Q1. INVERTED PENDULUM MODELLING (30 marks) Consider the inverted pendulum system presented in Fig. The pivot of the pendulum is mounted on a cart, which can move in a horizontal direction. The pendulum can be kept balanced at a specific position by applying a horizontal force to drive the carriage. Assume that the pendulum mass, m, is concentrated at the end of the massless rod. The horizontal displacement of the pivot on the cart is Ie, the rotational angle of the pendulum is (measured from the z-axes and increasing in a counter- clockwise direction), and the force driving the carriage is u. Assume that the friction coefficient of the cart is b. By using the Euler-Lagrange approach, show that when considering q1 = c and q2 = 0 the dynamic model for the inverted pendulum system in Fig. I can be described by: u = (M + m) + ba - mla sin(9) - mlq3 cos (92) 0 = 9 sin(92) - l2 - g cos(92) y 0 U b Ic l M 0 m mg Figure 1: Inverted Pendulum System. I (la) (1b)