Consider the following two$-$link planar RR manipulator that is controlled using a PD controller at Joint $1.$

Joint $1$ exhibits the following properties:

inertial load I at the joint varies between $3\mathrm{.}5$Kgm$^(2)$ and $15\mathrm{.}0$Kgm$^(2),$

damping load at the joint is constant b$=6$N$($s$)/($m$)$

motor's rotor inertia is I$\_($m$)=0\mathrm{.}25$Kgm$^(2),$

motor's rotor damping is b$\_($m$)=0\mathrm{.}27$N$($s$)/($m$),$

gear ratio between rotor and joint is $\backslash$eta $=7,$ and

system possesses unmodeled resonances at $17\mathrm{.}0$ra$($d$)/($sec$,21\mathrm{.}6$ra$($d$)/($sec$)),$ and $48\mathrm{.}2$ra$($d$)/($sec$).$

$(6$ pts$)$ Determine the load torque equation at Joint $1$ in the form $\backslash$tau $\_(1)=($I$\_($effective $))\backslash$theta $\_(1)^()+($b$\_($effective $))\backslash$theta $\_(1)^()$I$\_($effective $)$ and $\{$:b$\_($effective $)).$

$(4$ pts$)$ Determine the values of PD controller gains k$\_($p$)$ and k$\_($v$)$ such that the system response is never underdamped, never excites structural resonances,

but is as stiff as possible.