# Forward kinematics confusing point

I am studying robot arm kinematics. In order to construct a transformation from a global coordinate frame to the end effector frame, I have seen references to the Denavit–Hartenberg (DH) convention. In this convention, a single operator is constructed for each joint which takes its frame to the next and depends on the angle of the joint and three other parameters specific to the convention.

However, in this paper, section 4.1.1 page 104 implies that the transformation chain can be deconstructed into two transformations for each joint, one of which represents the rotation about an arbitrary axis (the joint axis) and the other is a constant affine transformation.

Does anyone know how to deconstruct the chain as this paper references? To me it seems that even what they call a constant affine transformation must still depend on the angle of the previous joint when compared to the DH method.

• A DDG search for "decompose transformation matrix into rotation and affine" turned up likely-looking hits. Skimming the first two was encouraging, but made it clear that my command of the relevant math is quite rusty. math.stackexchange and mathworks seem authoritative. Apr 6 at 14:46