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At the heart of many robotic systems is the robot arm, a quintessential component that symbolizes the confluence of engineering and design. The provided block diagram gives an insightful glimpse into the complexities behind controlling a robot arm's joint, ensuring its precise movement. This Robot arm joint control system starts with an input angle, θc, representing our desired position for the robot arm. The system then computes the error by comparing the desired and actual angles. This error undergoes processing in the Digital Controller, D(z), which produces a corresponding digital output. This digital signal is transformed into an analog one through a Zero-order hold (D/A (ZOH)), ensuring compatibility with the next stages. An Amplifier with a gain, K, further strengthens this analog signal to adequately drive the Servomotor. The Servomotor, characterized by its transfer function 200/(0.5s+1), then interprets this amplified signal into angular velocity, directing the arm's movement. To achieve the desired mechanical advantage, a gear system with a 1:100 ratio is integrated. The loop ensures that the arm consistently adjusts its position, aligning the actual angle, θa, with the intended angle, θc.

Such intricate control mechanisms, as represented in the diagram, highlight the profound engineering behind even a single robot arm joint, reflecting the broader sophistication of robotics in today's world.

Questions: 1. Let the plant include the amplifier gain $(K=1)$, servomotor, and gears, find $G(z)$ using your hand calculation and check the answer using MATLAB. Insert your answer here. 2. If $D(z)=K=1$, using hand calculation, find the damping ratio, natural frequency, and the time constant, then plot the step response using the MATLAB command. Insert your answer here. 3. If $D(z)=1$ and $K=100$, find the steady-state error for the unit step and ramp function. Insert your answer here. 4. If $D(z)=Kd$ and $K=1$; find the range of closed-loop stability using Jury's test (hand calculation) and root locus (using Matlab). Insert your answer here. 5. D(z) is a PID controller, using Simulink, plot the step response of the closed-loop system when. Use $Kp=0.1;Ki$ $=0.005$ and $Kd=0.0001$. Comment on the steady state error.