How can obtain the relative orientation between two quaternions?

Question

How do I obtain the relative orientation given two orientations (represented by quaternions q0 and q1)?

Answer 1

The relative orientation is obtained simply by division:

q = q0 / q1

Or, if division is not available:

q = q0 * inverse(q1)

Note that since the quaternions used to represent rotations are unit quaternions, the inverse of q1 is simply its conjugate q1*, and is obtained by flipping the sign of x, y, z but not w.

Answer 2

Quaternion is another representation of axis angle. The solution is to create a new quaternion from the original quaternion that only has the needed components.

An axis angle representation can be converted to a quaternion using the following formula

q[0] = cos(R/2);
q[1] = sin(R/2)*x;
q[2] = sin(R/2)*y;
q[3] = sin(R/2)*z;

Where R is the angle in radians, and (x,y,z) represents the axis, and quaternion is (R,x,y,z).

So in order to create a new quaternion with only the pitch component you just zero out the other components and normalize the quaternion:

Quaternion q; // this is your original quaternion
q.x = 0.0;
q.z = 0.0;
q.Normalize();

Edit based on your update:

Sensors AFAIK calculate orientation relative to gravity, in other words Y (or Z) is the direction of the gravity. You need take that into consideration. And I think you don't need to multiply it with the inverse initial orientation.