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Finding the relative pose of a robot gripper

I have a robot arm with a gripper. I know the gripper pose (relative to the robot base coordinate system) at any moment. At startup, I record the pose of the gripper and set this as the original pose O. Then, the gripper moves to its new pose N, again in the robot base coordinate system. What I want to calculate, is R, the relative pose of the gripper in the coordinate system of the original pose O, rather than in the coordinate system of the robot base, which would just be N.

So, O = original pose, N = new pose, R = relative pose

From my calculations:

O * R = N

Therefore:

R = O^-1 * N

However, from my observations, R is the pose of the hand relative to the original pose, but in the coordinate system of the robot base. I want it to be in the coordinate system of the original pose. How do I get this?