This assignment exercises your ability to do dynamics, control, and robot learning using an interesting test case, the cart pole system. The cart has a mass of 1kg and a horizontal rocket thruster to move in the horizontal X direction. A pole is attached to the cart with a joint. It has a mass of 0.1kg and is 1m long (and is infinitely thin).

**Part 1:**
Use a symbolic dynamics program to derive the inverse dynamics of
this system (we recommend Matlab). Compare the equations to the equations
you can find on the web. Turn in your input to the symbolic dynamics program.

**Part 2:**
Use your favorite simulator to implement the forward dynamics of this system.
ODE works well for this, but is not required.
Make a movie of the pendulum starting out and then falling over.

**Part 3:**
Try to find control gains manually that will keep this system upright.

**Part 4:**
Try to find control gains automatically that will keep this system upright.
Hint: LQR control design is a good place to start.

**Part 5:**
Try to find nonlinear gains to maximize the "basin of attraction", which
is the set of initial conditions that lead the system to move to
the upright equilibrium.

**Part 6:**
Design a trajectory that will drive the system from the stable pole down
configuration to the unstable pole up acceleration (swingup).
Make a movie of the pendulum swinging up.