I have begun a basic incursion into understanding quaternions. I have read the following: https://en.wikipedia.org/wiki/Quaternion and https://en.wikipedia.org/wiki/Quaternions_and_spatial_rotation. I understand that quaternions are used to identify the rotation of an object in 3-dimensional space. I have seen quaternions be converted into 3D space via turning the Quaternion into Euler rotations but from my understanding, these are not positional coordinates rather another representation of rotation. I am wondering if it is possible to use quaternions to plot an objects path or if I would need something separate. I am asking in the context of IMU data with 9df (Accelerometer, Magnetometer, Gyroscope). I have applied the Madgwick filter to this data and received quaternion output. However, I do not believe I can use this alone to plot a path. As I do not have a Mathematics background I was hoping to hear opinions from people with greater expertise.
You need to use Madgwicks filter and the algorithm described in detail here. Using Rivals theorem, this algorithm lets you calculate the acceleration of any point of the body (typically used for center of mass). Just integrate it twice to get the position (preferably, use Cauchy's formula for repeated integration to reduce the error of numerical integration). Don't forget to calibrate you IMU!