Displacement from accelerometer signal

[heitor]

Hello.

I'd like to have your advice about the integration of the acceleration signal to obtain the displacement.
I tried to integrate in the frequency domain by taking the fft of the acceleration signal, dividing it by -omega^2 and the displacement should be the inverse fft of the result. However the results are not good.
What do you think?

Thank you very much!

 

[EnglishMuffin]

It sounds as though you are trying to find the displacement in the time domain. If so, why are you getting involved with fft's ? All you need to do is to double integrate the acceleration data numerically. However, when you do this you will inevitably get drift errors, which can come, for example, from a combination of DC offset in the original or processed signal, and A/D conversion errors stemming from the use of a limited number of bits. If the object in question is just vibrating,with no mean translation, it is possible to write routines which correct for the drift to some extent, since one knows that the displacement should repeat after one or more cycles. I have managed to do this in some cases. However, if you are trying to find displacement from non-periodic acceleration data, as is done in inertial navigators for example, then as far as I am aware the only recourse is to use instrumentation which incorporates a very high accuracy A/D converter. The 12 bit devices often found in typical analyzers are probably nowhere near adequate. You do not say exactly why the results are not good, so perhaps your problem lies partially elsewhere. Also, I believe this sort of problem has come up a number of times before on Eng-Tips, and I am sure one of the MVP's will come up with some relevant thread numbers.

 

[heitor]

That's exactly what I'm looking for: from the time domain acceleration signal, I want to find the displacement signal.
The accelerometers are mounted near the bearings of a rotor and I need to have the orbits.

I've already done the numerical integration but the result is drifted. If I filter the signal before performing the numerical integration, do you think it would help?

I thought about frequency domain integration because it is the procedure used on the signal analyzers, isn't it?

Thank you!

 

[EnglishMuffin]

There is no point that I can see in integrating in the frequency domain - carrying out the fft and it's inverse simply introduces even more numerical errors. I repeat - you are better off just double integrating the raw time domain signal. I don't think "filtering" will help. However, the fact that the signal is periodic means that you have a chance to eliminate some of the drift. After the first integration, you will probably discover that there is a linearly increasing velocity superimposed on the data. Using, say, Excel, write a routine to cancel this out so that the final velocity after one cycle is still the same. Then integrate the corrected data again, after which you might be able to repeat the process. I can't guarantee that this will produce reasonable results - as I said, the only real solution is to have high precision data to begin with. You would probably be better off avoiding integration, by using displacement instrumentation, such as capacitance gages - these are often used for orbit investigations, but of course you would need somewhere to mount them which is fixed in space and this may not be practical in your case.

 

[cab1990]

more information please. I would simply comment that integration is an averaging procedure and we should not expect anything too precise from an average.

 

[EnglishMuffin]

Well, as I mentioned before, it is possible to get surprisingly precise results in the field of inertial navigation by double integrating accelerometer data, but even the best units exhibit drift. It used to be the only useable technique for navigating under the polar icecap in nuclear submarines. I vividly recall the Apollo astronauts having to reset theirs at intervals on their way to the moon, by taking sun sights with a sextant. But I don't know what percentage of the error is caused by angular drift of the gyrostabilized platform on which the accelerometers are mounted. My guess would be that whatever integration and instrumentation techniques are used in this field would represent the best achievable results at the current state of the art.

 

[heitor]

EnglishMuffin,

I've just finished to integrate my signal by using frequence domain integration and a time domain integration.

To validade the results I installed a proximity probe near the accelerometer and I took one set of measures.
The best results of integration was given by the frequency domain method, although there's an error of 10% on the amplitude.

The time domain integration was carried out with a trapeze integration and there was a lot of drift. Filtering the low frequencis did help but it was not enough.

I'm not writing this to go against your advice, please don't think so.

Thank you!

 

[EnglishMuffin]

Well, there are many more accurate numerical integration techniques than the trapezoidal method, so you could try one of those. But the results with any method are totally dependent on the quality of the basic data, which I believe is the ultimate limitation. If you get "better" results doing it your way - I can't argue with success. There is almost certainly going to be a DC offset error, which may account for the majority of the drift, and as I said, this is quite easy to eliminate if you can identify it. If you trust the prox probe, you can at least use that data to scale the results. But bear in mind, the prox probe results may be incorrect if the probe is not absolutely rigidly mounted and independent of any vibrations. If you do trust the prox probe data, why not use two and utilize that data instead?

 

[Franko]

Why are you trying to do it yourself? There are great instruments out there; one example:

http://www.endevco.com/main/news/2775b.php

Filter Integrator Module 35818 provides a capability for single or double integration of the input vibration signal.

Also contact:

http://www.wilcoxon.com

Be careful of very low frequency noise - see:

http://www.wilcoxon.com/technotes/lowfreq.pdf

There once was a consultant telling one of our customers that our Elliott Plant air compressor had high vibration with very low frequency for some reason. When I went to the plant, converting the signal to displacement (that really was just electrical noise) would have meant the compressor was jumping up and down about one foot. Proper isolation from ground of our own accelerometer eliminated the noise.

Can also doublecheck results by hand after obtaining acceleration frequency spectrum, using this calculator:

http://www.wilcoxon.com/knowdesk_calculator.cfm

 

[MikeyP]

I remember replying in a similar thread, but I can't seem to find it just now.

For my particular applications, I tend to use frequency domain integration wherever possible. However, I ensure that the signal I am integrating is in some sense "periodic" (e.g. starting and ending at zero). This means that you don't have to window the signal prior to the Fourier Transform and hence you avoid the corruption of the signal through convolution with the window function.

So in an experimental situation I would start acquiring data while the structutre is a rest, set the structure vibrating with my test signal, then switch off the test signal and allow the vibrations to decay away. The response would then be "leakage-free" as far as the Fourier Transform was concerned.

Of course, this may not be practicable in your situation and you may have no control over the excitation of the structure.

Remember that accelerometers (except for some specialist ones) have very poor response at very low frequencies, so you can filter out the first few (probably meaningless) spectral lines in the frequency domain to elimiate the problems of large DC terms.

M

--
Dr Michael F Platten

 

[EnglishMuffin]

Well, here is one existing thread on the subject
thread384-70681
I think there are others

 

[EnglishMuffin]

And here is a more precise time domain numerical integration scheme developed for use in inertial navigation:

http://twiki.psas.pdx.edu/pub/PSAS/ImuCalcs/numerical.pdf

As I said before, I believe it is possible to eliminate most of the drift after the first integration, since you are dealing with a cyclic situation of known frequency. In my view, this is a more accurate technique than using filters, which will inevitably destroy some good information.

 

[heitor]

EnglishMuffin, I got good results also by using boole's method of integration but I had to low pass filter the data before the integration. Without the filter, there was a linear drift and with the filter the drift seens to be a sine of period equals to the double of the duration of the signal.
I can't use the prox probes at all places that I have accelerometers because I have only one prox probe available.

 

[EnglishMuffin]

I suspect that some of the linear drift comes from DC offset which is usually present. But as I said, rather than use a filter, if the drift is linear, and since in your case (cyclic) you should be able to figure out it's magnitude exactly, just eliminate it mathematically after the first integration(that's what I did in a similar situation). Another thing I would do is debug your technique using something which is oscillating purely sinusoidally. Although I have quite a few numerical methods texts, I am not familiar with Boole's method (is that the Boole of Boolean algebra fame I wonder?). But good luck anyway.

 

[heitor]

The Boole's rule is explained at the site:

http://mathworld.wolfram.com/BoolesRule.html

It's a very good site.
I've just found another way to integrate the signal, this one is giving the best results so far: I fit a sin function to my data and then I integrate it analitically.

I will try what you told me about removing the drift after the first integration

Thank you!

 

[ophain]

A few years ago, I did this type of numerical integration in Labview as part of a real-time package. Top simple to do.

First of all, you must make sure that you don't have aliasing frequencies in your numeric acceleration signals. Otherwise you are bound to look at very high frequencies as very low ones that will in turn give rise to huge displacements. Modern delta-sigma ADC are very appropriate for this job unless you have high-quality antialiasing analog filters. No LP digital filters will help you here.

Second, you have to pass your numeric signal through a high-pass digital filter
Finally you integrate the output of this HP digital filter. One could place the HP filter after the integration in case your simple integration scheme produces LF compoents, which I doubt if your digital HP filters was good in the first place. If your transients are short, the integration scheme ought not be very sophisticated. Sophicitcated integration schemes also curtail the frequency band (I suspect).
Drift won't be an issue.
Never dream of getting displacement from dc on with seismic sensors like accelerometers. This violates the basic law of physics. There is no way to know how you move in the stellar space with an accelerometer. You need a frame of reference to do that like eddy probes do (their anchor point)
Hope this helps.
Gérard.