AN4508 Application note Parameters and calibration of a low-g 3-axis accelerometer Introduction This application note describes the parameters and calibration of a low-g 3-axis accelerometer. In general, the procedures described here may also be applied to 3-axis analog or digital accelerometers, depending on their respective specifications. Section 1 of this application note introduces the terminology and parameters related to the accelerometer, while Section 2 presents the accelerometer calibration techniques. June 2014 DocID026444 Rev 1 1/13 www.st.com Contents AN4508 Contents 1 2 Terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.1 Accelerometer datasheet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.2 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Calibrating the accelerometer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Appendix A Least square method. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 Revision history . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2/13 DocID026444 Rev 1 AN4508 List of figures List of figures Figure 1. Figure 2. Figure 3. Accelerometer inside a handheld device . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Pitch definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Roll definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 DocID026444 Rev 1 3/13 13 Terminology AN4508 1 Terminology 1.1 Accelerometer datasheet The first step is to examine the accelerometer specifications and understand the meaning of each parameter. Understanding the parameters 4/13 Vdd - Power supply: This parameter defines the accelerometer operating DC power supply. Correct operation of the accelerometer using a power supply voltage outside of this range is not guaranteed. The parameters are provided by the accelerometer manufacturer under Vdd = +2.5 V at a room temperature of T = 25 °C. It is recommended to keep Vdd clean, with minimum ripple. One possible way to do this is to use an ultra-low-noise low-dropout regulator to power the accelerometer. Idd - Current consumption in normal mode: lower ODR corresponds to lower current consumption. ODR - Output data rate in normal mode: This parameter shows the possible output data rates in normal mode from which the user may select. BW - System bandwidth: This parameter defines the bandwidth of the system. When ODR = 100 Hz, BW is typically 50 Hz with a built-in low pass filter. The system recognizes any motion below 50 Hz. If the system has dynamic motion higher than 50 Hz, then the ODR needs to be increased to a higher setting in order to cover all useful system signals. Ton - Turn-on time: This parameter defines the time required before the accelerometer is ready to output measured acceleration data after exiting power-down mode. Top - operating temperature range: This parameter defines the operating temperature range. When the device is operated inside the specified range, proper behavior of the sensor is guaranteed. FS - Full-scale measurement range: For tilt sensing applications, a ±2.0 g range is sufficient because the Earth’s gravity is ±1 g only. If the application requires measurement of higher g acceleration, the user can set the device to a higher full-scale range which results in lower sensitivity. So - Sensitivity: This parameter defines the value of 1 LSB with respect to mg in the digital representation. TCSo - Sensitivity change vs. temperature: This parameter defines how sensitivity changes with temperature. For example, at a ±2.0 g full-scale range, the sensitivity changes within ±0.01%/°C. Therefore, if the environmental temperature changes 40 °C, from 25 °C to 65 °C, then the sensitivity changes within the range of ±0.01% * 40 = ±0.4%, which means the sensitivity change over 40 °C is within 0.996 mg/LSB and 1.004 mg/LSB, which shows that the sensitivity is very stable versus temperature change. Thus, temperature compensation for sensitivity can be ignored. TyOff - Typical zero-g level offset accuracy: This parameter defines the zero-g accuracy at room temperature of 25 °C. For example, at a ±2.0 g full-scale range, the DocID026444 Rev 1 AN4508 Terminology 1.2 zero-g accuracy of ±20 mg means that the zero-g output varies typically in the range of ±20 mg around the expected ideal value. TCOff - Zero-g level change vs. temperature: This parameter defines how much the zero-g level is affected by temperature variations. An - Acceleration noise density: This parameter defines the standard resolution the user can obtain from the accelerometer (once the desired BW is selected). 1 resolution = A n g Hz BW Hz . The higher BW leads to lower resolution. NL - Non linearity: This parameter defines the maximum error between the outputs and the best-fit straight line. For example, at ±2.0 g full-scale range, the non-linearity of 0.5% of FS means the largest error is 0.5% * 4000 mg = 2 mg, which corresponds to 0.1°. When the application requires measurements around the 0 g condition (as with tilt measurement), the non-linearity effect is negligible and can be ignored. CrossAx - Cross-axis sensitivity: The cross-axis effect arises due to natural misalignment of die positioning on the package substrate. Even if negligible in most applications, for very accurate tilt measurement the cross-axis sensitivity effect can be easily compensated for by following the procedure in Section 2: Calibrating the accelerometer. Moreover, when the device is placed on the final application board, the accelerometer calibration compensates both the device cross-axis sensitivity, and the misalignment between the accelerometer sensing axes and the board axes. Definitions Assume that the accelerometer is installed in a handheld device, such as a cell phone, a PDA or simply on a PCB board as shown in Figure 1. Figure 1. Accelerometer inside a handheld device ;E 5ROO =$ ;$ 3LWFK <$ <E +DQGKHOGGHYLFH =E $0Y Xb, Yb and Zb are the handheld device body axes with a forward-right-down configuration. XA, YA and ZA are the accelerometer sensing axes, respectively. Note that the sign of YA and ZA from the sensor measurements need to be reversed to have the sensing axes in the same direction as the device body axes. Pitch and roll angles are referenced to the local horizontal plane, which is perpendicular to the Earth's gravity. Pitch () is defined as the angle between the Xb axis and the horizontal plane. Assume that the pitch angle resolution is 0.1°, then it goes from 0° to +179.9° when rotating around the Yb axis with the Xb axis moving upwards from a flat level, and then keeps DocID026444 Rev 1 5/13 13 Terminology AN4508 moving from a vertical position (+90°) back to a flat level again. The pitch angle goes from 0° to -180° when the Xb axis is moving downwards from a flat level, and then keeps moving from a vertical position (-90°) back to a flat level again. For example, in Figure 2, Yb is fixed, Xb is rotating from Pitch = 0° to +30°, +90°, +150° and +179.9° for a positive direction. Roll () is defined as the angle between the Yb axis and the horizontal plane. Assume that the roll angle resolution is 0.1°, then it goes from 0° to +179.9° when rotating around the Xb axis with the Yb axis moving downwards from a flat level, and then keeps moving from a vertical position (+90°) back to a flat level again. The roll angle goes from 0° to -180° when the Yb axis is moving upwards from a flat level, and then keeps moving from a vertical position (-90°) back to a flat level again. For example, in Figure 3, Xb is fixed, Yb is rotating from roll = 0° to +30°, +90°, +150° and +179.9° for a positive direction. Figure 2. Pitch definition 3LWFK ;E 3LWFK 3LWFK 3LWFK <E 3LWFK 3LWFK 3LWFK 3LWFK $0Y Figure 3. Roll definition 5ROO 3LWFK ;E 5ROO 5ROO <E 5ROO 5ROO 5ROO 3LWFK 5ROO $0Y Assume Ax, Ay, Az is the accelerometer raw measurement in the format of LSBs. Table 1 shows the sign definition of the raw sensor data at 6 stationary positions with respect to the known Earth gravity vector. For example, in Figure 1, Xb and Yb are level and Zb is pointing down. Therefore, Ax = Ay = 0, Az = +1 g. 6/13 DocID026444 Rev 1 AN4508 Terminology Table 1. Sign definition of sensor raw measurements Accelerometer (signed integer) Stationary position Ax Ay Az Zb down 0 0 +1 g Zb up 0 0 -1 g Yb down 0 +1 g 0 Yb up 0 -1 g 0 Xb down +1 g 0 0 Xb up -1 g 0 0 DocID026444 Rev 1 7/13 13 Calibrating the accelerometer 2 AN4508 Calibrating the accelerometer Section 1 describes the accelerometer parameters, the next step is to calibrate the accelerometer. Please note that all accelerometers from ST have been factory-calibrated. For most applications, such as screen portrait/landscape rotation and laptop lid open/close detection, accelerometer calibration is not necessary. This means that users can use the zero-g level and sensitivity parameters from the datasheet directly to convert raw measurements Ax, Ay and Az to normalized measurements Ax1, Ay1 and Az1. For applications that require better than 1° tilt-measurement accuracy, such as automobile alert systems, tilt-compensated electronic compasses and level monitoring systems, accelerometer calibration is recommended. The relationship between the normalized Ax1, Ay1 and Az1 and the accelerometer raw measurements Ax, Ay and Az can be expressed as, Equation 1 where [A_m] is the 3 x 3 misalignment matrix between the accelerometer sensing axes and the device body axes, A_SCi (i = x, y, z) is the sensitivity (or scale factor) and A_OSi is the zero-g level (or offset). The goal of accelerometer calibration is to determine 12 parameters from ACC10 to ACC33, so that with any given raw measurements at arbitrary positions, the normalized values Ax1, Ay1 and Az1 can be obtained, resulting in: Equation 2 Calibration can be performed at 6 stationary positions as shown in Table 1. Collect 5 to 10 seconds of accelerometer raw data with ODR = 100 Hz at each position with known Ax1, Ay1 and Az1. Then apply the least square method to obtain the 12 accelerometer calibration parameters. Refer to Appendix A for additional details. 8/13 DocID026444 Rev 1 AN4508 Least square method Appendix A Least square method Let's consider accelerometer calibration at the 6 stationary positions shown in Table 1. Equation 1 can be rewritten as: Equation 3 Or Equation 4 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), , and assume that at Zb down position, n1 sets of accelerometer raw data Ax, Ay and Az have been collected. Then, Equation 5 where: Matrix Y1 has the same row of [0 0 1]. Matrix w1 contains raw data in the format of LSBs. At Zb up position (P2 position), , and assume that at Zb up position, n2 sets of accelerometer raw data Ax, Ay and Az have been collected. Then, Equation 6 At Yb down position (P3 position), , and assume that at Yb down position, n3 sets of accelerometer raw data Ax, Ay and Az have been collected. Then, Equation 7 DocID026444 Rev 1 9/13 13 Least square method AN4508 At Yb up position (P4 position), , and assume that at Yb up position, n4 sets of accelerometer raw data Ax, Ay and Az have been collected. Then, Equation 8 At Xb down position (P5 position), , and assume that at Xb down position, n5 sets of accelerometer raw data Ax, Ay and Az have been collected. Then, Equation 9 At Xb up position (P6 position), , and assume that at Xb up position, n6 sets of accelerometer raw data Ax, Ay and Az have been collected. Then, Equation 10 Combine Equation 5 to 10 and let n = n1 + n2 + n3 + n4 + n5 + n6, then Equation 4 becomes: Equation 11 where: Equation 12 10/13 DocID026444 Rev 1 AN4508 Least square method Therefore, the calibration parameter matrix X can be determined by the least square method as: Equation 13 where: Equation 14 means matrix transpose means matrix inverse DocID026444 Rev 1 11/13 13 Revision history AN4508 Revision history Table 2. Document revision history 12/13 Date Revision 10-Jun-2014 1 Changes Initial release. DocID026444 Rev 1 AN4508 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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