# 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, 2021 at 14:46
• A little confused about your question. The page you reference shows that the constant DH transform (T_n) get's multiplied by the joint transform (J_n). This should make sense, as you move down the chain link-by-link, you offset according to the rigid link (T_n), then offset according to the joint angle (J_n). DH is just one method to get (T_n). Could you clarify what's confusing you? Sep 29, 2021 at 21:15
• Peter Corke's book goes through a pretty straightforward treatment of this, if I recall. Feb 25, 2022 at 12:55

The author of the referenced thesis is using exponential coordinates and screw theory via the Product of Exponentials formulation to generate the sequence of transformations from one link to the next.

They have separated out the constant offset from one joint to the next for a given link from the effects of each joint, but it is effectively using the same product of exponentials formulation which is explained very nicely in Chapter 4 of the Modern Robotics (MR) textbook by Lynch and Park (which is fully available online at http://modernrobotics.org, along with slides and videos explaining the key details).

To translate between the two formulations, the scalar thesis variable $$j_i$$ is the MR variable $$\theta$$, the vector $$v_i$$ is the MR vector $$\hat \omega$$, and the combined matrix product $$T_iJ_i$$ is the MR matrix $$e^{[S]\theta}$$, which combines both the constant transformation and the joint transformation via the screw axis $$S$$.