Q$4.$ Consider a simplified model for a human leg mechanism shown in Fig$\mathrm{.}1($a$).$ The model assumes an applied muscular torque $T_m,$ viscous damping $D,$ at the hip joint, and inertia $J,$ around the hip joint. The weight of the leg is $Mg,$ where $M$ is the mass of the leg and $g$ is the acceleration due to gravity. $($PS: The leg is of uniform density, the weight can be applied at $\frac{L}{2},$ where $L$ is the length of the leg.$)$

$($A$).$ Please derive the nonlinear dynamic equation of the system. $(5\%)$

$($B$).$ Apply system linearization at $=0$ and find the transfer function $G_P(s)=\frac{s}{T_m}(s).(5\%)$

$($D$).$ Let $J=0\mathrm{.}5(Kg-m^2),D=0\mathrm{.}1\frac{N-m}{r}a\frac{d}{s},M=1(Kg),L=0\mathrm{.}2(m)$ and $g=9\mathrm{.}8(\frac{m}{s^2}).$

Please design a PD$-$controller as shown in Fig. $1($b$)$ such that the closed$-$loop system is with $10\%$ overshoot and $0\mathrm{.}1($sec$)$ setting time. $(10\%).$

$($E$).$ What is the steady state output when a step input $R(s)=\frac{1}{s}$ is applied?$(5\%)$