Sign Errors in the Z′Y′X′ Euler-Angle Rotation Matrix in Appendix B of Craig's "Introduction to Robotics" (3rd, 4th, and Global Editions): An Independent Erratum Note with SymPy Verification

This technical note reports and corrects two sign errors in the rotation matrix printed for the Z′Y′X′ Euler-angle set (the yaw–pitch–roll convention) in Appendix B, "The 24 Angle-Set Conventions," of John J. Craig's Introduction to Robotics: Mechanics and Control. In entries (2,2) and (2,3), the product term containing sin(α)·sin(β) carries the wrong sign. The misprint appears identically in the 3rd Edition (2004), the 4th Edition (2018), and the Global Edition (2022). The note establishes the erratum by three independent arguments — failure of orthonormality (det R = 1 − 2sin²α sin²β), direct symbolic expansion ofR_Z(α)R_Y(β)R_X(γ) under the book's own definitions, and internal inconsistency with the book's X–Y–Z fixed-angles result — and explains four mechanisms by which the defect evades common testing (single-axis sweeps, row-norm checks, and angle round-trip tests all pass). All 24 angle-set matrices of Appendix B were transcribed and verified symbolically; the Z′Y′X′ matrix is the only one in error. The archive includes the SymPy script that reproduces every claim.