Question

Problem 3 (70 pts): Consider the mechanical system in Figure , the so-called "cart pendulum" system. The cart has a moving mass M, and is connected to a linear motor via a flexible coupling with stiffness K and damping B. An inverted pendulum of length1, negligible inertia and mass m is attached to the cart via a rotary actuator. If the pendulum damping coefficient is b, the linear actuator force is F and the rotary actuator torque is t 1) Form the system Lagrangean 2) Write the explicit dynamical equations of motion. 3) Find a possible LFPB trajectory from rest (no velocity)x-0,0-0 (pendulum down) to rest (no velocity) x-1, 0 (pendulum up) that this system can follow 4) Suppose you want to implement a computed torque controller with PD feedback to follow your trajectory, write the equations for the two actuator inputs as a function of the cart states in symbolic form. Draw a control diagram showing how you would implement a discrete controller Simulate and animate the system in MATLAB, using M=1, K=1, B=1, m=1, 0.1, b1, by using a PD controller (chose your gains!) to following the trajectory in part 3), Plot the resulting error signal, assuming that the initial (start) state is x=-05. theta /4 5) 6) 7) If the pendulum mass changes to m 2, but the controller is "not aware" of this change, plot the resulting error signal 0 Figure 1: The cart-pendulum system with state variables x and e

Answer 1

acthu 8pte Craft with molo

Th aetial forw

aud

lo iV 0 ouut Velocki

-O.SSI a.qug .841

h @ (K) gauss w.ร์(x, [0-8 , 의〉 lConv y2 Conv (kCx),fCx) Plot (xp, y2)

lot (xp,99 Close all; clear all f= 50, f- Zeroes CN,, no 50 for nN e n d ク Su sww

Save as Subs e.Vi. print f en d N length (x) ; Or* . ew