(Solved): For the single link robot arm system given in Figure 1 mass of the system is m, and length is l ...

For the single link robot arm system given in Figure 1 mass of the system is $m$, and length is $l$. The angle between the robot and ground is $\theta$. The angular velocity is $\dot{\theta }$. Control torque applied at the joint to the system is $\tau$. Also additional torque has the damping effect in the opposite rotation of the system due to air friction, which can be modeled as $-b\dot{\theta }$. Physical parameters for the given system are $m=0.5\mathrm{kg}$, Figure 1: Single Link Robot Arm $l=0.3m,g=9.8\mathrm{m}/{\mathrm{s}}^2,b=0.01\mathrm{Nms}$. The torque is limited by $|\tau |\le 1\mathrm{Nm}$. Considering the given system it is expected to solve the following questions. a) For the given system, resulting linearized equation of motion is given as follows: $\frac{m{l}^2}{3}\ddot{\theta }=\tau -b\dot{\theta }$ 1 Evaluate the transfer function of the system giving the relation between input $\tau$ and output $\theta$. Please find your resulting transfer function with system parameters, then substitute the given system parameters for final result. Plot and explain the step response and pole zero map. b) Design a controller that results to place the poles of the closed loop system to -3 and -4 . Plot the resulting closed loop step response and pole zero map. Use a design methodology covered during the course and explain the reason for the design methodology you have chosen. c) Consider that a PID controller is applied to the system; $F(s)=K_p+\frac{K_1}{s}+K_{d}s$ By using the open loop transfer function obtained in parametric form, apply the PID controller, and determine the type of the system (Do not design controller, only evaluate the system type).