Now that I've managed to reduce noise levels in the sensors to a mostly manageable level, it's time to record some data and try and model the board. It's worth reiterating what it is I'm attempting to achieve here: a hand-held underwater metal detector. Magnetometers can be a great way to achieve this, but this particular device is measuring the strength of the magnetic field.
I'm fairly confident that linear movement of the magnetometer is not going to significantly change the sensor measurements. This is because Earth's magnetic "lines of force" don't change particularly rapidly. You would probably have to move hundreds of miles to be able to see any significant change in the magnetometer's measurements, and can you really say you've moved in a "straight line" by then? More than likely you'll have been moving in an arc relative to gravitational center of the earth, which would be even less likely to show a significant change in measurement. Therefore, I'm going to ignore ADXL measurements.
The drawing above shows the board and the axes of measurement for each device. Unfortunately, sparkfun mounted the magnetometer such that it is 90° out of phase with the other devices. This means that the positive X axis of the accelerometer and gyro is the positive Y axis of the magnetometer, and the positive Y axis of the accelerometer and gyro is the
negative X axis of the magnetometer. An inconvenience, but hopefully just a minor one.
Data capture was achieved using a common spice rack. I wanted something that didn't have much, if any, metal in it, while having sufficient space to throw the entire test apparatus on. The sensor board is roughly centered (it's a little hard to see given that the tape holding it down is a similar color to the spice rack, but you can see the connector to it above the battery pack), with the power pack and the Arduino UNO, XBee shield and XBee around the edges. Data was transferred via the XBee radio to a PC running Linux. Data loss made it necessary to make the data processing software a bit more robust.
Data capture was successful, and the graph to the right is a plot of the gyro measurements. As you can see in the graph, the rotation was primarily occurring around the Z axis, which matches the diagram of the axes and the picture of the apparatus. What you can't really see very well in the graph is that the Y and X axes were also showing measurable rotation. Those measurements show the same oscillation that can be seen on the Z-axis measurements. This is always going to be the case in any real-world measurements. You're simply never going to be able to get all the axes completely lined up and the axis of rotation perfectly aligned with the Z-axis (or whichever axis you're picking as "up"). You might get a lot closer than what I achieved with some really expensive lab equipment. The Z rotation is negative, indicating that the rotation was in a counter-clockwise direction around the Z axis.
Before looking at the magnetometer data, it's time to take a quick excursion back (for me at least) to Euclid and his geometry, though for the sake of simplicity, I'm going to stick with a 2-dimensional geometry for now. When translating between polar coordinates (angle and radius) and Euclidian coordinates, the following formulae apply:
$\begin{array}{rcl}x_{1} & = & r \cos \alpha,\\y_{1} & = & r \sin \alpha.\end{array}$
where
r is the radius and
α is the angle. In Euclidian geometry, your axes are always orthogonal, meaning they're at angles of 90° to each other. As such, the trigonometric operations sine and cosine are also 90° out of phase with each other. This phase offset shows up clearly in the recorded data. Given that the experiment was intended to rotate the board at a reasonably constant rate around the Z axis, the X and Y magnetometer measurements will look like sinusoidal waves 90° out of phase with each other. The following two graphs demonstrate. The first graph is the actual data, while the second graph is a simulation of the data set using only cos and sin functions, which are scaled to the appropriate mean and amplitude.
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| Recorded Measurements |
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| Simulated with Trig functions |
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The "simulation" in the second plot is of the following functions in
standard form:
$\begin{array}{rcl}y_{1}(t) & = & A_{1}\cdot \cos (\omega t + \varphi ) + D_{1},\\y_{2}(t) & = & A_{2}\cdot \sin (\omega t + \varphi ) + D_{2},\\A_{1} & = & 82,\\D_{1} & = & 15.73,\\A_{2} & = & 93.5,\\D_{2} & = & -75.6,\\\omega & = & \frac{1}{4.6},\\\varphi & = & -6.\end{array}$
Note that the frequency (
ω) and phase (
φ) are the same for both equations. Only the amplitude (
A) and center amplitude (
D) are different.
One important thing to note (in fact, pretty much the whole point of this post) is that
ω is almost certainly a function of the Z-measurement of the gyro. In the simplified environment of a perfect rotation about a co-axial Z axis for the two sensors, this might be expressed as:
$\omega = \gamma z_{g}$
where
γ is some constant. The reason for this is that the gyro is measuring some
Δα, that is, the rotation rate around a given axis. This is, in fact, the frequency
ω. It only needs some scale factor applied to it to match the measurements to the model. The value of that constant scale factor can probably be derived from the data, but I probably need to take a few more sample runs at various speeds of rotation before I can feel comfortable quantifying it.
More to come...
