Article citation information:
Matyja, T.,
Kubik, A., Stanik, Z. The MEMS-based barometric altimeter inaccuracy
and drift phenomenon. Scientific Journal
of Silesian University of Technology. Series Transport. 2022, 116, 141-162. ISSN: 0209-3324. DOI: https://doi.org/10.20858/sjsutst.2022.116.9.
Tomasz MATYJA[1], Andrzej
KUBIK[2], Zbigniew
STANIK[3]
THE MEMS-BASED BAROMETRIC ALTIMETER INACCURACY
AND DRIFT PHENOMENON
Keywords: barometric altimeter, MEMS, error correction, drift
phenomenon, bicycle computer
1. INTRODUCTION
In common opinion, a
barometric altimeter is a device used in aviation and parachute jumping, as
well as in mountain climbing. However, the uses of the barometric altimeter
extend well beyond these areas. This is due to the availability and low price
of atmospheric pressure sensors made in the MEMS technology. Today, increasingly, electronic devices are
equipped with a barometric altimeter module, which usually works with the GPS
system. Thanks to this, it is possible to simultaneously record not only the GPS
coordinates of the route but also its vertical profile. Altimeter systems can
be found, among others, in smartphones, watches and sports bands, as well as in
bicycle computers and devices for tracking and recording the movement of motor
vehicles. It is commonly used in drone traffic control and control systems and
aviation modeling.
The accuracy of altitude
measurements with a barometric altimeter is very important in aviation. This
also applies to flights performed by drones (generally UAH). Indeed, instruments used in
aviation are usually several classes more accurate than those used, for
example, to record routes. For overland travel, the accuracy of the route
altitude measurement has no effect on safety but may be significant if the purpose of the tests
is to assess the energy needed to travel the route.
This paper discusses the sources and types of errors that
occur during altitude measurements with the use of a barometric altimeter. In
general, altimeter errors are classified into a fundamental error caused by
non-standard sea level conditions: error caused by external conditions, for
example, the
movement of the pressure sensor to air, and the error resulting from the
barometer drift. The authors proposed formulas to estimate the fundamental and
external errors. As part of the numerical experiment, data on the
altitudes of the route recorded by a bicycle computer were analyzed. These data were compared
with the altitudes of the route obtained based on the numerical terrain
model, which was considered a reference, and at the same time, accurate. The
calculations and simulations were performed in the Matlab environment.
The use of the route
recorder on the land was beneficial as it allowed obtaining information about the
real altitude; however, on the other hand, it was associated with
additional factors that disturb and distort the measurement. It should be noted
that a serious problem when measuring pressure while moving near the ground
surface is the significant changes in temperature caused by variable
insolation and thermal radiation. Various elements of road infrastructure and
nearby buildings heat up strongly and disturb the temperature distribution. The
temperature of the asphalt is usually higher than the perceived air
temperature, which is forecast in the shade and at a height of 2 m from the
ground level. The pressure sensor on the cycle computer is approximately
halfway up.
2. BAROMETRIC ALTIMETER
ERRORS
The
barometric altimeter converts the information about pressure into information
about the geopotential altitude above sea
level based on
the well-known formula [1, 3] (Appendix - markings and formulas (A1) - (A5)):
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(1) |
In formula (1),
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(2) |
where:
The
basic element of the barometric altimeter in MEMS technology is a closed
aneroid capsule with air under standard pressure. The flexible membrane of the
capsule deforms under the influence of external pressure. The capsule material
has a defined internal damping and the rate of deformation is not in phase with
the deformation itself. The operating temperature of the sensor is also of great importance. Due to this
mechanical element, each pressure sensor has a certain inertia and a response
delay. Hence, barometers
made in MEMS technology can operate on different physical principles for
measuring deformation. Piezoresistive strain gauge sensor measures changes in the electrical
resistance of resistors mounted on a diaphragm. Capacitive pressure sensor
measures changes in
electrical capacitance caused by the movement of a diaphragm. In piezoelectric
sensors, special
materials, that is, quartz crystals or ceramics,
generate a charge when pressure is applied [5].
An example of a typical digital barometric pressure
transmitter is the Bosch BMP180 sensor based on piezo-resistive technology. The
manufacturer states that his device can operate in the pressure range from 300 to 1100 hPa with a maximum
pressure resolution of 0.01 hPa. The noise
in the high-resolution mode is not less than 0.02 hPa. Figure
1 shows the resolution of the altitude measurement as a function of the pressure
measurement resolution, assuming standard atmosphere conditions. It is visible
at altitudes of up to 2,000 m above sea level, and the
resolution is from 17 to 20 cm. Presently,
it is difficult to find a
mass-produced sensor with higher resolution in the market.
The
measurement error of the barometric altimeter is its characteristic and is
inevitable. A well-known phenomenon is the
altimeter drift, which is manifested, inter alia, by a continuous
and slow change of indications under constant pressure and temperature
conditions [14]. Hence, in the literature, there are various attempts to
describe this error and various methods of correcting it. Usually, a barometric altimeter is
combined with a GPS system. Altitude measurement with the use of GPS is burdened
with a significant error and the obtained data without additional information
are of little use. After the connection, both systems complement each other,
and the appropriate algorithm, based on the Kalman filter [7, 10] or other
methods [13], generates the most probable altitude. Another proposed method is
to correct the altimeter indications with the data from the accelerometer and
the gyro sensors in combination with the GPS data [8].
Fig. 1. Altitude resolution
corresponding to 0.01 hPa
and 0.02 hPa pressure
resolution
The authors [9] proposed a stochastic approach to
the altimeter error modeling, distinguishing long and short-term errors. In
[14], the error of the geopotential
height determined by the barometric altimeter was divided into three
components:
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(3) |
where:
The
principal error
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(4) |
In
[14], a similar relationship was
obtained by expanding the functions of two variables
The
principal error is a function of the measured height. It can also be expressed
in the form of dependence on pressure and temperature increases:
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(5) |
where:
The
results of the calculation of the basic error coefficients (Figure 2) suggest the possibility of
neglecting the influence of the last factor related to the product
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(5a) |
Diston
[3], proposed a formula correcting
the altitude that considers
only the temperature difference
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(6) |
Interestingly,
this formula is formally valid for all layers of the standard atmosphere (ISA).
The derivation of formula (6) consists in comparing the two differential
equations of the hydrostatic
equilibrium of the standard and non-standard (for example, warmer) atmosphere. The assumption about the
equality of pressure differentials in these equations seems problematic. In
equation (6), the value of the measured pressure
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(7) |
Fig. 2. Values of the gross error
coefficients as a function of geopotential height
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Fig. 3. Principal error in selected
non-standard conditions and effects of corrective formulas
It
can be assumed that the maximum deviations from the standard temperature are
within limits
Many potential external factors can interfere with pressure sensor
measurements (temperature, vibration, air movement, etc.). One of the
relatively easily measurable causes of the appearance of external disturbance
error
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(8) |
where:
Considering the changes in
density with a change in height leads to a slightly different, surprisingly
simple, correction formula (derivation in the Appendix - formulas (A9) -
(A15)):
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(9) |
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Fig. 4. Numerical solutions and
approximate formula for calculations
The
approximation (9) was obtained based
on the exact dependence (Appendix - formula (A12)), which can be solved
numerically in terms of temperature, and then on its basis, the height error can be calculated. The numerical
solution is a function of speed and pressure (altitude). Figure 4 shows the error
For
comparison, Figure
5 shows the error
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Fig. 5. External disturbance error
Drift
error is hard to describe theoretically. It is a periodically changing signal
correlated with slow changes in pressure that may be the result of weather
phenomena. Drift also occurs with long-term measurements, and the drift trend
may also change over time. The drift signal is non-stationary, but its first
derivative shows white noise characteristics. Drift error can be modeled as a
stochastic random walk process:
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(10) |
where:
3. NUMERICAL EXPERIMENT
Data recorded by the bicycle computer during the route shown in Figure 6 were analyzed. The
journey along the route with a total length of 120 km took just over 5 hours.
On that day, the forecast temperature was 29oC, and the pressure was
around 1010 hPa, there was a weak wind from the west, the influence of which on
the course of the experiment was ignored.
According to the
manufacturer of the bicycle computer, the barometric altimeter is initiated
when data recording is started and then adjusted with the data from the GPS system.
The resolution of data
recording with an altitude is 0.2 m, which corresponds to the resolution
of typical pressure sensors in the MEMS technology. The recording rate for all
data (including GPS coordinates, altitude and temperature) is 1 Hz.
Recorded GPS coordinates
were projected onto the map plane in rectangular coordinates using the PUWG
1992 (EPSG2180) reference system [11]. Based on the transformed coordinates of
the route, it is possible to determine map sheets on the scale of 1: 5000,
which correspond to the digital terrain model (DTM) data sets. These collections
are currently available to the public and can be downloaded from the website
(https://mapy.geoportal.gov.pl/). According to the information on the website,
the height data was obtained through laser scanning (LIDAR) with a
resolution of a regular grid of 1 m x 1 m and a declared accuracy
of 0.2 m. The data is in text format and requires transformation
to matrix form to facilitate computer processing. Files only contain points
belonging to the area on the surface of the ellipsoid. Hence, after the projection
onto the map plane, the effect of missing information appears on some edges of
the rectangular area. As the sheets partially overlap, it is possible to
supplement data from adjacent sheets.
Fig. 6. Recorded route
Fig. 7. Determination of height
map sheets based on route coordinates
Knowing the trajectory of
the bicycle's movement, it is possible to designate the identification number of
the required map sheets and the associated DTM data. The automatic procedure for
searching for the necessary sheets is illustrated in Figure 7. For example, the first sheet
has an identification number: M-34-51-C-c-2-4. Based on the obtained set of the
necessary sheets, the appropriate files with the DTM data should be downloaded.
It is the minimum set. Due to the above-mentioned feature of
the DTM data,
there may be a need to download additional adjacent sheets to complete the data
at the edges.
The height maps of
selected exemplary sheets are shown in Figure 8. Preliminary analysis
of the DTM data shows (Figure 9) that in some cases, the data appears to be
completely raw, and in others, it appears to be averaged. Perhaps
this is because the DTM library was created gradually and at different
times, probably with the use of measuring equipment with different technical
parameters.
The position of each point
on the trajectory in rectangular coordinates can now be referenced to the
regular grid DTW, which allows the height to be calculated using Lagrange
interpolation. Due to the aforementioned noise of the DTM data, the received
height signal is averaged by the median filter of length eight.
Figure 10 shows the route
heights recorded by the bicycle computer and obtained from the DTM. As seen, the differences are quite large,
although the trend of changes is similar. Treating the data from the DTM as
accurate, the "real" error of the altitude measurement by the barometric altimeter
installed in the bicycle computer can be determined.
An attempt was made to
reconstruct the basic error and the error caused by the movement of the
pressure sensor to isolate the drift. The speed at which the bicycle was moving
was assumed as the relative speed of the sensor and the air, disregarding the
influence of the wind. The speed data recorded by the computer has been
dispensed with. Instead, they were determined from the route coordinates by
subjecting the obtained signal to smoothing with the median filter (Figure 11). The pressure pickup
slots are located on the underside of the cycle computer, where there is
positive pressure. As they are closer to the trailing edge, the dynamic
pressure will be lower than on the leading edge.
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Fig. 8. Sample height maps
based on data for sheets 1 and 5
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Fig. 9. Smaller fragments of
the surface (sheets 1 and 5)
Fig. 10. The altitude of the
route recorded and determined from the DTM
Fig. 11. The speed of the
bicycle with a pressure sensor is determined from GPS coordinates
It was assumed that the
sum of the measured height
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(11) |
The drift will then be
equal to:
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(12) |
The least squares method
and the minimization of the functions with constraints were used to reconstruct
the non-standard conditions and the impact of speed:
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(13) |
where:
Note that for about a
minute, the cycling computer recorded zero altitude, presumably getting a GPS
reference altitude during this time. This part of the signal was omitted.
Restrictions have been
assumed for the variables
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(14) |
The following results were
obtained:
Fig. 12. Total height error and
estimated error
The barometer drift
obtained with the proposed method is quite large and clearly increases during
both uphill
and downhill slopes. The derivative signal from drift fulfills in part the characteristic features of
white noise. However, there are peaks in it, which may be caused by the
altitude adjustment with the GPS signal by the internal algorithm of the
cycling computer. However, this cannot be checked, as the bicycle computer is a
black box for the user in terms of data processing algorithms.
The effect of correcting
the height measured with the estimated error is shown in Figure 14. It is not
satisfactory. Probably, due to the long journey time, constant sea level
conditions cannot be assumed. This is suggested by the temperature profile
measured by the computer during the trip (Figure 10). The thermal radiation of the hot asphalt
probably had a great influence on the temperature measurement.
The method proposed above
was used for shorter 30-minute sections of height measurement. In this case,
the results of the correction turned out to be much better, which means that
the increments of
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(15) |
where:
Fig. 13. Barometer drift and
time drift derivative
Fig. 14. Effect of correcting
the altitude with the estimated error
Fig. 15. The effect of
correcting the altitude assuming the variability of
temperature and pressure over time
Fig. 16. Comparison of the
total error and the error estimated with the assumption of temperature and pressure
variability over time
The defined optimization
problem (15) requires a bit more computational effort compared to (13).
Additionally, in this case, the gradient of the function cannot
be easily calculated. Even so, the standard fmincon
function from the Matlab environment library easily found the local minimum.
The effect of correcting the height after the applied changes is shown in Figure 15. It is definitely
better than before, which is also visible when comparing height errors (Figures 16). The drift amplitude
also decreased (Figure 17). The values of pressure and temperature correcting
the fundamental error, interpolated from the optimization results, are shown in
Figure
18. The coefficient of the influence of the sensor speed on the altitude error
was
Fig. 17. Drift error (assuming
temperature and pressure variability over time)
Fig. 18. Changes in pressure
and temperature correcting the fundamental error
4. DISCUSSION OF RESULTS
The
approximate formula for correcting the fundamental error (4) derived in this paper is basically only a more
accurate version of the formula given in the literature [14]. A different
derivation technique has been used,
which, however, leads to the same goal. Formula (4) differs by an additional
non-linear factor,
which, as shown by numerical simulations, is much smaller than the linear
factors (Figure
2). At the same time, the simulation showed that the height correction using
formula (4) is very effective (Figure
3). If for some reason, greater accuracy of the fundamental error correction
would be needed, a different, more precise expansion into a series of
exponential functions in the exact formula (A7) can be used.
Formula (6) correcting the height only
considering the
deviation from the standard temperature [3] is shown to be much less effective
(Figure 3). Its
possible advantage is that it can be used in all layers of the standard
atmosphere, practically up to an altitude of 86 km above sea level. In the case
of this formula, the authors'
assumption about the equality of the differentials
of the functions of two different pressure distributions is debatable. By measuring the altitude, and additionally, the temperature at this altitude, it is possible to
use formula (6) to iteratively calculate the temperature above sea level
(assuming a constant temperature gradient in the troposphere). For correcting the principal error
at low heights, formula (6) seems to be of little use.
In
the case of an external error caused by the movement of the pressure sensor,
the approximate formula (9)
presented in the appendix is more accurate than formula (8), in which [14] was assumed constant air density
regardless of height. It is also much simpler. The numerical simulations (Figures 4 and 5) also showed that formula (9) is also sufficiently
accurate compared to the non-linear relationship (A12) from which it was
derived.
The
advantage of moving on the ground was the possibility of comparing the
registered altitude with the altitude determined based on GPS route coordinates and DTM data. The total error
determined in this way turned out to be very large (Figure 10), reaching even 22 m.
However, it was not
possible to make a correction using formula (4) because the pressure and
temperature conditions related to the sea level were not exactly known. In the
case of an external error (9), it was
possible to calculate the speed with which the bicycle was moving (the wind
speed was omitted, as a weak wind was forecast and
the wind speed was not measured). At the same time, it was not known what
fraction of the maximum dynamic pressure was measured by the cycling computer.
To separate the drift error from
the total error of the altimeter, an optimization method (13) was proposed,
which was to recreate the pressure and temperature conditions above sea level
and estimate the influence of dynamic pressure on the measurement. The
calculation results showed that this method is not very effective if the entire
recorded drive is considered
(Figures 12 and 14). The method was much better
for shorter periods of time. It was concluded that the temperature and pressure increases cannot be treated
as constant over time. There is a "conflict
of interest" - too short data segments adversely affect the effectiveness
of the least squares method, and
too long limits the change in pressure and temperature values above sea level. Based on trials, it has been estimated
that the acceptable length of the time period is about 30 minutes.
The
modified new optimization method (15) allowed to solve this problem using interpolation of additional
discrete functions, the values of which were also the searched variables of the
minimized function. In this case, it was possible to achieve a significant
improvement in the efficiency of correcting the height error (Figures 15 and 16). At the same time, profiles of changes in
pressure and temperature increases during the trip were obtained (Figure 18).
Subtracting
the principal and external errors from the total error made it possible to
estimate the remaining error - barometer drift. Time derivative analysis of
this signal substantially confirmed the hypothesis [14] that drift is a
stochastic walk process (Figure
17). Interestingly, the same conclusion applied to the drift error determined
by a less accurate method (Figure
13). However, in this case, the average of the derivative signal was twice as
large, albeit at the ten-thousandth level. Characteristic peaks are observed in
the drift derivative signal, which cannot be present in the white noise signal.
They are most likely a result of the cycling computer algorithms that adjust
the altitude measured by the barometer using GPS data.
The
conducted research should be repeated in the future using a barometric
altimeter, or better,
a barometer that records only raw data, without any correction. Of course,
recreating the conditions above sea level only
makes sense if the actual weather data was not available or recorded. This
method can never give completely accurate results. The algorithm determines the
local minimum in the arbitrarily imposed allowable area, and without a more
detailed examination of the minimized function (which is difficult), there is
no certainty whether there is a better (optimal) solution.
5.
CONCLUSIONS
The theoretical
considerations about the sources of errors in the barometric altimeter and the methods of correcting them
presented in this paper may be used in practice in the process of
designing systems that minimize such errors, supported by additional
information, for example, from weather stations, accelerometers and GPS. It is
known from the literature that such systems are already in use and that
development works are still carried out to improve the effectiveness of altitude measurements. According
to the manufacturer's information, the bicycle computer used for the tests
has such an altimeter and GPS coupling system.
The numerical experiment
performed in this work, which consisted in comparing the measured height with the
route height obtained based on DTM data, showed a large measurement
error. The proposed method of reconstructing non-standard conditions at sea
level and the impact of the sensor movement speed allowed to isolate the
barometer drift from the total error signal based on the collected data. In
the case of long travel times, it was necessary to consider the variability of conditions
at sea level. The determined barometer drift signal satisfies the hypothesis of
the stochastic walk process quite well. The observed peaks of the drift
derivative signal may indicate the interference of the cycling computer and the
altitude adjustment based on GPS data. The method of retrospective
reconstruction of sea level conditions alone appears to be moderately
effective. It is much better and easier to correct the basic error and the error
caused by the sensor movement on a regular basis, which, however, requires
access to additional information.
The numerical experiment
showed that the speed of movement of the pressure sensor has practically no
effect on errors. This may be a sign of the correct positioning of the pressure
intake slots on the casing of the cycle computer. The manufacturer claims that
he used the results
of wind tunnel tests in the design of the computer housing and holder. The
evaluation of the proposed formula for correcting the external error still
requires independent bench tests in controlled air flow conditions.
Correctly read altitude
above sea level is very important during flights. It seems that in the case of the analysis of
the travel route on the ground surface, it is of secondary importance. However,
when the aim is to test the energy efficiency of vehicles, this parameter,
together with the slope of the route, is of crucial importance.
Appendix
Equation (1)
can be derived using three relationships: the equation of static air
equilibrium:
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(A1) |
ideal gas equations of state,
from which the formula for dry air density follows:
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(A2) |
and assumptions about a constant
temperature gradient in the troposphere:
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(A3) |
where:
Based on
equations (A1) - (A3), can be written:
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(A4) |
and after integration within the limits from
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(A5) |
which, after reconsidering (A3),
leads to formula (1).
The fundamental error of the altimeter can be
estimated as follows. Based on formula (1), and considering the pressure and
temperature modifications at sea level
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(A6) |
Convert (A6)
to form (A7):
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(A7) |
considering an approximation
(A8):
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(A8) |
and eliminating the measured pressure p from the
formula by replacing it with the measured height
If the location of the air inlet to the pressure
sensor does not guarantee the measurement of static pressure only, then the
sensor does not measure the dynamic pressure resulting from the relative
movement of the sensor and the air mass (which may be caused by both the wind
blowing and the movement of the sensor in space). It can be assumed that in
typical weather conditions, the relative movement of the air and pressure
sensor (compound of wind speed and sensor speed) has a direction parallel to
the Earth's surface. On the other hand, the static pressure of the air column
acts perpendicular to the Earth's surface. The device measures the total
pressure, which is the sum of the static and dynamic pressures:
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(A9) |
|
The formula considers the maximum dynamic
pressure at the stagnation point. In fact, the positive pressure at the
point of pressure may be less. The air density depends on the static pressure
and temperature (
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(A10) |
At the same time, from the pressure-temperature
equation (A5), the formula for the measured (total) pressure can be obtained:
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(A11) |
And after the transformations:
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(A12) |
Equation (A12) has no analytical solution
because of T. It can be solved numerically. Since the factor
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(A13) |
Fig. A1. Dependence of the factor V2/2RT on
speed
After considering the model of the constant
temperature gradient in the standard atmosphere and transformations, the
formula for height is obtained for the measured pressure and velocity:
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(A14) |
Formula (A14) differs from the standard formula
(1) in a factor that can be interpreted as the height of the speed. It is
also an error resulting from the measurement by the total pressure sensor:
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(A15) |
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Received 11.03.2022; accepted in
revised form 04.05.2022
Scientific Journal of Silesian University of Technology. Series
Transport is licensed under a Creative Commons Attribution 4.0 International
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[1]
Faculty of Transport and Aviation
Engineering, The Silesian University of Technology, Krasińskiego 8 Street,
40-019 Katowice, Poland. Email: tomasz.matyja@polsl.pl. ORCID: https://orcid.org/0000-0001-6364-619X
[2] Faculty of Transport and Aviation
Engineering, The Silesian University of Technology, Krasińskiego 8 Street,
40-019 Katowice, Poland. Email: andrzej.kubik@polsl.pl. ORCID: https://orcid.org/0000-0002-9765-6078
[3] Faculty of Transport and Aviation Engineering, The Silesian University of Technology, Krasińskiego 8 Street, 40-019 Katowice, Poland. Email: zbigniew.stanik@polsl.pl. ORCID: https://orcid.org/0000-0003-1965-4090