$3.(25$ points$)$ Robots can be used in manufacturing and assembly operations that require accu$-$

rate, fast, and versatile manipulation. The loop transfer function of a direct$-$drive arm is given

by

Gc$($s$)$G$($s$)=$ K$($s $+10)$

s$($s $+3)($s$2+4$s $+8)$

$($i$)$ Determine the value of the gain K when the system is oscillate,

$($ii$)$ Calculate the roots of the closed$-$loop systems for the K determined in part $($i$).$