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Application Note

Tilt Sensing Using Linear Accelerometers

by: Kimberly Tuck

Accelerometer Systems and Applications Engineering

Tempe, AZ

INTRODUCTION Calibration Technique for Accelerometers. Even though the

offset is trimmed, offset can shift due to packaging stresses,

This application note explains the importance of

aging and external mechanical stresses due to mounting and

understanding how to acquire a reliable and accurate tilt

orientation.This results in offset calibration error. It is important

reading for accelerometer applications by comparing the

to implement a 0g calibration routine for the accelerometer to

advantages and disadvantages of various tilt measurement

compensate for the 0g offset.

techniques. Accelerometers used for tilt sensing require high

resolution to meet the demands of many new emerging MEASUREMENT TECHNIQUES

applications such as tilt enabled computer mouse/pointers,

This section discusses the different ways to implement tilt

motion enabled video game solutions and PDA-cell phone/

comparing different ways to measure the corresponding angle

mp3 player screen navigation.

from the acceleration output.

The overall benefit of the accelerometer for tilting

applications used in PDAs for screen navigations is a new

Measuring Tilt using One Axis

method to view, scroll, select and move with a minimum

In the case of a dual-axis accelerometer (XY) mounted

number of buttons required. This concept affords a PDA with

perpendicular to gravity the tilt algorithm is limited to one axis

a larger screen area for viewing. Navigation through menus is

of sensitivity. As shown in Figure 1 the accelerometer is tilted

made easier with the ability to make selections based on tilt.

along the X-axis. The Y-axis remains at 0g output throughout

The choices are highlighted and then can be selected either

the full rotation of the X-axis in this case.

by using a physical execute button on the PDA or by using

click or double click tap detection of the accelerometer. The

user can make selections in a menu driven environment this

way. Also the accelerometer can also be used to sense the tilt +x

of the PDA to change from landscape to portrait using gravity

to change the screen orientation for viewing.

Interactive video games are becoming increasingly

popular. Accelerometers are used to detect the tilting motions

of the joystick for the game.This has created games where the

user can feel more immersed in the game.

Tilt is a static measurement. The force of gravity is used as Figure 1. Dual-Axis Accelerometer with One Axis of Tilt

an input to determine the orientation of an object calculating

If one axis (X-axis) is used to calculate the tilted angle of the

the degree of tilt.The accelerometer will experience

accelerometer the following trigonometry relationship is used:

acceleration in the range from -1g to +1g through 180 of tilt.

V OUTX = V OFF + S sin

1g = - 9.8 m/s2

Where: VOUTx is the voltage output from the X-axis of the

0G OFFSET CALIBRATION

accelerometer, VOFF is the offset voltage, and S is the

Accuracy and repeatability is a general concern for nearly sensitivity of the accelerometer.

all accelerometer applications. The accuracy of the tilt

The acceleration output on the X-axis due to gravity is

measurement can be improved by using a 0g-offset

equal to the following:

calibration technique to compensate for offset errors. Refer to

V OUTX V OFF

Freescale application note AN3447, Implementing Auto-Zero

A X =

Measuring Tilt using a Two Axis Solution

In order to solve for the angle of tilt the equation becomes

the following:

The resolution problems and tilt orientation difficulties can

= sin ( AX ) be addressed by mounting the accelerometer vertically so that

the Y-axis is parallel to gravity, or by using a tri-axis

accelerometer using at least 2 of the 3 axis. Using more than

one axis to calculate tilt produces a more accurate solution.

Figure 3. Using a (Dual- or Tri-Axis) Accelerometer with

Two Axes for Measuring Tilt

Figure 2. Accelerometer Output (g's) Tilting from -90 to

+90 with a One Axis Measurement

This graph shows the output in g s of the accelerometer as

it tilts from -90 to +90 . Notice that the tilt sensitivity

diminishes between -90 and -45 and between +45 to

+90 . This resolution problem between these values makes

this method of calculating the angle of tilt inaccurate when the

accelerometer output is near the +1g or -1g range. A dual-axis

accelerometer horizontally mounted would be limited by this

Figure 4. Sine Function of the X Output and Cosine

method of calculating tilt and would not be accurate over a

Function of the Y Output

360 rotation. It would only be useful for angle measurements

between -45 to +45 of tilt. The graph above shows that when using a two axis solution

Another disadvantage of the single axis measurement tilt the component due to gravity on the X-axis follows the sine

technique is that it is impossible to know the difference function while the component due to gravity acting on the

between two tilt angles that result in the same sensor output. Y-axis follows the cosine function. Notice that the tilt sensitivity

The output is a sine function, so for example it would be (slope of the line) in the X-direction is at its maximum while the

impossible to know from a 0.5g output reading if the Y-sensitivity is at its minimum and visa versa. Therefore the

accelerometer was tilted 30 or 150 by looking at the maximum tilt sensitivity can be maintained if both the X and

accelerometer output. One would have to be aware of the the Y outputs are combined.

correct orientation of the accelerometer and have a sense for Table 1 displays 360 of tilt with the acceleration output of

the quadrant of tilt. This disadvantage is overcome by using a the X component and Y components due to gravity. Also the

two axis measurement tilt technique and is explained in the change in gravity with the change in angle is analyzed through

next section. the full rotation for both components. The two sensitivities are

combined which results in a constant output of 17.45mg/ .

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Table 1. Tilt using the X and Y-axis

dg/dDeg AX dg/dDeg AY sqrt(TSX^2+TSY sqrt(AX^2+AY^2

Angle AX (g s) AY (g s)

TSX(mg/ ) TSY(mg/ ) ^2) (mg) ) (g)

0 0.000 17.452 1.000 -0.152 17.45 1.00

30 0.500 15.038 0.866 -8.858 17.45 1.00

60 0.866 8.594 0.500 -15.190 17.45 1.00

90 1.000 -0.152 0.000 -17.452 17.45 1.00

120 0.866 -8.858 -0.500 -15.038 17.45 1.00

150 0.500 -15.190 -0.866 -8.594 17.45 1.00

180 0.000 -17.452 -1.000 0.152 17.45 1.00

210 -0.500 -15.038 -0.866 8.858 17.45 1.00

240 -0.866 -8.594 -0.500 15.190 17.45 1.00

270 -1.000 0.152 0.000 17.452 17.45 1.00

300 -0.866 8.858 0.500 15.038 17.45 1.00

330 -0.500 15.190 0.866 8.594 17.45 1.00

Basic Trigonometry X-axis experiences a 0g acceleration. The combined

acceleration is always 1g.

2 2

A= A X + A Y = 1g

AY The sensor is most responsive to changes in tilt when the

sensitive axis is perpendicular to the force of gravity. When

perpendicular to the force of gravity the accelerometer

experiences approximately 17.45mg per degree tilt. It is least

responsive when the sensitive axis is parallel to the force of

gravity in the +1g or -1g orientation, with a responsiveness of

Figure 5. Basic Trigonometry only 0.15mg per degree of tilt. This is clearly displayed in

Figure 6 where the absolute value of the tilt sensitivity was

The acceleration in the X-axis in Table 1 is calculated by the

taken. As the X-axis is at its minimum tilt sensitivity the Y-axis

following equation:

is at its maximum tilt sensitivity. By combining the X and

A X = sin Y-axis solving for the tilt angle using arctan (AX/AY), a constant

tilt sensitivity of 17.45mg can be maintained through a 360

The acceleration on the Y-axis is calculated with:

rotation.

A Y = cos

If the combination of the X acceleration and the Y acceleration

is used:

= tan

The tilt sensitivity equation mg/ was calculated by taking

the difference between the acceleration output between

1 degree at that point. For example, the tilt sensitivity at 15 is

calculated by the following:

sin ( 16 ) sin ( 15 ) = 16.818

The Y-axis is 90 from the X-axis and therefore it makes

sense that the Y-axis experiences a 1g acceleration while the

Figure 6. Tilt Sensitivity versus Tilt Angle

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Quadrant Orientation Now the acceleration due to gravity on the X-axis,

Y-axis and Z-axis are combined. The resultant sum of the

accelerations from the three axes is equal to 1g when the

accelerometer is static.

Q2 Q1 2 2 2

A X + A Y + A Z = 1g

AX + AX +

AY - AY +

180

180 0 A/D Converter Resolution Limitations

Q3 Q4

Discrete values are used when the signal is digitized and

AX - AX -

AY - AY + therefore the resolution is limited by the number of bits in the

A/D converter. Table 2 displays the 8-bit A/D converter values

for the X and Z-axis assuming an ideal rotation about the

270

y axis.

Figure 7. Quadrants of a 360 Degree Rotation

The 3.3V supply voltage is divided by 255 (28-1) steps from

It is important to know the sign of the X and Y accelerations the A/D converter. This value is divided by the sensitivity of

to determine the quadrant of tilt that is applicable because the 0.8V/g to solve for the acceleration due to gravity at each step.

outputs from the first and third quadrant will be the same and

3.3V

the outputs from the second and fourth quadrant will also be = 16.176mg

255 0.8mV l g

the same. For example tan (45) = 1 and tan (225) = 1. When

taking the arctan of a positive value the tilt angle is in either the Therefore each increasing bit will account for an additional

first or third quadrant. Knowing the sign of AX and AY will 16.176mg.

determine exactly which quadrant. When taking the arctan of From Table 2 it can be seen that a single axis solution will

a negative value the tilt angle is in either the second or fourth produce a decreasing resolution as the device is tilted from 0

quadrant. Knowing the sign of AX and AY will determine to 90, but a two axis solution will produce a fairly steady

exactly which quadrant the accelerometer is tilting through. resolution throughout the entire tilt range.

The angle calculation based on acceleration of a single axis

If in Quadrant 1 = arctan (AX/AY)

is the following:

If in Quadrant 2 = arctan (AX/AY) + 180

= sin 1 ( A X )

If in Quadrant 3 = arctan (AX/AY) + 180

If in Quadrant 4 = arctan (AX/AY) + 360

The resolution goes from 0.927 degrees to 9.332, which is

unacceptable for a tilt application.The resolution gets

Measuring Tilt using a Three Axis Solution

increasingly worse through the tilt.

In order to define the angles of the accelerometer in three The angle calculation based on acceleration of two axes is

dimensions the pitch, roll and theta are sensed using all three the following:

outputs of the accelerometer. Pitch is defined as the angle AX

= tan 1

A Z

of the X-axis relative to ground. Roll is defined as the angle

of the Y-axis relative to the ground. Theta is the angle of

The resolution is between 0.748 - 1.317 throughout the

the Z axis relative to gravity.

entire tilt range. Again this shows the improved accuracy of

using two axes to calculate tilt. Figure 6 displays the

z z

comparison of these two methods using the 8-bit A/D

converter.

x x

NOTE: The same analysis applies for angles from 91 to

x x

360 in the other three quadrants.

Using a 10-bit A/D converter the 3.3V supply voltage is

Figure 8. Three Axis for Measuring Tilt

divided by 1023 (210-1) steps from the A/D converter. This

AX value is then divided by the sensitivity of 0.8V/g to solve for the

= arc tan

acceleration due to gravity at each step.

A2 + A 2

Y Z

3.3V

= 4.032mg

AY 1023 0.8mV l g

= arc tan

A + A2 2 Using a 10-bit A/D converter with a 2 axis solution the

X Z

resolution is between 0.171 and 0.327 throughout the tilt

range, while the 1 axis solution resolution starts out at 0.231

2 2

A X + A Y

= arc tan at 0 and increases to 5.147 as it approaches 90 . A higher

AZ resolution is achievable with a bigger A/D converter. The

AN3461

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comparison using the 10-bit A/D converter is shown in

Figure 10.

Figure 10. Tilt Resolution for a One or Two Axis Tilt

Figure 9. Tilt Resolution for a One or Two axis Tilt

Algorithm Using a 10-Bit A/D Converter

Algorithm Using an 8-Bit A/D Converter

Table 2. A/D converter values for AX and AZ for tilt from 0 to 90

Angle Angle Resolution Resolution

A/D AX Ax- g's A/D AZ Az-g's 1-Axis 2-Axes 1-Axis 2=Axis

128 0.0000 190 1.0029 0.0000 0.0000 0.9269 0.9240

129 0.0162 190 1.0029 0.9269 0.9240 0.9269 0.9240

130 0.0324 190 1.0029 1.8540 1.8476 0.9271 0.9236

131 0.0485 190 1.0029 2.7816 2.7702 0.9276 0.9226

132 0.0647 190 1.0029 3.7100 3.6914 0.9283 0.9212

133 0.0809 190 1.0029 4.6393 4.6106 0.9293 0.9193

134 0.0971 190 1.0029 5.5698 5.5275 0.9305 0.9169

135 0.1132 189 0.9868 6.5018 6.5463 0.9320 1.0188

136 0.1294 189 0.9868 7.4356 7.4716 0.9338 0.9253

137 0.1456 189 0.9868 8.3713 8.3929 0.9357 0.9214

138 0.1618 189 0.9868 9.3093 9.3099 0.9380 0.9170

139 0.1779 189 0.9868 10.2499 10.2222 0.9405 0.9122

140 0.1941 189 0.9868 11.1932 11.1292 0.9433 0.9070

141 0.2103 188 0.9706 12.1396 12.2251 0.9464 1.0959

142 0.2265 188 0.9706 13.0894 13.1340 0.9498 0.9089

143 0.2426 188 0.9706 14.0428 14.0362 0.9535 0.9022

144 0.2588 188 0.9706 15.0003 14.9314 0.9574 0.8952

145 0.2750 187 0.9544 15.9620 16.0736 0.9617 1.1422

146 0.2912 187 0.9544 16.9284 16.9661 0.9664 0.8926

147 0.3074 187 0.9544 17.8998 17.8503 0.9714 0.8842

148 0.3235 187 0.9544 18.8765 18.7258 0.9767 0.8755

149 0.3397 186 0.9382 19.8590 19.9037 0.9825 1.1780

150 0.3559 186 0.9382 20.8475 20.7723 0.9886 0.8685

151 0.3721 185 0.9221 21.8426 21.9745 0.9951 1.2023

152 0.3882 185 0.9221 22.8447 22.8337 1.0021 0.8591

153 0.4044 185 0.9221 23.8543 23.6821 1.0095 0.8484

154 0.4206 184 0.9059 24.8717 24.9048 1.0175 1.2227

155 0.4368 184 0.9059 25.8976 25.7407 1.0259 0.8359

156 0.4529 183 0.8897 26.9325 26.9802 1.0349 1.2395

157 0.4691 183 0.8897 27.9770 27.8015 1.0445 0.8212

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Table 2. A/D converter values for AX and AZ for tilt from 0 to 90 (continued)

Angle Angle Resolution Resolution

A/D AX Ax- g's A/D AZ Az-g's

1-Axis 2-Axes 1-Axis 2=Axis

158 0.4853 182 0.8735 29.0317 29.0546 1.0547 1.2531

159 0.5015 181 0.8574 30.0973 30.3236 1.0656 1.2690

160 0.5176 181 0.8574 31.1746 31.1225 1.0772 0.7989

161 0.5338 180 0.8412 32.2642 32.3998 1.0896 1.2774

162 0.5500 180 0.8412 33.3670 33.1785 1.1029 0.7787

163 0.5662 179 0.8250 34.4840 34.4608 1.1170 1.2823

164 0.5824 178 0.8088 35.6162 35.7539 1.1322 1.2931

165 0.5985 178 0.8088 36.7646 36.5014 1.1484 0.7476

166 0.6147 177 0.7926 37.9306 37.7939 1.1659 1.2925

167 0.6309 176 0.7765 39.1153 39.0939 1.1847 1.2999

168 0.6471 175 0.7603 40.3202 40.3999 1.2050 1.3060

169 0.6632 174 0.7441 41.5471 41.7108 1.2269 1.3109

170 0.6794 173 0.7279 42.7977 43.0251 1.2506 1.3143

171 0.6956 172 0.7118 44.0741 44.3415 1.2764 1.3164

172 0.7118 171 0.6956 45.3787 45.6585 1.3046 1.3171

173 0.7279 170 0.6794 46.7141 46.9749 1.3354 1.3164

174 0.7441 169 0.6632 48.0834 48.2892 1.3693 1.3143

175 0.7603 168 0.6471 49.4901 49.6001 1.4068 1.3109

176 0.7765 167 0.6309 50.9386 50.9061 1.4484 1.3060

177 0.7926 166 0.6147 52.4336 52.2061 1.4950 1.2999

178 0.8088 165 0.5985 53.9811 53.4986 1.5476 1.2925

178 0.8088 164 0.5824 53.9811 54.2461 1.5476 0.7476

179 0.8250 163 0.5662 55.5885 55.5392 1.6073 1.2931

180 0.8412 162 0.5500 57.2646 56.8215 1.6761 1.2823

180 0.8412 161 0.5338 57.2646 57.6002 1.6761 0.7787

181 0.8574 160 0.5176 59.0207 58.8775 1.7561 1.2774

181 0.8574 159 0.5015 59.0207 59.6764 1.7561 0.7989

182 0.8735 158 0.4853 60.8714 60.9454 1.8507 1.2690

183 0.8897 157 0.4691 62.8363 62.1985 1.9649 1.2531

183 0.8897 156 0.4529 62.8363 63.0198 1.9649 0.8212

184 0.9059 155 0.4368 64.9424 64.2593 2.1061 1.2395

184 0.9059 154 0.4206 64.9424 65.0952 2.1061 0.8359

185 0.9221 153 0.4044 67.2289 66.3179 2.2866 1.2227

185 0.9221 152 0.3882 67.2289 67.1663 2.2866 0.8484

185 0.9221 151 0.3721 67.2289 68.0255 2.2866 0.8591

186 0.9382 150 0.3559 69.7573 69.2277 2.5283 1.2023

186 0.9382 149 0.3397 69.7573 70.0963 2.5283 0.8685

187 0.9544 148 0.3235 72.6329 71.2742 2.8756 1.1780

187 0.9544 147 0.3074 72.6329 72.1497 2.8756 0.8755

187 0.9544 146 0.2912 72.6329 73.0339 2.8756 0.8842

187 0.9544 145 0.2750 72.6329 73.9264 2.8756 0.8926

188 0.9706 144 0.2588 76.0694 75.0686 3.4366 1.1422

188 0.9706 143 0.2426 76.0694 75.9638 3.4366 0.8952

188 0.9706 142 0.2265 76.0694 76.8660 3.4366 0.9022

188 0.9706 141 0.2103 76.0694 77.7749 3.4366 0.9089

189 0.9868 140 0.1941 80.6678 78.8708 4.5983 1.0959

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Table 2. A/D converter values for AX and AZ for tilt from 0 to 90 (continued)

Angle Angle Resolution Resolution

A/D AX Ax- g's A/D AZ Az-g's

1-Axis 2-Axes 1-Axis 2=Axis

189 0.9868 139 0.1779 80.6678 79.7778 4.5983 0.9070

189 0.9868 138 0.1618 80.6678 80.6901 4.5983 0.9122

189 0.9868 137 0.1456 80.6678 81.6071 4.5983 0.9170

189 0.9868 136 0.1294 80.6678 82.5284 4.5983 0.9214

189 0.9868 135 0.1132 80.6678 83.4537 4.5983 0.9253

190 1.0029 134 0.0971 90.0000 84.4725 9.3322 1.0188

190 1.0029 133 0.0809 90.0000 85.3894 9.3322 0.9169

190 1.0029 132 0.0647 90.0000 86.3086 9.3322 0.9193

190 1.0029 131 0.0485 90.0000 87.2298 9.3322 0.9212

190 1.0029 130 0.0324 90.0000 88.1524 9.3322 0.9226

190 1.0029 129 0.0162 90.0000 89.0760 9.3322 0.9236

190 1.0029 128 0.0000 90.0000 90.0000 9.3322 0.9240

AN3461

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Freescale Semiconductor 7

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AN3461

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