How to convert accelerometer values to path/coordinates

To get path or coordinates from accelerometer values you need to integrates them. From physics we know:

v = dr/dt
a = dv/dt = d²r/dt²

If a (acceleration) is constant, motion is called uniform accelerated:

v(t)=v(0)+at
r(t)=r(0)+v(0)t+0.5*at²

v(0), r(0) - initial velocity and displacement. If a != const, these formulas can not be used. If we know how a depends on t (for ex., a(t) = b + c*t), we can use:

v(t) = v(0) + b*t + 1/2*c*t²
r(t) = r(0) + v(0)*t + 1/2*b*t² + 1/6*c*t³

But if we don't know a(t), we can only use common rules:

          t
r(t)=r(0)+∫v(t)dt
          0

          t
v(t)=v(0)+∫a(t)dt
          0

and calculate this with some numerical integration method
Examples (models with plots) of such methods (in SciLab) you can found there [in integral.sci file: Simpson's rule, Trapezoid rule, Leo Tick's rule]. But we should know about 2 problems:

1. If there is some constant "component" in acceleration (or noise), this will be reason of BIG error in result
2. Each numerical integration method is filter and has transfer function, which give distortions on 0.2 - 0.4 Nyquist's frequency even [see R. W. Hamming, Digitals filters, 1977]

So, you can use more smart methods. One of them is complex (chaining) integration with filters. Idea is to filter each dataflow: between each integrator and to filter output result too:

a(t)->FILT->af(t)->INT->v(t)->FILT->vf(t)->INT->x(t)->FILT->xf(t)

   where FILT - is a filter, INT - is an integrator.

It can be implemented on dataflow model (as I done it in hmc-demo). You can found example of such model in SciLab source file "path.sci" where you can combain different integration methods with different filters and see result in plots.
And remember, eventually, you never get correct result. Use sensors of velocity instead.