[Math] 3D rotation of an object with respect to another object’s rotation

3dgeometryrotationstransformation

I am writing a python code to translate and rotate an object with respect to another object. Please take a look at the picture bellow:

object rotation with respect to another

The smiley face and the arrow have initial poses (position and orientation) as shown in the left picture. Then, the face rotates. I want the arrow to rotate and move to the pose shown on the right picture. I can imagine how to do that in 2D. However, I found it difficult to extend it to 3D.

So, the problem is as follows:
Given the initial and the current 3D pose of object x (say face) and the initial pose of object y (arrow), find a transformation T such that if we transform the initial pose of y using T, objects x and y keep their initial pose with respect to each other when the pose of object x changes.

The poses I get are quaternions, However, I have a python library which I can use to convert them to Euler angles and rotation matrices (and vice versa), so, that is not a problem.

Best Answer

Finally, I figured out how we can do that.

consider x_position and y_position are the initial 3D positions of x, and y respectively. And x_orientation and y_orientation are the initial orientations of x, and y respectively. x'_position and y'_position are the current position of x and y and x'_orientation , y'_orientation are the current orientations of x and y. 

y'_position = (x'_position - x_position) 
            - (y_position - x_position)
            + (y_position - x_position)*(x'_orientation - x_orientation)

y'_orientation = mat(x'_orientation - x_orientation)*mat(y_orientation)

mat converts Euler angels or quaternion to rotation matrix. For position, basically, we translate y into x, rotate it using the rotation matrix we got from the rotation of x, and translate it back to its new position.

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