Rot() and Tran() Operator

When we build a homogeneous transformation matrix $T$, we build it by multiplying two kinds of blocks together, $Rot()$ and $Tran()$. It is worth being clear on how these relate to the plain rotation matrix $R_k(\theta )$ that sits in the top left corner of $T$, since the two are easy to mix up.

The 3x3 rotation $R_k(\theta )$

$R_k(\theta )$ is a 3x3 pure rotation. It rotates by angle $\theta$ about axis $k$, where $k=1,2,3$ means the $x,y,z$ axes. It only carries orientation and knows nothing about position. This is the exact block that lives in the top left corner of $T$.

$$R_1(\theta )=\left[\begin{array}{ccc}1& 0& 0\\ 0& \cos \theta & -\sin \theta \\ 0& \sin \theta & \cos \theta \end{array}\right]$$

The 4x4 operator $Rot(k,\theta )$

$Rot(k,\theta )$ is the 4x4 homogeneous version of that same rotation. It is just $R_k(\theta )$ dropped into the top left corner, padded with a zero translation column on the right and a bottom row of $[0001]$:

$$Rot(k,\theta )=\left[\begin{array}{cc}{R}_k(\theta )& 0\\ 0& 1\end{array}\right]$$

The 4x4 operator $Tran(p)$

$Tran(p)$ is the matching pure translation, an identity rotation in the corner with the displacement $p$ sitting in the right column:

$$Tran(p)=\left[\begin{array}{cc}I& p\\ 0& 1\end{array}\right]=\left[\begin{array}{cccc}1& 0& 0& p_x\\ 0& 1& 0& p_y\\ 0& 0& 1& p_z\\ 0& 0& 0& 1\end{array}\right]$$

Why there are two sizes

To combine a rotation and a translation in one matrix multiply, both have to be $4\times 4$, because we cannot multiply a $3\times 3$ by a $4\times 4$. So we promote the small $R_k$ into a padded $4\times 4$, and that padded thing is what $Rot()$ means. Think of $R_k$ as the ingredient and $Rot()$ as the ingredient cut to the right size for the chain.

$Rot()$ is not "the one we use when there is no translation". $Rot()$ carries the rotation and $Tran()$ carries the translation. Each is pure on its own, but we multiply them to get a $T$ that has both.

Putting them together

$$T=Tran(p)\cdot Rot(3,\theta )=\left[\begin{array}{cc}{R}_3(\theta )& p\\ 0& 1\end{array}\right]$$

This $T$ rotates about $z$ by $\theta$ and translates by $p$. The rotations sitting in its top left corner is exactly the $R_3(\theta )$ that came from the $Rot()$ operator, and the right column is the $p$ that came from the $Tran()$ operator.

The $R$ in the top left of a general $T$ is whatever rotation comes out of all the $Rot()$ steps chained together. A single rotation gives one $R_k$, while several rotations (like yaw pitch roll) give a product such as $R_3{R}_2{R}_1$. The D-H matrix is this same pattern, $Rot(z,\theta ),Tran(z,d),Tran(x,a),Rot(x,\alpha )$, multiplied out, which is why its top left block is a product of two rotations and its right column is a translation. See Composite Homogeneous Transformations for how we chain these to build a full transform.