Pedro is designing software for a circular-shaped mobile robot that will move around within his home. The robot moves within a space that is a 3m by 3m square. The robot can rotate in place at a desired angular velocity and it can travel straight ahead at a desired linear velocity in the direction it is facing. Pedro's goal for the robot is to travel to certain positions at certain times.

Form a mathematical model for this system by defining a state space, an action space, and a transition function. Specifically, your model should include:

• An equation of the form $X=\underline{\hspace{0.5in}}$, to define the state space, along with an explanation for what a specific state means.
• An equation of the form $U=\underline{\hspace{0.5in}}$, to define the action space, along with an explanation for what a specific action means.
• An equation of the form $x_{k+1} =\underline{\hspace{0.5in}} $, or equivalently $f(x_k, u_k) = \underline{\hspace{0.5in}}$, to define the state transitions correctly. You may assume that the actions are executed perfectly, without noise or interference.