AN3192 Application note Using LSM303DLH for a tilt compensated electronic compass Introduction This application note describes the method for building a tilt compensated electronic compass using an LSM303DLH sensor module. The LSM303DLH is a 5 x 5 x 1 mm with LGA-28L package IC chip that includes a 3D digital linear acceleration and a 3D digital magnetic sensor. It has a selectable linear acceleration full scale range of ±2g / ±4g / ±8g and a selectable magnetic field full scale range of ±1.3 / ±1.9 / ±2.5 / ±4.0 / ±4.7 / ±5.6 / ±8.1 gauss. Both the magnetic sensor and the accelerometer parts can be powered down separately to reduce power consumption. Sensor measurements can be acquired by a microcontroller through an I2C serial bus interface. The key features of the system are: ■ One single chip solution ■ I2C communication interface ■ Tilt compensation Section 1 describes the basics of the electronic compass. Section 2 presents a typical hardware connection between the LSM303DLH and a microcontroller and sample code for sensor data acquisition. Section 3 focuses on the methods of the determination of sensor calibration parameters. Section 4 shows the methods of lab testing and field testing for checking the electronic compass performance. Section 5 gives recommendations for microcontroller firmware implementation when designing a standalone tilt compensated electronic compass. August 2010 Doc ID 17353 Rev 1 1/34 www.st.com Contents AN3192 Contents 1 2 3 Electronic compass basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.2 Heading calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.3 Tilt compensation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.4 Electronic compass system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.5 Getting started with LSM303DLH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 LSM303DLH configuration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.1 Typical hardware connection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.2 LSM303DLH accelerometer data acquisition . . . . . . . . . . . . . . . . . . . . . . . 9 2.3 LSM303DLH magnetic sensor data acquisition . . . . . . . . . . . . . . . . . . . . 10 Calibrating the LSM303DLH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 3.1 LSM303DLH accelerometer calibration . . . . . . . . . . . . . . . . . . . . . . . . . . 12 3.2 LSM303DLH magnetic sensor calibration . . . . . . . . . . . . . . . . . . . . . . . . 12 3.2.1 4 Testing the electronic compass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 4.1 4.2 5 Terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 Lab testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 4.1.1 Absolute heading testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 4.1.2 Tilt compensation testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 Field testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 4.2.1 Relative heading testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 4.2.2 Tilt compensation testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 Firmware implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 Appendix A LSM303DLH pitch/roll/heading calculation . . . . . . . . . . . . . . . . . . . 19 A.1 Pitch/roll calculation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 A.2 Heading calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 Appendix B Accelerometer calibration method. . . . . . . . . . . . . . . . . . . . . . . . . . 24 2/34 Doc ID 17353 Rev 1 AN3192 Contents Appendix C Magnetic sensor calibration method . . . . . . . . . . . . . . . . . . . . . . . . 27 C.1 Step 1: Soft-iron effect verification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 C.2 Step 2: Hard-iron, soft-iron and scale factor compensation . . . . . . . . . . . 27 C.3 Step 3: Misalignment error compensation. . . . . . . . . . . . . . . . . . . . . . . . . 30 6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 7 Revision history . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 Doc ID 17353 Rev 1 3/34 List of figures AN3192 List of figures Figure 1. Figure 2. Figure 3. Figure 4. Figure 5. Figure 6. Figure 7. Figure 8. Figure 9. Figure 10. Figure 11. Figure 12. Figure 13. 4/34 Body coordinates and attitude angles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Heading calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Handheld device at tilted position . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Block diagram of electronic compass system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Typical hardware connection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 Lab testing setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 Field testing setup. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 Electronic compass coordinate system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 Rotation procedures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 Heading calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 3D rotations plus three 2D full round rotations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 After hard-iron, soft-iron, and scale factor compensation . . . . . . . . . . . . . . . . . . . . . . . . . . 28 After misalignment compensation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 Doc ID 17353 Rev 1 AN3192 1 Electronic compass basics Electronic compass basics The strength of the earth's magnetic field is about 0.5 to 0.6 gauss and has a component parallel to the earth's surface that always points toward the magnetic north pole. In the northern hemisphere, this field points down. At the equator, it points horizontally and in the southern hemisphere, it points up. This angle between the earth’s magnetic field and the horizontal plane is defined as an inclination angle. Another angle between the earth's magnetic north and geographic north is defined as a declination angle in the range of ± 20º depending on the geographic location. A tilt compensated electronic compass system requires a 3-axis magnetic sensor and a 3axis accelerometer sensor. The accelerometer is used to measure the tilt angles of pitch and roll for tilt compensation. And the magnetic sensor is used to measure the earth’s magnetic field and then to determine the heading angle with respect to the magnetic north. If the heading with respect to the geographic north is required, the declination angle at the current geographic location should be compensated to the magnetic heading. 1.1 Definitions For compass applications in a handheld device such as a cell phone or a PDA, the aircraft convention is widely used to define the device body coordinates and three attitude angles pitch, roll and heading as shown in Figure 1. Figure 1. Body coordinates and attitude angles From Figure 1, the device (or aircraft) body coordinates Xb/Yb/Zb are defined as forward/right/down based on the right-hand rule. Three attitude angles are referenced to the local horizontal plane which is perpendicular to earth’s gravity. Heading is defined as the angle between the Xb axis and the magnetic north on the horizontal plane measured in a clockwise direction when viewing from the top of the device (or aircraft). Pitch is defined as the angle between the Xb axis and the horizontal plane. When rotating the device around the Yb axis with the Xb axis moving upwards, pitch is positive and increasing. Roll is defined as the angle between the Yb axis and the horizontal plane. When rotating the device around the Xb axis with the Yb axis moving downwards, roll is positive and increasing. Doc ID 17353 Rev 1 5/34 Electronic compass basics 1.2 AN3192 Heading calculation When the device is at a leveled position, pitch and roll angles are 0°. Then the heading angle can be determined as shown in Figure 2. Figure 2. Heading calculation ;E ;K +K (ORIZONTALCOMPONENTOF+ (EADING <E <K =E + ,OCALEARTHMAGNETICFIELD !-V Local earth magnetic field H has a fixed component Hh on the horizontal plane pointing to the earth’s magnetic north. This component can be measured by the magnetic sensor sensing axes XM and YM that are named as Xh and Yh. Then the heading angle is calculated as: Equation 1 Heading = arctan( Yh / X h ) In Figure 2, when the device body Xb axis is parallel to Hh which is pointing to the magnetic north, then Xh = max and Yh = 0 so that heading = 0°. Rotating the device clockwise on the horizontal plane, the heading increases. When Xh = 0 and Yh = min, then heading = 90°. Keep rotating until Xh = min and Yh = 0, then heading = 180°. And so on. After a full round 360° rotation, the user sees a centered circle if plotting Xh and Yh values coming from the magnetic sensor measurements. 1.3 Tilt compensation If the handheld device is tilted, then the pitch and roll angles are not equal to 0° as shown in Figure 3, where the pitch and roll can be measured by a 3-axis accelerometer. Therefore, the magnetic sensor measurements XM, YM, and ZM need to be compensated to obtain Xh and Yh as shown in Equation 2. And then apply Equation 1 for the heading calculation. 6/34 Doc ID 17353 Rev 1 AN3192 Electronic compass basics Figure 3. Handheld device at tilted position $FFHOHURPHWHU 3LWFK ; ;K +RUL]RQWDOSODQH < 5ROO <K !-V Equation 2 X h = X M cos Pitch + Z M sin Pitch Yh = X M sin Roll sin Pitch + YM cos Roll − Z M sin Roll cos Pitch Where, XM, YM, and ZM are magnetic sensor measurements. 1.4 Electronic compass system Figure 4 below shows the block diagram of an electronic compass system. A microcontroller (MCU) is used to collect the 3-axis accelerometer raw data for the pitch and roll calculation and collect the 3-axis magnetic sensor raw data for the heading calculation. The following is the procedure for building a working electronic compass system. ● Hardware design to make sure the MCU can get clean raw data from the accelerometer and the magnetic sensor ● Accelerometer calibration to obtain parameters to convert accelerometer raw data to normalized values for pitch and roll calculation ● Magnetic sensor calibration to obtain parameters to convert magnetic sensor raw data to normalized values for the heading calculation ● Test the performance of the electronic compass system. Figure 4. Block diagram of electronic compass system 3LWFK D[LV DFFHOHURPHWHU D[LV PDJQHWLFVHQVRU $QDORJWR GLJLWDO FRQYHUWHU 5ROO 0LFURFRQWUROOHU +HDGLQJ !-V Doc ID 17353 Rev 1 7/34 Electronic compass basics 1.5 AN3192 Getting started with LSM303DLH The LSM303DLH from STMicroelectronics is a 6D sensor module that contains a 3D accelerometer and a 3D magnetic sensor. It has an I2C digital interface so that the analog to digital converter is avoided. The MCU can collect 6D sensor data directly through the I2C interface. After understanding the electronic compass basics, it is time to get started on building a working electronic compass system based on the LSM303DLH 6D sensor module. The detailed definition and pitch/roll/heading calculation are described in Appendix A. 8/34 Doc ID 17353 Rev 1 AN3192 LSM303DLH configuration 2 LSM303DLH configuration 2.1 Typical hardware connection Figure 5 shows a typical hardware connection between the LSM303DLH and a microcontroller. The +3 V power supply is used to power the LSM303DLH and the +1.8 V power supply is used for digital IO lines. Figure 5. Typical hardware connection 9 & ȝ) 9 9GG 5HVHUYHG . 9&& 6&/B0 6&/ 6&/B$ & ȝ) 6'$B0 5HVHUYHG & ȝ) . 6(7 6'$B$ 6'$ 9GGB,2B$ 9GGBGLJB0 6(7 5HVHUYHG & '5'<B0 6$B$ ,17 ,17 *1' 9 0LFUR FRQWUROOHU & ȝ) 966 /60'/+ ,&6ODYH ,&0DVWHU !-V The LSM303DLH is an I2C slave and the microcontroller is an I2C master. When designing the hardware, the user should pay attention to the following: ● Some reserved pins not shown in Figure 5 should be connected to their corresponding pins according to the LSM303DLH datasheet ● Power supply decoupling ceramic capacitors C3 (10 µF) and C4 (0.1 µF) should be placed as near as possible to the Vdd pin 6 ● Choose a microcontroller that has a built-in I2C controller. If a bit-banging scheme implemented in the microcontroller's firmware is used, pay attention to the I2C communication timing specifications ● The +3 V and +1.8 V power supply should be regulated and clean to reduce the noise In addition, the handheld device may have high current active components, for example, an RF amplifier circuit. It may also have ferromagnetic metal materials that a magnet would stick to. These generate magnetic field distortion to the earth’s magnetic field. Even though the magnetic sensor calibration can compensate these hard-iron and soft-iron distortions, it is recommended to place the LSM303DLH onto the device PCB as far away as possible from the above distortions. 2.2 LSM303DLH accelerometer data acquisition SA0_A pin 4 in Figure 5 is tied to ground. So the 7-bit I2C slave address for the accelerometer, accordingly, is 0011000b or 0x18. For electronic compass applications, a full scale range of ±2 gauss and bandwidth of 10 Hz should be sufficient. Doc ID 17353 Rev 1 9/34 LSM303DLH configuration AN3192 After power-on of the LSM303DLH, two registers, CTRL_REG1_A (20h) and CTRL_REG4_A (23h) need to be configured. Write 0x27 to the CTRL_REG1_A register to bring the accelerometer into normal operation mode with ODR 50Hz. Write 0x40 to the CTRL_REG4_A register to keep a full scale range ±2 gauss in continuous data update mode and change the little-endian to a big-endian structure. The following sample C code shows how to acquire accelerometer data: unsigned char ACC_Data[6]; int Ax, Ay, Az; void main(void) { Write(0x18, 0x20, 0x27);//set CTRL_REG1_A register Write(0x18, 0x23, 0x40);//set CTRL_REG4_A register While (1) { ACC_Data[0] = Read(0x18, 0x28);//read OUT_X_L_A (MSB) ACC_Data[1] = Read(0x18, 0x29);//read OUT_X_H_A (LSB) ACC_Data[2] = Read(0x18, 0x2A);//read OUT_Y_L_A (MSB) ACC_Data[3] = Read(0x18, 0x2B);//read OUT_Y_H_A (LSB) ACC_Data[4] = Read(0x18, 0x2C);//read OUT_Z_L_A (MSB) ACC_Data[5] = Read(0x18, 0x2D);//read OUT_Z_H_A (LSB) Ax = (int) (ACC_Data[0] << 8) + ACC_Data[1]; Ay = (int) (ACC_Data[2] << 8) + ACC_Data[3]; Az = (int) (ACC_Data[4] << 8) + ACC_Data[5]; } } 2.3 LSM303DLH magnetic sensor data acquisition The 7-bit I2C slave address for the magnetic sensor is 0011110b or 0x1E. For electronic compass applications, a full scale range of ±1.3 gauss and ODR at 30Hz should be sufficient. After power-on of the LSM303DLH, two registers CRA_REG_M (00h) and MR_REG_M (02h), need to be configured. Write 0x14 to the CRA_REG_M register to change the ODR from 15 Hz to 30 Hz. Write 0x00 to the MR_REG_M register to put the magnetic sensor into continuous mode from sleep mode. The following sample C code shows how to acquire magnetic sensor data: unsigned char temp, MR_Data[6]; int Mx, My, Mz; void main(void) 10/34 Doc ID 17353 Rev 1 AN3192 LSM303DLH configuration { Write(0x1E, 0x00, 0x14);//set CRA_REG_M register Write(0x1E, 0x02, 0x00);//set MR_REG_M register While (1) { Temp = Read(0x1E, 0x02); //read MR_REG_M MR_Data[0] = ReadCurrentAddress(); //read OUT_X_H_M (MSB) MR_Data[1] = ReadCurrentAddress(); //read OUT_X_L_M (LSB) MR_Data[2] = ReadCurrentAddress(); //read OUT_Y_H_M (MSB) MR_Data[3] = ReadCurrentAddress(); //read OUT_Y_L_M (LSB) MR_Data[4] = ReadCurrentAddress();//read OUT_Z_H_M (MSB) MR_Data[5] = ReadCurrentAddress(); //read OUT_Z_L_M (LSB) Mx = (int) (MR_Data[0] << 8) + MR_Data[1]; My = (int) (MR_Data[2] << 8) + MR_Data[3]; Mz = (int) (MR_Data[4] << 8) + MR_Data[5]; } } The pointer address of the magnetic sensor of the LSM303DLH has an automatic update feature. After a read of the MR_REG_M (02h) register, the address pointer automatically increases 1 to 03h which is the OUT_X_H_M register. After a read of the OUT_X_H_M register, the address pointer increases 1 to the OUT_X_L_M register. So one function ReadCurrentAddress() can be used to read 6 bytes of magnetic sensor X-Y-Z data continuously. Please refer to the LSM303DLH datasheet; Sensor module: 3-axis accelerometer and 3-axis magnetometer, for more details. Doc ID 17353 Rev 1 11/34 Calibrating the LSM303DLH AN3192 3 Calibrating the LSM303DLH 3.1 LSM303DLH accelerometer calibration All ST MEMS accelerometers are factory calibrated, allowing the user to avoid any further calibration for most of the applications now present in the market. However, to reach a heading accuracy of below 2°, an easy calibration procedure is hereafter described. After the LSM303DLH is installed in the handheld device, it is necessary to calibrate the accelerometer part again at the handheld device's manufacturers in order to determine the offset, the scale factor, and the misalignment matrix with respect to the device body axes Xb/Yb/Zb. After the device is released to the market, end users don't need to perform further accelerometer calibration in field. The relationship between the normalized Ax1, Ay1, and Az1 and the accelerometer raw measurements Ax, Ay, and Az can be expressed as, Equation 3 0 0 ⎡ A x1 ⎤ ⎡1/ A _ SC x ⎤ ⎡A x − A _ OS x ⎤ ⎢ ⎥ [ ⎢ ⎥ ⎢ ⎥ ] = A A _ m 0 1 / A _ SC 0 y 3x 3 ⎢ ⎢ y1 ⎥ ⎥ ⋅ ⎢A y − A _ OS y ⎥ ⎢⎣ A z1 ⎥⎦ ⎢⎣ 0 0 1/ A _ SC z ⎥⎦ ⎢⎣ A z − A _ OS z ⎥⎦ ⎡ ACC11 ACC12 ACC13 ⎤ ⎡A x ⎤ ⎡ ACC10 ⎤ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ = ⎢ACC 21 ACC 22 ACC 23 ⎥ ⋅ ⎢A y ⎥ + ⎢ACC 20 ⎥ ⎢⎣ACC 31 ACC 32 ACC 33 ⎥⎦ ⎢⎣ A z ⎥⎦ ⎢⎣ACC 30 ⎥⎦ where [A_m] is a 3x3 misalignment matrix between the accelerometer sensing axes and the device body axes; A_SCi (i = x, y, z) is the scale factor and A_OSi is the offset. The goal of the accelerometer calibration is to determine 12 parameters from ACC10 to ACC33 so that with any given raw measurements at arbitrary positions, the normalized values can be obtained. Therefore, pitch and roll can be calculated by Equation 10 in Appendix A. The calibration can be performed at 6 stationary positions as shown in Table 2 in Appendix A. Collect 5 to 10 second accelerometer raw data at each position with known Ax1, Ay1, and Az1. Then apply the least square method to obtain the optimal 12 accelerometer calibration parameters. Please refer to Appendix B for the accelerometer calibration. In order to get better pitch/roll accuracy, the user can add 4 more stationary positions for accelerometer calibration. They are: 2 positions with Ax = 0, Ay = ±0.707g, Az = -0.707g, and 2 positions with Ax = ±0.707g, Ay = 0, Az = -0.707g. 3.2 LSM303DLH magnetic sensor calibration LSM303DLH has a magnetic field resolution of 8 mGauss at Vdd = +3 V. And the average magnitude of the horizontal magnetic field component is in about the 200 mGauss range (more at the equator, less near the magnetic poles). Therefore, the expected heading accuracy is about 2.3° [=arctan (8/200)]. At a full scale range of ±1.3 gauss, the sensitivity of the LSM303DLH's magnetic sensor is 1055 LSB/gauss for the X/Y axis and 950 LSB/gauss for the Z axis. For example in San Francisco, California, the earth’s magnetic field strength is 0.49932 gauss with a declination 12/34 Doc ID 17353 Rev 1 AN3192 Calibrating the LSM303DLH angle of 15.5 º and an inclination angle of 61.4º. Then the Zb component, in Figure 1, is 0.438 gauss. So the Mz raw measurement should be around 950 x 0.438 = +416 LSBs. The relationship between the normalized Mx1, My1, and Mz1 and the magnetic sensor raw measurements Mx, My, and Mz can be expressed as, Equation 4 0 0 ⎡M x1 ⎤ ⎡1/ M _ SC x ⎤ ⎡M x − M _ OS x ⎤ ⎢ ⎥ [ ⎢ ⎥ ⎢ ⎥ 0 1/ M _ SC y 0 ⎢M y1 ⎥ = M _ m]3 x 3 ⎢ ⎥ ⋅ [M _ si]3 x 3 ⎢M y − M _ OS y ⎥ ⎢⎣Mz1 ⎥⎦ ⎢⎣ ⎢⎣Mz − M _ OS z ⎥⎦ 0 0 1/ M _ SC z ⎥⎦ ⎡MR11 MR12 ⎢ = ⎢MR 21 MR 22 ⎢⎣MR 31 MR 32 MR13 ⎤ ⎡M x − MR10 ⎤ ⎥ ⎢ ⎥ MR 23 ⎥ ⋅ ⎢M y − MR 20 ⎥ MR 33 ⎥⎦ ⎢⎣Mz − MR 30 ⎥⎦ where [M_m] is a 3x3 misalignment matrix between the magnetic sensor sensing axes and the device body axes; M_SCi (i = x, y, z) is the scale factor and M_OSi is the offset caused by hard-iron distortion; [M_si] is a 3x3 matrix caused by soft-iron distortion. The goal of the magnetic sensor calibration is to determine the parameters from MR10 to MR33 so that with any given raw measurements at arbitrary positions, the normalized values can be obtained. Therefore, the heading can be calculated by Equation 12 and 13 in Appendix A for tilt compensation. 3.2.1 Terminology Hard-iron interference magnetic field is normally generated by ferromagnetic materials with permanent magnetic fields that are part of the handheld device structure. These materials could be permanent magnets or magnetized iron or steel. They are time invariant. These unwanted magnetic fields are superimposed on the output of the magnetic sensor measurements of the earth's magnetic field. The effect of this superposition is to bias the magnetic sensor outputs. It is described as M_OSi (i = x, y, z) or MR10, MR20 and MR30 in Equation 4. A soft-iron interference magnetic field is generated by the items inside the handheld device. They could be current carrying traces on the PCB or magnetically soft materials. They generate a time varying magnetic field that is superimposed on the magnetic sensor output in response to the earth's magnetic field. The effect of the soft-iron distortion is to make a full round rotation circle become a tilted ellipse. It is described as the [M_si] 3x3 matrix in Equation 4. Scale factor error is defined as the mismatch of the sensitivity of the magnetic sensor sensing axes. Ideally, the 3-axis magnetic sensors that make up the triad are identical. In reality, however, this may not be the case. Each magnetic sensor channel may have different sensitivities. The effect of the scale factor error causes the full round rotation circle to become an ellipse. It is described as M_SCi (i = x, y, z) in Equation 4. Misalignment error is defined as the angles between the magnetic sensor sensing axes and the device body axes. When assembling the LSM303DLH in the handheld device, these small angles always exist and need to be compensated. It is described as [M_m] 3x3 matrix in Equation 4. Magnetic sensor calibration can be performed by 3 full round rotations along with device body axis Zb down, Yb down, and Xb down respectively, at a leveled smooth surface without a nearby interference magnetic field. They are 2D rotations. The rotation speed should be Doc ID 17353 Rev 1 13/34 Calibrating the LSM303DLH AN3192 slow in order to collect as many data points as possible. But it does not require constant rotation speed and an accurate sampling time interval. The full round rotation can be clockwise or counterclockwise. Collected magnetic sensor raw data of 3 full round rotations is used to accurately determine the 12 magnetic sensor calibration parameters. 3D random rotations are performed by rotating the handheld device in random directions. If the handheld device doesn't have hard-iron and soft-iron interference magnetic fields, and the scale factor of each axis is identical, and the LSM303DLH magnetic sensor's sensing axes are aligned to the device body axes, then each full round rotation forms a centered circle with the same radius and the 3D rotations form a centered sphere. However, due to the hard-iron and soft-iron magnetic field distortions, and the errors of the scale factor and the misalignment, the centered sphere becomes a shifted, tilted ellipsoid when plotting the collected magnetic sensor raw data. The 3 steps for magnetic sensor calibration are presented in Appendix C. Please note that the magnetic sensor calibration can only compensate the hard-iron and soft-iron interference magnetic field generated by the handheld device itself. This means that during full round rotations of the calibration, the hard-iron and soft-iron fields also rotate with the device. The electronic compass is sensitive to environmental magnetic interference fields outside of the handheld device. A single Z-axis gyro chip can be used to aid the compass when environmental magnetic interference is detected. 14/34 Doc ID 17353 Rev 1 AN3192 4 Testing the electronic compass Testing the electronic compass After the calibration parameters for the accelerometer and the magnetic sensor of the LSM303DLH have been determined, it is necessary to check the performance of the electronic compass. This could be carried out with accurate lab testing and rough field testing. The expected pitch/roll/heading accuracy is shown in Table 1. Table 1. 4.1 Expected pitch/roll/heading accuracy Parameter Value Heading accuracy < 2° RMS (tilt within ±50°), range: 0° ~ 359.9° Pitch and roll accuracy < 1° RMS (tilt within ±50°), range: -90° ~ +90° Resolution 0.1° for heading, pitch, and roll Lab testing A convenient setup for accurate lab testing is a wooden platform with 3 degrees of rotation freedom, as shown in Figure 6. Figure 6. Lab testing setup +DQGKHOG GHYLFH 0DJQHWLF QRUWKPDUN :RRGHQ SODWIRUP !-V There should be no strong external interference magnetic field close to the wooden platform. CRT monitors, power cords, electrical equipment, and metallic frames, sometimes hidden in the structure below the wooden platform, have to be removed during testing. If the local magnetic north direction is known by means of a reference magnetometer, a sign can be placed on a side panel and the handheld device fixed to the top of the platform. An optical head may be used to align Xb to the mark. 4.1.1 Absolute heading testing ● Level the wooden platform ● Align the handheld device Xb axis to the local magnetic north direction mark on the wall ● Check the heading output of the electronic compass. It should be 0º ± 2º. This is the absolute heading accuracy at 0º ● Rotate the wooden platform horizontally clockwise or counterclockwise at a random angle which can be read from the marks on the platform. Then compare the compass Doc ID 17353 Rev 1 15/34 Testing the electronic compass AN3192 heading output with the known heading angle. The difference should be within ± 2º. This is the absolute heading accuracy at random position. ● 4.1.2 4.2 Fix a certain position. Rotate the platform back and forth and then stop at the same position. The repeatability error also should be within ± 2º. Tilt compensation testing ● Level the wooden platform ● Align the handheld device Xb axis to any direction. Record the compass heading output value ● Rotate the platform around the Yb axis to generate a plus or minus pitch angle which can be read from the marks on the wooden platform. Compare the compass pitch output with the known pitch angle. The difference should be within ± 1º (see Table 1). At the same time the change of the compass heading output should be within ± 2º which means the compass is tilt compensated ● Level the wooden platform ● Align the handheld device Xb axis to any direction. Record the compass heading output value ● Rotate the platform around the Xb axis to generate a plus or minus roll angle which can be read from the marks on the platform. Compare the compass roll output with the known roll angle. The difference should be within ± 1º (see Table 1). At the same time the change of the compass heading output should be within ± 2º which means the compass is tilt compensated. Field testing In any physical situations outside the lab, rough field testing can be performed. A wooden table with a smooth surface is required. The surface does not have to be leveled. Draw some lines, for example, 20º apart on a white sheet of paper as shown in Figure 7. Figure 7. Field testing setup 16/34 Doc ID 17353 Rev 1 !-V AN3192 4.2.1 4.2.2 Testing the electronic compass Relative heading testing ● Place the handheld device on the paper which is taped to the wooden table ● Align the device Xb axis to any direction ● Align the left edge of the device to any line in Figure 7 ● Record the compass heading output value ● Use one hand or both hands to rotate the device to align its left edge to any other line. Then the difference between the compass new heading output and the previous one should be the same as the difference of the line degree change. The error should be within ± 2º ● Another quick and easy way to check the relative heading accuracy is to align the left edge of the device to the wooden table edge. Record the compass heading output value. Then rotate the device and align the top edge, bottom edge, or right edge to the same table edge. The difference between the new heading output and the previous output should be either ± 90º or 180º with an error of ± 2º. Tilt compensation testing ● Place the handheld device on the wooden table with the Xb axis pointing in any direction ● Use two hands to hold the device and tilt the device along its top or bottom edge carefully ● While the compass pitch output is changing, the heading output should remain the same with an error of ± 2º. This means the compass is tilt compensated ● Use two hands to hold the device and tilt the device along its left or right edge ● While the compass roll output is changing, the heading output should remain the same with an error of ± 2º. This means the compass is tilt compensated. Doc ID 17353 Rev 1 17/34 Firmware implementation 5 AN3192 Firmware implementation Some microcontrollers may not support floating point operation and are timing critical. In order to build a standalone tilt compensated electronic compass, the following recommendations may be helpful: 18/34 ● Use look-up tables for sin, cos, arcsin, and arctan functions to reduce clock cycles ● Use assembly code to implement signed integer multiplication and division subroutines to reduce clock cycles ● If some sensor calibration parameters are very small, the user can multiply the whole accelerometer and magnetic sensor calibration parameter matrix with a big constant integer, then divide it before the pitch/roll/heading calculation ● Use internal EEPROM to save sensor calibration parameters ● Implement some kind of digital filtering or simple moving average function onto the sensor raw measurements to reduce the noise level and improve the pitch/roll/heading accuracy Doc ID 17353 Rev 1 AN3192 LSM303DLH pitch/roll/heading calculation Appendix A LSM303DLH pitch/roll/heading calculation Assume that the LSM303DLH chip has been installed in a handheld device, such as a cell phone, or a PDA or simply on a PCB board as shown in Figure 8. Figure 8. Electronic compass coordinate system ;E 5ROO =$0 /60'/+ ;$0 3LWFK <$0 <E +HDGLQJ +DQGKHOGGHYLFH =E !-V Xb, Yb, and Zb are the handheld device body axes with forward-right-down configuration. XA,M, YA,M, and ZA,M are the accelerometer and magnetic sensor sensing axes, respectively. Note that the sign of YA,M and ZA,M from the sensor measurements needs to be reversed to make the sensing axes the same direction as the device body axes. Pitch/roll/heading angles are referenced to the local horizontal plane which is perpendicular to the earth's gravity. ● Heading (ψ) or azimuth is defined as the angle with respect to the magnetic north pole. It is always positive from 0º to +359º when rotating around the Zb axis clockwise topview with right-hand rule. If the heading with respect to the geographic north is required, then the declination angle at the user's current geographic location should be added or subtracted from the magnetic heading ● Pitch (ρ) is defined as the angle between the Xb axis and the horizontal plane. It goes from 0º to +90º when rotating around the Yb axis with the Xb axis moving upwards. When the Xb axis is moving downwards, the pitch angle goes from 0º to -90º ● Roll (γ) is defined as the angle between the Yb axis and the horizontal plane. It goes from 0º to +90º when rotating around the Xb axis with the Yb axis moving downwards. When the Yb axis is moving upwards, the roll angle goes from 0º to -90º. Let Ax, Ay, Az be the accelerometer raw measurement, and Mx, My, Mz the magnetic sensor raw measurement. Table 2 shows the sign definition of the sensor raw data at 6 stationary positions. For example, in Figure 8, Xb and Yb are level, Zb is pointing down. Therefore, Ax = Ay = 0, Az = +1g. Mx and My can be positive (+) or negative (-), while Mz should be positive if the current location is in the northern hemisphere. Doc ID 17353 Rev 1 19/34 LSM303DLH pitch/roll/heading calculation Table 2. Sign definition of LSM303DLH sensor raw measurements Accelerometer (signed integer) Stationary position A.1 AN3192 Magnetic sensor (signed integer) Ax Ay Az Mx My Mz Zb down 0 0 +1g + or - + or - + Zb up 0 0 -1g + or - + or - - Yb down 0 +1g 0 + or - + + or - Yb up 0 -1g 0 + or - - + or - Xb down +1g 0 0 + + or - + or - Xb up -1g 0 0 - + or - + or - Pitch/roll calculation When the device is at an arbitrary 3D position X'b, Y'b, and Z'b, there are a few rotation procedures to rotate the device from the local level frame Xb, Yb, and Zb, shown in Figure 8, to that 3D position. Different rotation procedures result in different rotation matrix. The aircraft convention of angle rotation is used in this application note. Firstly, rotate the handheld device around the Zb axis clockwise at an angle (ψ) with the view from the origin to downwards. Then rotate the device around Yb at an angle (ρ) with Xb moving upwards. Then rotate the device around Xb at an angle (γ) with Yb moving downwards. The new device body axes become X'b, Y'b, and Z'b, as shown in Figure 9. Figure 9. Rotation procedures ;E +HDGLQJ 3LWFK ; 5ROO ;;¶E ; ; <E < << < =E= <¶E = = = =¶E !-V Then each rotation matrix is: Equation 5 ⎡ cos ψ sin ψ 0⎤ ⎢ ⎥ R ψ = ⎢− sin ψ cos ψ 0⎥ ⎢⎣ 0 0 1⎥⎦ 20/34 Doc ID 17353 Rev 1 AN3192 LSM303DLH pitch/roll/heading calculation Equation 6 ⎡cos ρ 0 − sin ρ⎤ ⎢ ⎥ Rρ = ⎢ 0 1 0 ⎥ ⎢⎣ sin ρ 0 cos ρ ⎥⎦ Equation 7 0 0 ⎤ ⎡1 ⎢ ⎥ R γ = ⎢0 cos γ sin γ ⎥ ⎢⎣0 − sin γ cos γ ⎥⎦ And the relationship between X'b/Y'b/Z'b and Xb/Yb/Zb is: Equation 8 ⎡X'b ⎤ ⎡X b ⎤ ⎢ ⎥ ⎢ ⎥ ⎢Y'b ⎥ = R γRρR ψ ⎢ Yb ⎥ ⎢⎣ Z'b ⎥⎦ ⎢⎣ Z b ⎥⎦ cos ρ cos ψ cos ρ sin ψ − sin ρ ⎤ ⎡X b ⎤ ⎡ ⎥ ⎢ ⎥ ⎢ = ⎢cos ψ sin ρ sin γ − cos γ sin ψ cos γ cos ψ + sin ρ sin γ sin ψ cos ρ sin γ ⎥ ⋅ ⎢ Yb ⎥ ⎢⎣cos ψ sin ρ cos γ + sin γ sin ψ − sin γ cos ψ + sin ρ cos γ sin ψ cos ρ cos γ ⎥⎦ ⎢⎣ Z b ⎥⎦ In the local horizontal plane, as shown in Figure 8, Xb = Yb = 0, Zb = +1g. At X'b/Y'b/Z'b, the LSM303DLH accelerometer raw measurements are Ax, Ay, and Az which are signed integer in terms of LSBs. Let Ax1, Ay1, and Az1 be the normalized values after applying accelerometer calibration parameters into Ax, Ay, and Az. So Ax1, Ay1, and Az1 become floating point values less than 1 in terms of g (earth gravity), and the root sum of their squared values should be equal to 1 when the accelerometer is still. Then Equation 8 becomes: Equation 9 cos ρ cos ψ cos ρ sin ψ − sin ρ ⎤ ⎡0⎤ ⎡ A x1 ⎤ ⎡ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ A cos sin sin cos sin cos cos sin sin sin cos = ψ ρ γ − γ ψ γ ψ + ρ γ ψ ρ sin γ ⎥ ⋅ ⎢0⎥ ⎢ y1 ⎥ ⎢ ⎢⎣ A z1 ⎥⎦ ⎢⎣cos ψ sin ρ cos γ + sin γ sin ψ − sin γ cos ψ + sin ρ cos γ sin ψ cos ρ cos γ ⎥⎦ ⎢⎣ 1⎥⎦ Therefore, pitch and roll angle can be calculated as: Equation 10 Pitch = ρ = arcsin(− A x1 ) Roll = γ = arcsin(A y1 / cos ρ) Note: When pitch = ±90º, roll should be set to 0º to avoid singularity The arcsin function has good linearity between about -45º to +45º, so the accuracy of the pitch and roll calculation degrades when tilt angles exceed this range Normalized accelerometer measurement Az1 is not used for the pitch and roll calculation. But it can be used to check if the magnitude A = A 2x1 + A 2y1 + A 2z1 is equal to 1. If not, then it means linear acceleration or angular acceleration is detected. Doc ID 17353 Rev 1 21/34 LSM303DLH pitch/roll/heading calculation A.2 AN3192 Heading calculation For the heading calculation, 3-axis magnetic sensor measurements need to be normalized by applying magnetic sensor calibration parameters and then reflected onto the horizontal plane by tilt compensation, as shown in Figure 10. Figure 10. Heading calculation ;E (ISTHEHORIZONTALCOMPONENTOFTHE 0[ %ARTHSMAGNETICFIELDTOGETHERWITHTHE NEARBYINTERFERENCEMAGNETICFIELD ȥ + =EPOINTINGINTOPAGE 0\ <E !-V If the device rotates from Xb/Yb/Zb to X''b/Y''b/Z''b by roll angle rotation followed by pitch angle rotation, then Equation 11 0 sin ρ ⎡X b ⎤ ⎡X' 'b ⎤ ⎡ cos ρ ⎤ ⎡X' 'b ⎤ ⎢ ⎥ ⎥ ⎢ ⎥ ⎢ ⎥ −1 −1 ⎢ Y R R Y ' ' sin sin cos sin cos = = γ ρ γ − γ ρ γ ρ ⎢ b⎥ ⎢ b⎥ ⎢ ⎥ ⋅ ⎢Y' 'b ⎥ ⎢⎣ Z b ⎥⎦ ⎢⎣ Z' 'b ⎥⎦ ⎢⎣− cos γ sin ρ sin γ cos γ cos ρ ⎥⎦ ⎢⎣ Z' 'b ⎥⎦ Let Mx1, My1, and Mz1 be the normalized magnetic sensor measurements after applying calibration parameters correction into magnetic sensor raw measurements Mx, My, and Mz at new positions X''b/Y''b/Z''b. Mx, My and Mz which are signed integer in terms of LSBs, while Mx1, My1, and Mz1 are floating point values less than 1 in terms of the magnetic field strength, and the square root of the sum squared values should be equal to 1 when the there is no external interference magnetic field. Then from Equation 11, tilt compensated magnetic sensor measurements Mx2, My2, and Mz2 can be obtained as: Equation 12 M x 2 = M x1 cos ρ + Mz1 sin ρ M y 2 = M x1 sin γ sin ρ + M y1 cos γ − Mz1 sin γ cos ρ Mz 2 = −M x1 cos γ sin ρ + M y1 sin γ + Mz1 cos γ cos ρ 22/34 Doc ID 17353 Rev 1 AN3192 LSM303DLH pitch/roll/heading calculation Therefore, Equation 13 ⎛M ⎞ Heading = ψ = arctan⎜ y 2 for M x2 > 0 and M y2 >= 0 ⎟ M x 2 ⎝ ⎠ ⎛M ⎞ = 180° + arctan⎜ y 2 ⎟ for M x2 < 0 M x 2 ⎝ ⎠ ⎛My2 ⎞ = 360° + arctan⎜ ⎟ for M x2 > 0 and M y2 <= 0 M x 2 ⎝ ⎠ = 90° for M x2 = 0 and M y2 < 0 = 270° for M x2 = 0 and M y2 > 0 The magnitude M = M2 + M2 + M2 should also be equal to 1. If not, it means that the x2 y2 z2 external magnetic interference field is detected or a pitch/roll error is present. Because the accelerometer measurements cannot distinguish earth’s gravity from linear acceleration or angular acceleration, fast motion causes pitch/roll calculation error which then directly introduces an error to the heading calculation. In most cases, the fast motion doesn't last long and the device goes back to a stationary position. So the heading accuracy in an electronic compass means static accuracy when the device is still or without acceleration. Doc ID 17353 Rev 1 23/34 Accelerometer calibration method Appendix B AN3192 Accelerometer calibration method Let's consider accelerometer calibration at 6 stationary positions, as shown in Table 2. Equation 3 can be rewritten as: Equation 14 [A x1 ] [ A y1 A z1 = A x Ay Az ⎡ ACC11 ⎢ ACC12 1⋅⎢ ⎢ACC13 ⎢ ⎣ACC10 ACC 21 ACC 22 ACC 23 ACC 20 ] ACC 31 ⎤ ⎥ ACC 32 ⎥ ACC 33 ⎥ ⎥ ACC 30 ⎦ Or Equation 15 Y = w⋅X Where, Matrix X is the 12 calibration parameters that need to be determined Matrix w is sensor raw data LSBs collected at 6 stationary positions Matrix Y is the known normalized earth gravity vector For example, ● At Zb down position (P1 position), [A x1 A y1 A z1 ] = [0 0 1] and assume that at Zb down position, n1 sets of accelerometer raw data Ax, Ay, and Az have been collected. Then, Equation 16 Y1 = [0 0 1]n1x 3 [ ] w1 = A xP1 A yP1 A zP1 1 n1x 4 Where, The matrix Y1 has the same row of [ 0 0 1 ] The matrix W1 contains raw data in the format of LSBs ● At Zb up position (P2 position), [A x1 A y1 A z1 ] = [0 0 − 1] and assume that at Zb up position, n2 sets of accelerometer raw data Ax, Ay, and Az have been collected. Then, Equation 17 Y2 = [0 0 − 1]n2 x 3 [ w 2 = A xP2 ● A yP2 ] A zP2 1 n2 x 4 At Yb down position (P3 position), [A x1 A y1 A z1 ] = [0 1 0] and assume that at Yb down position, n3 sets of accelerometer raw data Ax, Ay, and Az have been collected. Then, Equation 18 Y3 = [0 1 0]n3 x 3 [ w 3 = A xP3 24/34 A yP3 Doc ID 17353 Rev 1 ] A zP3 1 n3 x 4 AN3192 Accelerometer calibration method ● At Yb up position (P4 position), [A x1 A y1 A z1] = [0 − 1 0] and assume that at Yb up position, n4 sets of accelerometer raw data Ax, Ay, and Az have been collected. Then, Equation 19 Y4 = [0 − 1 0]n4 x 4 [ w 4 = A xP4 ● ] A XP4 1 n4 x 4 At Xb down position (P5 position), [Ax1 A y1 Az1 ] = [1 0 0 ] and assume that at Xb down position, n5 sets of accelerometer raw data Ax, Ay, and Az have been collected. Then, Equation 20 Y5 = [1 0 0]n5 x 3 [ w 5 = A xP5 ● A yP4 A yP5 ] A zP5 1 n5 x 4 At Xb up position (P6 position), [A x1 A y1 A z1 ] = [− 1 0 0] and assume that at Xb up position, n6 sets of accelerometer raw data Ax, Ay, and Az have been collected. Then, Equation 21 Y6 = [− 1 0 0]n6 x 3 [ w 6 = A xP6 A yP6 ] A zP6 1 n6 x 4 Combine Equation 16 with Equation 21 and let n = n1 + n2 + n3 + n4 + n5 + n6, then Equation 15 becomes, Equation 22 Ynx 3 = w nx 4 ⋅ X 4 x 3 Where, Equation 23 ⎡Y1 ⎤ ⎢ ⎥ ⎢Y2 ⎥ ⎢Y3 ⎥ Y=⎢ ⎥ ⎢Y4 ⎥ ⎢Y ⎥ ⎢ 5⎥ ⎣⎢Y6 ⎦⎥ nx 3 ⎡w1 ⎤ ⎢ ⎥ ⎢w 2 ⎥ ⎢w 3 ⎥ w=⎢ ⎥ ⎢w 4 ⎥ ⎢w ⎥ ⎢ 5⎥ ⎢⎣w 6 ⎥⎦ nx 4 Therefore, the calibration parameter matrix X can be determined by the least square method as, Doc ID 17353 Rev 1 25/34 Accelerometer calibration method AN3192 Equation 24 [ X = wT ⋅ w ] −1 ⋅ wT ⋅ Y Where, means matrix transpose wT [w 26/34 T ⋅w ] −1 means matrix inverse Doc ID 17353 Rev 1 AN3192 Magnetic sensor calibration method Appendix C C.1 Magnetic sensor calibration method Step 1: Soft-iron effect verification It is always good to know if the device has soft-iron interference before choosing which model for the identification of the calibration parameters, tilted ellipsoid, or non-tilted ellipsoid. This can be done by performing 3D rotations in a clean environmental area. Then plot the collected magnetic sensor raw data to check if there is a soft-iron interference field inside the device. This set of data is not used for the following magnetic sensor calibration. However, in practical situations, three 2D full round rotations may not be easy to perform. Then an amount of 3D rotations data can be used for rough field calibration. An example plot of 3D rotations data and three 2D full round rotations data is shown in Figure 11. It is obvious that this electronic compass has a built-in soft-iron effect. Figure 11. 3D rotations plus three 2D full round rotations C.2 Step 2: Hard-iron, soft-iron and scale factor compensation If there is soft-iron distortion, the 3D rotations show a tilt ellipsoid which can be described as the following equation: Equation 25 (x − x 0)2 + (y − y0)2 + (z − z0)2 + (x − x 0)(y − y0) + (x − x 0)(z − z0) + (y − y0)(z − z0) = R2 a2 b2 c2 d2 e2 f2 where: ● x0, y0, z0 are the offsets M_OSi (i = x, y, z) caused by hard-iron distortion ● x, y, z are magnetic sensor raw data Mx, My and Mz ● a, b, c are the semi-axes lengths, ● d, e, f are cross axis effect to make the ellipsoid tilted, ● R is a constant of the earth’s magnetic field strength. Doc ID 17353 Rev 1 27/34 Magnetic sensor calibration method AN3192 If there is no soft-iron distortion inside the device, or the soft-iron effect is very small and can be ignored, then the ellipsoid from 3D rotations is not tilted. So the soft-iron matrix [M_si] is a 3x3 identity matrix and Equation 25 can be simplified as: Equation 26 (x − x 0)2 + (y − y0)2 + (z − z0)2 a 2 b 2 c 2 = R2 Therefore, the least square fitting ellipsoid method can be used to discover the parameters of M_SCi, M_OSi (i = x, y, z) and [M_si]. The magnetic sensor raw data used here could be three 2D full round rotations, or 3D rotations, or both. Applying these parameters to the collected 3D rotations data and three 2D full round rotations, the plot is shown in Figure 12. Now the shifted tilted ellipsoid becomes a centered unit sphere. Figure 12. After hard-iron, soft-iron, and scale factor compensation Let's assume there is no soft-iron distortion. The soft-iron matrix [M_si] is a 3x3 identity matrix. Then Equation 26 can be rewritten as: Equation 27 [ x2 = x y z − y2 − z2 2x 0 ⎡ ⎤ ⎢ ⎥ a2 ⎢ ⎥ 2y 0 2 ⎢ ⎥ b 2 ⎢ ⎥ a 2z 0 ⎢ ⎥ 2 c ⎢ ⎥ 2 ⎥ 1⋅⎢ a ⎢ ⎥ 2 b ⎢ ⎥ 2 a ⎢ ⎥ ⎢ ⎥ 2 c ⎢ ⎥ 2 2 ⎢a 2R 2 − x 2 − a y 2 − a z 2 ⎥ 0 0 0 ⎢⎣ b2 c 2 ⎥⎦ ] After three 2D full round rotations magnetic sensor raw data have been collected, it is possible to combine Mx, My, and Mz as column vector and row vector. Then Equation 27 becomes, 28/34 Doc ID 17353 Rev 1 AN3192 Magnetic sensor calibration method Equation 28 w nx1 = [H]nx 6 ⋅ X 6 x1 The least square method can be applied to determine the parameters X vector as: Equation 29 [ ] −1 X = H T H HT ⋅ w Then, Equation 30 M _ OS x = x 0 = X(1) / 2 M _ OS y = y 0 = X(2) /(2 ⋅ X(4)) M _ OS z = z 0 = X(3) /(2 ⋅ X(5)) And, Equation 31 A = a 2R 2 = X(6) + x 20 + X(4) ⋅ y 20 + X(5) ⋅ z 20 B = A / X(4) C = A / X(5) Let, Equation 32 xx = M x − M _ OS x yy = M y − M _ OS y zz = Mz − M _ OS z Then Equation 26 becomes, Equation 33 xx 2 yy 2 zz 2 + + =1 A B C Therefore, Equation 34 M _ SC x = A M _ SC y = B M _ SC z = C Doc ID 17353 Rev 1 29/34 Magnetic sensor calibration method AN3192 Let, Equation 35 xxx = xx / M _ SC x yyy = yy / M _ SC y zzz = zz / M _ SC z Then, Equation 36 xxx 2 + yyy 2 + zzz 2 = 1 Up to now, M_SCi (i = x, y, z), the scale factor, M_OSi, the offset caused by hard-iron distortion, and [M_si], 3x3 matrix caused by soft-iron distortion have been determined. Applying these parameters to three 2D full round rotations and 3D random rotations, the 3D random rotations become a centered unit sphere with three 2D full round rotations sitting on its surface. Similarly, the least square method can be used to determine the [M_si] 3x3 matrix when there is soft-iron distortion. C.3 Step 3: Misalignment error compensation Misalignment error compensation is to align the magnetic sensor sensing axes to the device body axes based on three 2D full round rotations, as shown in Figure 12. The vector dotproduct method can be used to find each normalized vector that rotates, corrected, three 2D full round rotation circles to their corresponding body axes. The normalized vector means the magnitude is equal to 1. These 3 vectors form the [M_m] 3x3 misalignment matrix described in Equation 4. Applying the [M_m] 3x3 misalignment matrix to the above unit sphere and three 2D circles, the plot is shown in Figure 13. Now three 2D full round rotations are aligned to the device body axes. For example, the red color Zb down rotation is parallel to Xb - Yb plane. Let's consider the Zb down rotation. After hard-iron and scale factor correction, the rotation becomes a centered circle. If the circle is aligned to the device Zb axis, then all zzzz values of the Zb down rotation circle should be the same, less than 1. If it is not the case, then a vector can be found to rotate this circle to align to the device Zb axis. Let, Equation 37 Hmx 3 = [xxx z yyy z zzz z ] be the Zb down rotation circle data after scale factor, hard-iron and soft-iron correction. Equation 38 w mx1 = (xxx 2 z + yyy 2z + zzz 2z Then, Equation 39 [ ] −1 X 3 x1 = H T H H T ⋅ w 30/34 Doc ID 17353 Rev 1 ) AN3192 Magnetic sensor calibration method So the normalized rotation vector for Zb down rotation is, Equation 40 Rz = X / (X(1) 2 + X(2)2 + X(3)2 ) Similarly, the normalized rotation vectors Rx and Ry for Xb down rotation and Yb down rotation can be found. Then the final misalignment compensation matrix is, Equation 41 [ M _ m3x 3 = R x Ry Rz ] Figure 13. After misalignment compensation Doc ID 17353 Rev 1 31/34 References 6 AN3192 References 1. STMicroelectronics, Inc. LSM303DLH sensor module datasheet http://www.st.com/stonline/products/families/sensors/motion_sensors/lsm303dlh.htm 2. Honeywell, Inc. Applications of Magnetoresistive Sensors in Navigation Systems 3. Honeywell, Inc. Applications of Magnetic Sensors for Low Cost Compass Systems http://www.ssec.honeywell.com/position-sensors/datasheets/sae.pdf http://www.ssec.honeywell.com/magnetic/datasheets/lowcost.pdf 32/34 Doc ID 17353 Rev 1 AN3192 7 Revision history Revision history Table 3. Document revision history Date Revision 02-Aug-2010 1 Changes Initial release. Doc ID 17353 Rev 1 33/34 AN3192 Please Read Carefully: Information in this document is provided solely in connection with ST products. STMicroelectronics NV and its subsidiaries (“ST”) reserve the right to make changes, corrections, modifications or improvements, to this document, and the products and services described herein at any time, without notice. All ST products are sold pursuant to ST’s terms and conditions of sale. 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