# Double integration of the acceleration signals to obtain displacment signals

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UTS on 17 Dec 2014
Commented: Samuel Hudson on 14 Jul 2022
Hi,
In order to obtain the displacement signals from the acceleration data, The following steps are used to convert the acceleration data to achieve the displacement values:
1- The acceleration signals are filtered [High pass filter]
2- The cumtrapz is applied to integrate the displacement to obtain the velocity
3- The velocity signals are filtered [High pass filter]
4- Finally, the filtered signals of velocity is integrated to get the displacement signals.
The concern is still about the correct methods and the accuracy of the results which are obtained.
Please have look at the Matlab Program as stated below; If there any comments and contribution regarding the proposed methodology please do not hesitate to shear us your experience, Thank you
%%accelerations are integrated twice to produce displacements
clear all
close all
clc
figure
plot(time,acc)
xlabel('Time (sec)')
ylabel('Acceleration (mm/sec^2)')
%%Design High Pass Filter
fs = 8000; % Sampling Rate
fc = 0.1/30; % Cut off Frequency
order = 6; % 6th Order Filter
%%Filter Acceleration Signals
[b1 a1] = butter(order,fc,'high');
accf=filtfilt(b1,a1,acc);
figure (2)
plot(time,accf,'r'); hold on
plot(time,acc)
xlabel('Time (sec)')
ylabel('Acceleration (mm/sec^2)')
%%First Integration (Acceleration - Veloicty)
velocity=cumtrapz(time,accf);
figure (3)
plot(time,velocity)
xlabel('Time (sec)')
ylabel('Velocity (mm/sec)')
%%Filter Veloicty Signals
[b2 a2] = butter(order,fc,'high');
velf = filtfilt(b2,a2,velocity);
%%Second Integration (Velocity - Displacement)
Displacement=cumtrapz(time, velf);
figure(4)
plot(time,Displacement)
xlabel('Time (sec)')
ylabel('Displacement (mm)')
Samuel Hudson on 14 Jul 2022
Would you be willing to provide a sample of data that works in this code?

UTS on 18 Dec 2014
Matthew Nijsten on 27 Mar 2018
Hi there, so if i put an array of 2498x1 acceleration points into this code it will output me 2498x1 respective data points

as k on 3 Apr 2016
Its working ıts working , Thank you so much my friend . Really saved my life today. I hope the sun may always shine upon your path.

as k on 3 Apr 2016
One more thing , we usually measure disp. in mm and those calculation above give us result ,generally, in meters so this is just a reminder.

Hasan Siddiqui on 14 Jul 2016
What units does the original accelerometer data need to be in originally? I have data collected from an arduino that is a 10 bit voltage reading from 0-1023. Does it need to be converted to the range of the accelerometer in gs first (-3g to 3g)?

Adi Negoro on 24 Apr 2017
I'm newbie,
how to understand the "fc=0.1/30; % Cut off Frequency"?
Vincent den Ouden on 26 May 2022
the term butterworth says more something about the idea/characteristics behind the filter, and in the case of the butterworth the idea is to have the most flat frequency response possible in the passband (frequencies that you want). The filter itself can be lowpass/highpass or bandpass.

Aashish Sah on 19 May 2020
Edited: Aashish Sah on 19 May 2020
Hi,
I have one question regarding this code. I have acceleration data in terms of g, so I multiply by 9.8 to get in terms of ms^-2. The displacement should then be in terms of m. However, when I apply the filter to my acceleration data, it normalizes the data. For example, the max and min of raw acceleration data is [10.5, 9.6] ms^-2 and after I apply the filter I get [0.68 -0.56]. It seems that filtering process normalizes the raw data between -1 and 1.
This introduces an issue for me since I wish to compare the actual displacement from different accelerometer sensors.
Any suggestions?
Broc Sommermeyer on 15 Nov 2020
This code is designed to measure the velocity and displacement that are a function of time dependent acceleration by applying the high pass filter to remove the constant gravity value of 1g. Fourier series representation of periodic signals would break down a(t) into a constant term and an infinite sum of sine and cosine terms. The filter removes the constant term and any sine and cosine terms that have a frequency lower than the cutoff frequency fc. If you want to look at the complete accelerometer data including gravity remove the filter from the code.

Joven Chou on 30 Dec 2020
awesome codes

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