Mixing intrinsic and extrinsic 3D rotations I am struggling with mixing two types of rotations; after much reading, I still feel a little confused. I am trying to mix intrinsic and extrinsic rotations.
I have three pieces of hardware that have a common point of rotation. The first motor rotates about the global z-axis. The second rotates about the global x-axis, but the third rotates about the local z-axis (resultant from the rotated x-axis). 
How do I mix the two sets of rotations, and can I combine them in to one rotation matrix?
My attempt in understanding this: using the notation from here (where ' denotes a rotation affected by the one before it), I figure it should look something like,
$R = Rz*(Rx*Rz')$
But should there be different rotation matrices for intrinsic and extrinsic rotations? Or is there a different way of applying them? I'm using standard 3D rotation matrices.
Edit: Is the difference between an extrinsic rotation and intrinsic rotation the difference between $R_x$ and $R_x^T$?
Any help is appreciated.
 A: 
Is the difference between an extrinsic rotation and intrinsic rotation the difference between $R_x$ and $R_x^{\;T}$?

No. The difference is the order in which the rotations or transformations chain
As noted in a comment, there is no standard. I've encountered almost every possible variant in my many decades of working. So first I'll specify the convention I'm using in this answer.


*

*For the most part, I'll be using transformation matrices as opposed to rotation matrices. My distinction: A rotation matrix describes the physical rotation of an object (e.g., the hand of a clock). A transformation matrix transforms the representation of a vector in one coordinate system to the representation of the same vector in another coordinate system. When needed, I'll distinguish between the two by using $\mathrm T$ for a transformation matrix, $\mathrm R$ for a rotation matrix.
The two concepts are obviously related. Think of a transformation matrix as being the consequence of physically rotating each the three principal axes by the same rotation operation.

*I'll be using matrices for transforming (or rotating) column vectors as opposed to row vectors. A n×1 column vector transforms via $\mathrm T\mathbf v$ while a 1×n row vector transforms via $\mathbf v \mathrm T$.

*I'll be using transformation matrices that transform a vector as represented in some parent (original) frame to the vector's representation in some child frame (a frame that derives from the original frame by a sequence of one or more primitive rotations).

*I'll be using transformations that follow the very standard right hand rotation rule. (Yes, there are some people/organizations who use left handed rotations, or left handed frames. Microsoft, for example.)
With this, the primitive transformations are
$$
\begin{aligned}
T_x(\theta) = \begin{bmatrix}
  1&0&0 \\
  0&\phantom{-}\cos\theta&\sin\theta \\
  0 & -\sin\theta&\cos\theta
\end{bmatrix} \\
T_y(\theta) = \begin{bmatrix}
  \cos\theta & 0 & -\sin\theta \\
  0&1&0 \\
  \sin\theta & 0 & \phantom{-}\cos\theta
\end{bmatrix} \\
T_z(\theta) = \begin{bmatrix}
  \phantom{-}\cos\theta&\sin\theta&0 \\
  -\sin\theta&\cos\theta&0\\
  0&0&1
\end{bmatrix} \\
\end{aligned}
$$
With this convention, the rules for a sequence of intrinsic transformations $a$ followed by $b$ followed by ... followed by $z$ is $\mathrm T_{a,b,\ldots,z} = \mathrm T_z \cdots \mathrm T_b \mathrm T_a$: They chain right to left. The order is reversed with a sequence of extrinsic transformations: Extrinsic transformations chain left to right as opposed to the right to left chain for a sequence of intrinsic transformations.
What about a mix? The rule is still simple. Write down with the first operation in the sequence. The next operation goes on the left if that operation is intrinsic, but on the right if it's extrinsic. Rinse and repeat.
Your devices started with a rotation about global Z ($T_z(\theta_1)$). This is followed by a rotation about global X ($T_x(\theta_2)$). This is extrinsic, so the combined sequence becomes $T_z(\theta_1)T_x(\theta_2)$. This is in turn followed by a rotation about the Z axis ($T_z(\theta_3)$) that results after the first two rotations. This is intrinsic, so the full sequence is $T_z(\theta_3)T_z(\theta_1)T_x(\theta_2)$.
And now there are two big problems with your device. The first is that you can toss one of those Z joints. The second is that your device does not span the space of all possible orientations in three dimensional space.
