity error (RMS)	North velocity error (RMS)	Position error (CEP)
With improved method	1.0678 m/s	0.6514 m/s	0.3717 n mile/h
With traditional method	7.9038 m/s	9.9562 m/s	0.7279 n mile/h

Sensor performace	Gyroscope	Accelerometer	Photoelectric encoder
Full scale	$- 300 ~ + 300 (\circ / s)$	$- 20 ~ + 20 (g)$	$0 ~ 360 (\circ)$
Bias stability	$\leq 0.01 (\circ / h)$	$\leq 20 (μ g)$	$\pm 2.97 (a r c - s e c o n d)$
Scale factor stability	$\leq 20 (p p m)$	$\leq 50 (p p m)$	$17 (p p m /^{\circ} C)$
Sensitivity	$\leq 0.005 (\circ / h)$	$\leq 5 (μ g)$	$\leq 0.82 (a r c - s e c o n d)$

Tilt Angle(°)	$β_{G x}^{Y} (^{″})$	$θ_{G z}^{Y} (^{″})$	$θ_{G z}^{X} (^{″})$	$θ_{G x}^{X} (^{″})$	$γ (^{″})$
-20	185	35.3	433.7	388.9	−265.1
-15	181	35.9	433.8	388.8	−261.9
-10	182	35.2	433.8	388.8	−262.2
-5	185	39.6	434.7	388.9	−270.4
0	179	42.5	434.7	388.9	−267.3
+ 5	182	42.6	434.2	389.0	−269.8
+ 10	181	45.3	434.2	388.9	−271.6
+ 15	183	46.4	433.9	388.9	−274.4
+ 20	183	40.8	434.7	388.8	−269.7

Innovative self-calibration method for accelerometer scale factor of the missile-borne RINS with fiber optic gyro

Abstract

1. Introduction

2. Theoretical analysis

2.1 Configuration of RINS and frame definition

2.2 Analysis of calibration error

3. Calibration algorithm

3.1 Calibration of non-orthogonality angle

3.2 Calibration of non-horizontal angle

4. Experimental results

4.1 Laboratory static experiment

4.2 Dynamic vehicle experiment

5. Conclusion

Funding

References and links

Cited By

Figures (16)

Tables (3)

Equations (26)

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