# Classical DH and Euler angles parametrization

Let DH convention is carried out as $$A_i = \text{Rot}_{z,\theta_i}\text{Trans}_{z,d_i}\text{Trans}_{x,a_i}\text{Rot}_{x,\alpha_i}$$ This is the classical DH method stated in Spong's book. The homogenous transformation matrix $$T^0_n=A_1 A_2\cdots A_n$$ is $$T^0_n= \begin{bmatrix} R^0_n &p^0_n \\ 0&1\end{bmatrix}\in \text{SE}(3)$$

The question is what is the Euler parameterization convention embedded in the rotation matrix $$R^0_n$$ (i.e. ZYZ,XYZ, ...,etc)? In the Spong's book, does the following occur accidentally? I'm asking this because I need to derive the analytic Jacobian for the angular part (i.e. $$J_w$$) which requires which Euler angles parameterization is being used.

It appears that $$R^{3}_{6}$$ in equation 3.15 likely arises from the fact that the specific robot in the example has a "spherical wrist" structure. Considering joints $$n-2$$, $$n-1$$, and $$n$$, this structure occurs when joints $$n-2$$ and $$n$$ have colinear axes of rotation when joint $$n-1$$ is at some angle $$\theta$$ (this is typically the case at $$\theta = 0$$). Pictorially, this looks like:
We can see that the rotation of $$\beta$$ by another $$\frac{\pi}{2}$$ radians would align the first and third joints' rotational axes.
So, to answer your question the given excerpts from the Spong book do not happen accidentally - rather they occur purposefully. The kinematic structure of the spherical wrist is essentially a one-to-one mapping to the ZYZ euler angles. From a physical perspective, consider the fact that $$\beta$$ in the above picture is along the Y-axis of joint 1 and consider the case where joint 1, joint 2, and joint 3 all start at angles of $$0$$ (the end-effector pointing straight up). Thus, the end-effector's current Z axis will be aligned with joint 1. Any rotation along joint 1 will not disassociate the joint 2 axis of rotation with the current Y axis of the end-effector. Thus, any rotation of joint 2 will be equivalent to a current Y rotation. Finally, the formulation of the spherical wrist is such that the final joint always rotates along the current Z axis of the end-effector. Thus, joint 3 is equivalent to the final current Z rotation of the end-effector.