Composite Homogeneous Transformations

Step by step

$$F=I$$

First, initialize homogeneuous transformation matrix $T$ to identity matrix $I$.

The full form of $T$ will be something like $Rot()\times Tran()$ and vice versa depending on the question, but this just means we have to split the rotation and translation up.

If mobile $M$ is to be rotated about or translated along unit axis of fixed $F$, we need to pre-multiply, meaning on the left side.

If Mobile $M$ is to be rotated about or translated along its own unit axis, we will post-multiply, meaning on right side.

Question Example

• in the above $Tran(p)$, we should write it as $Tran(3i_2)$ instead, which is better

Whichever comes first, goes into the $T$ equation first. So in the above example, we translate along $f$ first then rotate about $f$, so $Tran(p)$ will be closer to $[m_1{]}^M$ than $Rot$