In the book Introduction to Robotics by John J. Craig, the explanation of multiplying matrices for fixed-angle rotation is a bit confusing. I understand fixed-angle rotation order $XYZ$ is equal to Euler angle $ZYX$, but I am wondering why the equation is written as following, or what mathematical convention describes this way of writing so I know in the future what it means.

It is described as:

The derivation of the equivalent rotation matrix, $_B^AR_{XYZ}(\gamma,\beta,\alpha)$ is straight-forward, because all rotations occur about axes of the reference frame; that is, $$ _B^AR_{XYZ}(\gamma,\beta,\alpha) = R_Z(\alpha)R_Y(\beta)R_X(\gamma) $$

Since rotation operations do not commute, the order is important, so I would imagine it should be obvious for a notation to imply it is multiplied in reverse order.

  • $\begingroup$ @OnceUponACrinoid I am asking if there is a convention for why the order is then written backwords. Reading right to left or left to right is the same, since it is still the matrix RxRyRz not RzRyRx, and since ABC = A(BC), the order of reading is irrelevant. Maybe the answer is just 'how to think about it', but i was looking if there was a specific explanation $\endgroup$ – user-2147482637 Jun 15 '15 at 6:58
  • $\begingroup$ Well, when describing Euler rotations, it says Rx'y'z' = RxRyRz , and subsequent operations are added to the right side, in both cases (so if you rotate first, then translate based on the rotation, translation is appended to right side) $\endgroup$ – user-2147482637 Jun 15 '15 at 7:07

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