Monday, November 19, 2012

Things that there are never enough of, part 1

There is never enough bandwidth.

I can get an MPU6000 from Newark for about $39, or I can get an MPU6050 from Sparkfun for about $20. The only issue with the 6050 is that it runs on I2C, and maxes out at 400kHz. In burst mode, I can get a byte across in 9 clock cycles. An MPU6050 readout has three 16-bit gyro registers, three 16-bit acc registers, and a 16-bit temperature readout, a total of 14 bytes, plus addressing the part, for 15 bytes. Nine cycles each gives 135 cycles, plus a couple more for starts and stops, call it 140 cycles.

Is this enough? Of course not. There is never enough bandwidth. But is it enough? In 1 millisecond, there are 400 cycles, so in theory the MPU6050 being read at 1000Hz will use up 33% of the available bandwidth.

Now there is also the high-accelerometer, read out using an I2C ADC. The ADC has three 16-bit data registers, so 7 bytes counting address. 7 bytes at 1000Hz, call it 70 more cycles. Now we are at 52% bus usage. We need to throw in a compass read and pressure sensor read every once in a while, but it looks like enough to run the sensor at whatever rate I want.

So maybe there is enough. One way to get more bandwidth is to use the other I2C bus on the chip, but this involves a fairly heavy redesign of the board. We would put the MPU and compass on one bus, and the HighAcc and pressure sensor on the other bus.

Onto the measurements. I put together a program which reads the sensors with no delay, in effect as fast as possible with blocking. The program records the tick count (TC), which increments at 60MHz, before each reading, then reads the ADXL345, HMC5883, MPU6050, and L3G4200D in that order. So, the time spent doing the ADXL345 read is the TC of each HMC packet minus the TC of the corresponding ADXL packet, and so forth. The ADXL345 consistently takes 131 microseconds with a variation of less than 1 microsecond. The HMC takes 483 microseconds. And for the moment of truth, the MPU6050 takes 539 microseconds. All is fine and good, right? Nope. We spend most of our time waiting for the card to write out. On a good write, it takes 9 milliseconds, 18 times longer than an MPU read, to write a sector to the SD card. A lot of that could be gotten around by not busy-waiting for the card to finish.

In any case, there is plenty of bandwidth for the sensors, so there is no point in splitting up the traffic to two I2C buses. Reading twice as many sensors as necessary and producing more data than necessary (the real rocketometer will only carry one set of gyros), it still is reading out at about 300Hz.

Dual (or dueling) gyroscopes

The experiment that I wrote about yesterday was actually carrying two gyroscopes: The one in the MPU6050 discussed then, and the L3G4200D on board the 11DoF. This gyro was connected to the Loginator by SPI running at 1MHz. The 11DoF tab was oriented such that during the South integration, +X was East, +Y was Up, and +Z was South. In the same integration, the MPU tab was oriented such that +X was West, +Y was South, and +Z was Up (not Down as marked on the board -- the silkscreen is wrong, proven by the positive signal on the +Z MPU signal.)

So, we match up axes as follows:

Therefore we just perform the calculation using +Z on the 11DoF just like yesterday we used +Y.

The result doesn't look so good. First, data from yesterday, showing what a detection looks like:

This shows 1 minute of data from before and after the rotation. The noisy part on both ends is 10 seconds of raw data, and the smoothed bit in the middle shows a 2000-sample (roughly 20 second) boxcar smooth. The white data was with +Y pointing north, and the red data was with +Y pointing south. In that middle region, the difference is due to the rotation of the Earth.

Now, the data from the L3G, also taken yesterday, but not reduced until today.

Since the signals cross, there is no clear detection, even with a 20 second boxcar average. Why not? As it turns out, this sensor was set to 2000°/s, its least sensitive setting, with 1/8 the resolution of the MPU. So,  it's not a fair test.

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?

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....


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.