Integration of nonlinear numerical acceleration data to find displacement

I want to find displacement of a horizontal object subjected to an impact loading. I am using an b&k accelerometer, along with a signal conditionar and labview data acquisition module. Attached is part of the random time-acceleration data and the matlab code I am using currently. I have used cumtrapz, which is not good for integrating nonlinear data. Is there any better method I can adopt? TIA

t = data(:,1);
dt = mean(diff(t));      
fs = 1/dt;                 
% Acceleration data
acc = data(:,2);        
acc = detrend(acc);     
N = 2;
fc = 14.5; % Hz
[B,A] = butter(N,2*fc/fs,'high');
acc = filter(B,A,acc);
velocity = cumtrapz(dt,acc);
velocity= detrend(velocity);    
disp     = cumtrapz(dt,velocity);
disp = detrend(disp);  
figure(1)
subplot(311),plot(t,acc);
title('acceleration'); ylabel('m/s^2');xlabel('time (s)');
subplot(312),plot(t,velocity);
title('velocity'); ylabel('m/s');xlabel('time (s)');
subplot(313),plot(t,disp);
title('disp'); ylabel('m');xlabel('time (s)');

Here is the part of random data:

 Time  Avg Acc 
 
  6.451325  0.003012 
 
  6.453325  0.037054 
 
  6.455323  0.056475 
 
  6.457322  0.054647 
 
  6.459321  0.037797 
 
  6.46132  0.092066 
 
  6.463318  0.092933 
 
  6.465317  -0.00944 
 
  6.467317  0.301396 
 
  6.469316  0.288696 
 
  6.473313  0.121895 
 
  6.475312  0.503231 
 
  6.477312  0.373444 
 
  6.479311  0.317782 
 
  6.481308  0.387228 
 
  6.483307  0.165198 
 
  6.485306  0.385958 
 
  6.487307  0.316326 
 
  6.489304  0.259053 
 
  6.491302  0.329088 
 
  6.493301  0.241521 
 
  6.4953  0.304463 
 
  6.497298  -0.00699 
 
  6.499298  -0.12854 
 
  6.501297  0.459772 
 
  6.504295  -0.09465 
 
  6.506294  0.114058 
 
  6.506294  0.077879 
 
  6.508293  0.10712 
 
  6.510293  -0.08205 
 
  6.515289  -0.46168 
 
  6.517288  -0.24058 
 
  6.518288  -0.5196 
 
  6.520286  -0.61733 
 
  6.520286  -0.39084 
 
  6.522285  -0.72429 
 
  6.524284  -0.48633 
 
  6.526285  -0.48621 
 
  6.528282  -0.59481 
 
  6.53028  -0.87241 
 
  6.532278  -4.9192 
 
  6.534278  -183.414 
 
  6.536277  -854.641 
 
  6.538277  -424.772 
 
  6.540274  -452.27 
 
  6.542274  -591.437 
 
  6.544272  -646.52 
 
  6.546271  -722.656 
 
  6.548269  -683.694 
 
  6.550269  -681.5 
 
  6.552267  -703.934 
 
  6.557265  -718.37 
 
  6.559265  -707.032 
 
  6.560265  -700.216 
 
  6.562263  -652.975 
 
  6.562263  -646.437 
 
  6.56426  -635.972 
 
  6.566259  -636.907 
 
  6.568258  -651.909 
 
  6.570258  -645.747 
 
  6.572256  -636.96 
 
  6.574255  -628.006 
 
  6.576254  -619.058 
 
  6.578253  -610.372 
 
  6.580251  -601.722 
 
  6.58225  -593.129 
 
  6.585248  -584.606 
 
  6.587247  -576.156 
 
  6.589246  -567.795 
 
  6.592245  -559.53 
 
  6.592245  -551.362 
 
  6.594244  -543.3 
 
  6.60024  -535.322 
 
  6.60124  -527.423 
 
  6.60124  -519.569 
 
  6.604239  -511.794 
 
  6.604239  -504.081 
 
  6.606236  -496.454 
 
  6.608236  -488.941 
 
  6.610237  -481.509 
 
  6.612234  -474.169 
 
  6.614233  -466.906 
 
  6.61623  -459.686 
 
  6.618229  -452.529 
 
  6.620229  -444.29 
 
  6.622227  -438.431 
 
  6.624226  -418.453 
 
  6.626225  -376.641 
 
  6.628224  -293.612 
 
  6.630222  -372.172 
 
  6.632222  -370.207 
 
  6.634222  -342.622 
 
  6.63622  -387.704 
 
  6.638218  -388.784 
 
  6.640217  -382.948 
 
  6.642216  -376.25 
 
  6.644216  -370.363 
 
  6.646214  -364.2 
 
  6.648213  -358.086 
 
  6.650211  -352.007 
 
  6.652211  -345.991 
 
  6.65421  -340.022 
 
  6.656208  -334.119 
 
  6.658206  -328.317 
 
  6.660205  -322.595 
 
  6.662205  -316.902 
 
  6.664203  -311.301 
 
  6.666202  -305.742 
 
  6.668201  -300.225 
 
  6.6702  -294.75 
 
  6.672198  -289.323 
 
  6.674199  -283.993 
 
  6.676197  -278.724 
 
  6.678196  -273.545 
 
  6.680193  -268.413 
 
  6.682193  -263.319 
 
  6.684193  -258.272 
 
  6.687191  -253.271 
 
  6.689188  -248.292 
 
  6.691189  -243.394 
 
  6.695187  -238.568 
 
  6.695187  -233.808 
 
  6.697185  -229.108 
 
  6.699185  -224.46 
 
  6.701182  -219.839 
 
  6.703181  -215.258 
 
  6.70518  -210.723 
 
  6.707179  -206.232 
 
  6.709177  -201.805 
 
  6.711176  -197.453 
 
  6.713174  -193.161 
 
  6.715174  -188.925 
 
  6.717173  -184.707 
 
  6.719172  -180.517 
 
  6.721171  -176.358 
 
  6.723169  -172.246 
 
  6.725168  -168.209 
 
  6.730165  -164.222 
 
  6.734165  -160.285 
 
  6.734165  -156.382 
 
  6.735164  -154.328 
 
  6.737163  -151.844 
 
  6.737163  -147.996 
 
  6.739161  -144.119 
 
  6.741159  -140.215 
 
  6.743158  -136.106 
 
  6.745158  -131.595 
 
  6.747156  -127.462 
 
  6.749155  -123.92 
 
  6.751153  -120.148 
 
  6.753152  -115.721 
 
  6.755154  -112.003 
 
  6.75715  -107.719 
 
  6.759149  -103.546 
 
  6.761148  -99.7241 
 
  6.763149  -96.165 
 
  6.765145  -93.0234 
 
  6.767146  -88.5902 
 
  6.767146  -84.4235 
 
  6.773141  -79.1224 
 
  6.77614  -72.6124 
 
  6.77614  -68.0654 
 
  6.778139  -62.9195 
 
  6.778139  -59.5198 
 
  6.780138  -56.6111 
 
  6.782135  -53.937 
 
  6.784135  -50.3906 
 
  6.786133  -46.2572 
 
  6.788134  -42.3364 
 
  6.790131  -34.9261 
 
  6.79213  -26.0749 
 
  6.794128  -21.8548 
 
  6.796127  -18.9241 
 
  6.798126  -16.1219 
 
  6.800126  -12.2873 
 
  6.802125  -9.54045 
 
  6.805122  -5.08956 
 
  6.807121  -2.08658 
 
  6.809122  -0.04531 
 
  6.809122  4.366211 
 
  6.811119  7.647571 
 
  6.813118  10.77139 
 
  6.815117  13.486 
 
  6.817116  16.37893 
 
  6.819114  19.87792 
 
  6.821114  22.49086 
 
  6.823112  25.43032 
 
  6.825112  27.59332 
 
  6.82711  30.51719 
 
  6.82911  33.26129 
 
  6.831109  34.98137 
 
  6.833107  37.28277 
 
  6.835105  40.44568 
 
  6.837104  44.28815 
 
  6.839103  46.3599 
 
  6.841102  49.35477 
 
  6.843102  52.05956 
 
  6.8451  54.62616 
 
  6.8471  56.4586 
 
  6.850097  59.63188 
 
  6.853096  61.88437 
 
  6.855093  64.40225 
 
  6.861091  66.85651 
 
  6.86309  68.85804 
 
  6.864089  70.77302 
 
  6.864089  73.11574 
 
  6.866088  75.82446 
 
  6.866088  78.38586 
 
  6.868086  80.85267 
 
  6.870086  83.50539 
 
  6.872084  85.50301 
 
  6.874083  87.47917 
 
  6.876081  89.76604 
 
  6.878081  91.43018 
 
  6.880079  93.5584 
 
  6.882078  95.26148 
 
  6.884078  97.05516 
 
  6.886076  99.08549 
 
  6.888075  100.8353 
 
  6.890073  102.5426 
 
  6.892073  104.4462 
 
  6.894071  105.7351 
 
  6.89607  107.2711 
 
  6.89807  108.8058 
 
  6.903066  110.5969 
 
  6.905066  112.0763 
 
  6.905066  113.6704 
 
  6.907065  115.3641 
 
  6.909064  116.8518 
 
  6.909064  118.2229 
 
  6.911062  119.6783 
 
  6.913061  120.9934 
 
  6.91506  122.3841 
 
  6.917058  123.9788 
 
  6.919057  125.464 
 
  6.921056  126.631 
 
  6.923054  128.0014 
 
  6.925054  129.6026 
 
  6.927052  130.7827 
 
  6.929051  131.7979 
 
  6.93105  133.159 
 
  6.933049  134.3399 
 
  6.935048  135.2165 
 
  6.937048  136.1109 
 
  6.939045  137.2935 
 
  6.945042  138.235 
 
  6.948041  139.0824 
 
  6.948041  140.0449 
 
  6.95004  141.2205 
 
  6.95004  142.0929 
 
  6.952038  143.099 
 
  6.954038  144.2232 
 
  6.956038  145.0893 
 
  6.958037  145.9516 
 
  6.96005  146.9087 
 
  6.962034  147.5583 
 
  6.964034  148.3394 
 
  6.966032  149.2911 
 
  6.968032  150.0796 
 
  6.970028  150.7998 
 
  6.972027  151.6231 
 
  6.974026  152.4815 
 
  6.976025  153.1206 
 
  6.978024  153.533 
 
  6.980022  154.1727 
 
  6.986019  154.7068 
 
  6.989019  155.1737 
 
  6.989019  155.7696 
 
  6.990017  156.294 
 
  6.992015  156.8771 
 
  6.992015  157.4982 
 
  6.994014  157.9492 
 
  6.996013  158.5362 
 
  6.998012  159.0485 
 
  7.000011  159.5271 
 
  7.002009  160.0849 
 
  7.004009  160.6587 
 
  7.006008  161.0866 
 
  7.008006  161.5438 
 
  7.010005  162.0543 
 
  7.012004  162.3111 
 
  7.014002  162.6459 
 
  7.016003  163.197 
 
  7.018003  163.6098 
 
  7.019999  163.8961 
 
  7.021999  164.4491 
 
  7.026995  164.8175 
 
  7.029994  165.119 
 
  7.030993  165.6028 
 
  7.032994  165.9381 
 
  7.032994  166.1743 
 
  7.034991  166.6661 
 
  7.03699  167.0216 
 
  7.038989  167.1757 
 
  7.040989  167.5963 
 
  7.042985  167.9184 
 
  7.044985  168.1421 
 
  7.046985  168.3151 
 
  7.048982  168.4674 
 
  7.050982  168.6788 
 
  7.052981  168.8595 
 
  7.054979  169.1347 
 
  7.056979  169.3541 
 
  7.059976  169.5881 
 
  7.059976  169.8421 
 
  7.061975  169.8756 
 
  7.063975  169.8835 
 
  7.069971  170.0201 
 
  7.072969  170.081 
 
  7.073971  169.8862 
 
  7.073971  170.2223 
 
  7.075968  170.4187 
 
  7.075968  169.9634 
 
  7.077966  170.2005 
 
  7.079967  170.3453 
 
  7.081963  170.1 
 
  7.083963  170.0746 
 
  7.085961  170.1612 
 
  7.08796  170.2195 
 
  7.089958  170.0519 
 
  7.091958  169.9185 
 
  7.093956  170.0157 
 
  7.095956  169.6901 
 
  7.098953  169.6671 
 
  7.100953  169.8258 
 
  7.102952  169.6786 
 
  7.10495  169.4224 
 
  7.10695  169.5625 
 
  7.111946  169.4362 
 
  7.113945  169.1236 
 
  7.114945  168.9096 
 
  7.114945  168.823 
 
  7.116946  168.4672 
 
  7.118942  168.4161 
 
  7.120942  168.3383 
 
  7.12294  168.1032 
 
  7.124939  168.0498 
 
  7.126937  167.9284 
 
  7.128938  167.5371 
 
  7.130936  167.4433 
 
  7.132934  167.3786 
 
  7.134933  166.9073 
 
  7.136932  166.5702 
 
  7.138931  166.6945 
 
  7.14093  166.3292 
 
  7.142929  165.9293 
 
  7.144927  165.8595 
 
  7.146926  165.6535 
 
  7.148925  165.2087 
 
  7.153922  165.0613 
 
  7.156921  164.8679 
 
  7.156921  164.3869 
 
  7.156921  164.176 
 
  7.158922  163.939 
 
  7.160919  163.4177 
 
  7.162918  163.2695 
 
  7.164918  163.213 
 
  7.166915  162.6047 
 
  7.168913  162.3941 
 
  7.170913  162.3912 
 
  7.172911  161.8986 
 
  7.17491  161.4153 
 
  7.176908  161.3523 
 
  7.178908  160.9821 
 
  7.180907  160.3998 
 
  7.182906  160.278 
 
  7.184905  160.0473 
 
  7.186904  159.4666 
 
  7.188902  159.2784 
 
  7.190902  159.0804 
 
  7.195898  158.5516 
 
  7.197897  158.1809 
 
  7.197897  157.9026 
 
  7.199897  157.422 
 
  7.201896  157.0084 
 
  7.201896  156.7611 
 
  7.203894  156.3454 
 
  7.205893  155.9246 
 
  7.207891  155.6399 
 
  7.209889  155.2374 
 
  7.211889  154.7015 
 
  7.213888  154.3584 
 
  7.215886  154.0497 
 
  7.217887  153.6112 
 
  7.219884  153.185 
 
  7.221883  152.9069 
 
  7.223883  152.5624 
 
  7.225882  152.0973 
 
  7.22788  151.6786 
 
  7.229878  151.3029 
 
  7.231877  150.8857 
 
  7.233878  150.4329 
 
  7.238873  150.0656 
 
  7.242872  149.6794 
 
  7.242872  149.2617 
 
  7.242872  148.8169 
 
  7.244871  148.3509 
 
  7.24687  147.8903 
 
  7.24887  147.4539 
 
  7.250867  147.0196 
 
  7.252866  146.5736 
 
  7.254865  146.1978 
 
  7.256864  145.7908 
 
  7.258862  145.3437 
 
  7.260861  144.8409 
 
  7.262861  144.3478 
 
  7.264859  143.8838 
 
  7.266858  143.341 
 
  7.269856  142.8814 
 
  7.269856  142.5035 
 
  7.271854  141.9434 
 
  7.273857  141.4785 
 
  7.275854  141.1005 
 
  7.27985  140.5386 
 
  7.28385  139.9541 
 
  7.28485  139.5345 
 
  7.286847  139.025 
 
  7.286847  138.458 
 
  7.288846  138.048 
 
  7.290845  137.6063 
 
  7.292843  137.0192 
 
  7.294841  136.5222 
 
  7.296842  136.0165 
 
  7.29884  135.4328 
 
  7.300838  134.953 
 
  7.302838  134.5215 
 
  7.304836  134.0005 
 
  7.306834  133.5269 
 
  7.308834  133.1048 
 
  7.310833  132.5513 
 
  7.312831  131.992 
 
  7.31483  131.5622 
 
  7.31683  131.0893 
 
  7.318829  130.6033 
 
  7.322825  130.1884 
 
  7.324825  129.8079 
 
  7.324825  129.2779 
 
  7.326824  128.7917 
 
  7.328823  128.3614 
 
  7.330822  127.8725 
 
  7.332823  127.371 
 
  7.334819  126.9179 
 
  7.336817  126.4276 
 
  7.338817  125.9606 
 
  7.340816  125.5183 
 
  7.342814  124.9966 
 
  7.344813  124.4469 
 
  7.346812  123.9549 
 
  7.34881  123.4337 
 
  7.35081  122.8687 
 
  7.352808  122.3893 
 
  7.354807  121.9347 
 
  7.356805  121.4249 
 
  7.358805  120.9284 
 
  7.360806  120.4473 
 
  7.364801  119.9055 
 
  7.3688  119.3628 
 
  7.3688  118.8858 
 
  7.3688  118.3684 
 
  7.370799  117.856 
 
  7.372797  117.3925 
 
  7.374796  116.872 
 
  7.376796  116.3047 
 
  7.378794  115.774 
 
  7.380793  115.233 
 
  7.382793  114.6945 
 
  7.384789  114.1907 
 
  7.386789  113.6828 
 
  7.388788  113.1834 
 
  7.390786  112.6814 
 
  7.392786  112.1621 
 
  7.394784  111.6135 
 
  7.396783  111.0996 
 
  7.398781  110.5704 
 
  7.400781  110.0443 
 
  7.402781  109.5678 
 
  7.409777  109.1059 
 
  7.409777  108.6323 
 
  7.409777  108.1536 
 
  7.412774  107.6277 
 
  7.412774  107.1109 
 
  7.414773  106.6228 
 
  7.416772  106.1419 
 
  7.418771  105.6723 
 
  7.42077  105.2416 
 
  7.422768  104.834 
 
  7.424767  104.3555 
 
  7.426766  103.8537 
 
  7.428765  103.3712 
 
  7.430763  102.8606 
 
  7.432762  102.3579 
 
  7.434762  101.887 
 
  7.43676  101.408 
 
  7.438759  100.9246 
 
  7.441758  100.4705 
 
  7.443758  99.98105 
 
  7.447754  99.43892 
 
  7.449753  98.9369 
 
  7.449753  98.46317 
 
  7.449753  97.94021 
 
  7.452752  97.45898 
 
  7.454751  97.04469 
 
  7.456748  96.56067 
 
  7.458749  96.03065 
 
  7.460746  95.56767 
 
  7.462745  95.0264 
 
  7.464744  94.49927 
 
  7.466743  94.00122 
 
  7.468742  93.52367 

[Torsten]

Google

displacement from acceleration & matlab

You'll get many hits, e.g.

https://de.mathworks.com/matlabcentral/answers/21700-finding-the-velocity-from-displacement

[MAB]

Thanks for your suggestions. Almost all the suggedstions from the google search uses 'cumtrapz", integration method, which i already used and turned out to be inefficient for computing diplacement from acceleration of impact loading. I am looking for some other accurate integration methodology.

[Sam Chak]

@MAB, have you applied the system identication technique using the available data to estimate the motion dynamics?

[Jan]

@MAB: The format, you have posted your data in, is not useable. It impedes the reading of the question only.

Please explain, what this exactly means: "cumtrapz, which is not good for integrating nonlinear data". You are not only using the cumtrapz function, but apply some filters in different steps. Do you have good physical reasons for the coice of the parameters?

If you do not have an underlying model, the pure measurement data are the best you have. The noise in the results cannot vanish magically and you cannot simple obtain clean data, if the input is noisy.

[Walter Roberson]

cumtrapz is quite vectorized. If it is too "inefficient" for your purposes, then you are not likely going to be able to find an faster integration method (that does not involve skipping data without examining it.)

Answer 1

[Binaya]

Hi,

Based on the provided description, it appears that you are seeking an alternative method to perform numerical integration on non-linear acceleration data, distinct from the utilization of the "cumtrapz" function. Please find the below suggestions for your query:

• To evaluate non-linear acceleration data,one possible approach is to employ a spline interpolant.
• By utilizing a spline interpolant, the data can be represented by a smoother curve, facilitating the subsequent integration using the "cumtrapz" function.
• Integrating a spline interpolant can yield more accurate results compared to directly integrating the original non-linear data.
• This is due to the nature of spline interpolation, which constructs a piecewise polynomial representation that closely approximates the data points.
• By employing this two-step process, involving spline interpolation followed by integration using "cumtrapz," you can potentially achieve improved accuracy and reliability in the estimation of the displacement from non-linear acceleration data.

Please find the below documentation for more details on ‘Data interpolation with “spline”,”pchip” and “makima”’:https://www.mathworks.com/help/matlab/ref/pchip.html#:~:text=Data%20Interpolation%20with%20spline%2C%20pchip%2C%20and%20makima

I hope this helps.

Regards 

Binaya