Analysis of the Temperature Field Characteristics and Thermal-Induced Errors of Miniature Interferometric Fiber Optic Gyroscopes in a Vacuum Environment

Abstract

This paper investigates the mechanism of thermal-induced errors in interferometric fiber optic gyroscopes (IFOGs) caused by temperature changes in a vacuum environment, proposing a method for calculating thermal-induced errors in small fiber coils. Firstly, based on the Shupe effect and the thermal stress caused by temperature changes around the fiber coil, a three-dimensional thermal-induced error model for small fiber coils is established. Secondly, a spatial fiber optic inertial measurement unit (IMU) model is designed using the Creo 3D modeling software (creo 7.0.0). The model is then imported into the Ansys finite element simulation software (ANSYS Workbench 15.0), where a temperature field is applied to the IMU based on actual temperature profiles to obtain the temperature distribution of the fiber coil at different times in a vacuum state. These data are then used in the three-dimensional thermal-induced error model to calculate the thermal-induced error of the FOG. Finally, a thermal vacuum experimental platform is set up to collect temperature variation data from the inertial measurement components. The experimental data are compared with the three-dimensional error model proposed in this paper as well as traditional error models. The root mean square error is approximately 33% lower than that of traditional error calculation methods, which also proves the theoretical accuracy.

1. Introduction

With the rapid development of the aerospace industry, aviation-related technologies have advanced significantly. Among the most challenging technologies in the aviation field are spacecraft launch, orbital change, and in-orbit operation, all of which rely heavily on inertial navigation technology. Inertial technology, with its high integration, autonomy, comprehensive coverage, and real-time continuous information, can provide real-time position and attitude information for spacecraft and plays an irreplaceable role in various vehicles. The most crucial instrument in inertial technology is the gyroscope, whose measurement accuracy directly affects the positioning accuracy of inertial technology [1,2,3]. Advancements in gyroscope technology are of paramount importance for driving the development of inertial technology.

From an engineering application perspective, the most suitable model for the aerospace field is currently the interferometric fiber optic gyroscope (IFOG), which is also one of the most widely used fiber optic sensing instruments [4,5,6]. The IFOG is an angular rate measuring device based on the Sagnac effect, acquiring a system’s angular velocity through the phase difference produced by two light beams transmitting in a fiber loop. Due to its operational principles, the IFOG exhibits significant advantages in harsh space environments, demonstrating excellent performance in resistance to vacuum, impact, and radiation. The length of the optical fiber in IFOGs typically ranges from a few hundred meters to several kilometers, making it challenging for IFOGs to achieve miniaturization and lightweight characteristics.

On 26 January 2004, after a seven-month flight mission, the American “Opportunity” and “Spirit” rovers successfully landed on Mars. The positioning system carried by the rovers was an fiber optic gyroscope navigation system produced by Northrop Grumman [7,8]. This entire system provided the rovers with the necessary linear acceleration and angular acceleration information for attitude measurement during the flight. It determined the optimal parachute deployment time after entering the atmosphere and provided attitude and speed information during the rovers’ movement on the Martian surface. Based on the results of the 12 space missions executed by the spacecraft, the fiber optic gyroscope (FOG), with its light weight, high reliability, and strong resistance to space radiation, proved capable of adapting to the harsh working environment of space, successfully undertaking important deep-space exploration tasks.

In 2013, Serdar Ogut and Berk Osunluk from Bilkent University developed a more detailed finite element model of the fiber coil based on its dimensions. Their analysis not only considered the variations in the temperature field but also focused on the fiber coil itself. They established a finite element model of the fiber coil using the representative volume element (RVE) method for fiber coils wound using the quadrupole symmetric winding method [9].

In 2015, the research group led by Zhang at Harbin Engineering University conducted extensive experimental validation on the impact of temperature change rates on the thermal-induced errors in FOGs [10]. They theoretically analyzed the effect of temperature change rates on the thermal-induced errors in FOGs, with temperature change rates representing the specific manifestation of a time-varying temperature field acting on the fiber coil. In the experiments, they designed continuous temperature change tests with different temperature change rates, collecting data from the FOGs under seven different temperature change rates. Based on the experimental data, they selected appropriate function parameters to fit a relationship between the temperature change rates and the thermal-induced errors in FOGs. Using the fitted function parameters, they evaluated the thermal performance of the FOGs.

Furthermore, in 2019, KVH Industries Inc., from the United States, applied multifunctional photonic-integrated circuits (PICs) to small fiber optic gyroscopes [11]. By utilizing high-throughput, high-yield wafer-level manufacturing technology, they achieved the miniaturization of fiber optic gyroscopes. The product used an optical fiber loop with a diameter of 6 cm and a total fiber length of 110 m. Experiments measured the fiber optic gyroscope’s random walk coefficient at approximately 0.59°/$\sqrt{\mathrm{h}}$ and the zero-bias stability at approximately 0.048 °/h. By integrating optical devices, this FOG effectively reduced the system’s size and overall weight, achieving the precision required for use in space environments and resulting in a practical, miniaturized fiber optic gyroscope. The scheme of the small fiber optic gyroscopes in shown in Figure 1.

The China Aerospace Times Electronics Company has been prominent in the development and engineering technology of various FOGs, making breakthrough progress in several key indicators, such as accuracy, measurement range, frequency characteristics, electromagnetic compatibility, reliability, and long-term stability [12,13]. High-precision FOGs have achieved an accuracy of 0.01°/h, and medium-to-low-precision FOGs have met performance requirements within 5 s in a working range of −40 °C to +65 °C, with a zero-bias stability of less than 2°/h over the entire temperature range. This paper lists FOGs that have representative significance at different stages, with their performance indicators shown in

Table 1. Performance indicators of three series of FOGs developed by the China Aerospace Times Electronics Company.

Fiber Optic Gyroscope Indicators	FOG-C01	FOG-R01	FOG-P01
Short-term zero-bias stability	10–1 °/h	1–0.1 °/h	0.01 °/h
Random walk coefficient	$0.1 °/\sqrt{\mathrm{h}}$	$0.01 °/\sqrt{\mathrm{h}}$	$0.001 °/\sqrt{\mathrm{h}}$
Scale factor stability	60 ppm	40 ppm	20 ppm
Measuring range	±1000 °/s	±400 °/s	±100 °/s
Working temperature range	−40 °C~+65 °C	−40 °C~+65 °C	−40 °C~+65 °C

.

At present, the most crucial problem brought by the miniaturization of IFOGs is the heat dissipation issue, especially in a vacuum environment where heat convection cannot occur, leading to local overheating of the gyroscope and resulting in thermal-induced errors [14,15,16]. This paper first derives the relationship between temperature changes and phase changes in light waves based on the Shupe effect and thermal stress caused by temperature changes around the fiber coil. Combining the distribution of the refractive index of the fiber coil and its winding characteristics, a three-dimensional thermal-induced error model based on a fine-sized fiber coil is established using a refractive index distribution model of the fiber coil to obtain the refractive index changes between different levels. Secondly, according to the design requirements for space-use IFOGs, a space-use fiber inertial unit model is designed using the Creo 3D modeling software. The completed model is then imported into the Ansys finite element simulation software to perform edge processing, meshing, and boundary condition application. Temperature fields are applied to the fiber inertial unit based on actual temperature profiles, obtaining temperature distributions of the fiber coil at different times in vacuum and non-vacuum states. The temperature change rate of the fiber coil is calculated from the fitted data, and the thermal-induced error size of the IFOG is simulated. Finally, an experimental prototype of the fiber inertial unit is built and placed in a thermal vacuum chamber for vacuum processing. The environment’s temperature is controlled using radiation heating and liquid nitrogen cooling methods, and the output of the IFOG under vacuum and variable temperature conditions is tested. The generality of the model is verified through comparison, proving the correctness of the theory.

2. Methods

2.1. Mechanism of Thermally Induced Errors

When a time-varying temperature field is present around the fiber, the two counter-propagating light beams experience the same segment of fiber at different times. The refractive index of the fiber at that location changes with the temperature, affecting the propagation characteristics of the light waves in the fiber and causing non-reciprocal phase shifts after the beams pass through that location. This effect was first proposed by D.M. Shupe in 1980 and is known as the Shupe effect [15]. The phase shift induced by the Shupe effect cannot be distinguished from the phase shift generated by the Sagnac effect in FOGs, directly impacting the accuracy of angular velocity measurements in the gyroscope. The influence of the temperature field on the fiber essentially alters the refractive index of the fiber core. The refractive index n of a medium is defined as the ratio of the speed of light in a vacuum to the speed of light in the medium. For glass fibers, the refractive index is determined by the molecular volume and molecular refraction of the material. The larger the molecular volume of the fiber, the lower its refractive index. Conversely, the higher the molecular refraction of the fiber, the higher its refractive index. Within the normal operating temperature range of the fiber, the refractive index is approximately linearly related to the temperature, as expressed by the following equation:

$$\mathsf{\Delta }{n}_{\mathrm{T}}={\displaystyle \frac{dn}{dT}}\cdot [T(t_1)-T(t_2)]={\displaystyle \frac{dn}{dT}}\cdot \mathsf{\Delta }T$$

(1)

where $dn/dT$ represents the thermo-optic coefficient, $T(t_1)$ represents the temperature at time $t_1$, $T(t_2)$ represents the temperature at time $t_2$, and $\Delta T$ represents the temperature difference between two adjacent time points.

When there is no temperature disturbance around the fiber, the phase change generated by the light beam after passing through a segment of the fiber can be expressed as follows:

$$\varphi \left(L\right)={\rho }_0{\displaystyle {\int }_0^{L}n(z)dz}$$

(2)

where $L$ represents the fiber length, $n(z)$ represents the refractive index of the fiber at position z, and ${\rho }_0$ represents the propagation constant. ${\rho }_0=2\pi /\lambda$, where $\lambda$ represents the wavelength of the light wave.

When the surrounding temperature of the fiber is time-varying, not only does the refractive index of the fiber change, but the fiber also undergoes thermal expansion, causing a change in its length. Considering the impact of both factors on the fiber, the changes in the refractive index and length can be expressed as follows:

$$\begin{array}{l}{n}^{\prime }(z)=n_0+\mathsf{\Delta }{n}_{\mathrm{T}}=n_0+{\displaystyle \frac{dn}{dT}}\cdot \mathsf{\Delta }T\\ d^{\prime }z=dz+\mathsf{\Delta }d{z}_{\mathrm{T}}=dz+\alpha \cdot \mathsf{\Delta }Tdz\end{array}$$

(3)

where $\alpha$ represents the coefficient of thermal expansion. Substituting into Equation (2), the relationship for the phase change in the light wave in the fiber due to the Shupe effect can be derived as follows:

$$\begin{array}{c}\varphi \left(L\right)={\rho }_0{\displaystyle {\int }_0^L{n}^{\prime }(z)d^{\prime }z}={\rho }_0{\displaystyle {\int }_0^L(n_0+\mathsf{\Delta }{n}_{\mathrm{T}})(dz+\mathsf{\Delta }d{z}_{\mathrm{T}})}\\ ={\rho }_0{\displaystyle {\int }_0^L[n_0+\frac{dn}{dT}\cdot \mathsf{\Delta }T+n_0\cdot \alpha \cdot \mathsf{\Delta }T+\alpha \cdot \frac{dn}{dT}\cdot \mathsf{\Delta }T\cdot \mathsf{\Delta }T]dz}\\ ={\rho }_0{n}_{0}L+{\rho }_0{\displaystyle {\int }_0^L[\frac{dn}{dT}\cdot \mathsf{\Delta }T+n_0\cdot \alpha \cdot \mathsf{\Delta }T+\alpha \cdot \frac{dn}{dT}\cdot \mathsf{\Delta }T\cdot \mathsf{\Delta }T]dz}\end{array}$$

(4)

The order of magnitude of the thermo-optic coefficient is ${10}^{-5}$, and the order of magnitude of the thermal expansion coefficient is ${10}^{-6}$. After multiplying the two, the resulting magnitude is significantly smaller compared to other terms and can therefore be neglected. When the IFOG is in a stationary state, the CW light beam and the CCW light beam travel through the fiber coil and are output from both ends of the fiber coil after time t, and then interfere in the coupler. The time at which the CW beam passes through point zzz in the fiber coil is denoted as $t_{cw}$. The phase change in the CW light wave, after passing through the entire fiber coil and returning to the multifunctional phase modulator at this time, is given by the following:

$${\varphi }_{\mathrm{CW}}(t)={\rho }_0{n}_{0}L+{\rho }_0{\displaystyle {\int }_0^L[(\frac{dn}{dT}+n_0\alpha )\cdot \mathsf{\Delta }T(z,t-\frac{L-z}{c_{\mathrm{m}}})]dz}$$

(5)

Similarly, the time at which the CCW beam passes through point zzz in the fiber coil is denoted as $t_{\mathrm{ccw}}=t-z/c_{\mathrm{m}}$. By substituting this into Equation (4), the phase change in the CCW light wave, after passing through the fiber coil and returning to the multifunctional integrated optical circuit (MIOC) at time t, can be determined as follows:

$${\varphi }_{\mathrm{CCW}}(t)={\rho }_0{n}_{o}L+{\rho }_0{\displaystyle {\int }_0^L[(\frac{dn}{dT}+n_0\alpha )\cdot \mathsf{\Delta }T(z,t-\frac{z}{c_{\mathrm{m}}})]dz}$$

(6)

By subtracting Equation (5) from Equation (6), the phase difference between the two beams at the coupler at time t can be obtained as follows:

$$\begin{array}{c}\mathsf{\Delta }\varphi (t)={\varphi }_{\mathrm{ccw}}(t)-{\varphi }_{\mathrm{cw}}(t)\\ ={\rho }_0{\displaystyle {\int }_0^L\left\{(\frac{dn}{dT}+n_0\alpha )\cdot [\mathsf{\Delta }T(z,t-\frac{z}{c_{\mathrm{m}}})-\mathsf{\Delta }T(L-z,t-\frac{z}{c_{\mathrm{m}}})]\right\}dz}\end{array}$$

(7)

According to the definition of differentiation in mathematics, we have the following: $\underset{\mathsf{\Delta }t\to 0}{\lim }[f(t+\mathsf{\Delta }t)-f(t)]/\mathsf{\Delta }t=\dot{f}(t)$. When $\Delta t$ is relatively small, we can obtain ${\lim }_{\mathsf{\Delta }t\to 0}[f(t+\mathsf{\Delta }t)-f(t)]=\dot{f}(t)\cdot \mathsf{\Delta }t$ [16]. Given that the length of the fiber is several hundred meters and light can travel hundreds of millions of meters per second in the fiber, the transmission time of light in the fiber is extremely short, allowing for approximation. According to the Sagnac effect, the relationship expression between the measured Sagnac phase shift and the angular velocity of the IFOG is the following:

$$\mathsf{\Delta }\varphi ={\displaystyle \frac{8{\pi }^2{R}^{2}N}{\lambda c_{\mathrm{m}}}}\mathsf{\Omega }={\displaystyle \frac{4\pi RL}{\lambda c_{\mathrm{m}}}}\mathsf{\Omega }$$

(8)

where R represents the radius of the fiber coil, N represents the number of turns, $c_m$ represents the speed of light in the fiber, and $p_{12}$ represents the photoelastic coefficient of the material. The error expression for IFOG caused by the Shupe effect can be obtained as follows:

$${\mathsf{\Omega }}_{\mathrm{T}}={\displaystyle \frac{n_0}{2RL}}{\displaystyle {\int }_0^L\left\{[\frac{dn}{dT}+(n_0+\frac{n_0^3}{2}(p_{11}-2v{p}_{12})\times [\frac{(7-6v)d^2}{16R^2(1-v)}])\cdot \alpha ]\cdot \dot{T}(z,t)\cdot (L-2z)\right\}dz}$$

(9)

2.2. The Mechanism of Thermal Stress in the Fiber Optic Coil

Since the fiber optic coil is a multi-layer, multi-turn ring structure, thermal expansion and contraction between molecules not only cause changes in the refractive index but also affect the stress experienced by the fibers in each layer. This stress, induced by temperature changes, is referred to as thermal stress. The deformation of the fiber due to temperature changes is elastic deformation and follows the basic theory of fiber bending. In addition to the refractive index changes caused by the temperature gradient discussed in the previous section, there are also refractive index changes induced by thermal stress. The variation in the propagation axial refractive index of the polarization-maintaining fiber caused by thermal stress $\Delta n_p$ is represented by the following:

$$\begin{array}{l}\mathsf{\Delta }{n}_{\mathrm{p}}=-{\displaystyle \frac{n^3}{2Y}}[(p_{11}-2\nu p_{12}){\sigma }_s+(p_{12}-\nu p_{11}-\nu p_{12}){\sigma }_f+(p_{12}-\nu p_{11}-\nu p_{12}){\sigma }_a]\\ =-{\displaystyle \frac{n^3}{2Y}}[(p_{11}-2\nu p_{12})F_{\mathrm{x}}+(p_{12}-\nu p_{11}-\nu p_{12})F_{\mathrm{y}}+(p_{12}-\nu p_{11}-\nu p_{12})F_{\mathrm{z}}]\end{array}$$

(10)

where $F_x$, $F_y$, and $F_z$ represent thermal stresses in the x, y, and z directions, respectively.

The component causes a change in the length of the fiber, thereby affecting the infinitesimal element $dz$. Based on the changes in the refractive index and length of the polarization-maintaining fiber caused by the temperature gradient, the expressions for the refractive index and length of the fiber due to thermal stress can be derived as follows:

$$\begin{array}{c}{n}^{\prime }(z)=n_0+\mathsf{\Delta }{n}_{\mathrm{P}}=n_0+{\displaystyle \frac{\partial n}{\partial F_{\mathrm{x}}}}\cdot F_{\mathrm{x}}+{\displaystyle \frac{\partial n}{\partial F_{\mathrm{y}}}}\cdot F_{\mathrm{y}}+{\displaystyle \frac{\partial n}{\partial F_{\mathrm{z}}}}\cdot F_{\mathrm{z}}\\ d^{\prime }z=dz+\mathsf{\Delta }d{z}_{\mathrm{F}}=(1+{\displaystyle \frac{\partial z}{\partial F_{\mathrm{x}}}}+{\displaystyle \frac{\partial z}{\partial F_{\mathrm{y}}}}+{\displaystyle \frac{\partial z}{\partial F_{\mathrm{z}}}})\cdot dz\end{array}$$

(11)

Similarly, by substituting Equation (11) into Equation (2), the relationship for the phase change in the optical wave in the fiber caused by thermal stress can be derived as follows:

$$\begin{array}{ll}\varphi (L)& ={\rho }_0{\displaystyle {\int }_0^L{n}^{\prime }(z)d^{\prime }z=}{\rho }_0{\displaystyle {\int }_0^L(n_0+\mathsf{\Delta }{n}_{\mathrm{p}})(dz+\mathsf{\Delta }d{z}_{\mathrm{F}})}\\ & ={\rho }_0{\displaystyle {\int }_0^L(n_0+\frac{\partial n}{\partial F_{\mathrm{x}}}+\frac{\partial n}{\partial F_{\mathrm{y}}}+\frac{\partial n}{\partial F_{\mathrm{z}}}+n_0\cdot \frac{\partial z}{\partial F_{\mathrm{x}}}\cdot F_{\mathrm{x}}+\frac{\partial n}{\partial F_{\mathrm{x}}}\cdot \frac{\partial z}{\partial F_{\mathrm{x}}}\cdot F_{\mathrm{x}}\cdot F_{\mathrm{x}}}\\ & +{\displaystyle \frac{\partial n}{\partial F_{\mathrm{y}}}}\cdot {\displaystyle \frac{\partial z}{\partial F_{\mathrm{x}}}}{F}_{\mathrm{y}}\cdot F_{\mathrm{x}}+{\displaystyle \frac{\partial n}{\partial F_{\mathrm{z}}}}\cdot {\displaystyle \frac{\partial z}{\partial F_{\mathrm{x}}}}{F}_{\mathrm{z}}\cdot F_{\mathrm{x}}+n_0\cdot {\displaystyle \frac{\partial z}{\partial F_{\mathrm{y}}}}\cdot F_{\mathrm{y}}+{\displaystyle \frac{\partial n}{\partial F_{\mathrm{x}}}}\cdot {\displaystyle \frac{\partial z}{\partial F_{\mathrm{y}}}}\cdot F_{\mathrm{x}}\cdot F_{\mathrm{y}}\\ & +{\displaystyle \frac{\partial n}{\partial F_y}}\cdot {\displaystyle \frac{\partial z}{\partial F_y}}{F}_y\cdot F_y+{\displaystyle \frac{\partial n}{\partial F_z}}\cdot {\displaystyle \frac{\partial z}{\partial F_y}}{F}_z\cdot F_y+n_0\cdot {\displaystyle \frac{\partial z}{\partial F_z}}\cdot F_z+{\displaystyle \frac{\partial n}{\partial F_x}}\cdot {\displaystyle \frac{\partial z}{\partial F_z}}\cdot F_x\cdot F_z\\ & +{\displaystyle \frac{\partial n}{\partial F_{\mathrm{y}}}}\cdot {\displaystyle \frac{\partial z}{\partial F_{\mathrm{z}}}}{F}_{\mathrm{y}}\cdot F_{\mathrm{z}}+{\displaystyle \frac{\partial n}{\partial F_{\mathrm{z}}}}\cdot {\displaystyle \frac{\partial z}{\partial F_{\mathrm{z}}}}{F}_{\mathrm{z}}\cdot F_{\mathrm{z}})dz\end{array}$$

(12)

Using the method for calculating the phase difference in CW and CCW optical waves caused by the Shupe effect from the previous subsection, it is not difficult to see from the above analysis that the Shupe effect and thermal stress induced by temperature gradients in polarization-maintaining fibers are related to the fiber length and refractive index. Therefore, the phase shift model of polarization-maintaining fibers established based on the Shupe effect is also applicable to the calculation of thermal stress.

The thermal stress experienced by the fiber is entirely due to the thermal expansion and contraction caused by temperature changes, and its magnitude is the product of the thermal expansion coefficient, temperature difference, and the material’s Young’s modulus at that location. We can derive that the thermal stress–strain coefficient in the x-axis is the negative ratio of Poisson’s ratio to Young’s modulus. The thermal stress–strain coefficient in the y-axis is the same as that in the x-axis, and the thermal stress–strain coefficient in the z-axis is the reciprocal of Young’s modulus.

When the fiber undergoes thermal expansion, it affects the fiber length only in the axial direction, and the deformation amount of the microelement due to heat depends on the temperature difference and the fiber’s thermal expansion coefficient. Additionally, the primary impact on the propagation of optical waves occurs within the fiber’s cross-section. Since the thermal expansion coefficients of the cladding and coating layers differ significantly, it is not feasible to calculate them using the thermal expansion coefficient of silica alone. After thermal expansion, the deformation and stress at the interface between the cladding and coating layers in the fiber remain continuous. Therefore, during temperature variations, the thermal stress at the interface between the cladding and coating layers is a pair of reaction forces. This can be represented by the force balance equation as follows:

$$\begin{array}{l}{\alpha }_{\mathrm{t}}\mathsf{\Delta }T{Y}_{\mathrm{t}}-\sigma ={\alpha }_{\mathrm{r}}\mathsf{\Delta }T{Y}_{\mathrm{t}}\\ {\alpha }_{\mathrm{b}}\mathsf{\Delta }T{Y}_{\mathrm{b}}-\sigma ={\alpha }_{\mathrm{r}}\mathsf{\Delta }T{Y}_{\mathrm{b}}\end{array}$$

(13)

where ${\alpha }_t$ represents the thermal expansion coefficient of the coating layer, ${\alpha }_r$ represents the radial thermal expansion coefficient of the fiber cross-section, ${\alpha }_b$ represents the thermal expansion coefficient of the cladding, $Y_t$ represents the Young modulus of acrylic resin, and $Y_b$ represents the Young modulus of silica.

Since the fiber cross-section is a centrally symmetric structure, the thermal expansion coefficients are the same in all directions. Therefore, the thermal expansion coefficient in the y direction is equal to that in the x direction. Based on this, the error expression for the IFOG caused by thermal stress can be derived as follows:

$$\begin{array}{l}{\mathsf{\Omega }}_{\mathrm{F}}={\displaystyle \frac{n_{0}Y}{2RL}}{\displaystyle {\int }_0^L\left\{[\frac{\partial n}{\partial F_{\mathrm{x}}}+(n_0+\frac{n_0^3}{2}(p_{11}-2\nu p_{12})\times [\frac{(7-6\nu )d^2}{16R^2(1-\nu )}])\frac{\partial z}{\partial F_{\mathrm{x}}}]\cdot {\alpha }_{\mathrm{r}}\dot{T}(z,t)\cdot (L-2z)\right.}\\ \begin{array}{cc}& \end{array}\hspace{1em}+[{\displaystyle \frac{\partial n}{\partial F_{\mathrm{y}}}}+(n_0+{\displaystyle \frac{n_0^3}{2}}(p_{11}-2\nu p_{12})\times [{\displaystyle \frac{(7-6\nu )d^2}{16R^2(1-\nu )}}]){\displaystyle \frac{\partial z}{\partial F_{\mathrm{y}}}}]\cdot {\alpha }_{\mathrm{r}}\dot{T}(z,t)\cdot (L-2z)\\ \begin{array}{cc}& \end{array}\left.\hspace{1em}+[{\displaystyle \frac{\partial n}{\partial F_z}}+(n_0+{\displaystyle \frac{n_0^3}{2}}(p_{11}-2\nu p_{12})\times [{\displaystyle \frac{(7-6\nu )d^2}{16R^2(1-\nu )}}]){\displaystyle \frac{\partial z}{\partial F_z}}]\cdot {\alpha }_z\dot{T}(z,t)\cdot (L-2z)\right\}dz\end{array}$$

(14)

2.3. Establishment of the 3D Thermally Induced Error Model for IFOG

According to the explanations of the Shupe effect and thermal stress in the previous sections, we analyzed the mechanisms of thermally induced errors and derived mathematical models for the thermally induced errors of IFOGs under two different mechanisms. By adding the angular velocity variation caused by the Shupe effect (${\Omega }_T$) and the angular velocity variation caused by thermal stress (${\Omega }_F$), we can obtain the thermally induced drift (${\Omega }_e$) caused by the temperature gradient in the fiber coil.

In the derivation process, the fiber coil is considered as a planar ring, resulting in a 2D model of thermally induced errors for IFOGs. However, in practice, the fiber coil is arranged in a circular structure with different layers. Simplifying it to a single inner-diameter coil will inevitably reduce the accuracy of the error model, especially for the tiny fiber coil error model, where the reduction in accuracy is more significant. Therefore, based on the fiber coil formed by the non-stepwise quadrupole winding method, we consider the changes in the bending radius between different layers to establish a 3D thermally induced error model for tiny fiber coils.

As the fiber is a thermally conductive material, the temperature field changes as it transmits through various layers of the fiber coil. Both the length of the fiber affected by the temperature field and the refractive index of the fiber itself will influence the magnitude of the thermally induced error. We need to calculate the temperature gradient of each layer, the corresponding refractive index, and the change in the fiber length for each position. By calculating the phase error of each turn of the fiber caused by the temperature gradient and summing them up, we can establish a 3D thermally induced error model.

The analysis approach is similar to the previous ones, as we separately model the phase changes in the CW and CCW directions in three dimensions. This yields the following results:

$$\begin{array}{l}{\varphi }_{\mathrm{Cw}}(t)={\displaystyle \sum _{i=1}^M{\displaystyle \sum _{j=1}^N{\rho }_0}}{n}_{\mathrm{i}}L+{\rho }_0{\displaystyle {\int }_{L_{i,j,1}}^{L_{i,j}}\left[\left(\frac{dn}{dT}+n_{\mathrm{i}}\cdot \alpha \right)\cdot \Delta T\left(z,t-\frac{z}{c_m}\right)+\left(\frac{\partial n}{\partial F_{\mathrm{x}}}+n_i\frac{\partial z}{\partial F_{\mathrm{x}}}\right)\right.}\hfill \\ \left.\cdot F_{\mathrm{x}}\left(z,t-{\displaystyle \frac{L-z}{c_{\mathrm{m}}}}\right)+\left({\displaystyle \frac{\partial n}{\partial F_{\mathrm{y}}}}+n_i{\displaystyle \frac{\partial z}{\partial F_{\mathrm{y}}}}\right)\cdot F_{\mathrm{y}}\left(z,t-{\displaystyle \frac{L-z}{c_{\mathrm{m}}}}\right)+\left({\displaystyle \frac{\partial n}{\partial F_{\mathrm{z}}}}+n_i{\displaystyle \frac{\partial z}{\partial F_{\mathrm{z}}}}\right)\cdot F_{\mathrm{z}}\left(z,t-{\displaystyle \frac{L-z}{c_{\mathrm{m}}}}\right)\right]dz\hfill \end{array}$$

(15)

$$\begin{array}{l}{\varphi }_{\mathrm{CCW}}(t)={\displaystyle \sum _{i=1}^M{\displaystyle \sum _{j=1}^N{\rho }_0}}{n}_{\mathrm{i}}L+{\rho }_0{\displaystyle {\int }_{L_{i,j-1}}^{L_{i,j}}\left[\left(\frac{dn}{dT}+n_{\mathrm{i}}\cdot \alpha \right)\cdot \Delta T\left(z,t-\frac{z}{c_m}\right)+\left(\frac{\partial n}{\partial F_{\mathrm{x}}}+n_i\frac{\partial z}{\partial F_{\mathrm{x}}}\right)\right.}\hfill \\ \left.\cdot F_{\mathrm{x}}\left(z,t-{\displaystyle \frac{z}{c_{\mathrm{m}}}}\right)+\left({\displaystyle \frac{\partial n}{\partial F_{\mathrm{y}}}}+n_i{\displaystyle \frac{\partial z}{\partial F_{\mathrm{y}}}}\right)\cdot F_{\mathrm{y}}\left(z,t-{\displaystyle \frac{z}{c_{\mathrm{m}}}}\right)+\left({\displaystyle \frac{\partial n}{\partial F_{\mathrm{z}}}}+n_i{\displaystyle \frac{\partial z}{\partial F_{\mathrm{z}}}}\right)\cdot F_{\mathrm{z}}\left(z,t-{\displaystyle \frac{z}{c_{\mathrm{m}}}}\right)\right]dz\hfill \end{array}$$

(16)

where i represents the corresponding layer number, and j represents the corresponding number of turns.

By subtracting the phase change in the CW direction from the phase change in the CCW direction, the calculation process is similar to the derivation of thermally induced errors caused by the Shupe effect. The formula calculates the total phase of the clockwise and counterclockwise light beams by summing the phase differences generated by each segment of the fiber. By subtracting these, the phase difference between the two light beams after passing through the fiber coil can be obtained, thereby calculating the thermally induced error of the IFOG. Thus, the mathematical model of the 3D thermally induced error based on tiny fiber coils can be derived as follows:

$$\begin{array}{ll}{\Omega }_{\mathrm{e}}& ={\displaystyle \sum _{i=1}^M{\displaystyle \sum _{j=1}^N\frac{n_0}{2R_{i}L}}}{\displaystyle {\int }_{L_{i,j-1}}^{L_{1,j}}\left\{\left(\frac{dn}{dT}+n_i\cdot {\alpha }_z\right)\cdot T(ij,t)\cdot (L-2z)+E\cdot \left(\frac{\partial n}{\partial F_{\mathrm{x}}}+n_i\cdot \frac{\partial z}{\partial F_x}\right)\right.}\\ & \hspace{1em}+E\cdot \left[\left({\displaystyle \frac{\partial n}{\partial F_{\mathrm{x}}}}+n_i\cdot {\displaystyle \frac{\partial z}{\partial F_x}}\right)\cdot {\alpha }_{r}T(ij,t)\cdot (L-2z)+\left({\displaystyle \frac{\partial n}{\partial F_{\mathrm{y}}}}+n_i\cdot {\displaystyle \frac{\partial z}{\partial F_y}}\right]\right)\cdot {\alpha }_r\dot{T}(ij,t)\cdot (L-2z)\\ & \left.\left.\hspace{1em}+\left({\displaystyle \frac{\partial n}{\partial F_z}}+n_i\cdot {\displaystyle \frac{\partial z}{\partial F_z}}\right)\cdot {\alpha }_{z}T(ij,t)\cdot (L-2z)\right]\right\}dz\end{array}$$

(17)

3. Establishment and Analysis of the IFOG Thermal Vacuum Finite Element Model

3.1. Physical Field Solving Method

When the ambient temperature remains constant, meaning that the internal temperature of the IFOG, the internal temperature of the spacecraft, and the temperature around the fiber coil are all in thermal equilibrium, there is no temperature gradient in the fiber coil and, thus, no zero-point drift caused by thermally induced errors. However, during the spacecraft’s orbital operation, changes in solar radiation intensity and the heat generated by internal components of the spacecraft and the IFOG result in the IFOG being in a variable temperature state for most of the time. When the ambient temperature changes, the heat transfer between the fiber coil and the external environment creates a temperature gradient within the fiber coil, which, in turn, generates thermally induced errors. To determine the magnitude of the thermally induced errors in the fiber coil, the temperature field distribution of the fiber coil must first be solved. To establish the transmission conditions between the interior of the fiber coil and the external environment, the heat transfer laws of the overall system must first be determined.

When the IFOG operates in space, its working environment is a vacuum with no air medium present. Additionally, the fiber coil is not a heat-generating component; the fiber does not heat up during operation. Therefore, the temperature changes around the fiber are due to heat transfer from external components that generate heat to the fiber coil. According to thermodynamics, whenever there is a temperature difference between materials, heat will transfer from the higher-temperature substance to the lower-temperature substance [17,18,19]. The three basic methods of heat transfer are conduction, convection, and radiation. Heat conduction occurs due to the random movement of free electrons in conductive solids and the vibration of the lattice in non-conductive solids, facilitating heat transfer, while heat convection involves the heat exchange caused by the molecular movement between fluids of different temperatures. In non-vacuum conditions, heat convection is one of the primary modes of heat transfer. However, since the spacecraft operates in a dry vacuum environment with no air or other fluid mediums, there is no heat convection. Heat radiation refers to the emission of heat from a body with a temperature above absolute zero, which is an ongoing process and does not rely on a medium. Therefore, this mode of transfer is still present in a vacuum environment. Consequently, the modes of heat transfer for IFOGs in a vacuum are conduction and radiation. According to the theory of heat conduction, the primary mode of heat exchange within the fiber coil is heat conduction [18]. The fiber is an isotropic material, and the internal temperature field transfer of the fiber coil in the Cartesian coordinate system satisfies the following heat conduction equation:

$$\frac{\partial u}{\partial t}=\frac{K_0}{c_{\mathrm{f}}\rho }(\frac{{\partial }^{2}u}{\partial x^2}+{\displaystyle \frac{{\partial }^{2}u}{\partial y^2}}+{\displaystyle \frac{{\partial }^{2}u}{\partial z^2}})$$

(18)

where $u$ represents internal temperature field of the fiber coil, $K_0$ represents the thermal conductivity of the fiber material, $c_f$ represents the specific heat capacity of the fiber material, and $\rho$ represents the density of the fiber material.

The above equation represents the balance between the heat required to raise the temperature of the fiber coil and the heat transferred into the fiber coil from external contact. The right side of the equation indicates the total amount of heat needed to raise the temperature of the fiber per unit of time, conforming to the principle of energy conservation. The operating range of the IFOG is typically from −40 °C to 80 °C, and the operating temperature range in the low Earth orbit is from −25 °C to 55 °C. Within this temperature range, the heat transfer efficiency of thermal conduction and convection is significantly greater than that of thermal radiation. For ease of calculation, thermal radiation is usually neglected in research. However, since there is no convection in a vacuum environment, thermal radiation cannot be ignored. The heat transfer between objects by radiation can be calculated using the Stefan–Boltzmann law [20]:

$$Q=\epsilon \sigma A_{\mathrm{a}}{F}_{\mathrm{a}}(T_{\mathrm{a}}^4-T_{\mathrm{b}}^4)$$

(19)

where $\epsilon$ represents absorptivity, and $\sigma$ represents the Stefan–Boltzmann constant. In summary, the heat transfer between a space IFOG and its working environment mainly consists of the following types: thermal radiation from the interior of the spacecraft to the outer surface of the gyro structure; thermal radiation within the fiber optic gyro cavity between various structural components and from these components to the fiber coil; conductive heat transfer at the bottom surface of the fiber optic gyro, heat conduction from various heat-generating electronic components in the PCB board to the PCB board, and thermal conduction between the light source, PCB board, and structural components; and heat conduction between the gyro structure and the fixed surface of the fiber coil.

3.2. Structural Design of Fiber IMU

The IFOG component should comply with the mechanical design requirements specified in the satellite construction standards, such as requirements for layout, weight, fixation, and alignment. The mounting surfaces of the structure should retain their metallic appearance, while the surfaces of other equipment enclosures, except for the mounting surfaces, should undergo black anodizing or be coated with black paint. The hemispherical emissivity must be no less than 0.85. The design should enhance internal heat transfer within the equipment, ensuring that the heat generated by components is transferred to the mounting surface with minimal thermal resistance.

Since the payload capacity of a spacecraft is precisely calculated based on its orbital path and the carrying capacity of the launch vehicle, there are stringent requirements regarding the weight, volume, and fixation methods of the IFOG and the inertial navigation system (INS) as a whole. This paper adopts the commonly used space INS in engineering as the standard, with a total mass of less than 300 g and an overall volume not exceeding 74.7 mm × 74.7 mm × 53.5 mm. The remaining IFOG components should comply with the mechanical design requirements specified in the satellite construction standards.

In the structural design of INS, a Mg-Li alloy with a low density, high strength-to-weight ratio, and high modulus-to-weight ratio is chosen as the supporting structural material. The density of the Mg-Li alloy is between one-third to one-half that of the Al alloy, and the rigidity of the Mg-Li alloy is 22.68 N/m compared to 8.19 N/m for the Al alloy. Thus, the Mg-Li alloy possesses outstanding vibration-damping and heat dissipation properties, along with excellent machinability, which ensures the high machining precision and long-term stability of the equipment. This material is widely used in the aviation sector and is a mature material in the aerospace industry.

To meet the spacecraft’s volume requirements, the fiber coil, as a core component, must be miniaturized, which means using small-diameter fibers and reducing the initial bending radius of the fiber coil. The fiber used is YOFC’s PM1310-80-16/135-PM1016-G polarization-maintaining fiber, wound into a fiber coil using a non-stepped quadrupole symmetric winding method. The parameters of the Mg-Li alloy and the fiber coil used in the finite element simulation experiments are shown in

Table 2. Material property parameters.

Parameters	Mg-Li Alloy	Fiber Coil
Density (kg/m3)	1550	1200
Specific heat capacity (w/(K·m)	517	1302
Thermal conductivity (w/K·m)	80	0.41
Poisson’s ratio	0.33	0.4
Young’s modulus (109 Pa)	45	1.42
Coefficient of thermal expansion (/K)	2.4 × 10−5	10 × 10−5

.

Based on the above design requirements, the design of the fiber optic INS was completed. Using the Creo 3D modeling software, the structural design process was carried out according to the physical dimensions, and the established 3D model of the fiber optic inertial measurement unit (IMU) was imported into the Ansys Workbench finite element simulation software. The mechanical structure of the fiber optic IMU is shown in Figure 2. Except for the bottom installation surface, the surfaces of other parts of the equipment are treated with black anodizing. Figure 2a shows the effect after surface treatment, and Figure 2b shows the effect after transparency processing, clearly revealing the arrangement of the internal components of the fiber optic IMU. The fiber optic IMU contains three IFOGs, which are installed on the bottom, left side, and right side and are mutually perpendicular. To optimize space and component usage, all IFOGs use the SLD at the bottom as the input light source. To optimize the heat dissipation process, heating components are installed at the bottom to accelerate heat dissipation through thermal conduction. In addition to the three IFOGs, the fiber optic IMU is also equipped with three MEMS accelerometers, three gyroscope circuit boards, and one system circuit board. For the fiber optic IMU, the most important optical components are the three IFOG fiber coils on the bottom, left, and right sides, which are also the main objects of analysis in the subsequent finite element simulation experiments.

In order to improve computational efficiency, after importing the completed fiber IMU structure into the finite element analysis software, it is necessary to de-edge the entire structure. Components and design elements that do not affect the simulation results, such as screws, chamfers, top engravings, and aviation connectors, should be removed. Some manufacturing processes required in actual applications can make the shapes of non-essential structural parts irregular, reducing mesh uniformity, significantly increasing finite element computation time, and having a minimal impact on the results. Therefore, these elements should be processed before constructing the analysis elements. According to the mesh division principles, the entire fiber IMU structure should be meshed. Since the temperature distribution of the fiber coils is the focus of finite element analysis, the three IFOG fiber coils should be finely meshed, with an average mesh size not exceeding 0.5 mm², ensuring a mesh correlation greater than 95%. The final overall total mesh node count is 1,103,075, with 262,703 mesh elements and an overall correlation greater than 80%. The mesh division results of the entire structure are shown in Figure 3a, and the optimized mesh results of the fiber coils are shown in Figure 3b.

Firstly, the base of the fiber inertial module is set as a fixed constraint and is in thermal contact with the environment. In a vacuum environment, the fiber inertial group exchanges heat with the ambient temperature through thermal radiation. Secondly, according to the technical requirements, the operating temperature range of the IFOG is from 55 °C to −25 °C. To obtain a complete temperature cycle, the initial environmental temperature is set to 55 °C. After reaching equilibrium, the temperature is decreased at a rate of 1.5 °C/min until it reaches −25 °C, and then this temperature is maintained for 140 min. Subsequently, the temperature is increased at the same rate of 1.5 °C/min until it reaches 55 °C, and then it is maintained for 280 min. The entire temperature variation process can be divided into four segments: cooling process, low-temperature holding process, heating process, and high-temperature holding process. To ensure a comprehensive analysis, this paper selects a moment in each segment for a detailed analysis.

Figure 4 shows the temperature distribution of the IFOG fiber coil at 2400 s. Due to the low heat transfer effect of thermal radiation and the presence of heat sources within the fiber inertial group, the cooling rate in a vacuum environment is slower during the cooling process. At the same time, the temperature of the fiber coil in a vacuum is higher than that in a non-vacuum environment. This is because there is thermal conduction between the bottom surface of the fiber coil and the structural components, and its heat conduction effect is much higher than that of thermal radiation. Figure 5 shows the temperature distribution of the IFOG fiber coil at 7200 s. In this case, the external environment’s temperature has reached −25 °C. Due to poor heat dissipation in the vacuum environment, the overall temperature of the fiber inertial group and the fiber coil is higher than −25 °C. The principles of other temperature distribution conditions are similar to those in the previous stage. Figure 6 shows the temperature distribution of the IFOG fiber coil at 9900 s. Due to convective heat dissipation between fluids, the internal temperature of the fiber inertial group rises sharply in a vacuum state. Since the system enters the following stage after 10,800 s, the analysis is conducted after 9900 s. Figure 7 shows the temperature distribution of the IFOG fiber coil after 14,400 s. After 14,400 s, the external environment’s temperature has reached 55 °C, and the internal temperature distribution of the fiber coil also tends to stabilize.

Figure 8 shows the high-temperature expansion and low-temperature contraction of the fiber coil. It can be observed that, since the thermal expansion coefficient of the Mg-Li alloy is smaller than that of the fiber and the bottom surface of the fiber coil is connected to the structural component, the deformation of the bottom surface of the fiber coil is less than that of the top surface. Additionally, the heat sources around the bottom fiber coil are more concentrated, resulting in a steady-state temperature higher than that of the left and right fiber coils. Therefore, during low temperatures, its contraction is less than that of the other two fiber coils, and during high temperatures, its expansion is greater. Because the heat sources at the bottom of the fiber inertial group are symmetrically distributed, the deformation of the bottom fiber coil is essentially centrosymmetric, whereas the deformation distribution of the left and right fiber coils is asymmetric.

By exporting the temperature data of the fiber coil throughout the entire temperature cycle from the simulation software and performing fitting processing, the temperature distribution of the fiber coil at fixed moments can be obtained through the fitted function. The midpoint of the fiber coil is selected as the reference point for output, and the fitted temperature cycle is shown in Figure 9.

Temperature output at the reference point. As shown in the figure, the temperature changes in the right and left fiber coils are essentially the same, while the bottom fiber coil shows a slightly different behavior. In the vacuum state, the minimum temperature of the fiber coils is −14.51 °C, and the maximum temperature is 64.59 °C. By deriving the fitted function to obtain the slope, which is the rate of temperature change, and substituting the rate of temperature change obtained from the simulation experiments into Formula (15), the distribution of thermally induced errors for the three IFOGs in both vacuum and non-vacuum environments can be obtained, as shown in Figure 10. It can be seen that the bottom fiber coil, due to thermal contact, experiences greater temperature variations compared to the left and right fiber coils, resulting in larger thermally induced errors. This also indicates that the thermally induced error is mainly caused by temperature changes rather than deformation. Additionally, since the fiber IMU itself generates heat, the rate of temperature change in the heating phase is higher than in the cooling phase, which leads to the peak thermally induced error during the cooling phase being lower than that during the heating phase, consistent with the theoretical analysis in the previous section.

4. Thermal Vacuum Experiment

To validate the theoretical correctness of the fiber coil’s thermally induced error model presented in Section 3, an IMU was assembled as the experimental subject. The overall appearance of the fiber IMU is shown in Figure 11, with dimensions identical to those designed for the simulation experiment. The fiber IMU consists of a three-axis integrated IFOG, three MEMS accelerometers, signal acquisition and interface circuits, secondary power supply circuits, a light source module, a base, a mounting base, and a top cover. In the overall structure, the circuit parts are fixed using a multi-point method to ensure uniform heat transfer, minimizing thermal resistance and transferring the heat from the components to the mounting surface efficiently.

The thermal vacuum test system uses an integrated structure, with a cylindrical shell that effectively meets the vacuum pressure requirements, completed with seamless welding technology. The door of the box opens using mechanical hinges, and its special design ensures both secure sealing and easy opening. The thermal vacuum environment test system can control the temperature between −180 °C and 200 °C according to actual needs and provide an environment with a vacuum degree as low as $1\times {10}^{-6}$ Pa. The system employs advanced international technology for heat sink temperature control to transfer heat within the test chamber, ensuring an excellent temperature uniformity which meets space testing standards.

After closing the tank door, the tank is vacuumed, reducing the internal pressure from 101 KPa to $3\times {10}^{-3}$ Pa, and then the temperature in the thermal vacuum chamber is raised to 55 °C. Since the IFOG requires a 120 min startup time to achieve stable operation, during which all internal components maintain a steady state, the IMU is powered on after the chamber is heated to 55 °C. After 120 min, data are collected from the fiber optic gyroscope using data acquisition software.

Simultaneously, the chamber is cooled at a rate of −1.5 °C/min to −25 °C, maintained at −25 °C for 140 min, and then reheated. The heating process follows a rate of 1.5 °C/min back to 55 °C, where it is held for 280 min before completing data collection. The output data from the triaxial IFOG is shown in Figure 12. For the convenience of comparing thermal-induced error data, the data from the triaxial IFOG is adjusted by subtracting the Earth’s rotation rate. Here, the x-axis IFOG corresponds to the left-side fiber optic gyroscope, the y-axis to the right-side IFOG, and the z-axis to the bottom IFOG.

As shown in Figure 12, due to the presence of white noise in the output of the IFOGs, it is not easy to make an accurate comparison with the simulation results. At the same time, the vibrations generated during the heating and cooling processes of the temperature chamber also affect the accuracy of the IFOGs. The smoothed data from the three-axis IFOGs are then compared with the simulation results, as shown in Figure 12, Figure 13 and Figure 14. Additionally, theoretical error values calculated using the equivalent radius method under the same conditions are included.

Since the z-axis IFOG fiber coil is closer to the heat source, its temperature change rate during the cooling process is slightly lower than that of the x-axis and y-axis IFOGs, resulting in the thermal-induced error of the z-axis IFOG being slightly smaller than that of the x-axis and y-axis IFOGs. Conversely, during the heating process, the temperature change rate of the z-axis IFOG is slightly higher than that of the x-axis and y-axis IFOGs, leading to the thermal-induced error of the z-axis IFOG being slightly greater than that of the x-axis and y-axis IFOGs. The experimental phenomena are generally consistent with the simulation results.

Comparing the experimental data of the three IFOGs with the simulation results indicates that the 3D error model’s simulation curve is closer to the experimental data than the equivalent radius method. Similar to the evaluation of magnetic-induced errors, the root mean square error (RMSE) is used to compare the two methods with the experimental data. The RMSE of the x-axis IFOG 3D error model is 0.109°/h, and the RMSE of the equivalent radius method model is 0.164°/h, reducing the RMSE by 33.6%. The RMSE of the y-axis IFOG 3D error model is 0.105°/h, and the RMSE of the equivalent radius method model is 0.160°/h, reducing the RMSE by 33.5%. The RMSE of the z-axis IFOG 3D error model is 0.104°/h, and the RMSE of the equivalent radius method model is 0.157°/h, reducing the RMSE by 33.8%.

The error data from the three IFOGs indicate that the accuracy of the 3D error model is higher than that of the equivalent radius error model, demonstrating that the 3D error model is more suitable. However, in practical experiments, temperature changes simultaneously introduce circuit errors such as half-wave voltage and scale factor in the IFOGs, causing some discrepancies between the experimental and simulation data.

5. Discussion

The three-dimensional gyro thermally induced error model proposed in this paper provides a theoretical basis for the temperature compensation of fiber optic gyroscopes. It is of great significance for the preparation of miniaturized high-precision fiber optic gyroscopes and offers potential for their application in various scenarios.

6. Conclusions

This paper investigated the mechanism of thermal-induced errors in IFOGs caused by temperature changes in a vacuum environment and established a model. Finite element analysis was used to conduct simulation experiments, calculating the temperature distribution and deformation of fiber coils under thermal vacuum conditions. Using the established thermal-induced error model, the thermal-induced errors of the IFOGs during heating and cooling under vacuum conditions were calculated. A thermal vacuum experimental platform was set up to collect variable temperature data from the IMU. The experimental data were compared with the three-dimensional error model proposed in this paper and the traditional error model. The root mean square error was about 33% better than the traditional error calculation method, thereby proving the correctness of the theory.