I am attempting to calculate an angular velocity that will get me from the current gameObject.transform.rotation to a given Quaternion rotation using angular velocity. I've tried several methods but all of them seem to cause oscillation (maybe due to the gimbal nature of Vector3 for angular velocity).

Here are the pieces of code that I have tried:

Quaternion rotationDelta = Quaternion.Inverse(gameObject.transform.rotation) * remoteRotation;
return (rotationDelta.eulerAngles / PlayerMovement.BROADCAST_INTERVAL);

Another attempt using toAngleAxis:

rotationDelta.ToAngleAxis(out angleInDegrees, out rotationAxis);
Vector3 angularDisplacement = rotationAxis * angleInDegrees * Mathf.Deg2Rad;
return angularDisplacement / PlayerMovement.BROADCAST_INTERVAL;

Using Eulers:

 Vector3 difference = new Vector3(Mathf.DeltaAngle(gameObject.transform.rotation.eulerAngles.x, remoteRotation.eulerAngles.x),
                                  Mathf.DeltaAngle(gameObject.transform.rotation.eulerAngles.y, remoteRotation.eulerAngles.y),
                                  Mathf.DeltaAngle(gameObject.transform.rotation.eulerAngles.z, remoteRotation.eulerAngles.z));

 Vector3 differenceInRadians = new Vector3(difference.x * Mathf.Deg2Rad,
                                           difference.y * Mathf.Deg2Rad,
                                           difference.z * Mathf.Deg2Rad);

 return differenceInRadians / PlayerMovement.BROADCAST_INTERVAL;

I am trying to compare the Z-axis and the X-axis of two different quaternions in a way that would give me the Euler angles about the X and Y axes to line up the two different axes. In my program, the first quaternion (whose X-axis we are trying to line up) only rotates about the z and y axes and is the one being rotated first. Whereas the second quaternion (whose Z-axis we are trying to line up) only rotates about the x and y axis and is the one being rotated to match the other quaternion. Currently, I've been able to deduce the direct distance to line up the axes though am struggling to break the angles into pitch and yaw.