Why not just integrate the accelerometer?

It seems natural to try to accomplish $6$-DOF tracking with an IMU alone. Recall from Figure 9.5 that the accelerometer measures the vector sum of true linear acceleration and acceleration due to gravity. If the gravity component is subtracted away from the output, as is heuristically accomplished for tilt correction, then it seems that the remaining part is pure body acceleration. Why not simply integrate this acceleration twice to obtain position estimates? The trouble is that the drift error rate is much larger than in the case of a gyroscope. A simple calibration error leads to linearly growing drift error in the gyroscope case because it is the result of a single integration. After a double integration, a calibration error leads to quadratically growing drift error. This becomes unbearable in practice after a fraction of a second. Furthermore, the true body acceleration cannot be accurately extracted, especially when the body quickly rotates. Finally, as drift accumulates, what sensors can be used to estimate the positional drift error? The IMU alone cannot help. Note that it cannot even distinguish motions at constant velocity, including zero motion; this is the same as our vestibular organs. Despite its shortcomings, modern IMUs remain an important part of $6$-DOF tracking systems because of their high sampling rates and ability to accurately handle the rotational component.