Lightweight Thermal Compensation Technique for MEMS Capacitive Accelerometer Oriented to Quasi-Static Measurements
Abstract
:1. Introduction
2. The MEMS Accelerometer
2.1. Working Principle
2.2. Triaxial Accelerometer Calibration
2.3. Tilt Measurement Techniques for Accelerometers
2.4. Thermal Behavior
Thermal Compensation
- = Bias in s at 0 °C.
- = Data variation proportional to temperature in s/°C. Same as .
- = Sensibility of the acceleration in s. Same as in Equation (1).
- = Second order . It does not have much effect, since it is three orders lower than .
- = Sensitivity change because of temperature. Same as .
- = Second order non-linearity sensibility in s. Typical order of .
3. Methodology
3.1. The Accelerometer—Device under Test
3.2. Hardware—TestBench
3.3. Tests Conditions
3.4. Signal Processing
4. Results
4.1. Thermal Drift
4.2. Self-Heating
4.3. Temporal Drift
4.4. System Noise
5. Analysis
Temperature Drift of Bias and Temperature Drift of Scale Factor
6. Compensation
6.1. Thermal Drift
6.2. Self-Heating Drift
6.3. Improvement as Inclinometer
7. Discussion
7.1. Methods Comparison
7.2. Application of the Thermal Calibration Algorithm
7.3. Typical Drifts
7.4. Self-Heating
7.5. Computing Time
8. Conclusions
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
Abbreviations
MEMS | MicroElectroMechanical System |
TDB | Temperature Drift of Bias |
TDSF | Temperature Drift of Scale Factor |
MCU | Microcontroller Unit |
DUT | Device Under Test |
Appendix A
Second Order Surface | Proposed Method | Goodness of Fit | Relative Maximum | ||||
---|---|---|---|---|---|---|---|
, , | , , | TDB | TDSF | RMSE | Error | ||
Case 1 | −1.64 | −70.7 | 0.9848 | 4.904 | 0.575% | ||
Case 2 | −1.61 | −70.5 | 0.9893 | 4.045 | 0.557% | ||
354 | |||||||
Case 3 | −0.86 | −183.8 | 0.9990 | 0.663 | 0.135% | ||
Case 4 | −0.91 | −154.4 | 0.9936 | 1.789 | 0.237% | ||
Appendix A
- Place the accelerometer with a pitch and roll of approximately 35°. In this way, the accelerations in the 3 axes are about 577 mg, achieving the maximum study range for the three axes.
- Generate a temperature of 25 °C and record the acceleration measurements. This is the for each axis.
- Generate a thermal variation, for example, 20 °C.
- Use these new accelerations to compute the with Equation (11) for each axis.
- Without lowering the temperature, invert the accelerometer: pitch and roll angles at approximately −35°.
- Record acceleration.
- Generate again the temperature of 25 °C, these accelerations are .
- Use these accelerations to obtain the .
- is calculated as the difference of and divided between the difference of and (see Equation (A1)).
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Parameter | Typical Value |
---|---|
Resolution | 16 bits |
Sensitivity | 0.06 mg |
Output data rate | 3.125 Hz to 1.6 kHz |
Sensitivity change vs temperature () | 0.01%/°C |
Typical zero-g level offset accuracy | ±40 mg |
Zero-g level change vs. temperature () | ±0.5 mg/°C |
Acceleration noise density (Data Rate = 100 Hz) | 150 μg/ |
Register | Value (HEX) | Details |
---|---|---|
CTRL_REG4 (20 h) | 17 | ODR: 3.125 Hz. All axes active |
CTRL_REG5 (24 h) | C0 | Antialiasing: 50 Hz. FS: ±2 g. No self-test. 4-wire SPI |
X | Y | Z | |||||||
---|---|---|---|---|---|---|---|---|---|
Time | Avg. | Worst | C.U. | Avg. | Worst | C.U. | Avg. | Worst | C.U. |
0–1 h | 1.24 | 3.43 | 2.31 | 1.68 | 2.21 | 4.79 | 1.18 | 2.75 | 3.58 |
1–3 h | 0.39 | 1.15 | 0.35 | 0.42 | 0.89 | 0.83 | 0.46 | 1.13 | 0.63 |
3–5 h | 0.20 | 0.55 | 0.06 | 0.23 | 0.48 | 0.12 | 0.28 | 0.81 | 0.12 |
5–10 h | 0.24 | 0.34 | 0.01 | 0.16 | 0.33 | 0.10 | 0.40 | 0.58 | 0.24 |
10–20 h | 0.10 | 0.25 | 0.07 | 0.13 | 0.21 | 0.02 | 0.16 | 0.33 | 0.10 |
X | Y | Z | ||||
---|---|---|---|---|---|---|
DUT | Standard | Max | Standard | Max | Standard | Max |
Deviation | Deviation | Deviation | Deviation | Deviation | Deviation | |
1 | 0.221 | 1.704 | 0.201 | 1.305 | 0.297 | 2.031 |
2 | 0.227 | 1.619 | 0.196 | 1.357 | 0.245 | 1.802 |
3 | 0.193 | 1.333 | 0.170 | 1.245 | 0.242 | 1.736 |
4 | 0.232 | 1.699 | 0.187 | 1.292 | 0.247 | 1.859 |
5 | 0.176 | 1.240 | 0.160 | 0.980 | 0.267 | 1.592 |
Control | 0.224 | 1.678 | 0.216 | 1.445 | 0.261 | 1.699 |
TDB (mg/°C) | TDSF (ppm/°C) | |||||
---|---|---|---|---|---|---|
DUT | X | Y | Z | X | Y | Z |
1 | 1.26 | 0.76 | −1.32 | −118 | −44 | −34 |
2 | 0.09 | 0.31 | 0.12 | −42 | −103 | −188 |
3 | 0.3 | −0.22 | −0.3 | −107 | −188 | −104 |
4 | −0.44 | 0.19 | −0.61 | −128 | −277 | −60 |
5 | 0.39 | −0.08 | −0.72 | −398 | −65 | −169 |
Control | −1.18 | −0.21 | −0.81 | −134 | −46 | −226 |
X | Y | Z | |||||||
---|---|---|---|---|---|---|---|---|---|
DUT | Uncomp. | Comp. | Impr. | Uncomp. | Comp. | Impr. | Uncomp. | Comp. | Impr. |
1 | 4.66 | 0.77 | 83.48% | 2.99 | 1.91 | 36.12% | 4.49 | 0.63 | 85.97% |
2 | 0.92 | 1.04 | −13.04% | 1.56 | 0.34 | 78.21% | 0.72 | 0.77 | −6.94% |
3 | 2.49 | 1.26 | 49.40% | 2.08 | 1.51 | 27.40% | 2.52 | 1.28 | 49.21% |
4 | 1.54 | 0.44 | 71.43% | 1.78 | 1.61 | 9.55% | 2.38 | 1.08 | 54.62% |
5 | 2.28 | 0.63 | 73.68% | 0.84 | 0.77 | 8.33% | 3.47 | 1.95 | 39.77% |
Control | 5.51 | 0.69 | 87.48% | 1.96 | 1.88 | 4.08% | 3.21 | 0.68 | 78.82% |
Avg. | 2.90 | 0.80 | 72.41% | 1.87 | 1.34 | 28.46% | 2.80 | 1.09 | 61.11% |
X | Y | Z | |||||||
---|---|---|---|---|---|---|---|---|---|
DUT | Uncomp. | Comp. | Impr. | Uncomp. | Comp. | Impr. | Uncomp. | Comp. | Impr. |
1 | 17.63 | 4.63 | 73.74% | 18.51 | 13.81 | 25.39% | 18.44 | 4.26 | 76.90% |
2 | 5.41 | 6.31 | −16.64% | 6.08 | 1.92 | 68.42% | 4.07 | 3.78 | 7.13% |
3 | 11.07 | 6.26 | 43.45% | 11.28 | 8.41 | 25.44% | 11.21 | 6.25 | 44.25% |
4 | 7.12 | 3.41 | 52.11% | 11.10 | 10.66 | 3.96% | 10.19 | 4.77 | 53.19% |
5 | 9.44 | 3.48 | 63.14% | 4.96 | 4.68 | 5.65% | 13.27 | 7.47 | 43.71% |
Control | 23.74 | 4.45 | 81.26% | 10.79 | 10.39 | 3.71% | 13.06 | 3.47 | 73.43% |
Avg. | 12.40 | 4.76 | 61.64% | 10.45 | 8.31 | 20.49% | 11.71 | 5.00 | 57.29% |
X | Y | Z | |||||||
---|---|---|---|---|---|---|---|---|---|
DUT | Uncomp. | Comp. | Impr. | Uncomp. | Comp. | Impr. | Uncomp. | Comp. | Impr. |
1 | 1.636 | 0.896 | 45.2% | 1.220 | 0.760 | 37.7% | −1.572 | −0.804 | 48.9% |
2 | −0.088 | −0.112 | −27.3% | 0.344 | 0.176 | 48.8% | −0.044 | −0.008 | 81.8% |
3 | 0.280 | 0.100 | 64.3% | −0.628 | −0.520 | 17.2% | −0.244 | 0.020 | 91.8% |
4 | −0.364 | −0.108 | 70.3% | 0.136 | 0.024 | 82.4% | −1.344 | −0.956 | 28.9% |
5 | 0.652 | 0.420 | 35.6% | −0.536 | −0.476 | 11.2% | −0.724 | −0.132 | 81.8% |
Control | −0.716 | 0.052 | 92.7% | −1.268 | −1.116 | 12.0% | −1.040 | −0.636 | 38.8% |
Avg. | 0.623 | 0.281 | 54.8% | 0.689 | 0.512 | 25.7% | 0.828 | 0.426 | 48.6% |
Pitch | Roll | |||||
---|---|---|---|---|---|---|
DUT | Uncomp. | Comp. | Impr. | Uncomp. | Comp. | Impr. |
1 | 0.268 | 0.040 | 85.0% | 0.164 | 0.091 | 42.2% |
2 | 0.026 | 0.036 | −36.9% | 0.084 | 0.016 | 80.0% |
3 | 0.119 | 0.058 | 50.7% | 0.111 | 0.071 | 35.6% |
4 | 0.085 | 0.019 | 77.6% | 0.081 | 0.067 | 17.1% |
5 | 0.104 | 0.027 | 73.7% | 0.035 | 0.023 | 33.3% |
Control | 0.302 | 0.036 | 88.1% | 0.076 | 0.085 | −11.8% |
Pitch | Roll | |||||
---|---|---|---|---|---|---|
DUT | Uncomp. | Comp. | Impr. | Uncomp. | Comp. | Impr. |
1 | 0.953 | 0.194 | 79.5% | 0.824 | 0.529 | 35.7% |
2 | 0.142 | 0.173 | −21.9% | 0.304 | 0.085 | 72.0% |
3 | 0.507 | 0.274 | 45.9% | 0.539 | 0.356 | 33.9% |
4 | 0.330 | 0.110 | 66.6% | 0.393 | 0.351 | 10.6% |
5 | 0.405 | 0.140 | 65.4% | 0.158 | 0.129 | 18.1% |
Control | 1.213 | 0.177 | 85.4% | 0.381 | 0.465 | −22.0% |
MCU | Proposed Method | Second Order Surface | Third Order Curve |
---|---|---|---|
ATmega328P | 65.6 μs | 117.6 μs | 63.7 μs |
ATSAMD21G18A | 27.7 μs | 49.1 μs | 34.1 μs |
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Martínez, J.; Asiain, D.; Beltrán, J.R. Lightweight Thermal Compensation Technique for MEMS Capacitive Accelerometer Oriented to Quasi-Static Measurements. Sensors 2021, 21, 3117. https://doi.org/10.3390/s21093117
Martínez J, Asiain D, Beltrán JR. Lightweight Thermal Compensation Technique for MEMS Capacitive Accelerometer Oriented to Quasi-Static Measurements. Sensors. 2021; 21(9):3117. https://doi.org/10.3390/s21093117
Chicago/Turabian StyleMartínez, Javier, David Asiain, and José Ramón Beltrán. 2021. "Lightweight Thermal Compensation Technique for MEMS Capacitive Accelerometer Oriented to Quasi-Static Measurements" Sensors 21, no. 9: 3117. https://doi.org/10.3390/s21093117
APA StyleMartínez, J., Asiain, D., & Beltrán, J. R. (2021). Lightweight Thermal Compensation Technique for MEMS Capacitive Accelerometer Oriented to Quasi-Static Measurements. Sensors, 21(9), 3117. https://doi.org/10.3390/s21093117