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Three axis rotation measurements with Kalman filter data-fusion.


Sensor of angular velocity is called gyroscope, sensor of linear acceleration is accelerometer and sensor of magnetic field is magnetometer. MEMS techlogy has been popular last few years. This technology allows creating mechanical parts of sensors (referential mass of gyroscope and accelerometer) on silicon chip with dimensions of the order of millimeters. The accuracy of these sensors is not very good yet. But it is possible to combine data from different types of sensors and achieve improve accuracy. The Kalman filter is suitable for this purpose. Magnetic sensors are very useful in navigation systems (Caruso, 2002). Most of these systems use some type of compass to determine heading direction. Nowadays there is much information about magnetic sensors (Caruso, 2003) as well as about inercial navigation (Titterton & Weston, 2004).


Accelerometer is used for measure tilt towards the horizontal axis in this application (as an inclinometer). The horizontal axis corresponds to zero measured acceleration. Vertical position corresponds to the measured value of 1g. The advantage is the absolute way to measure tilt (due to the gravity vector). The disadvantage is sensitivity to the other accelerations due to shake or movement and a dependence on temperature. Dependence on temperature is linear and it is easy to suppress it. Parasitic acceleration is possible to suppress by using a low-pas filter (for example FIR filter). It is possible to measure tilt with one accelerometer in horizontal axis using arcsin() function. Function sinus is very flat near 90[degrees] therefore measurement would be inaccurate. Hence we measure acceleration in axis perpendicular and in axis parallel to Earth's surface. The tilt angle is calculated using (1). This equation ensures a good accuracy throughout the range of 0[degrees] to 90[degrees] (ulike simple sine function).


Where z is the acceleration in perpendicular axis and x is in the parallel axis to the Earth's surface and sgn() is the signum function.


Output of the gyroscope is proportional to angular velocity. Information about the angle of rotation is obtained by integration. It is a relative measurement.

The main advantage of gyroscope is a small dependence on acceleration. It is possible to use a gyroscope to measure rotation around the axis with random orientation against the earth's surface. The disadvantage is a significant dependence on temperature and the error caused by integrating the output signal.

Both disadvantages are suppressed by using the Kalman filter.


Magnetometer is used for measure the Earth magnetic field like compass. The main advantage is the absolute way to measure (due to the Earth magnetic field vector). The disadvantage is the temperature dependence (offset changes), sensitivity to disruptive magnetic field, metal objects and the tilt influence.

Temperature dependence and sensitivity to disruptive magnetic field are suppressed by using the Kalman filter.

The measurement error dependence on the tilt angle is shown in Fig.1. Different curves represent different azimuth settings.


This error is compensated by using equation (2). With the knowing of tilt angles, the equation recalculates measured data into the horizontal planes using matrix of rotation (Solc & Zalud, 2002). Compensation result is shown in Fig.2.


Where [alpha], [beta] are tilt angles along x,y axis and [S.sub.x1], [S.sub.x2], [S.sub.x3] are the measured values of magnetic field.



The Kalman filter is a set of mathematical equations (3) that provides an efficient computational (recursive) means to estimate the state of a process, in a way that minimizes the mean of the squared error (Welsh & Bishop, 2006).


The process is represented by matrix A, B, C and D. In this case is assuming that matrix B and D are zero. Matrix Q is the process noise covariance and R is the measurement noise covariance. They are constant and their value can be measured or estimated. The matrix P represents estimate error and its initial value should be a big value. The input of the Kalman filter is Y matrix (measured data from output of a real process). Output is X matrix (estimated states of a real process).


In this paper the process is a linear system of second order (4). It is a model of gyroscope integral error.


Each axis has one gyroscope (relative measurement) and one sensor, based on absolute measurement of the rotation angle (accelerometers in the horizontal plane and a magnetometer in the vertical plane). The gyroscope is very accurate in short time period, but unusable in long time period due to integration error. The accelerometer and magnetometer error is stable over time, but is grater than the gyroscope error.

Both sensors are on the same axis. and measure same magnitude. Output value from gyroscope is subtracted by output value from absolute rotatory sensor. Product of this is signal equivalent to integral error of gyroscope with noise from absolute rotatory sensor. Kalman filter removes this noise and on the output of this filter is estimation of integral error. Output from gyroscope is subtracted by this estimation. The result of this is signal without integral error and without noise. Block diagram of data-fusion is shown in Fig. 3.



With this method we can achieve precise measuring of rotation with low noise from gyroscope and with time stability of absolute rotatory sensor (accelerometer or magnetometer). Reachable precision of data fusion is around [+ or -] 1[degrees]. Overall precision depends on declination error of magnetometer. Overall precision is within [+ or -] 2[degrees] interval for declination within [+ or -] 45[degrees] interval.


This work has been supported in part by Ministry of Education, Youth and Sports of the Czech Republic (Research Intent MSM0021630529 Intelligent systems in automation), Grant Agency of the Czech Republic (102/09/H081 SYNERGY --Mobile Sensoric Systems and Network) and by Brno University of Technology. Without kind support of the abovementioned agencies and institutions the presented research and development would not be possible.


Caruso, J. M. (2002). Applications of Magnetoresistive Sensors in Navigation Systems, Honeywell, Available from: http://www. Accessed: 2009-06-26

Caruso, J. M. (2003). Applications of Magnetic Sensors for Low Cost Compass Systems, Honeywell, Available from: Accessed: 2009-06-28

Solc, F. & Zalud, L. (2002). Robotics, lecture notes, Brno University of Technology (BUT), Brno

Titterton, D. H. & Weston, J. L. (2004). Strapdown inertial navigation technology, MIT Lincoln Laboratory, 1-56347693-2, Lexinton, Massachusetts

Welsh, G. & Bishop, G. (2006). An Introduction to the Kalman Filter, University of North Carolina at Chapel Hill, Available from: pdf/ kalman_intro.pdf Accessed: 2009-06-25
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Author:Skula, David; Vesely, Milos
Publication:Annals of DAAAM & Proceedings
Article Type:Report
Geographic Code:4EUAU
Date:Jan 1, 2009
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