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Another possible alternative method of attitude determination of the examined object is to use the theory of quaternion. This method of quaternion requires more processing time, but it solves potential singularity issues that can emerge during the algorithmic computations while converting angles to radians.
References
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1. Soloviyov U. A. Systems of satellite navigation. Moscow: Eco trends, 2000. - 270 p.
2. Genike А. А., Pobedinskiy G. G. Global satellite systems of positioning and their application in geodesy. Moscow: Mapgeocenter, 2004. - 355 p.
3. Аkhmedov D. Sh., Moldabekov М. M., Raskaliyev А. S., Shabelnikov Ye. А. A mathematical model of code-phase differential correction for signals of global navigation satellite system. Almaty: Herald of the National Engineering Academy of the Republic of Kazakhstan, 2012. - Volume 2, 44 p.
4. Аkhmedov D. Sh., Moldabekov М. M., Raskaliyev А. S., Shabelnikov Ye. А. Mathematical models of differential correction in high-precision navigation systems. Almaty: Applied space research in Kazakhstan, LLP «Dyke-Press», 2010. - Volume 6, pp. 227-231.
Methods and algorithms used for calibration of INS Raskaliyev A.
Методы и алгоритмы, используемые при калибровке ИНС
Раскалиев А. С.
Раскалиев Алмат Серикович /Raskaliyev Almat Serikovich - научный сотрудник, лаборатория спутниковых навигационных технологий, Институт космической техники и технологий, г. Алматы, Республика Казахстан
Abstract: this article describes some practical and theoretical aspects of mathematical modeling of Inertial Navigation System (INS). It provides overview of methods and algorithms used for calibration in micro-electro-mechanical systems (MEMS) sensors. The author of this article also set forth implementation issues of the complementary filter and exponential noise smoothing algorithm that were encountered when calibrating MEMS sensors by six-position method (SPM) method.
Аннотация: в данной статье описываются некоторые практические и теоретические аспекты математического моделирования инерциальной навигационной системы (ИНС). Статья проводит обзор методов и алгоритмов, используемых для калибровки датчиков на основе микро-электромеханических систем (МЭМС). Автор данной статьи также обсуждает вопросы реализации комплиментарного фильтра и экспоненциального алгоритма сглаживания шумов, которые возникли при калибровке МЭМС датчиков шестипозиционным методом (SPM).
Keywords: INS, MEMS, calibration, SPM method. Ключевые слова: ИНС, МЭМС, калибровка, метод SPM.
1. Introduction
The methods of inertial navigation and inertial navigation system (INS) on their own currently are widely used for navigation of mobile objects. Compared with other types of navigation systems, INS meets a whole range of important requirements, such as versatility, full autonomy, noise immunity and interference immunity, as well as furtiveness of its activity. Moreover, at the present level of technological development, these systems can provide a sufficiently high accuracy of navigation and orientation, which is limited only by the accuracy of input sensors.
Despite its small size, weight and power consumption, the practical application of MEMS sensors in inertial navigation and orientation is limited by low sensitivity, scale factor instability and high noise level of the output signal [1]. MEMS has the highest drift (up to 300 - 1000°/hr) among existing types of gyroscopes, which does not allow to use MEMS without periodic correction of angular coordinates and attitude angles.
If we consider the autonomous operation of the INS which consists of only one accelerometer and a gyroscope, the positioning error, taking into account all possible sources of sensor errors, can be put into the expression:
p(t) * p0 + v0At + ba A- + bg A- + ag A- + 4 -VAt + SFa • f A- + SFg • Ay At, (1)
where
p0 - positioning error at the initial time t0; v0 - error of the velocity at time t0; At - time interval from the start of data acquisition; ba - accelerometer zero offset error at time tg; bg - gyroscope zero offset error at time t0;
ag - INS axis misalignment error of the roll and pitch angles with the axes of the local coordinate system;
Az ■VAt - INS misalignment error of the azimuth angle with the local coordinate system, multiplied by the distance traveled;
SFa - scale factor for the accelerometer; SFg - scale factor for the gyroscope; f- an apparent acceleration; g - acceleration of gravity.
One can notice from the formula (1) that the total positioning error of an object increases proportionally to the square of time because of accelerometer parameters, which are zero offset (b) and the scale factor (SF), and a cube of the time because of the zero offset of the gyroscope.
We will use the following equation to model signals in the output of MEMS sensors: Iwa (t) = *(t)+ bwa + SFw ax(t)+ NWaX(t) + £w a (t), (2)
where
x(t) - the vector of acting physical quantity, which can be an angular velocity m(t) or acceleration a(t) ;
Iw,a(t) - output signals from the sensors (m(t) or a(t));
bma - zero offsets;
SFm,a - scale factor matrices,
Nm,a - non-orthogonal matrix of sensor axes with respect to X axis; sm,a(t) - noise errors caused by transducers.
2. Calibration of the sensor by SPM method
Information in inertial navigation systems is formed on the basis of measurements from the sensors. These measurements contain errors that will eventually lead to the accumulation
of positioning errors, velocity and orientation angles of the object. Thus, one way to improve the accuracy of navigation is evaluation of the instrumental errors and introduction of relevant corrections to the measurements obtained by the sensors. Determination of such kind of errors is called calibration.
Currently, the following methods of laboratory calibration have become widespread: six-position method (SPM), modified six-position method (MSPM), multi-position method (MPM) and a modified multi-position method (MMPM). We focus on SPM method below.
SPM method can be used to evaluate zero offset of the sensors (b) and their scale factors (SF) by means of simple equations given in [3] for obtaining measurements in six positions.
Accelerometers typically are calibrated by measuring the gravity. The accelerometer is mounted on the calibration swiveling rotary table and its sensitive axis is directed vertically upwards to collect measurements from the sensors for 10-15 minutes. It allows calculating the value of fup by averaging the data obtained. Then, we make similar measurements by directing sensitive axis vertically downwards. As a result, we find the average value of fdown correspondingly for this type of measurements.
The magnitudes offup and fdown may be represented as:
fup=ba + (1+SFa)g
fdown=ba-(1+SFa)g (3)
The accelerometer's zero offset ba is calculated by addition of the equations given above:
ba = fup+fdown)/2; (4)
Then scale factor SFa will be equal to:
SFa = 1-(fup-fdown)/2g (5)
Considered method of determining the zero offset of the accelerometer is not demanding to initial orientation of the sensitivity axis, which greatly simplifies its implementation. However this statement is not valid for the calculation of the scale factor. The described sequence of steps should be carried out for each of the sensitive axes of the accelerometer.
Determination of the drift and scale factor of the gyroscope is made in a similar manner. Earth rotation rate can be considered as a quite weak signal, so it can be used only for calibration of very precise and costly INS, because the noise level of its angular velocity sensor is below the reference signal. However, MEMS sensor is equipped with a gyro of INS of middle and low class of accuracy. So it becomes viable to use an auxiliary swivel device that can provide a stable rotation speed as a reference for its calibration [4].
3. Complementary filter
As noted above, MEMS gyroscope has one main drawback called zero drift. The essence of this shortcoming comes from the fact that when you stop the rotation of the gyroscope, it will still show a nonzero value. Another disadvantage of this solution is the use of discrete integration procedures, which by its nature is inaccurate. The third shortcoming is caused by the gradual accumulation of angular errors during the calculation because of the limited accuracy of microcontroller variables.
At rest, the accelerometer can also be used to determine the orientation angles of pitch and roll. To find the angles by using the accelerometer, it is sufficient to apply simple geometric transformations to its measurements. As long as it is not acted upon by external forces, we obtain on the output device the value of projection of the gravitational acceleration on observed axis. Let us assume to get accelerometer readings for x axis marked as Ax. Knowing g and Ax, one can calculate the deviation angle from the horizontal position of the accelerometer - a:
sin(a) = Ax /g, a = arcsin(Ax /g). )
Doing such calculations, it is important to take into account that X and G should be measured in the same units. For example, if you convert the accelerometer readings into
units of gravity, in other words, g =1 for Earth's gravity, then the expression for the angle a takes the form:
a = arcsin(Ax). (7)
Thus, it can also be quite easy to build an inclinometer on the basis of only one accelerometer. Unfortunately, any external force causes a mistake in these calculations. Such an external force can be the vibration of drone's engines, or a sudden gust of wind. In order to partially remove this effect, you can use low-pass filter, but a side effect of that kind of signal processing is a sufficient decrease in the speed of the inclinometer.
So we have considered two navigational units that can be used for determination of inclination angles of the object relative to the ground. However, the gyroscope's accuracy is reduced because of the zero drift errors and integration, while the accelerometer is featured by too large sensitivity to external influences.
A complementary filter provides combination of measurements obtained from these devices. This filter might be expressed by an equation:
a = (1-K)* agyr + K* aacc, (8)
Where
a — output of the filter, the resulting inclination angle;
agyr and a^c — inclination angles obtained by measurements from the gyroscope and the accelerometer accordingly;
K — factor of the complementary filter.
It is obvious that the final value of the inclination angle is the sum of the integrated values of instantaneous values of the gyro and the accelerometer. In fact, the main task of the complementary filter is to neutralize zero drift of the gyroscope and the errors of the discrete integration. The above expression does exactly that. At each step of the integration, we adjust the integral of inclination angle by means of the accelerometer's measurements. The speed of this correction is determined by the filter's factor K.
Selection of the factor K depends on zero drift of the gyroscope, the rate of error accumulation during calculations and environment of MEMS application. Thus, if value of K is too large, then the filter's efficiency will be strongly affected by vibration of the device enclosure. Too small value of K may be insufficient to eliminate the zero drift of the gyroscope. Typically, the complementary filter's factor is selected manually for each inclinometer on the basis of the above conditions. For example, K may take a value in the range from 0.05 to 0.1 for inertial navigation of the amateur drone [5].
The use of complementary filter does not require large computational power and allows achieving sufficiently high quality flight stabilization or balancing. Figures 1 and 2 illustrate gradual correction of the angular integral by means of readings obtained from one of the axes of the MEMS accelerometer.
Fig. 2. The result of using the complementary filter with the factor which is equal to 0.03
4. Exponential noise smoothing and conclusions
Measurements of MEMS accelerometers are subject to rather severe noise, so that there is an urgent need to eliminate it. One of the ways to deal with noisy data is using the filter. The problem of motion detection imposes an important requirement to the filter. Its computational efficiency should be sufficient to apply the filter in real time mode with minimal delays. Certainly it will also be advantageous if the final value of the filter's factor will be close to its initial value.
The exponential filter is the simplest and most common recursive algorithm which is widely used for the analysis of time series, in particular, for their prediction. An output parameter of the exponential filter is equal to a weighted sum of the output parameter of the filter in the previous time and the current value of the input parameter multiplied by certain weights. The main advantage of the predictive model based on an exponential smoothing is
that it is able to consequently adapt to the new level of the iteration without significant response to random deviations.
Exponential smoothing can be represented as data filter which has been consistently receiving the outgoing members of the series at the input, and the output values are formed by the exponential average.
The data from the inertial sensor can be represented in the form of time series:
a=(a1,...,at),
Exponential smoothing can be written as follows:
a'i=(1-y)a'i_1+ yai (9)
Where
a'i - acceleration (angular velocity) after the processing at the current time;
a'i-1 - acceleration (angular velocity) after the processing at the previous time;
at - initial acceleration (angular velocity);
Y - the filter's factor which is selected experimentally.
Figure 3 shows a flowchart algorithm for data filtering by applying the exponential smoothing, developed on the basis of the mathematical model shown in the equation (9).
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Fig. 3. Data filtering algorithm by applying exponential smoothing
Two issues arise in the practical application of the exponential smoothing method: selection of the smoothing coefficient y, which largely affects the output results, and determination of the initial value of a. In one hand, we should reduce the value of y to smooth out random deviations. In the other hand, the value of y should be increased in order to enlarge the weight of newly input measurements.
Exponential average a'i-1 has the same mathematical expectation as the original series, but it has smaller variance. With the high value of y, the variance of the exponential average differs slightly from the dispersion of the series a. The smaller y, the more reduced the variance of the exponential average that causes suppressed fluctuation of the original series. In other words, if we choose the minimal value of the filter factor, it eliminates more noise, but at the same time more sensitivity of the accelerometer is reduced [6].
Figure 4 a) and b) present a processing log of the actual inertial accelerometer, noise of which is smoothed by exponential filter. In this illustration, you can easily compare the noise level of the accelerometer signal before and after exponential smoothing by means of factors y = 0.1 and y = 0.5.
b)
Fig. 4. Accelerometer signals before and after application of the exponential smoothing: a) Filter's factor y=0.1, b) Filter's factor y=0.5
Thus, the developed algorithms of filtering and calibration of three-axis MEMS gyroscopes and accelerometers provide sufficient improvement in accuracy of attitude determination of mobile objects. Moreover, the use of complementary filters for aggregating angles, calculated by means of the data from the accelerometer and gyroscope, provides compensation for errors that appear during integration of pitch and roll angles. All the presented methods are fairly accurate, computationally stable and easy for implementation. The experimental data was obtained with an inertial navigation system, consisting of a three-axis MEMS accelerometer LSM303DLM and gyroscope L3G4200D.
References
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- Calgary, Canada: The University of Calgary, 2009. - 230 p.
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5. Determination of the angle of the accelerometer. [Internet source]: «Bitaks» company's website. URL: http://bitaks.com/resources/inclinometer/content.html accessed 22 March 2016.
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