Getting a trajectory from accelerometer and gyroscope (IMU)

Question

I am well aware of the existence of this question but mine will differ. I also know that there could be significant errors with this approach but I want to understand the config

Answer 1

Your conceptual framework is correct, but the equations need some work. The acceleration is measured in the platform frame, which can rotate very quickly, so it is not advisable to integrate acceleration in the platform frame and rotate the position change. Rather, the accelerations are transformed into a relatively slowly rotating frame and the integration to velocity change and position change is done there. Typically a locally-level frame (e.g. North-East-Down or Wander Aziumuth) or an Earth-centered frame (ECEF or ECI). Gravity and Coriolis force must be included in the acceleration.

Derivations from first principles can be found in many references, one of my favorites is Strapdown Inertial Navigation Technology by Titterton and Weston. Derivations of the inertial navigation equations in locally-level and Earth-fixed frames are given in Chapter 3.

As you've recognized in your question - the initial velocity is an unknown constant of integration. Without some estimate of initial velocity the trajectory resulting from integrating the inertial data can be wildly wrong.