2013 Seventh International Conference on Sensing Technology
Scale Factor in MEMS Gyroscopes
The Effect of Power Supply Voltage
M. Vágner, P. Beneš
Faculty of Electrical Engineering and Communication
Brno University of Technology
Brno, Czech Republic
[email protected], [email protected]
the design becomes very important, and the navigation will
require information about the reliability of the position.
Abstract—This paper discusses the behavior of MEMS
gyroscopes during power supply fluctuations, which is a problem
that has not been sufficiently analyzed to date. The focus is
placed on the scale factor. In the opening section, the authors
present the basic output configurations of the MEMS gyroscopes
and propose their general models. The following part of the
article has a practical character. Here, eight types of the abovedefined gyroscopes are examined to demonstrate the influence of
the applied supply voltage. Firstly, the measurement procedure is
described, and subsequently the results of this experiment are
presented. The outcome of the performed research consists in
that the scale factor error is smaller than 1% if the power supply
voltage fluctuates within the range of ±0.25 V around the
nominal value.
Not much data is available on the sensitivity of MEMS
gyroscopes to power supply fluctuations. Typically, there are
borders for the zero rate level and the scale factor; this range is
nevertheless very wide and contains more sources of errors.
The shape and repeatability of the transfer function are
completely unknown. Moreover, the method for the
measurement of this property is not clearly defined.
In our previously published paper [7], we investigated the
zero rate drift and the internal temperature sensor bias drift
that are caused by the variations of power supply voltage. In
this context, we also suggested a suitable measurement
procedure and performed tests on different types of
gyroscopes. The related analysis showed that both the zero
rate level and the temperature sensor bias drift are rather
sensitive to the supply voltage level. Our samples indicated
the zero rate drift up to 6 deg/s/V in the absolute value and up
to 4 %/V of the full scale range. In most cases, this error can
be sufficiently described by a linear function because a
polynomial of the second order brings an improvement of only
about 1%.
Keywords-gyroscope; model; scale factor; offset; scale factor
supply drift; zero rate supply drift; power supply voltage; singleended; ratiometric; digital.
I.
INTRODUCTION
The popularity of micro-machined gyroscopes stems from
the fact that they are small, light-weight, low-power, and
cheap. The gyroscopes are thus suitable for portable
applications such as cell phones, game controllers, car
stabilization systems, personal transportation systems
(Segway), wearable input devices [1], and hand gesture
recognizers [2]. MEMS gyroscopes seem to be a very
interesting solution for unmanned vehicles; however, their use
for inertial navigation [3] or north seeking [4] is a challenging
problem because of their insufficient performance.
Another important parameter of a gyroscope is the scale
factor (sensitivity). In a MEMS gyroscope, the scale factor is
probably susceptible also to the voltage level, but there is
almost no information available on this phenomenon. Thus,
we intend to check the real behavior on different gyroscopes
and extend the current model.
Therefore, if the best performance is required, a model of
the gyroscope must be incorporated in the system. The
primary role of the model is to compensate for errors.
Although deterministic errors (such as nonlinearity) can be
corrected by calibration, MEMS gyroscopes are rather
sensitive to other quantities, for example temperature. In this
case, the input value must be known (measured) to enable
compensation. Then there are also stochastic errors, which are
difficult to remove. In some instances, they can be reduced
using techniques such as Kalman filtering [5] or wavelet
denoising [6]; however, the model of the noise must be
known, and it is also necessary to consider a large number of
limiting conditions.
In strapdown inertial navigation systems (INS), the error
caused by an inaccurate scale factor is less important than the
constant bias. The reason lies in the fact that the attitude error
is a function of the angular rate in this case; thus, it depends
on the motion characteristics. Because the error has a
multiplicative character [8], there is none at the steady state;
the error thus grows with the angular rate and is more
significant for curved trajectories.
The scale factor is defined on several levels. Firstly, we
have to consider the mechanical properties of the sensing
element, for example the stiffness of the springs or beams and
the weight of the moving mass. This aspect of the scale factor
is referred to as intrinsic sensitivity, and it is usually not
defined in a datasheet. Secondly, the deflection or mechanical
stress produced by the Coriolis force can be sensed via
different principles. The most commonly used transducers are
If there is no possibility of compensation, the model can be
used to estimate the worst case scenario. Then the aspect of
The presented research was supported by the European Regional
Development Fund under project No. CZ.1.05/2.1.00/01.0014, and the authors
also received related financial assistance from the Internal Grant Foundation
of Brno University of Technology (grant No. FEKT-S-11-6).
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2013 Seventh International Conference on Sensing Technology
G r, x (u) (u / u 0 x / x 0 ) ˜100 capacitive and piezoresistive. So far, there has been no reason
for the scale factor to be voltage-dependent. However, a signal
from the transducer is processed by an electronic interface,
and the moving mass is driven using a feedback loop; the
output scale factor is therefore strongly determined by the
electronic gain [9], [10].
where Gr,X(u ) is the ratiometric error [%] of the parameter x at
the voltage u against the reference values u0 and x0.
MEMS gyroscopes can be equipped with different output
configurations, namely analog and digital. The analog output
is usually single-ended with an internal voltage reference or
ratiometric. In the second group, the most common interfaces
are the SPI and I2C.
Figure 1. Interfacing the single-ended output.
A. Single-ended Output
Usually, there is an internal voltage reference that defines
the zero rate level in this configuration. The sensors within
this class are generally non-ratiometric, which means that the
sensitivity should be a constant value independent of the
supply voltage level. The output of the sensor can be described
as follows:
~ K ˜ ω b ω
When the same reference voltage is used for both the
sensor and the ADC (Fig. 2), then any error in the reference
voltage is automatically compensated. Therefore, the output
value from the ADC should be independent of the reference
voltage:
>K0 KS (u)@˜ ω bS (u) b0 where K0 and b0 are the constant portions related to nominal
conditions. The functions Ks and bs represent the variability of
these parameters according to the supply voltage u.
>K0 KS (u)˜ ω bS (u) b0 @˜ 2N / u 0 TEST SETUP
In comparison with the first experiment [7], we
significantly improved the measurement chain; the related
diagram is indicated in Fig. 3. The DUT was rigidly fastened
to the top of a RMS SDL1401 rate table system, which is
equipped with an CTS T-50/60 temperature chamber.
The sensitive axis of the sensor was aligned with the output
axis of the rate table, which is perpendicular to the ground.
Moreover, the rotation axis passes through the center of the
DUT; thus, there is no centrifugal acceleration that could
produce the bias drift. The rate accuracy is better than 0,001%
in this configuration.
where the scale factor K0 and the zero rate level b0 are related
to the nominal voltage u0.
The difference between the actual change in the offset or
sensitivity and the ideal state is referred to as the ratiometric
error, and it can be evaluated by the equation
Figure 2. Interfacing the ratiometric single-ended output.
978-1-4673-5221-5/13/$31.00 ©2013 IEEE
Despite the benefits, the ratiometric output is used only in
gyroscopes made by Analog Devices and accelerometers
produced by Kionix and Murata.
II.
B. Ratiometric Single-ended Output
As outlined above, the second option is the ratiometric
output, which was introduced in patents [11], [12], and [13] to
suppress power supply voltage fluctuations. In this instance,
both the zero rate level and the scale factor should be ideally
proportional to the reference voltage. This condition is
described by the following formula:
~ (u) (K ˜ ω b ) ˜ u / u ω
0
0
0
ω̂(u)
C. Digital Output
The best option seems to consist in integrating an ADC
inside the sensor because the reference is internally shared and
all components are exposed to the same temperature.
However, if we need to compensate for the effects of the
supply drift, it can be difficult to assure the simultaneous
sampling.
Although this type of output is the most common one, it
may be rather more susceptible to power supply fluctuations
because there is no shared reference voltage for the sensor and
the A/D converter. If the reference voltage is externally
available, then a differential DAC is an advantage as this
configuration can suppress the reference drift. In other cases, a
single-ended DAC is sufficient but constitutes the worst
scenario. Both these possibilities are shown in Fig. 1.
where N is the resolution in bits. As (5) is similar to (1), the
model that considers the ratiometric error can be expressed as
follows:
where ω is the angular rate, K is the scale factor, b is the zero
rate level, and ω represents the output value. If we assume
that the parameters are sensitive to the supply voltage, then the
model can be written as
~ (u)
ω
ω̂ (K 0 ˜ ω b0 ) ˜ 2 N / u 0 248
2013 Seventh International Conference on Sensing Technology
Figure 3. The measurement chain.
The temperature inside the chamber was maintained at the
constant value of 25°C during the experiment, and the
temperature stability corresponded to ±0.5°C.
Another possible solution consists in a different data
acquisition unit. We replaced the digital multimeters and
power supply with a PXI system. The gyroscope was powered
by a PXI-4130 programmable power supply, which includes
the remote sensing feature used to eliminate voltage drop
across the power cable and the slip-ring unit.
Figure 4. Instruments and a DUT
Finally, three independent trial tests were made for each
sensor to estimate the uncertainty. The uncertainty is mostly
caused by the noise and the temperature variation. The noise
consists of the following components: angle random walk,
bias instability, and rate random walk. The angle random walk
is usually described as white noise. Thus, it can be reduced by
averaging, and for this reason the supply pattern is repeated
several times. The other stochastic errors are difficult to
compensate as they have the character of correlated noise. The
temperature variation affects the zero rate level and the scale
factor. We neglected these errors because, during the tests, the
actual variation was not only smaller than ±0.25°C but also
periodical.
The analog outputs were captured using a PXI-4462 fourchannel dynamic signal analyzer. The resolution of the device
is 24 bits, and analog and digital anti-aliasing filters are
embedded. Thus, the signals were sampled at 2 kHz to enable
the best performance, and the simultaneous sampling
prevented the occurrence of time shift between the input
channels.
The digital interface was based on a PXI-7854R
multifunction module, which is equipped with an FPGA. The
communication protocol was coded in the FPGA; therefore,
the requirements regarding the timing and synchronization
could be easily satisfied. As there are different logical levels
between the sensors and the I/O module, an additional level
translator was used. This also solved the capacitive load
problem because the outputs of a sensor can drive only a small
load; however, the cable was considerably long.
III.
Within the first step, the zero rate level and the
corresponding supply drift were estimated. All values were
defined for the nominal supply voltage. These results are
summarized in Tab. I. In this article, a new model (6) for the
ratiometric sensors is used; therefore, the zero rate supply drift
is not entirely comparable with our previous results.
Moreover, the values are expressed directly in the output
quantity because it is independent of the scale factor.
However, the results can be recalculated in the range of the
uncertainty; therefore, our first results were proved to be
reliable.
The timing of the experiment was almost the same as
described in [1]. In the opening phase, we provided for a delay
to heat up the sensor. This stage was necessary for the internal
temperature to reach a steady state; importantly, we extended
this phase because of the more complex fixture to the rate
table. Thus, the actual length was based on the temperature
gradient. The following stage included thirty periods of the
supply pattern to facilitate better uncertainty. Each step of the
supply pattern comprised a 100 ms delay before the recording
started. Although this delay was essential to stabilize the
power supply voltage, we reduced it because the new power
supply provided a better transient response. Finally, the
acquisition stage took 100 ms.
The ADXRS642 and ADXRS649 gyroscopes were
represented by three samples, and each sample provided a
different result. Thus, the zero rate level drift is individual in
each item and cannot be generalized for one type.
The “zig-zag” pattern [7], which alternates with a short
period around the mean value, was used instead of the
common stair-like pattern because it helps us to avoid the selfheating problem.
The smallest zero rate drift was detected in the examined
ADXRS450 gyroscope, which is equipped with a 16-bit ADC.
The output is almost insensitive to the supply voltage because
the value is smaller than the noise and probably even smaller
than the resolution of the converter.
These steps were repeated for twenty angular rates
uniformly distributed in the full-scale range of each sensor to
reveal the scale factor behavior.
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RESULTS
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2013 Seventh International Conference on Sensing Technology
TABLE I.
ZERO RATE LEVEL
Zero Rate Level
Out.
type
Supply
[V]
Offset
[V]
Supply drift
[mV/V]
LISY300AL
Fig. 1a
3.3
1.634 ± 0.007
-17 ± 3
MLX90609N2
Fig. 1b
5
0.035 ± 0.008
-154.3 ± 1.3
ADXRS300
Fig. 1b
5
-0.108 ± 0.002
-14.9 ± 0.4
ADXRS610
Fig. 2
5
2.487 ± 0.003
0.3 ± 0.5
ADXRS613
Fig. 2
5
2.417 ± 0.004
-2.6 ± 0.3
ADXRS642 (0)
Fig. 2
5
2.5094 ± 0.002
26.1 ± 0.3
ADXRS642 (1)
Fig. 2
5
2.531 ± 0.002
16.6 ± 0.2
ADXRS642 (2)
Fig. 2
5
2.479 ± 0.002
-7.9 ± 0.3
ADXRS649 (0)
Fig. 2
5
2.4979 ± 0.0003
4.17 ± 0.05
ADXRS649 (1)
Fig. 2
5
2.5063 ± 0.0004
3.39 ± 0.08
ADXRS649 (2)
Fig. 2
5
2.5093 ± 0.0003
5.31 ± 0.05
[LSB]
[LSB/V]
17 ± 11
0±2
Gyroscope
ADXRS450
SPI
5
eb
b̂S ˜ 'u / K 0 ˜ t
eK
17 ˜ 0.01 / 3.193 ˜ 100
K̂ S ˜ 'u / K 0 ˜ ω ˜ t
0.092 ˜ 0.01 / 3.193 ˜ 300 ˜ 100
VN
N˜ t
0.01 ˜ 100
0.1q As a matter of fact, all characteristics are slightly curved.
This trend can be completely obviated using a polynomial of
the second order; however, a linear approximation provides an
error smaller than 0.1%, which means that the benefit is
questionable.
In the performed comparison of ratiometric and nonratiometric sensors, none of the versions was found to be
significantly more beneficial than its counterpart. This
statement is obviously valid only when the ratiometric sensor
is interfaced according to Fig. 2. If there is no common
reference voltage for the sensor and the ADC, the error will be
approximately 20 %/V.
Figure 5. The non-ratiometric sensors.
To emphasize the influence of the scale factor drift, the
following example can be considered. Let us have a
LISY300AL gyroscope with the these parameters: the zero
rate level drift of b̂S -17 mV/V , the scale factor drift of
,
the
angle
random
walk
of
N 0.01 q / s , the angular rate of 300 °/s, and the voltage
difference of 'u = 0.01 V. After the interval of 100 s, we
obtain the following errors:
Figure 6. The ratiometric sensors.
978-1-4673-5221-5/13/$31.00 ©2013 IEEE
The angle random walk error is usually considered very
important in INS applications. The distribution of the endpoints exhibits the standard deviation of 0.1° in this case.
However, the error caused by the zero rate drift is about fifty
times higher, and this indicates that the stability of the power
supply is a very important parameter.
It is apparent that all examined sensors exhibit the scale
factor error of less than 1% in the voltage range of ±0.25 V
around the nominal value. The worst case is the LISY300AL,
whose scale factor drift corresponds to 2.9 %/V. By contrast,
the best result is provided by the ADXRS613. The
characteristic cannot be described by a linear function, but the
error is under 0.1 % in the whole range.
0.092 mV/ q / s/V
8.6q
The largest error is caused by the scale factor drift;
however, it is very unlikely that the rotation will be at the
maximum rate of the sensor during the entire time. In the
course of a typical motion, the angular rate is distributed
around the zero. Therefore, at a given time, the error is
proportional to the mean value of the angular rate from the
beginning of the integration period.
The scale factor was estimated using the least squares
method as the slope of the line fit to the input-output data
separately for each power supply level. The relative error as a
function of voltage difference is presented in Fig. 5 (for nonratiometric sensors) and Fig. 6 (for ratiometric sensors). The
typical supply voltage was used as the reference value. The
coefficients are summarized in Tab. II.
K̂ S
5.3q 250
2013 Seventh International Conference on Sensing Technology
TABLE II.
SCALE FACTOR
By contrast, the scale factor drift can be usually neglected
because the related error is proportional to the angular rate,
whose mean value is typically close to the zero. Thus, the
contribution is also very small.
Scale factor
Gyroscope
LISY300AL
MLX90609N2
ADXRS300
Typical
[mV/°/s]
Actual
[mV/°/s]
Supply drift
[μV/°/s/V]
3.3
3.193 ± 0.008
92 ± 2
26.67
26.604 ± 0.007
410 ± 5
5
4.9956 ± 0.0011
19.9 ± 0.4
ADXRS610
6
5.827 ± 0.002
-41.4 ± 0.3
ADXRS613
12.5
12.204 ± 0.003
-
ADXRS642 (0)
7
7.2102 ± 0.0013
-73.4 ± 0.5
ADXRS642 (1)
7
7.099 ± 0.003
-72.0 ± 0.5
ADXRS642 (2)
7
7.0663 ± 0.0010
-74.3 ± 0.8
ADXRS649 (0)
0.1
0.0983 ± 0.0003
-0.55 ± 0.08
ADXRS649 (1)
0.1
0.0984 ± 0.0002
-0.92 ± 0.08
ADXRS649 (2)
ADXRS450
0.1
0.1011 ± 0.0003
-1.05 ± 0.08
[LSB/°/s]
[LSB/deg/s]
[LSB/°/s/V]
80
79.98 ± 0.09
0.359 ± 0.010
IV.
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CONCLUSION
Our aim was to determine and verify how power supply
voltage affects the parameters of a low-cost MEMS
gyroscope. For this purpose, we set up an applicable
measurement method and examined several suitable samples.
[7]
[8]
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