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This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

Temperature changes have a strong effect on Hemispherical Resonator Gyro (HRG) output; therefore, it is of vital importance to observe their influence and then make necessary compensations. In this paper, a temperature compensation model for HRG based on the natural frequency of the resonator is established and then temperature drift compensations are accomplished. To begin with, a math model of the relationship between the temperature and the natural frequency of HRG is set up. Then, the math model is written into a Taylor expansion expression and the expansion coefficients are calibrated through temperature experiments. The experimental results show that the frequency changes correspond to temperature changes and each temperature only corresponds to one natural frequency, so the output of HRG can be compensated through the natural frequency of the resonator instead of the temperature itself. As a result, compensations are made for the output drift of HRG based on natural frequency through a stepwise linear regression method. The compensation results show that temperature-frequency method is valid and suitable for the gyroscope drift compensation, which would ensure HRG's application in a larger temperature range in the future.

The hemispherical resonator gyro (HRG) is a solid state gyroscope whose sensing property is based on a standing vibration wave precession. It has the features of high accuracy, long life span, inherent high reliability, natural radiation hardness and no parts that can wear out. With its excellent performance, the Scalable Space Inertial Reference Unit (Scalable SIRU) and its predecessor, the Space Inertial Reference Unit (SIRU), which all are made from HRGs, have been launched on more than 125 spacecraft missions for NASA, Department of Defense, commercial and international space applications [

The HRG contains three primary functional components: the hemispherical resonator, the forcer and the pickoff. They are all made of quartz and bonded together within a sealed vacuum housing [

The temperatures of both the inner house and resonator will change due to the heat produced through the vibration of the resonator and ambient temperature changes of the HRG. Moreover, owing to the uneven heat conduction, a temperature gradient will exist in the vacuum housing of the HRG. Since factors such as temperature changes and temperature gradient can strongly result in temperature drifts which seriously affect HRG's application in commercial and military areas, it is of great importance to suppress or compensate these temperature drifts. At present, there are mainly two methods to suppress the drift caused by temperature changes [

Temperature stabilization method: the HRG is placed in a controlled temperature chamber, that can keep the surrounding temperature constant and provide the best conditions for the gyro, which decreases the drift resulting from temperature changes.

Temperature compensation method based on a math drift model of the HRG: obtain a curve about the relationship between the output of the HRG and temperature and make compensations on-board through software.

As for the first method, the sensing components (resonator) of the gyro are encapsulated in a vacuum, so basic modes of heat exchange could only depend on the thermal radiation and the heat transfer through the sustaining pole between the resonator and the outside cover, causing the temperature of sensing components to change slowly. As a result, it takes a long time to make the gyro sensing components' temperature approach the pre-set temperature of the controlled temperature chamber before it could work, so it could not meet the needs of rapid reaction. Furthermore, the temperature control system will greatly increase the volume, weight and cost which would make the strapdown inertial navigation system much too expensive. One point worth mentioning is that volume and weight are two decisive factors in space applications, so big volume and weight are regarded as fatal limitations.

Compared with the first method, the latter one (temperature compensation method) is much easier to adopt since it doesn't require an increase in volume, weight or hardware cost. However, the resonator is sealed in a vacuum house and any accessories attached to it for temperature sensing would seriously deteriorate its performance, making it unrealistic to set up any temperature sensor on the resonator. Although the temperature sensor could be fixed on the inner vacuum housing, the heat exchange is very slow without air. Thus, the temperature sensor attached to the inner housing is not able to represent the real time temperature of the resonator. In a word, it would be very difficult to directly measure the temperature of the resonator.

Fortunately, as references [

The resonator itself could serve as a high precision temperature sensor for temperature compensation of the gyroscope. In reference [

This paper provides detailed descriptions of the relationship between the temperature and frequency of the HRG. As long as the frequency of resonator is obtained by the digital control loops of the HRG, temperature compensation for the output of the gyroscope can be realized in real time [

References such as [

The energy method can be applied to determine the natural frequency of the HRG resonator, and then the temperature coefficient of the natural frequency of HRG can be obtained. A model for a thin axis-symmetrical hemispherical shell with mean radius _{i}_{i}_{i}

When the materials are isotropic, Hooke's law is written as follows:
_{φ}_{θ}_{φθ}_{φ}_{θ}

For the case of a hemispherical shell, the middle surface strain and curvature changes in

As for free vibration of clamped-free hemispherical shell, under the condition of paucity displacement, the Lord Rayleigh inextentional condition is satisfied, so the normal stress and shear stress will be approximately reduced into zero, which is:

Substituting

Substituting

Substituting

Utilizing the condition _{max} = _{max}, the natural frequency of the hemispherical shell can be determined, which is [

Attention is paid to the

I and J are only relative to the shape of the hemispherical shell and

From

However, if all the terms which are affected by temperature are respectively taken into consideration, the relationship between temperature and natural frequency will be very difficult to obtain. Therefore, in this paper, a Taylor expansion method is employed to analyze the temperature coefficient of frequency of HRG for simplicity. The frequency temperature function _{0} can be described as a Taylor series which is:

Based on the theory of thermodynamics of materials, the natural frequency of quartz can be expressed as a three-order polynomial, so the high-order terms can be neglected:

Comparing

Under the FTR mode of the HRG, four control loops, which are reference-phase loop, amplitude-control loop, quadrature-control loop and rebalance control loop, are employed to ensure that the HRG works at a high performance status. Reference-phase loop and amplitude-control loop are employed to maintain the primary vibration pattern at its natural frequency and at constant amplitude. The quadrature control loop which changes the stiffness of the resonator is employed to eliminate the frequency split of the two vibration modes. Simultaneously, the rebalance loop is employed to nullify the response of the second mode, and the rotation rate can be obtained from the force which is used to nullify the response of the second mode. The information including rotation rate and the vibrating frequency of the resonator are all obtained in digital form. Since the reference phase loop is locked to the resonator, its frequency change traces the temperature changes of the resonator.

Experiments are designed to get the temperature-frequency coefficient of the HRG, as shown in

The frequency change of the gyroscope which was placed in the temperature chamber with the temperature setting −10 °C is shown in

Similarly,

The coefficient of

The second order coefficient is 9.702 × 10^{−4} and the third order coefficient is 1.528 × 10^{−5}, which means that the high order terms have little effect on the frequency. The fitting curve is shown in

Using

The largest deviation of the temperature error was below 0.1 °C in a temperature interval from −20 °C to 40 °C, which is close to the actual temperature. The experimental results indicate that the variation stability is very small, with a tolerance of less than 0.1 °C.

Based on the analysis above, it can be concluded that the natural frequency of the HRG is relative to the temperature and each temperature degree only corresponds to one natural frequency. Consequently, the natural frequency of the HRG, which can be easily obtained at any time, can be regarded as a high precision index of the temperature of the resonator. As a result, the frequency which is transmitted by the HRG system in real time can be used for HRG temperature compensation.

Based on the work above, we can compensate the output of HRG when the temperature is changing. Under FTR mode, the reference-phase loop and amplitude-control loop are employed to control the variation of the primary vibration pattern, which maintains the vibration at its natural frequency and at a constant amplitude [

Suppose y is an arbitrary variable, and its relation with the independent variables _{i},i_{0},_{1},…,_{n}_{0},_{1},…,_{n}_{i},i

In ^{2}_{n}

As a result,

and then:

The sufficient condition for the least error is:

As a systematic method, stepwise regression involves adding or removing terms from a model on the basis of their statistical status in a regression. If a term is not currently in the model, the null hypothesis is that the term would have a zero coefficient if added to the model. If there is sufficient evidence to reject this null hypothesis, the term will be then added to the model. Conversely, if a term is currently in the model, the null hypothesis is that the term has a zero coefficient, if evidence is not enough to reject the null hypothesis, the term will be removed from the model. To be brief, stepwise regression is widely used since it is unbiased and has a minimum variance among all unbiased estimators formed from linear combinations of the response data by the Gauss-Markov theorem.

In the real work conditions for the HRG, the frequency change rates are different from each other because the heat field is uncertain and the heat conduction is uneven. Therefore, a drift model based on the frequency changes and frequency change rates is applied to make temperature compensations for the HRG. Besides, considering the coupled terms of frequency changes and the frequency change rates, three order temperature model can be established:
_{i}_{0} represents the constant bias and it has no relation to the temperature; _{1} − _{3} represent the frequency change item coefficients, showing the change trend of the bias related to the frequency(temperature). _{4} − _{5} are the coefficients of frequency change rates; _{7} is the coefficient of frequency change coupled with frequency change rates, showing the effect on the bias of HRG by their combination.

We make compensations for the drift of HRG by using the model described above. The blue curve in

The external temperature changes have a strong effect on the HRG, for example, the material properties such as Young's modulus, the radius of the resonator and so on change because of the heat; the excite electrodes, resonator and pick-off electrodes displace irregularly due to the heat deformation, and all those phenomena result in bias drift decreasing the degree of precision of the HRG. As a result, it is of vital necessity to observe the influence on the HRG output by the temperature changes and then compensate for it. Only in this way, can the performance of the HRG be improved. In this paper, the relationship between temperature and frequency are firstly established, and then we compensate for the output of HRG by the frequency changes through its relation to the temperature changes. This method reduces the complexity of the compensation without using a temperature sensor. More importantly, it can be found that the experiments give a satisfactory result by using this compensation method, and it significantly improves the temperature stability of the HRG.

This work was carried out at the Laboratory of Inertial Technology, College of Mechanical Engineering and Automation, National University of Defense Technology. The authors are sincerely indebted to Mingming Jiang for his devotion to those temperature experiments. This work was supported in part by Program for New Century Excellent Talents (NCET-07-0225) in University of China.

The illustration of the mean shell of the hemispherical resonator.

The temperature experiments carried out by a controlled temperature chamber.

HRG natural frequency varies with temperature changes. (

Temperature-frequency fitting curve.

Temperature calculated from the frequency. (

The output of the gyro. (

Different natural frequencies of HRG in correspondence with different temperatures.

−20 | 4,416.97 |

−10 | 4,421.05 |

0 | 4,424.90 |

10 | 4,428.56 |

20 | 4,432.07 |

30 | 4,435.56 |

40 | 4,439.20 |

Temperature calculated from the frequency and the calculation errors.

Actual temperature(°C) | −20 | −10 | 0 | 10 | 20 | 30 | 40 |

Calculate temperature (°C) | −19.976 | −10.058 | −0.0026 | 10.070 | 20.011 | 29.924 | 40.031 |

Calculated error(°C) | −0.024 | 0.0578 | 0.0026 | −0.070 | −0.011 | 0.076 | −0.031 |