3. Stabilizing a rocket to a vertical position during the early stages of a launch is essentially the same task as stabilizing an inverted pendulum. Think of how you move your hand if trying to keep a pole on your hand upright. A simple 2dimensiona1 version of this problem is an inverted pendulum mounted on a moving can with mass 1'10 as shown. below. The sphere has mass Mig and diameter d and is connected to the cart by a massless bar with zero friction at the coupling pin. There is zero rolling resistance at the cart tvheels and the input to the system is the force Fi which is used to keep 3 as close to zero as possible. From tables= ch ofthe sphere is Madgllo. 3.]. The following modeling equations have been formulated for this system. Note: for now assume E is an unspecied input. Verify that the system is nonlinear and that the unknowns are 3': 3:, l9: 2 and E. (1) M56} g)Lsin3 + Msc'Lcos + [$5 = 0 Sum of moments at connection pin (2) x = z + Lsin horizontal position x of sphere cg (3) y = L 6053 vertical position y of sphere cg [4) M52 + F,- F} = I] di'Alembert equation for cart position z (5) - + M556 = 0 d'Alembert equation for horizontal displacement x of sphere 3.2 Verify that the substitution of the 2nd derivatives of (2) and (3) into (1) gives (1) (MSLZ +35! + Mchos(8)E MsgLsin(9) = o and using (5) and the 2nd derivative of (2)= equation (-1-) becomes (4) (Mc + Ms + Msi'Lcosm) MSLZ sin[9] + Ft- = o