How to get the relative rotation of 2 quaternions in Unity

Question

I'm trying to obtain the angle between the upperarm en lowerarm using two IMU's respectively. I calibrate both imu's and use Quaternions to get global orientation of the upperarm and lowerarm object in Unity 3D.

Qimu1 = Quaternion from the imu located on the lowerarm

Qimu2 = Quaternion from the imu located on upperarm.

I made two objects in Unity representing the upperarm en lowerarm. Where the lowerarm is a child object of the upperarm and is connected to the upperarm using Character joint.

This is the script for the upperarm:

if (Input.GetKeyDown(KeyCode.Space))
        {
            Qstart2 = Qimu2;
        }

        transform.localRotation = Qstart2 * Quaternion.Inverse(Qimu2);

Lowerarm script:

if (Input.GetKeyDown(KeyCode.Space))
        {
            Qstart1 = Qimu1;
        }

        Qimu1new = Qstart1 * Quaternion.Inverse(Qimu1);
        Quaternion Qdiff = Quaternion.Inverse(Qimu1new) * Qimu2;
        transform.localRotation = Qdiff;

What i would like to acquire is a visual representation of arm movement in Unity3D using 2 IMU's. Now if i keep my elbow extended the script works. But as soon as i start to flex my elbow the lowerarm object starts to continuously rotate around the same axis.

Answer 1

We can use some algebra to find the calculation for the lower arm transform's relative rotation:

The formula Qstartx * Quaternion.Inverse(Qimux) gives you a rotation for that part relative to the body:

upper arm's rotation relative to body = Qstart2 * Quaternion.Inverse(Qimu2)
lower arm's rotation relative to body = Qstart1 * Quaternion.Inverse(Qimu1)

We know that since lower arm is a child of the upper arm:

lower arm's rotation relative to body 
    = upper arm's rotation relative to body * lowerArmTransform.relativeRotation

With substitution, we can determine:

Qstart1 * Quaternion.Inverse(Qimu1) 
    = (Qstart2 * Quaternion.Inverse(Qimu2)) * lowerArmTransform.relativeRotation

Multiply both sides on the left by Quaternion.Inverse(Qstart2 * Quaternion.Inverse(Qimu2)):

Quaternion.Inverse(Qstart2 * Quaternion.Inverse(Qimu2)) * Qstart1 
* Quaternion.Inverse(Qimu1) 
    = Quaternion.Inverse(Qstart2 * Quaternion.Inverse(Qimu2)) 
      * ( Qstart2 * Quaternion.Inverse(Qimu2) )
      * lowerArmTransform.relativeRotation

Quaternion.Inverse(Qstart2 * Quaternion.Inverse(Qimu2)) * ( Qstart2 * Quaternion.Inverse(Qimu2) ) cancel out:

Quaternion.Inverse(Qstart2 * Quaternion.Inverse(Qimu2)) * Qstart1 
* Quaternion.Inverse(Qimu1) 
    = lowerArmTransform.relativeRotation

Since Qimu1new = Qstart1 * Quaternion.Inverse(Qimu1);, we can conclude:

lowerArmTransform.relativeRotation = Quaternion.Inverse(Qstart2 * Quaternion.Inverse(Qimu2)) * Qimu1new;