Question

. Consider the Furuta pendulum system; See Figure1 on the next page. The angle of the horizontal arm is denoted θ1 and the angle of the pendulum fron the vertically upward line is denoted θ2. Their corresponding angular velocities are denoted θ| and 02, respectively. The kinetic energy K and the potential energy V of the system are given by Vo COS in terms of some mechanical parameters Io, 111, 12, 112, Vo of the system that have all positive values such that Recall that the Lagrangian L is defined as L-K-V (a) Compute and畿 (b) Suppose that the external force F on the system is given by 01, where u is the control torque and the other terms are friction. Compute the equations of motion of the Furuta pendulum system using the Euler-Lagrange equations. In other words, for i 1,2 compute (c) Put the equations of motion in (first-order) state-space form with the state vector (d) Verify that (z, u) = (0,0) E R4 × R is an equilibrium point of the system. (e) Linearize the system at the equilibrium point. (f) Determine the stability of the linearized systern when u = 0 ald B.-B.-0. You may use a computer for this sub-question to do symbolic computations, but it should be straightforward for a KAIST student to compute the determinant of a 4 × 4 matrix by hand.

Answer 1