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 T P 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/s²] 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 sin a = a) Prove that the dynamic time response of the system is as follows: a(t) = 0.625 -0.3982e1.653t 0.2267e-2.903t Bonus: Calculate the angular position if t converges to infitinity. (The value of the output if we wait infinitely long) N k 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.