how to convert a accelerometer data to displacements

I have an accelerometer data which i have collected from gyroscope. I like to convert those acceleration to displacements as disturbances for calculating the dynamics of suspension. The problem is when i tried to convert the acceleration data to displacement by using two integrators in series the displacement data seems non realistic. In actual acceleration data i found there are decelerations as well but in displacement output i see all positive values. Please suggest me some techniques.......

[Sean de Wolski]

How are you doing the double integration?

[jawad]

i have just use two integrators in series.......

[Dr. Seis]

jawad, did the solution below help?

[Mahshid karimi]

How we can do spotting any trends in FFT peaks as a "function" of water depth Using function command?

Answer 1

[Dr. Seis]

Doing this sort of conversion is best done in the frequency domain. This should work now. Two things weren't working before:

1. Before I defined N as "nextpow2(N1)" instead of "2^nextpow2(N1)

2. I forgot to handle the case where f(i) = 0, we cannot divide by 0 !!

function disp_time_data = acc2disp(acc_time_data,dt)
% acc_time_data should be acceleration amplitudes evenly spaced in time
% dt in units of seconds per sample

N1 = length(acc_time_data);
  N = 2^nextpow2(N1);
  if N > N1
      acc_time_data(N1+1:N) = 0; % pad array with 0's
  end
  df = 1 / (N*dt); % frequency increment
  Nyq = 1 / (2*dt); % Nyquist frequency
  acc_freq_data = fftshift(fft(acc_time_data));
  disp_freq_data = zeros(size(acc_freq_data));
  f = -Nyq:df:Nyq-df; % I think this is how fftshift organizes it
  for i = 1 : N
      if f(i) ~= 0
          disp_freq_data(i) = acc_freq_data(i)/(2*pi*f(i)*sqrt(-1))^2;
      else
          disp_freq_data(i) = 0;
      end
  end
  disp_time_data = ifft(ifftshift(disp_freq_data));    
  disp_time_data = disp_time_data(1:N1);

return

Try my sine wave example:

dt = 0.01; % seconds per sample 
N = 512; % number of samples 
t = 0 : dt : (N-1)*dt; % in seconds 
wave_freq = 1; % in Hertz 
acc_time_data = sin(2*pi*wave_freq*t);

The numerical answer obtained should be close to the analytical result by double-integrating by hand, i.e.:

disp_time_data_analytical = -1/(2*pi*wave_freq)^2*sin(2*pi*wave_freq*t);

The difference comes from the fact that we are chopping off the sine wave abruptly. You will notice that the "true" answer is super-imposed on a low-frequency sine wave - this is why it may be important to run a high-pass filter on the displacement result. It is possible to avoid this by removing any trend from the data, tapering each side of your timeseries down to 0, and then padding each side evenly with 0's (up to the next power of 2) before "converting" from acceleration to displacement.

[Fangjun Jiang]

Don't be mis-led by the OP's inaccurate use of "convert". It's not really a conversion. It's a calculation. Acceleration be integrated will be velocity. Velocity be integrated will be position (or displacement). That is why the OP mentioned "two integrators in series".

[Sean de Wolski]

Out of curiosity, why does this work better than doing the integral numerically in the time domain?

[Dr. Seis]

Fangjun, I dealt with ground velocity and ground acceleration data during my graduate studies (I am an earthquake seismologist by "trade"). I routinely "convert" these various timeseries to displacement by the method I described above because is much more numerically stable than trying to integrate or double-integrate in the time domain (or differentiate twice if you were going from disp. to acc.).

Sean, I knew why back in graduate school. I have since forgotten the specifics/details why it is a more numerically stable result. I will have to look that up later.

[Sean de Wolski]

Yes, the double derivation would obviously be unstable, but the integration should be stable, and since the measurements were accelerations it shouldn't be an issue.

Why does this appear to produce low values?

acc_time_data = 0:99;

dt = 0.5;

sum(abs(disp_time_data)) %a significant imaginary component

ans =

77.108

Seems _very_ low.

[Fangjun Jiang]

Thank you for the clarification!

[Dr. Seis]

Yeah... the answer shouldn't have any imaginary parts. Trying to work that out now. Like I mentioned, I have the solution at home. I was trying to work from memory what this function would look like.

Also, the acc_time_data should be more like something I mentioned above.

[Dr. Seis]

Sean de... check out page 94 here:

http://books.google.com/books?id=GnfpELDfzmEC&pg=PA95&lpg=PA95&dq=integration+differentiation+time+domain+versus+multiplication+division+in+frequency&source=bl&ots=oysKAgsSG2&sig=xjMXX6dSxy7eP5s5FKJSgjS4OIM&hl=en&ei=6fSNTtWbIuf-sQL7j72VAQ&sa=X&oi=book_result&ct=result&resnum=6&ved=0CEoQ6AEwBQ#v=onepage&q&f=false

[Dr. Seis]

See my above answer for the corrected code. Should be able to handle my sine wave example.

[gopal]

Hi Elige Grant, I have similar requirement where I need to convert acceleration data from 3-axis accelerometer into displacements in the corresponding direction. Theoretically speaking, the above method only works if the signal is zero mean. In general, this data will not be zero mean. In such case how can we use this method?

Also, as you said above, I can see the true output is superimposed on a low frequency sinusoid. However I did not understand how high pass filtering of this result will help to overcome this problem. Appreciate your suggestions. Thanks.

[John]

I believe you need to set disp_freq_data(1) to zero also.

[MANTHAN SHAH]

Hello Dr. Seis, Thanks for answering this. I appreciate your efforts towards the community. I am new with Matlab and working with vibrational data too. I was wondering how to tune the displacement data we get? As the output from this code is completely different from what it supoosed to be.

Answer 2

[Fangjun Jiang]

There is really no solution for this problem except that you need to check the quality of your accelerometer signal data. Is it noisy?

You can make up an ideal acceleration signal first and feed that signal to you double-integrator model to verify that you have everything right, for example, sample time, initial value, coefficient, etc.

[Dr. Seis]

Not true. This conversion should be performed in the frequency domain, not the time domain. Will provide solution later.

[Fangjun Jiang]

@Elige, Why does it have to do with frequency domain or time domain?

[Sean de Wolski]

@Fangjun: I'm curious as well.

[Sean de Wolski]

And why the value intuitively seems low for:

acc_time_data = 0:99;

[jawad]

well what is the criteria to know about noise????......because i have just text file which are the measurement for road roughness.......

[Dr. Seis]

acc_time_data should be acceleration amplitudes. For example, if the measured acceleration is a 1Hz sine wave:

dt = 0.01; % seconds per sample

N = 512; % number of samples

t = 0 : dt : (N-1)*dt; % in seconds

wave_freq = 1; % in Hertz

acc_time_data = sin(2*pi*wave_freq*t);

[Fangjun Jiang]

You should assume any measurement has some kind of noise. The approach for you is to test your double-integrator with an ideal signal first.

[jawad]

why not i upload my plot so that you have a look???? Please tell me how i can attach a file to this conversations.....

[Sean de Wolski]

a free web hosting site somewhere. Make sure it's saves as .mat file, .txt file or .rtf file or we won't open it.

[Fangjun Jiang]

That's not really necessary. If you don't have a clue what is the noise, we won't have any better idea. Again, test your algorithm first with an idea signal. It won't be that hard to prove this simple algorithm. Then, if the result from measured data doesn't seem right, you need to trace back to find out the quality of the measurement. Elige Grant's answer seems to be promising since he had lots of similar experience before. You could also test out that function first with an ideal signal and then test it out on your measured data.

[Dr. Seis]

I'm working on getting the code to work for my sine wave now.

[jawad]

http://www.4shared.com/file/9jbdpGR3/test123.html

will be waiting for comments

[Dr. Seis]

See my answer for corrected Matlab code.

[Mohamed Sayed]

could you please re-share the uploaded file again with us, as i think the (4shared) link you are sharing is expired. thanks in advance.