- 3 A Spring Loaded Inverted Pendulum With A Constant Rod Length R Is Mechanically Attached To A Mechanism With A Dc Mo 1 (75.97 KiB) Viewed 8 times
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3- A spring-loaded inverted pendulum with a constant rod length (r) is mechanically attached to a mechanism with a DC motor at point P.A point mass (m) is attached to the pendulum tip; its rotational inertia is neglected. The motor can generate rotational motion along the a axis, and its torque is symbolized with t. The gear and bearings introduces a torsional dissipation with a damping coefficient of b. A torsional spring with a stiffness constant of k acts along the a axis. The spring is in rest condition when a = 0. a=0 m Z Fig. 1: A spring-loaded pendulum. Concerning the pendulum model, the tip mass is K2 [kg], rod length is K1 [m], torsional spring stiffness is K3 [Nm/rad), damping coefficient is K4 (Nms/rad). Gravitational acceleration is assumed to be 9.8 [m/s] We implement a step input of motor torque with an amplitude of -6 Nm from zero initial conditions. (the initial angular position is 0.0 (rad) and initial angular velocity is 0.0 (rad/s]. (Tip: linearize the system by considering sina a) Prove that the dynamic time response of the system is as follows: alt) = 0.625 -0.3982e1653+ - 0.2267e-2.9034 Bonus: Calculate the angular position if t converges to infitinity. (The value of the output if we wait infinitely long) 4- Obtain the state-state space representation of the equation of motion given in Question 3. Check whether the linearized system is stable or not, in the sense of Lyapunov. 5. Design a state feedback controller such that the system can be stabilized and the controlled system could produce only 1.5% overshoot and settles within 1.25 seconds. The system parameters can be obtained from Question 3.