Exercise $2$: $2$D Transformations as Affine Matrices

Transformations between coordinate frames play an important role in robotics. As background for exercises $2$ and $3$ on this sheet, please refer to the linear algebra slides on affine transformations and transformation combination.

The $2$D pose of a robot w$.$r$.$t$.$ a global coordinate frame is commonly written as $x=(x,y,{)}^T$ where $(x,y)$ denotes its position in the $xy-$plane and $$ its orientation. The homogeneous transformation matrix that represents a pose $x=(x,y,{)}^T$ w$.$r$.$t to the origin $(0,0,0{)}^T$ of the global coordinate system is given by

$T=([R(),t],[0,1]),R()=([cos,-sin],[sin,cos]),t=(\frac{x}{y})$

a$)$ While being at pose $x_1=(x_1,y_1,{}_1{)}^T,$ the robot senses a landmark $l$ at position $(l_x,l_y)$ w$.$r$.$t$.$ to its local frame. Use the matrix $T$ to calculate the coordinates of I w$.$r$.$t$.$ the global frame.

b$)$ Now imagine that you are given the landmark's coordinates w$.$r$.$t$.$ the global frame. How can you calculate the coordinates that the will sense in its local frame?

c$)$ The robot moves to a new pose $x_2=(x_2,y_2,{}_2{)}^T$ w$.$r$.$t$.$ the global frame. Find the transformation matrix $T_{12}$ that represents the new pose w$.$r$.$t$.$ to $x_1.$ Hint: Write $T_{12}$ as a product of homogeneous transformation matrices.

d$)$ The robot is at position $x_2.$ Where is the landmark $I=(I_x,I_y)$ w$.$r$.$t$.$ the robot's local frame now?