Wheel inverted pendulums (WIP) are being widely used for short and rapid mobile transfer applications in pedestrian places. A simplified model for the WIP is shown with the following parameters: The mass of the wheel =0.75kg The mass of the inverted pendulum =0.5kg The mass of the connecting rod =0.2kg The coefficient of friction between the wheel and floor =0.05Ns/m The length of the armL=0.6m. A control system is required to keep the angle to minimum. Your task will be as follows: 1. Derive a mathematical model for WIP assuming small angle . 2. Determine the transfer function between the inverted pendulum angle , and the wheel moment M. 3. Check the system stability for simple proportional gain using Routh-Hurwitz criteria and confirm your findings with root locus. 4. Design a PID control system such that the angle is always less than 12, and the pendulum settles to the upright position in 6 seconds for unit impulse force F. 5. Compare the transient responses and steady-state errors of the two controlled systems and comment on the results (For groups of 3 students)