I am using Tsai's Robot Analysis (The Mechanics of Serial and Parallel Manipulators) and I am working on deriving the homogeneous transformation matrices for the SCARA robot with the assigned frames as given in the diagram. From my understanding of the DH convention, to obtain θ1 (row 1 in the table) we rotate the x0-axis about the z0-axis such that x0 aligns with x1, in this case wouldn't θ1 (row 1 in the table) be = 90° + θ1 (joint variable)? But according to the example's solution (and many other scientific journals that I've been through) θ1 (row 1 in the table) is just equal to θ1 (joint variable), regardless of the frame configuration, how is this value determined? Wouldn't θ1 (row 1 in the table) only be equal to θ1 (joint variable) if the x0 and y0 axes swap directions because in that case x0 will already be aligned with x1 and so x0 would only be required to be rotated by θ1 (joint variable) to remain along the same direction as x1. I tried substituting my DH parameters into the formulae for the transformation matrices but I didn't get the same values, even though I assigned the axes in the same frame configuration as the example (my solution is in the image below this paragraph). Am I missing/overlooking something or making some mistake in my analysis? Is this an error in the textbook or am I wrong?
For instance, in this research paper that I found [Inverse kinematics for a Humanoid Robot : a mix between closed form and geometric solutions, Fabrice R. Noreils], the author applies the rules that I've mentioned (for joints 1, 2, 3 and 6 in the table), which looks like a similar situation to joint 1 in my table for the SCARA manipulator. However, even in this paper, the author seems to have used cos(θ1) and sin(θ1) instead of cos(θ1 - 90°) = sin(θ1) and sin(θ1 - 90°) = -cos(θ1) (for row 1 of the table and similarly for rows 2, 3 and 6) when substituting the DH parameters in the homogeneous transformation matrix formula (0T1).
In yet another set of slides that I found [https://profesores.utec.edu.pe/oramos/teaching/18/robotics/lectures.html], the professor applies the rules I've mentioned AND substitutes the DH parameters from the table accordingly. This is all so confusing, which is the correct technique for deriving the transformations? If the constant angles are sometimes omitted then consistent results are not obtained throughout robotics articles and the transformations will not comply with the assigned robot joint coordinates.