Consider a simplified two$-$joint robotic arm where the two joints are connected by $$rigid links.

Each joint has a mass, and the links are modeled as $$idealized rods. The motion of $$each joint

affects the other through the coupling of $$forces$.$ The system can be $$modeled by $$the following

second$-$order coupled ODEs:

m$1$$1"+$k$1($$1-$rest$)+$k$3($$1-$$2)=0($equation$1)$

m$2$$2"+$k$2($$2-$rest$)+$k$3($$2-$$1)=0($equation$2)$

where $1($t$)$and $2($t$)$represent the angular positions of $$the first and second joints, respec$-$

tively. m$1$and m$2$are the effective masses at $$each joint. k$1,$k$2$and k$3$are the stiffness

constants representing the elastic restoring forces in $$the joints and links. rest$=0$represents

the equilibrium position for each joint.

The parameters are:

m$1=1,$m$2=1,$rest$=0,$k$1=1,$k$2=2,$k$3=3,$rest$=0$

The initial conditions are:

$1(0)=1,$$2(0)=2$

Please solve by $$hand using Laplace. Show all step and calculations, and provide a written description of $$each step. Then do $$the same using different initial conditions of $$your choice.