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I know that integrating acceleration twice will give me position (acceleration-->velocity-->position) but how can I do all this when I all I have are a set of data points (ex: 1 second = some # m/s^2); how can I take these data points and integrate them?

Does this have something to do with numerical integration?

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Even if you also have gyroscopes, the errors from them and from the accelerometers will be too large when integrated over any useful period of time and the results for position will have huge errors.

What you would be constructing is an inertial navigation system (INS). This requires highly accurate instruments such as you find in the INS used in military and other aircraft. An accurate INS is much bigger than you could fit in a smartphone - even those ridiculously large ones they are marketing nowadays!

Even a military-grade INS has errors and its data is usually mixed with GPS data to provide a highly accurate and resilient navigation system. This integrated system is also used in commercial aircraft.

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Theoretically these calculations are possible. But it gets complicated by the fact that each of the three accelerometers (one each in an axis perpendicular to the other two) are constantly changing in orientation. You could track orientation in a similar way with data from electronic gyros. I'm not sure if android devices include gyros. You are also limited by the resolution of your data and errors would creep in rather quickly limiting the accuracy for fairly short durations.

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