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physics - How to calculate position from 3 dimensional acceleration data?


I discussed this problem yesterday with Pickett here. We concluded that the best solution may be just Interpolate and then NIntegrate to get the position vector. I also suggested smoothing but as Pickett commented you can lose important details with it.


How to calculate the position from acceleration data in Mathematica?


Example where moving to one direction and other directions about the same



Import["http://pastebin.com/raw.php?i=jZ57mqZT"]

or


time_tick,acc_X_value,acc_Y_value,acc_Z_value
0.008387,-7.051625,-0.432767,-6.701011
0.041984,-7.308113,-0.712300,-6.199558
0.074841,-6.989672,-1.105712,-6.235771
0.108211,-7.580313,-0.931228,-5.587518
0.141834,-7.547990,-0.979114,-5.437576
0.174273,-7.075867,-1.138783,-6.130123

0.208107,-7.554275,-0.835906,-5.692567
0.240980,-7.329661,-0.960558,-5.710225
0.274663,-7.546344,-0.690752,-5.827994
0.308129,-6.949119,-0.860447,-6.631278
0.341972,-6.716275,-0.842939,-6.727797
0.374906,-7.340585,-0.757642,-6.366709
0.408325,-6.646092,-0.905340,-6.730790
0.440989,-6.814441,-1.069348,-6.686945
0.474559,-7.092926,-0.681474,-6.726151
0.508442,-6.090618,-0.887832,-7.290455

0.541162,-7.464041,-0.712600,-6.281712
0.574674,-6.314932,-0.937214,-6.787804
0.608281,-6.961689,-0.820642,-6.581596
0.641125,-7.042347,-0.777395,-6.264054
0.674738,-6.658662,-0.941104,-6.425518
0.708979,-6.956152,-0.764526,-6.639957
0.741439,-6.618408,-0.791462,-6.837934
0.774397,-6.924129,-0.730108,-6.721961

Answer



Assuming that all the initial positions (x[0], y[0] and z[0]) and the initial velocities (x'[0], y'[0] and z'[0]) are equal to 0 you can do:



adat = Rest@Import["http://pastebin.com/raw.php?i=jZ57mqZT"];
{ax, ay, az} = Interpolation /@ (adat[[All, {1, #}]] & /@ {2, 3, 4});
{xt, yt, zt} = (x /. Quiet@First@NDSolve[{
        x[0] == 0, x'[0] == 0,
        x''[t] == #[t]
        }, x, {t, First@adat[[All, 1]], Last@adat[[All, 1]]}] & /@ {ax, ay, az});


Plot[{ax[t], ay[t], az[t]}, {t, First@adat[[All, 1]],Last@adat[[All, 1]]}, 
 PlotLabel -> "Acceleration", Frame -> True, 

 FrameLabel -> {"Time", "Aceleration"}]


Mathematica graphics



Plot[{xt[t], yt[t], zt[t]}, {t, First@adat[[All, 1]], Last@adat[[All, 1]]}, 
 PlotLabel -> "Positions", Frame -> True, 
 FrameLabel -> {"Time", "Position"}]



Mathematica graphics



ParametricPlot3D[{xt[t], yt[t], zt[t]}, {t, First@adat[[All, 1]], Last@adat[[All, 1]]}, 
 PlotRange -> {{0, -2}, {-1, 1}, {0, -2}}, 
 AxesLabel -> {"X", "Y", "Z"}]


Mathematica graphics



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