OpenWave #6 : Getting vertical displacement working

In post #5 I outlined the steps used to go from vertical acceleration to vertical displacement and how the results were completely off. I think I’ve figured out what was going wrong. The reason why each of the displacement graphs at the end of the post look so totally different just by changing the integral reset points is because of a negative offset to the velocity signal. I think of it as a negative DC offset, as in the average value of the signal is not zero. This causes huge problems when you go to integrate the signal. Lets take a look at the vertical acceleration signal again:

Z-axis acceleration signal

You may notice that there is a negative offset of somewhere around -0.15 or thereabouts. I was actually correcting for this because without the correction, the first integration to obtain velocity goes completely mad. So I was offset correcting this z-axis acceleration and then double integrating to get displacement. This was the cause of all of my issues. It turns out that even if I offset correct the acceleration signal, the velocity signal may still have an offset present in it. If it does have an offset and you integrate again to get displacement, the displacement is totally off.

Lets take a look at an example graph with which does not offset correct the velocity signal before integrating to get displacement:

The blue vertical lines indicate the seperate chunks of data, with everything being split based on the minima of the acceleration signal. You may notice that the velocity signal has a slight positive bias. When this signal is integrated to get the bottom graph of displacement, the displacement starts at zero but does not return to zero as you would expect as the rotating arm is rotating in a circle.

Now lets take a look at the graphs when you do offset correct before each integration step:

Displacement with offset correction of acceleration and velocity

This plot finally makes some sense, and to make things better, the average displacement for the 5 chunks of data is 41.7cm. This measurement has an error of 4.25 %. The results from offset correcting are very encouraging but more testing needs to be done to verify that this hasn’t been a lucky one off where the end result happens to match the expected displacement of 40cm. I’ve drilled extra mounting holes into the stick at a radius of 20cm, 40cm and 60cm. I’ll repeat the above experiment for each of those heights to verify that the process actually works.

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