International Journal of Advanced Robotic Systems A Novel Temperature Compensation Method for a MEMS Gyroscope Oriented on a Periphery Circuit Regular Paper

This paper investigates temperature compensation methods used for the scale factor and bias of the MEMS gyroscope within the temperature range from 40°C to 60°C. The structure and periphery monitor circuit are introduced. Then the determinant elements of the MEMS gyroscope’s scale factor are analysed and the results indicate that scale factor is directly proportional to drive amplitude and sense loop gain and is inversely proportional to the frequency gap between two modes. After that, the compensation methods are proposed, the thermal resistor’s positive temperature coefficient (tempco) is utilized to calibrate the scale factor’s tempco through regulating the drive mode amplitude and the sense loop gain, and each method is applied respectively and the results are contrasted. The test results of the two specimens express that the most effective compensation method could decrease the scale factor’s tempco from 693ppm/°C (640ppm/°C) to 250ppm/°C (257ppm/°C), improving it by 63.9% (59.8%). Finally, a method of declining bias’s tempco is investigated and implemented after scale factor compensation. A summator with a thermal resistor is utilized at the output level and decreases the bias’s tempco from 103.89o/h/°C (100.78o/h/°C) to 9.70o/h/°C (12o/h/°C) improving it by 90.7% (88%). Repeat tests are performed based on two specimens to prove the repeatability and reproducibility of the methods.


Introduction
MEMS gyroscope is a silicon-based sensor which can detect input angular rate information through the Coriolis acceleration. Due to the advantages of its low cost and low power consumption, micro volume, tiny weight and excellent shock survival capability, the MEMS gyroscope has been applied in many areas, such as: angular velocity measuring systems, micro inertial navigation systems, automobile security systems, consumer electronics and robot control systems (Fei et al. [1]- [2]). The architecture of the MEMS gyroscope is fabricated with silicon, which is a high temperature-sensitive material and its physical characteristics vary greatly with ambient temperature, meanwhile, the mechanical-thermal noise in the structure can also influence the gyro's performance (Leland [3]). The MEMS gyroscope's temperature performance is described in several references: Ferguson et al. [4] indicated that the drive and sense-resonant frequency have a linear relationship with temperature, and the bias's tempco is 13mV/°C over a range of 35°C to 65°C. Patel et al. [5] designed temperature experiments from -25°C to 125°C with different angular rates and repeated them 500 times; the results showed that the MEMS gyroscope has a significant shift when it works in thermal cycles with a long duration. Sun et al. [6] employed the architecture model of a linear vibrating gyroscope to analyse the influence of slow-changing temperature upon the amplitude and the phase of the drive and sense modes. Liu et al. [7] investigated the relationships between ambient temperature and the MEMS gyroscope's detection capacities, the dynamic characteristics of which are calculated based on a model. Joo [8] described the influence of temperature on the package of the gyroscope.
A large amount of literatures focuses on improving the gyroscope's temperature characteristics. The methods in these papers can be summarized as belonging to three different areas: 1. Architecture compensation and material improvement: Trusov et al. [9] presented an architecture which has a higher temperature stability and robustness. Ho [10] presented a temperature compensation method using a silicon I-shaped beam. Tsai [11] presented the structure compensation for a MEMS gyroscope with a disturbance estimator and indicated that the architecture's imperfect fabrication and asymmetry can also decrease the temperature characteristic. Cao [12] investigated how the gyroscope structure's mechanical model is affected by temperature and proposed methods to improve the silicon structure's temperature robustness. Hou et al. [13] attached two kinds of epoxy material between the architecture substrate and its package to decrease the resonant frequency and the tempco of the quality factor.
2. Software compensation: Zhu et al. [14] processed the output data with a linear compensation algorithm by using the relationship between the temperature inside the gyroscope's shell and the output data. After the compensation, the tempcos of the bias stability reduce from 229.1º/h/°C to 35.7º/h/°C and this method is fit for gyroscopes with good temperature repeatability, but which lack instantaneity. Zhang and Wang [15] eliminated the dynamic angular velocity error by using a RBF neural network system. Zhang et al. [16] used a high order polynomial to compensate the bias of a double H quartz tuning fork gyroscope on a digital signal processing platform. The variation of the bias decreases from 300mV to 0.2mV during -40°C to 80°C after the compensation. Fang [17] introduced an integrated electromechanical-thermal error model and employed a least-squares algorithm to compensate the bias drift which is caused by temperature.
3. Temperature-control: Xia et al. [18] proposed a temperature-control system to steady the ambient temperature, which utilizes a BP (Back Propagation) neural network to build the temperature model and a PID control algorithm to compensate and control the gyroscope's internal temperature. Using that method, the maximum absolute bias in the temperature range −40°C ~ +80°C is reduced to 0.608º/s from 12.331º/s, and the ambient temperature is restricted to within 0.3°C when the temperature changes from -20°C to 35°C. Lee et al. [19] investigated on-chip temperature control technology which is based on a micro thermal resister, a heater and a thermal isolate package. After on-chip temperature control, the drive resonant frequency's tempco decreases to 0.96 ppm/°C (0.22 ppm/°C with additional compensation) from 17.3 ppm/°C. This paper compensates the tempco of scale factor by changing the drive mode amplitude and sense loop gain, and employs a summator and a thermistor to decrease the bias's tempco. The method is not mentioned in previous literature and it requires that the devices have a good temperature repeatability and reliability. In consideration of the variation of the bias value after the compensation of the scale factor, the scale factor should be compensated before the bias and the process step for this is illustrated in Figure 1. Firstly, a temperature test is done to describe the scale factor's temperature curve which is the foundation of the compensation. Then, the parameter values are configured in the circuit with different compensation methods. Next, the temperature tests are arranged to verify the compensation results. Finally, the bias compensation method is implemented based on the most effective scale factor compensation circuit. The remainder of this article is organized as follows: section 2 introduces a fully decoupled double mass structure and a monitor peripheral circuit of the MEMS gyroscope, the processes of assembling and testing are mentioned in this section too. In section 3, the components influencing the scale factor are analysed and compensation methods are proposed with their experiment results. Bias compensation and tests are presented in section 4 to verify the method. Section 5 concludes the paper.

Structure of the MEMS gyroscope and periphery circuit
The gyroscope contains two parts: the silicon structure which can transform the angular rate information into Coriolis force; the other part is the monitor periphery circuit, which provides the electrostatic force to drive mode and detects the displacement of the sense mode caused by Coriolis force.

MEMS gyroscope's structure
The structure used in this paper is fully decoupled and has double mass, as shown in Figure 2.  The two masses are symmetrical and stimulated along the x axis by the electrostatic force from the drive comb electrodes. The amplitude of the drive frame is measured by the drive sense comb electrodes and the displacement caused by the Coriolis force on the y axis is detected by the sense comb electrodes. The two masses vibrate in the anti-phase mode to decrease the sensitivity of the acceleration. The slide-film damping comb can achieve a high quality factor (Q) and mechanical sensitivity. Furthermore, the push-pull drive method is employed to achieve a better drive effect. The structure is fabricated with DDSOG (Deep Dry Silicon On Glass) process technology and an overall photo and a partially enlarged drawing are shown in Figure 3 (Cao [12]).
The equivalent ideal mechanical model of a one mass system is described in Figure 4 (Fei et al. [20]). It includes drive and sense modes, and each mode is a spring-massdamping system. A fully-decoupled structure could isolate the influence between the two modes and ignore the mechanical quadrature error (Cao [12]). The movement formulas for the ideal model of the gyroscope architecture can be expressed as: where mx, my, cx, cy, kx and ky are the equivalent masses, effective damping and stiffness of the drive and sense modes; x and y are the displacement of the drive and sense frames; Fd is the drive force with amplitude AF and angular frequency ωd; mp is the Coriolis mass; Ωz is the angular rate around the z axis. Define: as the drive and sense modes' resonant angular frequencies and quality factors, usually let ωd=ωnx in order to achieve the largest amplitude of the drive mode, and assuming my≈mp, then x and y can be written as:

Metal electrode
Glass basement Silicon structure where Ax and Ay are the amplitude of x and y, then mechanical sensitivity can be found as: where the design value of ωnx and ωny are around 20000 rad/s, and the gap between them is about 700 rad/s, Qy is larger than 2000, so (4) can be simplified as: So, the structure's mechanical sensitivity is mostly decided by the drive mode's vibrating amplitude and the frequency gap between the two modes.

MEMS gyroscope's periphery circuit
As shown in Figure 5, the drive circuit is controlled by an AGC closed-loop with self-oscillation technology which can make the drive mode vibrate with constant amplitude at its resonant frequency (Cui et al. [21]). The sense circuit detects the displacement of the sense mode and processes the Coriolis signal.
In the drive loop, the drive frame displacement x(t) is detected by the drive sense combs and then linearly transformed into the voltage signal Vsdr by the X/V convertor, and then the Vsd is obtained once the Vsdr is amplified. Vsd's phase is delayed 90º to satisfy the phase requirements of the AC drive signal VdAC= VdACASin(ωdt).
After that, VdACA is picked up using a full-wave rectifier and a low pass filter, and, later, VdACA is compared with the reference voltage Vf. Next, the integrator controller generates the VdI utilizing the output of the comparator, and the drive AC signal VAC is generated by VdI modulated with VdAC. Finally, the drive DC signal VDC superposing VAC forms the force that stimulates the drive mode. In order to describe the drive loop's workings more clearly, the signal in the drive loop imitates the gyroscope drive mode from power-on to steady state and there is an initiating process in the loop. This paper employs the averaging method to analyse the drive loop's working steady state to find the determinant elements of the drive frame's amplitude.
The sense circuit is an open-loop and it employs the same interface as the drive circuit. The output signal of the amplifier Vs is demodulated with VdAC to generate Vm. After passing through a low pass filter, Vso is superposed by the bias compensation circuit (module "B") and then produces the final signal Vo. The sense loop's signal describes the response under the drive loop's steady state and has a constant angular rate signal input.

Assembling and test process
The gyroscope's silicon architecture is fixed in a vacuum ceramic package, which is fixed on the PCB board with a periphery circuit. The metal shell protects the gyroscope and restrains electromagnetic interference, just as shown in Figure 6. The power supply of gyroscope circuit is ±8V DC, and the output signal is collected by an Agilent digit multimeter, and the sample rate is 1 point per second.  The compensation component has a high-precision thermal resistance, the relationship of which to temperature can be described by the following equation: where Rt is the thermal resistance, t is the ambient temperature.

Scale factor compensation
Scale factor indicates the proportion between the output signal and the input angular rate, and it is analysed based on Figure 5 before the compensation.
As mentioned in the last section, double side push-pull stimulating technology is implemented in the drive mode along the x axis and the drive force can be expressed as: where, Cd is the capacitance formed by the drive combs on one side. Use inputs (7) to (1), expand VAC and combine with Figure 5, then get: Assuming the drive displacement is: where a(t) and φ(t) are the amplitude and phase of the drive frame's displacement, then its speed can be expressed as: According to the averaging method, there is another equation: then (12) can be simplified as: Furthermore, the acceleration of the drive frame is: By substituting (11), (14) and (15) (14) to (10) and applying the average method, considering the average value in one period (T=2π/ωnx) of (9), (10), (17) and (18): then: Let the right side of (23), (24), (25) be equal to zero, then get: So, the drive close system only has this one stable status. Under these conditions, the drive frame's vibrating amplitude a is governed by the drive mode's resonant frequency, reference voltage and drive loop gain.
The sense circuit is an open loop as shown in Figure 5, the signal after the demodulator can be expressed as: Substituting (2), (3), (5) and VdAC, it can then be expressed: After the LPF the high frequency's component is eliminated, before the compensation the output signal is: So, the scale factor can be expressed as: When the drive circuit works at its stable status, substituting AX with 0 a in (28) and letting VdACA=Vf, ωd=ωnx then: From (34), it is obvious that the scale factor has a positive coefficient relationship with Vf 2 , cos(φ y ), and sense loop gain; a negative coefficient with ω d , drive loop gain and the difference in the two modes' frequencies ∆ω=ω ny -ω d . ∆ω and ω d change very little (several Hz) during -40°C~60°C, so according to (3), cos(φ y ) can be considered as being constant when the temperature changes. Otherwise, ∆ω and ω d are difficult to change once they are fabricated, but the scale factor can be adjusted easily using the periphery circuit by varying the drive amplitude and the sense loop gain.
This paper would like to clarify that two specimens with the same structure and circuit are tested in this paper's work, and the results are reproducible. So, in order to make the paper simpler to read and understand, this paper only displays the tests curves of specimen A (quite similar to that of specimen B), and the test results of specimen B are listed at the end of the paper.  Just like the process steps mentioned in Figure 1, the scale factor values under different temperatures are tested before the compensation and these are shown in Figure 7.
Three repetitive experiments prove its repeatability and the average value is calculated using the data which indicates that the scale factor's tempco is 693ppm/°C. The red line is a linear fit (using the least square method) of the average data, which is considered to be the scale factor's temperature drift line:

Compensation using drive loop gain
As investigated in the above section, the scale factor is inversely proportional to the drive loop's pre-amplifier gain KDA (compensation point "SA" in Figure 5). Meanwhile, equation (35) shows that the scale factor has a negative tempco with KDA, so KDA should have the same tempco in order to decline the scale factor's variation. The circuit diagram of this modular is shown in Figure 8, and this circuit can achieve a high gain with an accurate phase (Graeme [22]).
The parameter names of the pre-amplifier in drive loop gain do not have dotted boxes above the components (RSt does not exist in the drive loop, so it is considered as a short cut here) and the transfer function of this modular can be expressed as: where, RDt is the thermal resistance mentioned in (6); RD1-RD4 are low tempco constant resistances. Configure the appropriate values of the resistances so as to make KDA have the same tempco as the scale factor and the temperature test results are shown in Figure 9. The scale factor's tempco is reduced to 257ppm/°C and three repetitive tests are utilized to verify the reliability of this compensation method.

Compensation through sense loop gain
The sense loop gain KSA is proved to influence the scale factor in (34), so the tempco of the scale factor can be declined by changing KSA's value. Since the scale factor has a negative tempco, KSA is expected to have a positive one to compensate it. The circuit in this modular ("SB" in Figure 5) is the same as the pre-amplifier in the drive circuit as shown in Figure 8. The parameter names have dotted boxes under the components (RDt is considered short cut here). So, KSA is governed by the equation: where, RS1-RS4 are precise resistances with a constant value and tiny tempco; RSt is the thermal resistance satisfied with equation (6). Adjust the parameters so that KSA has an appropriate tempco to compensate the scale factor in Figure 7. Temperature experiments are done and shown in Figure 8, the average data is calculated (the red dot line), which shows that the scale factor tempco is reduced to 250ppm/°C.
The results of these two compensation methods are nearly the same, and this paper chooses the sense loop gain compensation method (SB) to continue the bias compensation.

Bias compensation
Bias is the gyroscope's output signal, which is picked up by the navigation system directly, so bias stability and temperature character are some of the most crucial parameters for a gyroscope. According to the process flow in Figure 1, the temperature curve of the bias is obtained after scale factor compensation, described in Figure 11. The least square method is utilized to evaluate the relationship between temperature and bias (before bias compensation, the modular "B" does not exist): where, Vo is the output bias; Vso is the output of the low pass filter; Kso=-5.801mV/°C and bso=751.19mV are the Vo's tempco and 0°C value. Bias compensation is based on a summator with a thermal resister in the output module of the sense circuit (compensation point B in Figure 5), the schematic is illustrated in Figure 12. This module contains two parts: one is for temperature compensation and the other part corrects the bias' average value. The final output signal Vo is governed by: where, VCC and VDb are the reference voltage; Rbt is a thermal resistance described in (6); RS5-RS7 are constant value resistors. The parameters are calculated to compensate Kso and bso in (38) and the test results are depicted in Figure 13. The variation in the whole temperature range is reduced from 29.58 mV(103.89º/h/°C) to 2.80 mV(9.70 º/h/°C), three repetitive tests curves prove this compensation method's repeatability is exceedingly good (the red dot line is their average value).

Conclusion
This article investigates and summarizes the MEMS gyroscope temperature compensation methods proposed in the existing literature and presents a novel idea for compensating the scale factor and bias's tempco utilizing a thermal resistor based on the periphery analogue circuit over the range of -40°C to 60°C. The gyroscope's architecture is a symmetric dual proof mass with fullydecoupled, periphery circuits containing a drive selfoscillating closed-loop and sense open-loop. The scale factor is compensated in the first stage, a kind of high precision thermal resistor Rt is configured in a drive and a sense loop gain to calibrate the scale factor's tempco. The second stage is to employ a summator to compensate the bias's tempco and value. In order to verify the repeatability of the method, three identical experiments are arranged for every stage. Two specimens are tested and the results verify the compensation methods have excellent reproducibility. The crucial result of before and after compensation is contrasted in  [14] and [18]. With this method, the volume and power consumption of the circuit stays almost the same, meanwhile the compensation circuit is simple and is highly reliable (depending on the high capability thermal resistance).