Suppose an object starts at a location given by the homogeneous transform matrix $0^$H$\_1.$ We then

apply the following sequence of motions:

$($a$)$ relative motion T$1,$ followed by

$($b$)$ absolute motion T$2,$ followed by

$($c$)$ relative motion $($T$2)^1$

What is the homogeneous transform matrix T$\_$f that describes the final location of the object?

$($here $$relative motion$$ means the transformation is performed with respect to the current frame,

$$absolute motion$$ means that the transformation is performed with respect to the $\{0\}$ frame.