This is the accepted version of the manuscript: D. Nemec, A. Janota, M. Hruboš and V. Šimák, "Intelligent Real-Time MEMS Sensor Fusion and Calibration," in IEEE Sensors Journal, vol. 16, no. 19, pp. 7150-7160, Oct.1, 2016, doi: 10.1109/JSEN.2016.2597292, URL: http://ieeexplore.ieee.org/document/7529156 Intelligent real-time MEMS sensor fusion and calibration Dušan Nemec, Aleš Janota, Marián Hruboš, and Vojtech Šimák example of velocity measurement when the sensor is Abstract—This paper discusses an innovative adaptive measuring time derivative of desired variable. Main heterogeneous fusion algorithm based on estimation of the mean disadvantage of this method is great sensitivity of the output square error of all variables used in real time processing. The quality to the precision of the sensor measurements (especially algorithm is designed for a fusion between derivative and sensor bias will cause increasing drift of the integrated result). absolute sensors and is explained by the fusion of the 3-axial gyroscope, 3-axial accelerometer and 3-axial magnetometer into First step of elimination of the integrated error is calibration of attitude and heading estimation. Our algorithm has similar error the sensor. A standard way of calibration measures raw output performance in the steady state but much faster dynamic of the sensor as a response to stimulus with known amplitude. response compared to the fixed-gain fusion algorithm. In Relation between raw and real sensor outputs is formed into comparison with the extended Kalman filter the proposed the transfer function (calibration curve) and its parameters are algorithm converges faster and takes less computational time. On obtained from the measurements during calibration in offline the other hand, Kalman filter has smaller mean square output error in a steady state but becomes unstable if the estimated state mode. It is possible to calibrate by: changes too rapidly. Additionally, the noisy fusion deviation can --One point (zero order transfer function - bias only). be used in the process of calibration. The paper proposes and --Two points (first order transfer function - bias and explains a real-time calibration method based on machine gain), learning working in the online mode during run-time. This allows --Multiple points (calibration curve is a polyline or higher compensation of sensor thermal drift right in the sensor’s order curve). working environment without need of re-calibration in the laboratory. In order to eliminate influence of the sensor random noise each calibration point has to be computed as an average of Index Terms—calibration, inertial navigation, mean square multiple measurements in the same conditions [1]. This error methods, sensor fusion. requires special laboratory equipment which provides accurate and steady simulation of different sensor stimuli. Zhang et al. proposed a method of estimation of the calibration constants I. INTRODUCTION for the 3-axial inertial sensor (gyroscope, accelerometer) [2]. NERTIAL SENSORS manufactured by the MEMS (Micro Gyroscope bias is determined directly in a steady state and I Electro-Mechanical Systems) technology are the core of accelerometer bias is computed after multiple steps when the modern low-cost AHRSs (Attitude and Heading Reference acceleration sensor is oriented vertically along each of its axes Systems). The purpose of these systems is to determine one by one or the sensor has to be exposed to precisely known rotation of the measured object with respect to the horizontal stimuli [3][4]. All these methods are working in the offline plane and northern direction which is a crucial task in mobile mode. Wang and Hao proposed a method utilizing an artificial robotics, aviation, automated car navigation and many others. neural network combined with the Kalman filter for estimation These sensor systems use nonlinear discrete numerical of nonlinear calibration parameters [5]. For online calibration integration of the measured angular velocity which is a typical it is necessary to detect steady state of the object, e.g. by lower vibrations [6]. © 2016 IEEE. Personal use of this material is permitted. Permission from When MEMS sensors are used, their calibration parameters IEEE must be obtained for all other uses, in any current or future media, tend to drift with temperature [5] [7]. Transfer function is including reprinting/republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to servers therefore two-dimensional – one input corresponds to raw or lists, or reuse of any copyrighted component of this work in other works. sensor data and the second input is sensor temperature. Most of commercially available integrated MEMS sensors The paper was submitted for review in February 22, 2016. This work was supported by the European Regional Development Fund and the Ministry of incorporate a temperature sensor which allows usage of Education of the Slovak Republic, within the project ITMS 26220220089 advanced temperature compensation techniques. “New methods of measurement of physical dynamic parameters and In order to compensate integrated error during run-time it is interactions of motor vehicles, traffic flow and road”. Authors are with Dept. of Control and Information Systems, Faculty of necessary to use a secondary absolute sensor and provide a Electrical Engineering, University of Žilina, Univerzitná 8215/1, Žilina data fusion. The secondary sensor may be much noisier and 01026, Slovakia. e-mail: have slower response but its error has to be kept inside fixed {dusan.nemec; ales.janota; marian.hrubos; vojtech.simak}@fel.uniza.sk bounds. A modification of the Kalman filter can be used as a This is the accepted version of the manuscript: D. Nemec, A. Janota, M. Hruboš and V. Šimák, "Intelligent Real-Time MEMS Sensor Fusion and Calibration," in IEEE Sensors Journal, vol. 16, no. 19, pp. 7150-7160, Oct.1, 2016, doi: 10.1109/JSEN.2016.2597292, URL: http://ieeexplore.ieee.org/document/7529156 core of the sensor fusion algorithm [3][8][9][23], however it each axis and constant: might be difficult to estimate parameters of the filter MSE(i t) MSE(i ) Egyro const (3) (covariance matrix, state model) for a standalone sensor MSEs of the updated uncompensated rotational matrix Ru system because the Kalman filter parameters depend on the are: measured system. Another sensor fusion utilizes Bayesian 2 2 MSE(Ru1,k ) MSE(R1,k ) R2,k Egyro R3,k Egyro networks and the stochastic approach [10][11][12]. We have (4.1) 2 2 proposed a heterogeneous sensor fusion method for one y MSE(R3,k ) z MSE(R2,k ), differential sensor and one absolute sensor which requires MSE(R ) MSE(R ) R 2 E R 2 E only minimum count of parameters independently from the u2,k 2,k 3,k gyro 1,k gyro (4.2) 2 2 measured system. The performance of our algorithm will be z MSE(R1,k ) x MSE(R3,k ), compared with the performance of the extended Kalman filter 2 2 MSE(Ru3,k ) MSE(R3,k ) R1,k Egyro R2,k Egyro (EKF) used in direct form described in [23]. (4.3) 2 MSE(R ) 2MSE(R ), Our real-time calibration method utilizes error estimate x 2,k y 1,k obtained as a side output from the sensor fusion algorithm. where index k = 13. As can be seen, errors of all matrix This approach eliminates the need of steady state detection elements are increasing with new samples. Uncompensated and offline calibration. Since it can be running all the time Euler angles are then [15]: when the sensor is in use our method should compensate long- gyro atan2Ru2,3,Ru3,3 , (5.1) term drifts continuously. gyro arcsin Ru1,3 , (5.2) atan2 R , R . (5.3) II. HETEROGENEOUS FUSION ALGORITHM CONSIDERING gyro u1,2 u1,1 QUALITY Corresponding MSEs of the Euler angles are: R 2MSE(R ) R 2MSE(R ) The method will be explained on the example of the fusion MSE( ) u3,3 u2,3 u2,3 u3,3 , (6.1) of the 3-axial gyroscope (velocity sensor), 3-axial gyro 2 2 2 Ru2,3 Ru3,3 accelerometer (absolute attitude sensor) and 3-axial MSE(R ) magnetometer (absolute heading sensor). Sensor axes are MSE( ) u1,3 , (6.2) gyro 1 R 2 orientated according to the NED convention (x-North or u1,3 2 2 forward, y-East or right, z-Down), Euler angles are computed Ru1,1 MSE(Ru1,2 ) Ru1,2 MSE(Ru1,1 ) MSE( gyro) . (6.3) in the ZYX convention (α – Roll, β- Pitch, γ- Yaw). Attitude 2 2 2 Ru1,1 Ru1,2 of object is then expressed by roll and pitch angles; heading is expressed by yaw angle. These formulas are undefined at the gimbal lock (cos = 0 In order to express the quality of estimation we will use the and Ru1,3= 1). If such condition occurs it is impossible to mean square error (MSE). In general the error model of the determine both roll and yaw (one has to be chosen) and MSE attitude estimation is nonlinear [13]. MSE of the directly estimation is very imprecise. measured data is considered constant and depends on the used B. Estimating roll and pitch from accelerometer readings sensor; MSE of a computed variable y = f(x , x , …,x ) is 1 2 N Secondary, the attitude of the object can be obtained from approximated by: 2 acceleration readings by formulas [14]: N f (1) acc atan2(ay , az ) , (7) MSE(y) MSE(xk ). k1 x k 2 2 acc atan2(ax , ay az ) , (8) A. Estimating Euler angles from gyroscope readings where ax, ay, az are the components of the acceleration In the inertial navigation the object’s Euler angles are measured by the accelerometer bound with a moving object. primary computed from the angular velocity ω measured by MSE of attitude estimation depends on the dynamics of the the gyroscope.