$3\mathrm{.}2\mathrm{.}1$ Affine transformations and modelling joint positions and velocities

Suppose to have the three joints robot shown in the picture. Find the coordinates $x$

and $Y$ of the robot hand $($TCP: Tool Centre Point$)$ for given values of the joints $q1,q2,$

q$3.$ This problem is known in robotics as 'Forward kinematics'.

See for example robot$-$control$-$forward$-$transformation$-$matrices$/$ or Forward kinematics

Requirements for passing the subject, try to answer the following questions $($when

applicable$).$

Give an explanation of linear and affine transformations and how they relate to

the problem.

Explain how a translation in $2$ dimensions $($an affine transformation$)$ can be rep$-$

resented using a linear transformation $($a matrix$)$ in $3$ dimensions and why this is

relevant to the problem.

Use the matrix formulation of the problem to describe the coordinate of TCP if

there is at least $2$ joints with an example.

The following additions contribute to a higher grade:

Show how matrices can be used to describe complex movements in $3$D such as

the position of multiple fingerstips on a simple hand or the position of the moon

relative the sun in a simplified model.

Try to make a short animation where you illustrate the use of the method.