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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/).

In the analysis of the effects of temperature on the performance of microgyroscopes, it is found that the resonant frequency of the microgyroscope decreases linearly as the temperature increases, and the quality factor changes drastically at low temperatures. Moreover, the zero bias changes greatly with temperature variations. To reduce the temperature effects on the microgyroscope, temperature compensation-control methods are proposed. In the first place, a BP (Back Propagation) neural network and polynomial fitting are utilized for building the temperature model of the microgyroscope. Considering the simplicity and real-time requirements, piecewise polynomial fitting is applied in the temperature compensation system. Then, an integral-separated PID (Proportion Integration Differentiation) control algorithm is adopted in the temperature control system, which can stabilize the temperature inside the microgyrocope in pursuing its optimal performance. Experimental results reveal that the combination of microgyroscope temperature compensation and control methods is both realizable and effective in a miniaturized microgyroscope prototype.

In recent years, the silicon microgyroscope has been used as a kind of inertial device for measuring the angular velocity of an object's motion [

Owing to the expansion and centralization in material dimension over temperature, the stiffness of the silicon microgyroscope will change with temperature variation. According to the kinetic equation of the gyroscope, the resonant frequency of a microgyroscope is relevant to the sensitivity and stability as well as its dynamic characteristics, therefore temperature variation has significant impacts on both the output sensitivity, the stability and the dynamic characteristics of microgyroscopes, potentially resulting in a great temperature drift of the entire system.

In [

Above all, the performance of a microgyroscope is greatly affected by the power consumption and ambient temperature variation, and the effect of environmental temperature becomes one of the most important sources of error in microgyroscopes. Therefore some effective measures should be taken to reduce the influence of temperature, including temperature compensation and temperature control, which should be both efficient and feasible for improving the precision and stability of the microgyroscope.

Two different methods are proposed in this paper. The first one is a temperature compensation method, the other one is the temperature control method. For the temperature compensation method, a temperature drift model of the microgyroscope is constructed first. It is then used to estimate the current zero bias which is then subtracted from the actual output to obtain the compensated output of the microgyroscope. As for the temperature control method, a microprocessor is employed to stabilize the temperature inside the gyroscope casing.

In this paper, the symmetrical and decoupled microgyroscope used is essentially a two linear vibratory gyroscope to detect rotation angular rate with the oscillating components. Its simple model can be described as a total four DOFS system including proof mass, drive component, sense component, and damping elements.

The Coriolis dynamic equations of Z-axis microgyroscope is:

where _{x}_{y}_{x}_{y}_{x}_{y}_{z}_{e}_{d}_{d}t_{d}_{d}

According to the kinetic equations, ignoring the crosstalk interference terms,

where _{x}_{x}/_{x}ω_{x}_{y}_{y}/_{y}ω_{y}

Various factors can impact the performance of microgyroscope, including the changes of the characteristics of its material, and variations of the electrical characteristics of its peripheral circuit. Noticeably, temperature has a remarkable influence on various aspects, including the resonant frequency, the quality factor and the zero bias.

Considering that the substrate of the microgyroscope is glass, and its supporting beam is made of monocrystalline silicon, the Young's modulus changes with the temperature variation, so their relationship can be expressed as:

where _{E}^{−5} _{0} is the temperature coefficient of monocrystalline silicon at Kelvin temperature.

Meanwhile, a strain exists between the supporting beam and the substrate owing to the difference in thermal dilation coefficient among different materials, thus the strain quantum can be expressed as:

where _{s}, α_{g}

Therefore the variable quantity of residual stress is:

where ν denotes the Poisson ratio of the monocrystalline silicon.

The relationship between the natural frequency and the Young's modulus, as well as the residual stress of the supporting beam, can be expressed as [

where _{1} denote the height and the length of the structure respectively,

From the above analysis, the Young's modulus

According to [

where _{p}_{γ}^{2}/sec^{2}/K is the generalized mol constant of gas,

where

The vibrating displacement in drive mode of the microgyroscope is:

According to [_{x}_{1} in drive mode of microgyroscope are proportional to the temperature. Thereby the amplitude of vibrating displacement is negatively correlated to the temperature, that is, the amplitude of vibrating displacement in drive mode decreases when temperature increases, which can be seen from the simulation shown in

Similarly, the vibrating displacement in sense mode is:

where _{y}_{2} the damping coefficient. Similar to the simulation in drive mode, the vibrating amplitude in sense mode is negatively correlated to temperature.

In order to determine the actual effects of temperature on the performance of a gyroscope, a vacuum encapsulated microgyroscope named B34 was adopted, and relevant experiments designed and carried out to verify the above theoretical analysis.

First of all, an open-loop driving circuit was adopted to drive the microgyroscope for testing its resonant frequency and the quality factor under the circumstances of the temperature varying from −40 °C to 60 °C. The waveform generator provides the sinusoidal signal to drive the microgyroscope. _{d1}_{d}ε_{0}_{d2}_{d}ε_{0}_{d}_{0} denotes the dielectric constant; _{x}_{d}_{a}_{td}_{ts}

In _{td}_{x}_{d}_{d}_{−}_{3db}

During the course of measurement, the microgyroscope and the circuit are statically mounted. The temperature value in the temperature control box rises from −40 °C to 60 °C in 10 °C intervals. The temperature at each sampling point is maintained for sixty minutes before testing to ensure that the temperature in the box is uniformly distributed and the microgyroscope is fully heated. In this way, any difference between the inner and outer temperature of the microgyroscope casing can be effectively reduced. The testing results are provided in

The changing trends of gyroscope's resonant frequency and quality factor with temperature variation can be seen in

As shown in the above figures, it is obvious that the testing results shown in

According to the data in

The second step is to test other performances of the same microgyroscope by the closed-loop driving circuit. According to the prior art [_{d}_{out}_{d}

Firstly, five groups of zero microgyroscope bias are recorded, respectively, at normal temperature under the same conditions. The test began from the start-up time once the microgyroscope is powered on, and a 30-minute interval is set between each successive group. From the results shown in

During the temperature experiments, the mirogyroscope is placed still in the temperature control box. At each temperature point it takes one hour to reach thermal balance before the recording of the zero bias of the microgyroscope. Each sampling point is recorded for sixty minutes. Then the average output of the microgyroscope at each temperature point is calculated, so the overall trend of closed-loop drive amplitude and zero bias over temperature changes can be shown as in

According to the analysis discussed in Section 2 and the testing results in Section 3, the temperature variation has multiple effects on the performance of a microgyroscope. Consequently the zero drift and the scale factor of gyroscope's output fluctuate with temperature variation, resulting in impairments of the precision and the stability. Therefore, it is necessary to conduct the research on temperature compensation and temperature control for the microgyroscope. In this paper, the BP neural network model and the polynomial fitting for compensation are proposed, then the former method is simulated by Matlab tools, and the latter one is applied in the actual system because of its simplicity and effectiveness. Finally, considering that the temperature compensation could not suppress the zero bias drift completely, an effective temperature control system is adopted to minimize temperature impacts on the gyroscope's performance.

BP (Back Propagation) neural networks are widely applied in function approximation, pattern recognition, classification and data compression [

The number of the adopted neurons and the hidden layers mainly depends on the complexity degree of the issue to be solved because BP neural networks require that the transfer function can be differential everywhere. The Sigmoid activation functions (tan-Sigmoid, log-Sigmoid and linear-Sigmoid) are usually adopted as the transfer function, and the linear function is regularly used as the output layer. Furthermore, each node in this structure has close-value function, and the weighted effects of upper layer are transmitted to lower layer through the transfer function.

The essence of nonlinear fitting by BP neural networks is that through self-studying the BP neural network can determine the corresponding relationship between the input and output, which is memorized as a connecting weight value in the network. Therefore the BP neural network structure adopted for temperature modeling and compensation can be selected as follows: first, the original sampled data, i.e., the zero bias over the temperature is normalized. Next, since it aims to build the model between temperature T and zero bias _{bias}, each input and output layer is selected with only one neuron. To ensure the precision of the curve fitting, more hidden layers with more neurons are required, however, on the contrary, this will add excessive computing to reach the learning and testing phases. Besides, it may affect the convergence rate and cause unintended fitting errors in case of improper training. There is no exact rule governing the selection of a particular number of hidden layers and neurons, and three hidden layers are chosen through various simulation trials and each layer has ten neurons for trade off. The transfer function is connecting input layer with hidden layers, while that connecting hidden layer and output layer are pure linear Sigmoid function. This structure can provide a satisfactory convergence effects while speeding up the training procedure at the same time.

The structure is shown in _{i}_{1i}_{2i}_{j}_{ij}

After the network model is built, the current temperature is used as the input value to get the corresponding zero bias. Then it is subtracted from the actual output to attain the compensated zero bias of the gyroscope. In the Matlab simulation, the original temperature is substituted to get the compensated zero bias curve, which is obviously near zero and almost becomes a straight line approaching zero in

In order to verify the effectiveness of the compensating model, the gyroscope is put in the thermal control box, and the ambient temperature is controlled to increase from −40 °C to 80 °C by 10 °C step. At each temperature point the zero bias is recorded for thirty minutes, thus the compensated zero bias curve is attained through the built up line compensating model in the computer. As can be seen in

Theoretically the BP neural networks modeling provides excellent results for the temperature compensation. However in the first place, massive calculation is incorporated so that it requires high processing capability of the microprocessor and a large database. Secondly, the network needs re-trained to update parameters once the input samples increase due to its poor generalization. Moreover it is relatively complex to implement real-time processing in current chosen microprocessors. Another way of building temperature model of microgyroscope is based on the numerical analysis of its actual output and corresponding temperature. Polynomial fitting has been widely applied in terms of numerical analysis, and the least mean square curve fitting is liable to get overall optimal results. Therefore the polynomial fitting method is proposed for simplicity and effectiveness, and compared to the BP network method, it can provide similarly satisfying compensating effects.

The main idea of the polynomial fitting compensation is as follows: the correlation between the temperature and the zero bias of microgyroscope can be found through experiments, and its mathematical expression can be obtained through polynomial fitting as the temperature is changed from −40 °C to 80 °C. The mathematical function is memorized in the microprocessor. Thus the corresponding compensation value for each real time temperature tested is calculated and subtracted in the actual output of gyroscope. Ultimately the final compensated zero bias of the gyroscope is attained.

While constructing the compensation model, special consideration must be paid to the precision, practicability and complexity of the model to satisfy the engineering requirements. With respect to the real-time performance of the compensating system and the control and operation capability of the C8051F360, the least mean squares (LMS) curve fitting is utilized as it is simple and easy for constituting the temperature model of the zero bias of gyroscope.

Using the high-order polynomial to describe the approximate function (regression equation) relationship of the experimental data (_{i}, _{i}) (where _{i} denotes the input, and _{i} denotes the corresponding output,

where _{i} denotes the error between the tested value and the result calculated by the regression equation. According to the LMS theory, the square of _{i} should be set to the minimum to obtain the optimum value of the coefficient _{j}:

From which we can deduce the following canonical

Then the linear equation for computing _{0}, _{1}, _{m}:

The solution for _{j}(

In the temperature experiments, the zero bias output of the gyroscope is recorded while the temperature increases from −40 °C to 80 °C in 5 °C intervals, which can be seen in

After fitting the experimental curve with the LMS theory, we can separate it into three segments to reduce the order of polynomial and minimize the error.

When −40 °C ≤

When −20 °C <

When 50 °C <

The fitted curve is shown in

The temperature compensation scheme is shown in

Firstly the piecewise polynomial fitting equations are memorized in the microprocessor. Secondly the zero bias _{bias}_{bias}

Firstly the uncompensated zero bias is recorded while the ambient temperature rises up from −40 °C to +80 °C. Secondly, the polynomial fitting is accomplished and the programs with the fitting equations are downloaded to the microprocessor. To verify the compensating effects, the same microgyroscope is placed in the temperature control box with no rotation and several groups of temperature experiments are carried out. In the first and second group, the ambient temperature is controlled to rise from −40 °C to +80 °C. Nevertheless in the third group the temperature descends from +80 °C to −40 °C to test the effectiveness of the model in case that ambient temperature decreases, the results show that the model can be effective when the ambient temperature descends. The compensation effects of each group are shown in

As shown in

Note: the scale factor of the tested microgyroscope is 8.774mV/(°/s).

As shown in

Through abundant experiments and previous experience, it is found that the optimal working temperature for the microgyroscope is about 55 °C. It is proposed that temperature inside the packaged gyroscope should be controlled around this optimal value in order to achieve the best performance of the microgyroscope. Therefore a temperature controlling system is proposed and its effects are analyzed in detail. The temperature control system adopts a single closed-loop method. The current temperature measured by a DS18B20 temperature sensor is sent to the microprocessor. Next the tested value is compared with the set temperature value to attain the deviation value. Then the PID adjuster calculates the control value, which is subsequently transformed by the driving circuit and applied on the thermoelectric cooler (a semiconductor chip). Finally the temperature inside the gyroscope cavity is controlled around the set value. The block diagram of temperature control system is shown in

A high-current integrated driving chip OPA548 (Burr-Brown Company) is employed for driving a TEC based on the Peltier effect. Its bipolar power supply mode is used to further amplify the power for the D/A converter AD5060, implementing power driving of the thermoelectric cooler (TEC) chip.

The Integral-separated PID algorithm is employed for the temperature control to ensure the integral action and decrease the overshoot. Hence the controlling performance can be improved remarkably [

To realize an efficient control, a threshold value

In the math expression, the integral part is multiplied by

where _{p}_{I}_{D}_{P}_{D}_{P}T_{D}_{p}

When |

where _{p}_{I}_{D}_{P}_{D}

Once the control system starts operation, the microprocessor will first send out the order to read the temperature. Next, the current temperature of the gyroscope T1 measured by the DS18B20 is transmitted to be compared with the set value T. Their difference is transferred in time to the PID controller. Then the control quantity calculated by the PID controller is then transformed to an analog voltage signal, which is subsequently amplified by the high current driving circuit to enhance its driving capability. Ultimately it is applied to the TEC to realize the heating or cooling function.

In order to improve the control speed, integral-separated PID control was adopted in the program development. Namely a threshold value ΔT is set beforehand. When the difference between the current temperature tested in the gyroscope is higher than or equal to ΔT (0.5 °C), the PD control method will activate, otherwise the PID control mode is used instead.

Due to the lag effect of the complex thermal inertial properties and inaccurate transfer function of the inner housing of the controlled gyroscope, through lots of experimental trial, the optimal control parameters of the controller module can be decided. In this case, when the proportional coefficient _{p}_{I}_{p}T_{I}_{D}_{p}T_{D}

To validate the effects of the temperature control system, the control circuit is integrated with the microgyroscope B34 investigated previously. The thermoelectric cooler (TEC) is wall-imbedded in the outside shell of the gyroscope. During the experiments, the ambient temperature is controlled by temperature control box, which descends from room temperature (16 °C) to −30 °C before rising up to 45 °C. The actual temperature inside the gyroscope and the environmental temperature are recorded and shown in

As shown in

Because the start-up procedure needs some time and there exists constant zero bias even at the set temperature of 55 °C, the combination of the controlling and compensation methods should be jointly employed to improve the overall performance in full temperature range. Under the temperature controlling regime, the polynomial fitting compensation program is also imbedded into the C8051F360, so the compensation-control methods are working under time-sharing mode to produce nearly null zero bias in the microgyroscope. To test the temperature compensation-control effects of the system, the gyroscope prototype is placed still in the temperature control box, the ambient temperature is controlled to descend from the room temperature 16 °C to −30 °C before rising up to 45 °C. The zero bias of microgyroscope at each temperature point is recorded for sixty minutes.

The performance of a microgyroscope is greatly affected by temperature variations. In this paper the temperature testing results uncover the temperature characteristics of a microgyrocope, and validate the theoretical analysis. To improve the performance of the microgyroscope, two methods are simulated and carried out for comparison, and the experiment results demonstrate that polynominal fitting can meet the performance requirement when its term order is high enough.

In the actual developed miniaturized prototype, due to the simplicity and real-time advantage, the polynomial fitting is adopted to build the zero bias temperature model of the microgyroscope. The microprocessor compensates the actual output with the estimated value calculated by the model. The experimental results show the effectiveness of the temperature compensation system, which reduces the maximum zero bias from 12.3310°/s before compensation to 0.608°/s after compensation, which has the same order of magnitude obtained with the previous BP neural network.

In order to get the ideal working state of microgyroscope, a temperature control system is proposed to stabilize the temperature inside the casing of microgyroscope. Experimental results show that the temperature control method can effectively stabilize the temperature around 55 °C inside the integrated microgyroscope while the ambient is within −20 °C ∼ +35 °C, and the maximum stable-state error can be smaller than 0.3 °C. However, to resolve the start-up time issue, a combination of the temperature compensation and controlling methods is used to ensure the overall performance over the full temperature range. By this effective way the gyroscope and its peripheral circuits are not subject to the ambient temperature fluctuations, so the entire output of the microgyroscope can be kept at a relatively stable state. Final experimental results validate the effectiveness of the temperature compensation-control methods of the microgyroscope.

The authors gratefully acknowledge Chinese Hi-Tech Research and Development Program's financial support (Contract No. 2002AA812038).

The package and SEM photos of a microgyrosocpe.

Simulation of the relationship between Q and temperature.

Simulation of relationship between the output amplitude and the temperature.

Temperature testing schemes and setup of microgyroscope.

Trend of resonant frequency with temperature in drive mode.

Trend of quality factor change with temperature in drive mode.

Trend of resonant frequency change with temperature in sense mode.

Trend of quality factor change with temperature in sense mode.

Closed-loop drive circuit test of zero bias of microgyroscope at normal temperature.

Closed-loop test of drive amplitude of microgyroscope with temperature changes.

Trend of drive amplitude change with temperature.

Closed-loop test of zero bias of microscope with temperature changes.

Trend of zero bias change with temperature.

Structure of the BP neural network adopted for modeling.

Training of network.

BP model compensation.

Verification of BP neural networks compensation effects.

Fitted curve of zero bias of microgyroscope with temperature.

Diagram of the temperature compensation system.

Zero bias of microgyroscope before and after compensation.

Zero bias of microgyroscope after compensation.

Zero bias of microgyroscope after compensation.

Compensated zero bias of the microgyroscope.

Block diagram of temperature control system.

Gyroscope casing design and system software flow chart.

Results of controlled temperature inside the integrated microgyroscope over ambient temperature changes.

A---When ambient temperature remains at normal temperature of 16 °C, temperature control system starts working.

B---When ambient temperature remains at 16 °C, the temperature is recorded inside the gyroscope after it reaches 55 °C in 30 minutes.

C---When ambient temperature decrease from 16 °C to 0 °C, the temperature is recorded inside the gyroscope.

D---When ambient temperature is kept at 0 °C for 20 minutes, the temperature is recorded inside the gyroscope.

E---When ambient temperature decrease from 0 °C to −10 °C, the temperature is recorded inside the gyroscope.

F---When ambient temperature is kept at −10 °C for 20 minutes, the temperature is recorded inside the gyroscope.

G---When ambient temperature decrease from −10 °C to −20 °C, the temperature is recorded inside the gyroscope.

H---When ambient temperature is kept at −20 °C for 20 minutes, the temperature is recorded inside the gyroscope.

I---When ambient temperature decrease from -20 °C to −30 °C, the temperature is recorded inside the gyroscope.

J---When ambient temperature is kept at −30 °C for 20 minutes, the temperature is recorded inside the gyroscope.

K---When ambient temperature increases from −30 °C to 16 °C, the temperature is recorded inside the gyroscope.

L---When ambient temperature is kept at 16 °C for 20 minutes, the temperature is recorded inside the gyroscope.

M---When ambient temperature increase from 16 °C to 35 °C, the temperature is recorded inside the gyroscope.

N---When ambient temperature is kept at 35 °C for 20 minutes, the temperature is recorded inside the gyroscope.

O---When ambient temperature increase from 35 °C to 45 °C, the temperature is recorded inside the gyroscope.

P---When ambient temperature is kept at 45 °C for 20 minutes, the temperature is recorded inside the gyroscope.

Zero bias of Microgyrosope.

Mean zero bias of the microgyrosope under temperature compensation-control.

Testing results of resonant frequency and quality factor.

| ||||
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−40 | ||||

−30 | ||||

−20 | ||||

−10 | ||||

0 | ||||

10 | ||||

20 | ||||

30 | ||||

40 | ||||

50 | ||||

60 |

Uncompensated zero bias of microgyroscope over temperature.

80 | 52.7 | 6.01 | 15 | 88.3 | 10.06 |

75 | 52.2 | 5.95 | 10 | 94.2 | 10.74 |

70 | 53.2 | 6.06 | 5 | 98.4 | 11.21 |

65 | 53.5 | 6.10 | 0 | 102.7 | 11.71 |

60 | 52.4 | 5.97 | −5 | 103.5 | 11.80 |

55 | 50.3 | 5.73 | −10 | 105.7 | 12.05 |

50 | 51.6 | 5.88 | −15 | 108.9 | 12.41 |

45 | 54.5 | 6.21 | −20 | 109.1 | 12.43 |

40 | 60.6 | 6.91 | −25 | 108.2 | 12.33 |

35 | 68.4 | 7.79 | −30 | 107.5 | 12.25 |

30 | 74.4 | 8.48 | −35 | 105.9 | 12.07 |

25 | 79.9 | 9.11 | −40 | 98.1 | 11.18 |

20 | 85.8 | 9.77 |

Note: the scale factor of the tested microgyroscope is 8.774mV/(°/s).

Results of the compensated zero bias of microgyroscope.

80 | 0.746 | 0.085 | 15 | 1.948 | 0.222 |

75 | −1.843 | −0.210 | 10 | 2.983 | 0.340 |

70 | −1.316 | −0.150 | 5 | 2.124 | 0.242 |

65 | −2.983 | −0.399 | 0 | 3.246 | 0.369 |

60 | −5.334 | −0.608 | −5 | 1.597 | 0.182 |

55 | −3.965 | −0.452 | −10 | 1.667 | 0.190 |

50 | −1.483 | −0.169 | −15 | −2.194 | −0.250 |

45 | −1.930 | −0.219 | −20 | −3.772 | −0.429 |

40 | −2.773 | −0.316 | −25 | −3.581 | −0.408 |

35 | −2.719 | −0.309 | −30 | −2.844 | −0.324 |

30 | −1.071 | −0.122 | −35 | 1.703 | 0.194 |

25 | −1.299 | −0.148 | −40 | 0.721 | 0.082 |

20 | 0.895 | 0.102 |