Using Two Tri-Axis Accelerometers for Rotational Measurements

Transcription

1 Using Two Tri-Axis Accelerometers for Rotational Measurements Introduction In man applications, customers would like to measure rotational motions (angular velocit, angular acceleration) in addition to linear motions. Most often, groscopes are added to their end product to obtain the rotational information. In some instances, customers alread have a sstem that contains two or more accelerometers. With some understanding of fundamental phsics, the can extract more than just linear acceleration data from their sstem. This application note describes how to use two Kionix MEMS low-g accelerometers to enable rotational measurements. Applicable theor, plots and equations are provided with this note as guidelines. The Theor Two accelerometers are rigidl mounted on a spinning object at distances r 1 and r from the center of rotation as shown in Figure 1. r x r 1 x r r 1 CR Fig. 1) Locations of two accelerometers on a rotating object The radial acceleration measured b accelerometer 1 is: a x 1 = ω r1 (1) where ω is the angular velocit. The radial acceleration measured b accelerometer is: a x = ω r () Taking the difference between the two measurements: a ( r r ) x ax1 = ω 1 = ω (3) where = r r 1 is the fixed separation between the two accelerometers. From this equation the magnitude of the angular velocit can be determined: ax ax1 ω = (4) The tangential acceleration measured at accelerometer 1 is: 36 Thornwood r. - Ithaca, NY tel: fax: info@kionix.com 10 Januar 008 Page 1 of 1

2 a 1 = αr 1 (5) where α is the angular acceleration. The tangential acceleration measured at accelerometer is: Taking the difference between the two measurements: a = α r (6) ( r r ) α a a1 = α 1 = (7) Re-arranging the terms to determine the angular acceleration: ( a a ) 1 α = (8) Looking at Eqns. (4) and (8), a couple of observations can be made. First, it is not necessar to know where the center of rotation is to determine the angular velocit or angular acceleration. The separation between the accelerometers, established at the design of the sstem, is the onl parameter needed. Second, this separation and the resolution of the accelerometers determines the resolution that can be obtained in the angular measurements. For accelerometers with a given resolution, better resolution of the angular measurements can be obtained b separating the accelerometers b a larger distance. Now consider two tri-axis accelerometers separated b distance with X-axes aligned as shown in Figure. The radial and tangential accelerations should be able to determine angular velocit around the Y-axis (ω, roll rate) and the Z-axis (ω, aw rate). x Z 1 ω ω x Fig. ) Two tri-axis accelerometers on a rotating rigid bod 10 Januar 008 Page of

3 The Algorithm To calculate the rotational rates, we start b sampling the X-, Y-, and Z-axis accelerations of both accelerometer #1 and accelerometer #. Keeping a running average (around 5 samples at 60 H) of the accelerations will reduce noise. The total rotation rate magnitude (ω) is calculated directl from radial acceleration as shown in Eqn. 4. An ω that is below a threshold (ω t ) is set to 0. a a x x1 ω = when ω ωt ω = 0 when ω < ωt (9) Fig. 3) An example of the total rotation rate magnitude calculated using Eqn. 9. The total rotation rate is now known, but the direction (sign) and the rotation rate about the Y-axis (roll) and the Z-axis (aw) are still unknown. Using Eqn. 8, the angular accelerations about the -axis and -axis are calculated from the Y-axis and Z-axis accelerations: ( a a ) 1 α = (10) α = ( a a ) 1 (11) Integrating α and α when ω is nonero allows us to determine ω and ω. Thus, the relative magnitude and direction of rotation are calculated from the integrals of the tangential accelerations. 10 Januar 008 Page 3 of 3

4 ω = t α 0 dt (1) t ω = α 0 dt (13) Since the angular accelerations, α and α, rise before ω, it is necessar to add the integral of α and α from several time steps before ω crosses the ω t threshold. Fig. 4) Timing of α and ω showing α preceeding ω. The vector sum of ω and ω from Eqns. 1 and 13 is equal to the magnitude of the total angular rotation rate, ω total. total = ω ω (14) ω + Using the total rotation rate magnitude calculated in Eqn. 9 and the relative magnitude and direction of rotation obtained from Eqn. 1, we can determine the aw: aw ω = ω * (15) ωtotal and the roll: roll ω = ω * (16) ωtotal 10 Januar 008 Page 4 of 4

5 Fig. 5) Examples of aw and roll rates calculated b Eqns. 15 and Januar 008 Page 5 of 5

6 Test Cases To test the algorithm, Kionix mounted two KXP accelerometers separated b a distance of 9.cm on a rigid bod. A gro was also mounted on the rigid bod so that the angular rate determined b the two accelerometers could be directl compared to the angular rate output of the gro. The following plots show the results of the testing Angular velocit (deg/s) Time (s) Gro Accels Fig. 6a) Measurement of angular velocit using a gro and accelerometers Angular velocit (deg/s) Time (s) Gro Accels Fig. 6b) Another measurement of angular velocit using a gro and accelerometers. As seen in the measurements, this method works ver well for certain motions short (< seconds), medium magnitude (140 /sec to 100 /sec) rotations. Low magnitude (<140 /sec) motions are not sensed because the are below the sstem 10 Januar 008 Page 6 of 6

7 resolution. uring long motions (> seconds), the integration error becomes large. Ver large magnitude (>100 /sec) motions will cause error because the accelerometer outputs saturate. We also found that the accelerometer sensitivities had to be well matched and an offset errors had to be corrected through a start-up calibration. Source code is available for running this algorithm using a Silabs 8051F31 and two Kionix KXP accelerometers. Send an to rotation@kionix.com for more information. Conclusions The measurements obtained in this stud verified two linear accelerometers can be used to determine angular rotation rates. This method works best when the rotational motions are quick with large angular accelerations. In this case, there is no chance of dividing b ero. Integration onl takes place for short time, so drift is not a problem. Angular accelerations need to be lower than the accelerometer sensing range. This method does not work as well when angular accelerations (α and α ) are ver small, since the algorithm relies on knowing the sign confidentl. There is some sensitivit to offset and sensitivit differences between the two sensors. Therefore, some method is needed to compensate for sensor mismatch, such as a calibration on start-up. Selection of the ω threshold (ω t ) is dependent on sensor performance and the expected motions in the application. Using two accelerometers in this manner can work well in applications where the expected motions align with the capabilities. Calculation of angular rate with accelerometers can be performed in existing microprocessors and ma provide significant cost and power consumption advantages over gro solutions. AN 019 The Kionix Advantage Kionix technolog provides for X, Y, and Z-axis sensing on a single, silicon chip. One accelerometer can be used to enable a variet of simultaneous features including, but not limited to: Hard isk rive protection Vibration analsis Tilt screen navigation Sports modeling Theft, man-down, accident alarm Image stabilit, screen orientation & scrolling Computer pointer Navigation, mapping Game plaing Automatic sleep mode 10 Januar 008 Page 7 of 7

8 Theor of Operation Kionix MEMS linear tri-axis accelerometers function on the principle of differential capacitance. Acceleration causes displacement of a silicon structure resulting in a change in capacitance. A signal-conditioning CMOS technolog ASIC detects and transforms changes in capacitance into an analog output voltage, which is proportional to acceleration. These outputs can then be sent to a micro-controller for integration into various applications. For product summaries, specifications, and schematics, please refer to the Kionix MEMS accelerometer product sheets at 10 Januar 008 Page 8 of 8