3. 2D vehicle is moving at a speed of v such that its velocity vector makes an angle with x-axis (called heading). The coordinates of vehicle's center with respect to x

and y axes are x and y, and the steering angle is p. The wheelbase L can be assumed to be 1m, and speed v to be constant 1 m/s. Then kinematic equations of the vehicle are below. They are linearized using the small angle formulae, so that you can use the linearized (LTI) control theory of this class. * = v cos 0 ≈ v = 1 v0 = 0 ý v sin 8 V 8 = -tan V === y, axis (x, y). x, axis ii. iii. Plot the position of the car y(t). iv. Plot the steering angle (t). 10 Let's say you want to control the vehicle to move it on a straight line (x axis) I.e. you want the y coordinate to be zero in steady state. Design a controller that uses the measurement of y and provides the correct steering angle $ to make the vehicle go in a straight line. Assume the vehicle started on x axis (yo = 0) with an initial angle 0o = 0.1 rad. a. What variable will be the input to the plant u? What variable will be the output of the plant y? What will be the transfer function of the plant from input to the output G(s)? Note: Finding what is u(t) and what is y(t) is the first step in modeling. What will be the value of the reference command r(t) for this system? b. c. Draw the block diagram of the system, which shows the initial condition, reference command, and all the relevant blocks. Reduce this block diagram so that the initial condition ultimately appears as a unit step disturbance du acting before the plant. d. Design the controller, so that if the car starts at an initial angle of 0.1 radian or less, on the x-axis, the controller brings it back to correct motion along x-axis within 5-ish seconds, while making sure that the steering angle remains within the bounds of +0.3rad. i. Show all the steps followed in sisotool, as a numbered list along with the snapshots of your rlocus at various stages. Write down the transfer function of your final controller.