Sunday, November 18, 2012

Detection of the Rotation of the Earth

Abstract: An MPU6050 6DoF MEMS sensor is used to measure the rotation of the Earth. The device is run pointed north for 1 minute, then south for 1 minute, taking 98samples/s during the runs. Actual rotation difference between north and south is 0.0063837°/s, taking into account the cosine(latitude) effect. Measurement is 0.0064822°/s±0.0015299°/s, representing a clear detection of the rotation.  The MPU6050 gyroscope is extremely accurate, probably sufficient for the Rocketometer mission.

This is also a review of the MPU6050 6DoF sensor. This part has a three-axis accelerometer with ranges ±2g, ±4g, ±8g, and ±16g, and a three-axis gyroscope with ranges ±250°/s, ±500°/s, ±1000°/s, and ±2000°/s. Since the intended mission is flying on a rocket, and certain rockets that may fly with this have been observed to accelerate at around 25g, it will need to be supplemented with a high-accelerometer, but the gyroscope will handle the 4.5Hz rotation expected.

The earth rotates 360° in 24h*60m*60s=86400s, or 1° in 240 seconds, or 0.00416666°/s. The gyroscope's most sensitive setting is ±250°/s and reads out at 16 bit resolution, resulting in 500°/s/65536DN=0.0076294°/s/DN, not quite enough to distinguish earth rotation from a standing stop, but enough to distinguish when one axis is pointing north, then south. Unfortunately, that presumes that the gyro is noise-free, which we will soon see is false.

The current experimental setup is on a breadboard connected to a version 1.1 Loginator by I2C at 400kHz, carefully aligned to true north by the following extremely accurate procedure: Since the walls of the secret underground laboratory are not aligned with true north, I pulled up the Google map of the lab and oriented the map such that the walls on the map were parallel to the real walls. The case of the phone was then aligned to true north. I put down a line of tape to mark this orientation. In this setup, the MPU6050 axes were aligned with +Z pointing down (thus we expect to see the 1g field read negative), the +Y axis pointing north first then south, and the +X axis pointing east first then west.

Of course, all measurements are contaminated with noise. This can often be beaten back by taking many measurements of the same thing. If you take into account a number of simplifying assumptions, the noise of the average of \(N\) measurements of the same quantity is \(\sigma/\sqrt{N}\). In short, if you take 100 measurements of the same thing, the average of them is expected to have 1/10 the noise of the original measurements. My previous efforts have taken data over very long stretches of time, hundreds of samples per second over hours. This time I decided to take data for 1 minute at a time. I sampled at 98samples/s (2samples/s were spent reading the pressure sensor, a topic for another day) for 1 minute with the Y axis pointing north, then 1 minute with the Y axis pointing south.

The estimated noise on each measurement, calculated with the standard deviation of all the measurements, was 11.02DN, or 0.084°/s, about 20 times that of the rotation of the earth, so it is obviously impossible to measure with one sample. But, I took 5880 samples, giving a predicted noise on the average of 0.14DN or 0.001°/s, plenty small enough to measure the rotation of the Earth. But did I? And if I did, why did I fail before?

Data:
Y north: -0.2217195°/s±0.0010972°/s (1σ)
Y south: -0.2282107°/s±0.0010972°/s (1σ)

Difference: 0.0064822°/s±0.0015299°/s (North is greater than South by this much)

Now, what is the expected value? First, is it positive or negative? Relative to inertial space, the device is rotating according to the right-hand rule around the Earth's axis. The device measures a right-handed rotation around an axis as positive, so the device should read more positive when pointing north than when pointing south, as it does.

Also, as mentioned above, the earth rotates at 0.00416°/s, but my Y axis is inclined 40° relative to the earth's rotation axis, so I should only expect to see \(\cos 40^\circ\) as much. Also, I would expect to see twice as much as that, because I am not comparing a standing stop to rotation, but rotation one way to rotation the other way.

Taking all this into account, the predicted measurement is....

0.0063837°/s.

My measurement clearly brackets this. In fact, the measurement is much better than I have any right to expect, being only 0.06σ above the expected value. I would have accepted any measurement with the proper sign and an error of less than 1σ.

Why did I fail before, with measurements taken over hours and hours? Gyro drift. Ideally, the zero point of the measurement would be zero DN, but we are happy if the zero point is just constant. But it's not. Temperature changes and other unmeasured effects cause the zero point to drift, and when measuring for hours and hours, this gyro drift swamps the signal I am trying to measure. To get a good rotation measurement, you want as many samples as possible, but taken over a relatively short time such that gyro drift is small.

So, back to the review. The MPU sensor is kind of noisy, with 11DN noise per sample, but can be read out sufficiently quickly and has sufficiently small gyro drift that it can measure the rotation of the Earth.

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