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I am studying robotics as an undergraduate. One of the things we're doing is rotational matrices. As a general principle I understand them, and if given a point in A I can find the given point in B with a rotational matrix. However, as an exercise for my own deeper understanding I want to derive a rotational matrix. I mostly understand how the common rotational matrices $X,Y,Z$ are determined, but not entirely. I have this image below, and I want to design a rotation matrix from the $(x_1, y_1, z_1)$ frame to the $(x_2, y_2, z_2)$ frame. I'd really like some help with this topic. Make note: This is not a school assignment. I really want to understand how to do this. Thanks!

robot

Peter Phipps
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jgauthier
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  • It seems to me that the diagram is mislabeled: there is no $z_1$ (which is OK: presumably it's normal to the $(x_1, y_1)$ plane), but surprisingly, there is no $x_2$ - shouldn't $z_2$ be $x_2$ instead? That would make for a simpler rotation matrix. Similarly the 3-frame should be just a translation of the 2-frame, so $z_3$ should be $x_3$. – NickD Sep 27 '18 at 19:40
  • Possible duplicate of Calculate Rotation Matrix to align Vector A to Vector B in 3d? – Adrian Keister Sep 27 '18 at 19:59
  • It’s not clear from your diagram what the known quantities are. In particular, do you already know what $x_2$, $y_2$ and $z_2$ are (in which case it’s almost trivial to construct the rotation), or do you have only the starting coordinate frame and the axis and angle of the rotation? – amd Sep 27 '18 at 20:02
  • z1 is facing away from the screen. x2 is facing into the screen.

    There are no known quantities. I want to build a matrix. Then when I have vector in body frame 1, I can use the matrix to determine it's location in body frame 2 (given an angle). It's a conceptual matrix, not a numerical one. For instance, I know that r11 of a matrix is how x1 projects onto x2, and that r12 is y1 projected on x2. What i can't figure out is what those projections are.

     – jgauthier Sep 27 '18 at 21:53

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