In this paper we present a systematic method to determine sets of close to optimal sensor calibration points for a polynomial approximation.

For each set of calibration points a polynomial is used to fit the nonlinear sensor response to the calibration reference. The polynomial parameters are calculated using ordinary least square fit. To determine the quality of each calibration, reference sensor data is measured at discrete test conditions. As an error indicator for the quality of a calibration the root mean square deviation between the calibration polynomial and the reference measurement is calculated. The calibration polynomials and the error indicators are calculated for all possible calibration point sets. To find close to optimal calibration point sets, the worst 99 % of the calibration options are dismissed. This results in a multi-dimensional probability distribution of the probably best calibration point sets.

In an experiment, barometric MEMS (micro-electromechanical systems) pressure sensors are calibrated using the proposed calibration method at several temperatures and pressures. The framework is applied to a batch of six of each of the following sensor types: Bosch BMP085, Bosch BMP180, and EPCOS T5400. Results indicate which set of calibration points should be chosen to achieve good calibration results.

MEMS (micro-electromechanical systems) sensors are calibrated at one or several points at the end of the production process in order to fulfill the product specifications. Various techniques can be applied to compensate sensor nonlinearities (see Brignell, 1987). In many cases (e.g., humidity sensors, gyroscopes, pressure sensors, barometric pressure sensors) calibration relies on a polynomial for fitting the raw sensor data readout to reference signals (van der Horn and Hujising, 1997; Lyahou et al., 1996; Cerry et al., 1990; Bolk, 1985). Usually production issues limit the number of calibration points available for calibrating sensors against a reference. An adequate choice of calibration points can reduce the number of calibration points and thus sensor calibration time and cost (see Dickow and Feiertag, 2014). As far as we know, no systematic technique for MEMS sensor calibration has been published, including both optimal calibration point selection and calibration using as few points as possible. In this paper a polynomial approach for MEMS sensor calibration with only a few calibration points is proposed. An algebraic framework is used to determine all possible calibration point combinations. In an experiment, the proposed method is applied to barometric MEMS pressure sensors. Recommendations are made for an optimal choice of calibration points. The paper is structured as follows: Sect. 2 introduces the concept of polynomial calibration with parameter extrapolation; in Sect. 3 a combinatorial framework is proposed for the selection of calibration point sets; in Sect. 4 the proposed framework is applied to barometric MEMS pressure sensors; and in Sect. 5 the results are discussed in a conclusion.

As sensors often show nonlinear behavior in their measurement range, calibration functions are used to linearize the raw digital sensor readout. To calibrate these sensors reference values are needed. Reference values and the sensors raw values are recorded in a calibration device at defined test conditions or calibration points. State of the art calibration uses multiple ordinary least square fits (multiple OLS) (see Seber and Lee, 2003; Martens and Naes, 2002) to calculate the sensor calibration parameters. For further reading about calibration methods refer to Martens and Naes, 2002, and Varmuza and Filzmoser, 2009. As multiple OLS is a linear regression method, calibration functions have to be linear in their parameters. In this paper a multivariable polynomial is chosen as the calibration function to linearize the sensor behavior over specification. The polynomial function is capable of linearizing continuous, differentiable sensor readouts to a given specification. In some cases, polynomial functions of very high order are needed to reach the desired calibration accuracy. Then, it is useful to switch to other functions, reaching accuracy goals with fewer parameters, or even switch to other regression methods, e.g., support vector regression (see Smola and Scholkopf, 2004).

A calibration function gives the sensor output signal as a function of the
sensor's raw values. Multiple regression calibration is used to generate an
appropriate calibration function for given reference values. The OLS
calibration method is an integral part of research and production in many
fields of applications (shown in Eriksson et al., 2006). A linear
regression model for

In matrix notation, the calibration problem transforms to

For example, MEMS pressure sensors with raw temperature

In this case,

In the following

To decide which sets of calibration points give the best sensor accuracy a
reference data set is necessary. This reference data is measured at discrete
values within the measurement range of the sensor. In the following we
assume that all calibration points match the discrete values of the
reference data. This leads to a finite number of possible sets of
calibration point combinations

Six selected calibration points for digital barometric pressure
sensors, chosen from a reference data set (marked with

For each

For a batch of

To retrieve more information about fields of attraction for best calibration
options in a multidimensional optimization problem, a selection criteria
weaker than Eq. (12) is proposed. The criteria

The proposed sensor calibration approach was implemented in a framework,
written in Python language, to investigate commercial MEMS sensors with
digital data readout. It uses the Fortran package LAPACK (Linear Algebra PACKage) to solve linear
equations using LU (lower, upper) decomposition (Strang, 1980) with partial pivoting and
row interchange. A typical calibration workflow is depicted in Table 1. Four
steps are needed to retrieve meaningful statistical data out of given
digital raw sensor readout and reference values: data processing,
calibration, comparison and statistical evaluation. If the number of
required calibration points

A typical calibration workflow using the proposed framework; data processing, calibration and evaluation are implemented in independent modules.

In this section, the new framework is applied to three types of commercial
barometric MEMS pressure sensors, which are calibrated against temperature
and barometric pressure using polynomials (see Kim et al., 2012; Köster
et al., 2003; Bosch Sensortec, 2008, 2013; EPCOS, 2013). Three batches of
six barometric MEMS pressure sensors each, of the type EPCOS T5400, Bosch
BMP 180 and Bosch BMP 085, are used to calculate calibration point
recommendations for a multiparametrical second-order calibration polynomial,

Recommended six calibration points for second order polynomial barometric pressure sensor calibration; frequency of the first three calibration points shown in the upper part; frequency of the last three calibration points shown in the lower part of the figure.

The test equipment uses a pressurized climate chamber, a General Electric PACE 5000 pressure controller as pressure calibration reference and a combination of Peltier elements and a type K thermocouple for reference temperature control, attached close to the sensor site. Data is recorded using a National Instruments USB-8451 I2C device, connected to the digital barometric MEMS pressure sensors, soldered on to a printed circuit board.

For all further investigations, it is assumed that the sensors deliver reproducible results. As barometric MEMS pressure sensors suffer from temperature hysteresis (see Waber et al., 2013), data was recorded in ascending temperature order to minimize the hysteresis effect.

Sensors used in the experiment have a measurement range from

Data is recorded according to the description in Sect. 4.2. For the calibration polynomial from Eq. (16), the most likely best calibration point combinations were calculated using the procedure taken from Table 1. The experimental results are shown in a distribution landscape plot in Fig. 2. The plot exemplarily separates the first 3 calibration points (upper row in Fig. 2) from the calibration points 4–6 (second row in Fig. 2). This clustering into two distribution plots provides information about how to combine temperature points, when calibration is restricted to only two possible pressure conditions, as it can happen in industrial sensor production to save calibration time. Figure 2 shows, that the investigated T5400 sensor should be calibrated at points at the borders of pressure range (300 and 1100 hPa), while the other sensor types investigated show a more heterogeneous calibration point recommendation (300, 900 and 1100 hPa).

The proposed method can be applied to all sorts of sensors, having one
output signal and a at minimum one input signal. For computational
complexity reasons, the number of measurement points for searching the
optimal calibration option should be low. In the pressure sensor example
there are 9 pressure points and 14 temperature points, resulting in a grid
of 126 measurement points for calibration. This is already a demanding task,
for a polynomial order 3 in both parameters (16 parameter polynomial),
according to Eq. (10). The proposed method can be applied to sensors like
gyroscopes (see Aggarwal et al., 2006) or any other kind of sensor, as long
as the number of parameters for calibration and the number of measurement
points investigated is comparable to the numbers from the barometric
pressure sensor example. For the gyroscope example in Aggarwal et al. (2006),
the parameter

The proposed framework was used to calibrate barometric MEMS pressure sensors with six calibration points and a second-order calibration polynomial in temperature and first-order in pressure. For all sensors investigated, points selected at the upper and lower borders of temperature and pressure range increase the likelihood of appearing within the best 1 % of the calibration options. The proposed framework determines all possible calibration options for a given set of sensor measurement data using a linear polynomial regression approach and then applies the rms error over measured test conditions as calibration benchmark. The worst 99 % of the calibrations are dismissed to show areas of attraction for good calibration options. For further research, calibration point extrapolation will be implemented to reduce the amount of calibration points measured. This is used to achieve the sensor specification required by statistically estimating offset values for higher-order calibration polynomial parameters. Further calibration uncertainty considerations will be evaluated according to Waber et al. (2013), Heydorn and Anglov (2002) and Brüggemann and Wennrich (2002).

This work was supported by the Bavarian ministry of economic affairs within the research project MEMSBaro.Edited by: N.-T. Nguyen Reviewed by: two anonymous referees