june 2014 docid026445 rev 1 1/10 AN4509 application note tilt measurement using a low-g 3-axis accelerometer introduction this application note describes tilt sensing theory and the methods of determining tilt angle measurement 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. www.st.com
contents AN4509 2/10 docid026445 rev 1 contents 1 calculating tilt angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.1 theory of operation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.2 tilt sensing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.2.1 single-axis tilt sensing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.2.2 dual-axis tilt sensing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.2.3 tri-axis tilt sensing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2 revision history . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
docid026445 rev 1 3/10 AN4509 list of figures 10 list of figures figure 1. tilt measurement using a single axis of the accelerometer . . . . . . . . . . . . . . . . . . . . . . . . . 4 figure 2. 360 rotation of a single axis of the accelerometer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 figure 3. plot of 360 rotation of a single axis of the accelerometer . . . . . . . . . . . . . . . . . . . . . . . . . . 5 figure 4. tilt sensitivity of a dual-axis accelerometer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .6 figure 5. tilt angles from a tri-axis accelerometer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .7 figure 6. tilt sensitivity of a tri-axis accelerometer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
calculating tilt angles AN4509 4/10 docid026445 rev 1 1 calculating tilt angles low- g mems accelerometers are widely used for tilt sensing in consumer electronics and industrial applications, such as screen rotation and automobile security alert systems. another popular application for low- g accelerometers is tilt-compensated electronic compasses for map rotation and personal navigation devices. this application note describes how to obtain accurate tilt measurements with respect to local earth horizontal plane, by compensating for a few non idealities that may cause angular tilt calculation error. for detailed information and device specifications, refer to the respective accelerometer datasheet available at http://www.st.com . in general, 3-axis analog or digital accelerometers may also be used, in accordance with their respective specifications. 1.1 theory of operation figure 1 shows the single sensing axis of the accelerometer for tilt measurement. figure 1. tilt measurement using a single axis of the accelerometer the accelerometer measures the projection of the gravity vector on the sensing axis. the amplitude of the sensed acceleration changes according to the sine of the angle ? between the sensing axis and the horizontal plane. equation 1 using equation 1 , it is possible to estimate the tilt angle, equation 2 where: ? a = acceleration measured ? g = earth?s gravity vector $ 0 y j $ . + r u l ] r q w d o s o d q h 6 h q v l q j $ [ l v r i w k h d f f h o h u r p h w h u . .
docid026445 rev 1 5/10 AN4509 calculating tilt angles 10 a single axis of the accelerometer with 360 rotation is shown in figure 2 and 3 . figure 2. 360 rotation of a single axis of the accelerometer figure 3. plot of 360 rotation of a single axis of the accelerometer 1.2 tilt sensing 1.2.1 single-axis tilt sensing from figure 2 and 3 , it can be observed that the sensor is most responsive to changes in tilt angle when the sensing axis is perpendicular to the force of gravity. in this case, the sensitivity is approximately 17.45 m g / [= sin(1) - sin(0)]. due to the derivate function of the sine function, the sensor has lower sensitivity (less responsive to tilt angle changes) when the sensing axis is close to its +1 g or -1 g position. in this case, sensitivity is only 0.15 mg/ [= sin(90) - sin(89)]. table 1 shows the sensitivity at different tilt angles. in other words, the sine function has good linearity at [0 45], [135 225] and [315 360] as shown in figure 3 . $ 0 y . ; j d [ j d [ j d [ ? ? ? $ 0 y $ f f h o h u d w l r q > j @ $ q j o h > ? @
calculating tilt angles AN4509 6/10 docid026445 rev 1 1.2.2 dual-axis tilt sensing when a dual-axis tilt sensing approach is used, the user should be aware of two different situations in which this approach could limit overall accuracy or even inhibit tilt calculation. ? figure 4 , example a: rotate the accelerometer counter-clockwise around the dotted arrow with ? angle. when ? is less than 45, the x-axis has higher sensitivity, while the y-axis has lower sensitivity. and when ? is greater than 45, the x-axis has lower sensitivity while the y-axis has higher sensitivity. therefore, when the two-axis approach is used, it is always recommended to calculate the angle based on the orthogonal axis to a 1 g condition. ? figure 4 , example b: at this position, both the x and y axes have high sensitivity. however, without the help of a third axis (for example the z-axis), it is impossible to distinguish a tilt angle of 30 from one of 150 because the x-axis has the same outputs at these two tilt angles. figure 4. tilt sensitivity of a dual-axis accelerometer table 1. tilt sensitivity of single axis accelerometer tilt [] acceleration [ g ] ? g / [m g /] 0 0.000 17.452 15 0.259 16.818 30 0.500 15.038 45 0.707 12.233 60 0.866 8.594 75 0.966 4.37 90 1.000 0.152 $ 0 y < ; j ; < ( [ d p s o h % ( [ d p s o h $ j ? 6 \ p j ? ! ? 6 \ ! p j ? ? ?
docid026445 rev 1 7/10 AN4509 calculating tilt angles 10 1.2.3 tri-axis tilt sensing with a 3-axis accelerometer, the user can use the z-axis to combine with the x and y axes for tilt sensing in order to improve tilt sensitivity and accuracy (see figure 5 ). there are two ways to calculate 3 tilt angles in figure 5 . the first is to use basic trigonometric equation 3 , 4 and 5 , where a x1 , a y1 and a z1 are the values obtained after applying accelerometer calibration on raw measurement data (a x , a y , a z ): equation 3 equation 4 equation 5 figure 5. tilt angles from a tri-axis accelerometer the second way is to use trigonometric equation 6 and 7 to calculate pitch and roll tilt angle, which produces constant sensitivity over 360 of rotation, as shown in figure 6 . equation 6 equation 7 $ 0 y . ; = <
calculating tilt angles AN4509 8/10 docid026445 rev 1 figure 6. tilt sensitivity of a tri-axis accelerometer $ 0 y 7 l o w l q j > ? @ 7 l o w v h q v l w l y l w \ > p j ? @ $ u f v l q ; $ u f f r v = $ u f w d q ; =
docid026445 rev 1 9/10 AN4509 revision history 10 2 revision history table 2. document revision history date revision changes 10-jun-2014 1 initial release.
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