## Abstract

Traditional compensation methods using temperature-related parameters have little effect when the ring laser gyroscope (RLG) bias changes rapidly. To solve this problem, a novel RLG bias temperature compensation method using readout signals is proposed in this paper. Combined with the least squares support vector machine (LS-SVM) algorithm, the novel method can improve the precision of the RLG bias. Experiments show that by utilizing the readout signals in the LS-SVM model, the RLG bias stability can be significantly raised compared to the original data. The novel method proposed in this paper is shown to be feasible, even when the RLG bias changes rapidly.

© 2015 Optical Society of America

## 1. Introduction

Accompanied by the extensive demand of high precision, light weight and compact navigation system both in military and civilian use, the ring laser gyroscope (RLG) has become a mainstream inertial sensor [1,2]. There are three main sources of error in the RLG: null shift, mode locking, and scale-factor error [3,4]. In order to eliminate these errors, several technologies have been successfully applied. Firstly, using mechanical dither frequency bias technology, the RLG can detect at low to zero rotation rates in order to avoid the mode-locking phenomenon. Furthermore, the noise signals added in the dither control system may perfectly eliminate the dynamic lock-in [3]. Secondly, by choosing ultra-low expansion glass and utilizing a path-length control method, the scale factor error has little effect on the RLG precision. Thirdly, for the RLG null shift (or bias drift) error, which is the most significant source of error for RLG precision, many kinds of information sampled from the RLG are used to compensate it. The most convenient information comes from the temperatures acquired from the platinum resistors, which are placed on the surface of the RLG block and inside the installation shell [5–7]. Many researchers have performed in-depth analyses on this issue. In particular, the positions of the thermometers in the RLG have been discussed [5]. More specifically, the temperature sensors pasted on the different points of the RLG play different roles and reflect various characteristics. By utilizing the sum of squared errors (SSE), a temperature compensation model for the RLG was built, and the optimum combination of temperature sensors was selected for practical compensation [5]. In order to erase the limitation of least-squares (LS) fitting and neural networks in temperature compensation, a new scheme of system-level temperature compensation of the RLG bias – based on least squares-support vector machine (LS-SVM) – was proposed [8,9]. Moreover, a novel modeling method using a particle swarm optimization tuning support vector machine (PSO-SVM) with multiple input temperature variables was proposed in order to satisfy the requirements of precision and generalization in the RLG temperature compensation method [2].

However, for the sources of heat due to gas discharge, ballast resistance, and the environment – such as the air or the preamplifier circuit inside the installation shell – bias hysteresis demonstrates a complex relationship with temperature change [6]. Furthermore, in some circumstances, the RLG bias changes rapidly and bears little relation to the temperature or the variation of temperature. To solve this problem, another parameter, the light intensity of the RLG, was used to enhance the precision of RLG bias stability [10]. The RLG bias compensation method using the light intensity of the RLG is a effective method because it instantly reflects the actual movement of the path of light in the optical harmonic cavity. However, the light intensity, which is mainly used for path length control, only reflects the effect of one working beam of the RLG to the RLG bias. Accordingly, the information from the readout signals, which are also the sample sources of the RLG bias, is neglected.

In this paper, the readout signals from the silicon biplanar photodiode detector, which receives the interference pattern, are firstly used for RLG bias compensation. Instead of using only one type of RLG parameter (i.e., temperature), other parameters such as light intensity and the readout signals are also included as the input signals in the RLG bias compensation model. By comparing different compensation methods with or without the readout signals, we conclude that the introduction of readout signals greatly contributes to the RLG bias compensation, especially in the bias mutation situations. Moreover, combined with the LS-SVM algorithm, the novel method can improve the precision of RLG bias even further.

This paper is organized as follows: In Section 2, the RLG bias compensation model using readout signals of the RLG and the LS-SVM method is described in detail. In Section 3, the results of the temperature experiment to validate the effectiveness of the proposed method are analyzed. Lastly, some conclusions are drawn in Section 4.

## 2. RLG bias compensation model

#### 2.1 Readout signals used for RLG bias compensation

The principle of the RLG is based on the Sagnac effect [3]. The phase difference of two opposing, coherent, electromagnetic light waves (CW: clockwise; CCW: counter-clockwise) propagating in a rotating optical resonant cavity is proportional to the rotation angle rate. The difference of light traveling time introduces the optical path difference:

where $A$ is the area enclosed by the light path, $\Omega $ is the rotation rate with the direction perpendicular to the plane of the light path, and $c$ is the velocity of light.The frequency difference is proportional to the rotation rate of the RLG along the sensitive axis:

where $K=4A/\lambda L$ is called the scale factor of the RLG, and $\lambda $ is wavelength of the light.The RLG readout system is shown in Fig. 1. The two laser beams are combined with a prism and interfere to form a fringe pattern. Assuming that the two beams have the same amplitude ${I}_{0},$ and the distribution of light intensity on the detector is:

where ${\varphi}_{0}$ is a constant phase difference, $\gamma $ is the angle between the two beams, and $x$ is a distance along the detector [3,11].The fringe spacing is$\Delta x=\lambda /\gamma $ and $\gamma =2n\alpha ,$ where $n$ is the refractive index and $\alpha $ is the angle parameter of the combining prism. When the RLG is rotated, the fringe pattern’s movement corresponds to the rotation rate and direction. For a given integration time $t,$ the number of fringes are:

As already mentioned above, the RLG bias instability is the main source of RLG error, and there have been many efforts to compensate for it by using different models with input parameters – such as the temperature of the RLG block sampled by thermometers or light intensity, which is also used for path length control of the RLG. The readout signals of the RLG used for bias compensation here share the same output signals of the biplanar photodiode, which senses the light of the interference fringe pattern with a phase difference of about 90°. In an optimal situation, the CW and CCW beams should be coherent through combining prism, but under many circumstance, these two beams are slightly incoherent. Figure 2 shows the scheme of readout signals used for RLG bias compensation.

After the RLG is powered on, if the CW and CCW beams are coherent, the fringe patterns that the two photo detectors receive have the same light intensity, which is shown in Fig. 2(a). However, the fact is that there are many elements in the RLG assembly, with the changing of temperature caused by the assembly itself or the environment (in addition to other variations such as vibrations). As a result, the instability of these elements will cause the two beams to move. This will eventually affect the readout signals, a result that is illustrated in Fig. 2(b) [12]. The fringe patterns that the two array photo detectors receive have different light intensities. Therefore, the difference reflects the relative change of the two beams working in the optical resonant cavity.

In order to measure this change and find the relationship between the readout signals and the RLG bias error, a circuit detects the light intensities of fringe patterns received by the two-array photodiode. Figure 3 shows the scheme of this circuit.

Because the photodiode changes the light intensity signals into current signals, which cannot be sampled directly by the microprocessor, the preamplifier circuit is needed for converting the current signals to voltage signals. The voltage bias, which is marked as V_2.5 in Fig. 3, can raise the voltage level to the input range of the A/D converter. The output signals from the circuit are the root mean square (RMS) of the corresponding voltages, which are proportional to the light intensities of the fringe pattern received by the photodiodes.

After the tasks mentioned above have been completed, these two readout signals sampled by the A/D converter are transmitted to the microprocessor, and their deviation is used as the input vector for the RLG bias compensation model. The structure of the hardware is shown in Fig. 4.

#### 2.2 Least squares support vector machine model

For nonlinear systems, the support vector machine (SVM) has become a popular tool, based on the theory of statistics, for machine-learning tasks. It can minimize the structural risk level of the system model [13,14]. For multi-input and single-output applications, SVM is supposed to be a better solution of machine learning compared to neural network (NN) technology [2,8]. Suppose the data sampled from the RLG form a linearly separable training set:$({x}_{i},{y}_{i}),i=1,\mathrm{...},n.$ $y=\{+1,-1\}$ is the classification label and the function of the hyperplane is:

where $w$ and $b$ are parameters of the hyperplane that should correctly classify all the sample data. This plane separates the sample data without errors and has the largest distance to the nearest sample data, which is equivalent to minimize ${\Vert w\Vert}^{2}.$ The restraint condition is:The hyperplane, which satisfies this condition and minimizes the value of ${\Vert w\Vert}^{2},$ is the optimal hyperplane. In order to construct such an optimal hyperplane, only a small fraction of the training sample data that determine the margin is needed, and these training data are called support vectors. The optimal hyperplane can also be expressed as a function to find the minimum value:

The problem discussed above is the optimal and general classification function. Furthermore, the only thing we should know is the inner product operation in the characteristic space where the optimal linear classification problem can be solved. As for the non-linear problem, the criteria of general linear discriminant functions can be used to translate a non-linear problem into a linear problem in another space where the optimal or general hyperplane can be found.

According to Mercer’s condition, for an arbitrary asymmetric function$K(x,x\text{'}),$ it is sufficient and necessary that the condition:

is satisfied for all $\phi (x)$ such that:Obviously, this condition is easy to achieve; and if the optimal hyperplane dot products are replaced by the inner products $K(x,y),$ then the original characteristic space is transformed to a new characteristic space where the SVM is:Compared to the standard SVM, the least squares-support vector machine (LS-SVM) can be less influenced by the number of training samples [8,9]. The convex constrained optimization problem of LS-SVM can be expressed as:

According to Karush-Kuhn-Tucker (KKT) conditions, the optimal non-negative Lagrange multiplier vector $a={[{a}_{1},\mathrm{...},{a}_{n}]}^{T}$ can be analyzed and reduced according to the following procedure:

At this stage, the non-linear problem is transformed into the following set of linear equations:

## 3. Experiment and analysis of results

#### 3.1 Experimental configuration

The experiment is performed by utilizing the aircraft strapdown inertial navigation system (INS) designed in our laboratory, which is shown in Fig. 5. As the key components in the INS, three RLGs and three pendulous accelerometers are mounted perpendicularly to each other on the inertial sensor assembly (ISA). Through eight vibration absorbers, the ISA is suspended among the installation case, which also contains the signals processing circuit and system’s power supply circuit. As is well known, the performance of the RLG determines the performance of the INS in a sense, so it is important to discover the relationships between the RLG bias and the RLG working parameters – such as temperature [11,15,16], light intensity [10], and readout signals. The experiment is carried out in a temperature chamber, and the program is set up as follows: 1) the INS is fixed with bolts in the temperature chamber and powered on; 2) the temperature of chamber is set at 20 °C and maintained there for 2 h; 3) the temperature is set to decrease at a rate of 1 °C/min until it arrives at −40 °C, and is maintained there for 2 h; 4) the chamber temperature is increased at a rate of 1 °C/min to 60 °C and maintained there for 2 h; 5) the chamber temperature is decreased to 20 °C at a rate of 1 °C/min and maintained there for 2 h. Throughout the experiment, the RLG bias data and working parameters – such as temperature, light intensity, and readout signals – are all recorded with a sampling frequency of 1 Hz. The entire procedure is performed twice: one set of sample data is used for training and validation, whereas the other is used for testing the RLG bias compensation through the LS-SVM model. Owing to the software of LS-SVMlab, each parameter is selected by utilizing the Gaussian kernel and can be cross validated via parallel grid search.

The original data sampled from the RLG include RLG bias output, temperature, light intensity, and the difference of readout signals. The relationship between the RLG bias and temperature is shown in Fig. 6 as an example.

As shown in Fig. 6, the trend of the RLG bias cannot be distinguished clearly. The most general method in this field is to average the sample data within a range of 100 s. When the sample frequency is 1 Hz, each segment of 100 s of data is averaged, and the variation of the RLG bias with different parameters is thereby shown clearly in Fig. 7.

As is shown in Fig. 7, the RLG bias exhibits a very complex non-linear drift with the temperature change, which can greatly derogate the performance of navigation. Different parameters acquired from the RLG also show various features according to Fig. 7. The temperature of the RLG is a slow-changing parameter with a smooth curve, but has little correlation to the RLG bias variation. The light intensity of the RLG exhibits a mild correlation to the RLG bias at the beginning, but quite low – even opposite – correlation to the RLG bias as the temperature raises from −40 °C to 60 °C. The difference of readout signals, on the contrary, shows a strong correlation to the RLG bias throughout the entire experiment. The correlation coefficients of these three parameters to the RLG bias are shown in Table 1.

Obviously, the difference of readout signals has a strong correlation with the RLG bias, which implies that the RLG bias compensation model using readout signals can achieve a better result.

#### 3.2 RLG bias compensation using the LS fitting model

In order to further the analysis, the traditional modeling method of LS fitting for the RLG bias is given by:

where $B$ is the RLG bias, $X$ is the RLG working parameter, and ${c}_{i}(i=0,1,2,3)$ are LS fitting coefficients. The compensation results of the RLG bias based on LS fitting using different working parameters are shown in Figs. 8(a)–8(d).From Figs. 8(a)–8(d) we can see that the LS fitting models of the RLG bias compensation using temperature[Fig. 8(a)] and light intensity [Fig. 8(c)] shows improvement little upon the raw data compared to the models using temperature variation rate [Fig. 8(b)] and difference of readout signals [Fig. 8(d)]. The fitting curves of Figs. 8(a) and 8(c) are even negatively correlated during the 200–300 time points, not to mention the mutations after the 100 time points. As for the LS fitting model of the RLG bias compensation using the temperature variation rate, the trend of the fitting curve coincides with the raw data well, but the mutation part does not. On the contrary, the LS fitting model of the RLG bias compensation using the difference of readout signals performs well in the mutation part of raw data, whereas the curve between the 200 to 300 time points deviates from the actual curve. In addition, the standard deviation (STD) estimations for the original and compensated data are listed in Table 2.

#### 3.3 The RLG bias compensation using the LS-SVM model

As discussed above, the LS-SVM can be suitable for the non-linear regression problem even with a small training data set. The compensation model is expressed in Eq. (17), and the compensation procedure operates in two steps: training and testing. Firstly, the training data are divided into several parts: some of them are used for the validation set, and others for training. Through the searching grid space we can get the grid point of $(c,\sigma )$ and the cross validation accuracy. Secondly, the best parameters are used to train the entire training set and produce the black-box model for the RLG bias compensation. Finally, the experimental data are applied to this model. The effects of compensation are shown in Figs. 9(a)–9(d).

From Figs. 9(a)–9(d) we can also see that the LS-SVM models of RLG bias compensation using temperature variation [Fig. 9(b)] and difference of readout signals [Fig. 9(d)] are better than the models of RLG bias compensation using temperature [Fig. 9(a)] and light intensity [Fig. 9(c)]. Compared to the LS fitting models, the LS-SVM models achieve better fitting results even for the model only using the temperature parameter. However, the fitting result using the difference of readout signals prove to be more effective. The STD estimations for the original and compensated data are listed in Table 3.

From Table 3, it can be clearly seen that the introduction of the difference of readout signals can dramatically improve the compensation precision of the RLG bias. The bias stability of the RLG output after it has been compensated for by the LS fitting model is 0.0073°/h, which has decreased by 0.0084°/h from that before compensation: 0.0157°/h. On the other hand, the bias stability of the RLG output after it has been compensated for by the LS-SVM model is 0.0035°/h, which has decreased by 0.0122°/h from that before compensation. The compensation results between different methods with and without the difference of readout signals are compared in Table 4, which powerfully shows that the proposed difference of readout signals used for RLG bias compensation is more feasible and effective than the traditional parameters, such as temperature and light intensity. Moreover, Table 4 illustrates that the LS-SVM compensation method can also greatly improve the RLG bias precision.

Finally, all parameters are introduced in the LS-SVM model of the RLG bias compensation. The result is shown in Fig. 10, and the bias stability of the RLG output after it has been compensated for is 0.0023°/h, which increases by a factor of 6.8 compared to the original precision.

## 4. Conclusion

In this paper, a novel RLG bias temperature compensation method using readout signals is proposed. This method brings four main advantages: 1) the parameter used for the RLG bias compensation has the same source of variation as the RLG, which means that the redress ability can be improved significantly; 2) the time delay is dismissed during the compensation process; 3) the readout signal can be easily translated to the voltage signal sampled by the A/D converter of the microprocessor by utilizing the operator amplifier; and 4) the other parameters also can be used as the input vectors for the RLG bias compensation model. The experimental results show that the RLG bias compensation models using the difference of readout signals are feasible and reliable even in the extreme bias mutation situation. By utilizing the difference of readout signals, the precision of the RLG bias compensation is improved by 29.3%. In addition, the proposed LS-SVM method was shown to be a powerful method as regards the RLG bias compensation application. By utilizing this method, the RLG bias precision increased by 52.1%. When all parameters are introduced to the LS-SVM model, the compensation effect for the original RLG bias is optimal, and the precision is increased by a factor of 6.8.

However, the optimization that can be achieved in future research will depend on how much the type of kernel function and choice of parameters in the SVM model can affect the accuracy and generalization ability of the RLG bias compensation model. The same experiment will be performed for more RLGs and the different kernel functions, such as the radial basis function (RBF) $K({x}_{i},{x}_{j})=\mathrm{exp}(-\Vert {x}_{i}-{x}_{j}\Vert /2{\sigma}^{2})$ ($\sigma $ is the parameter of RBF), the polynomial function $K({x}_{i},{x}_{j})={({x}_{i}{x}_{j}+b)}^{d}$, and the S function $K({x}_{i},{x}_{j})=\mathrm{tan}h[K({x}_{i},{x}_{j})+v](k>0,v<0)$, will be used for different types of RLG. Regarding the practical applications involving RLGs, the filter can also be used for de-noising the original output data to be modeled during the preprocessing of the training data.

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