ANSWER FULLY AND COMPLETELY, WITH YOUR OWN WORK $($DO NOT COPY ANOTHER CHEGG RESPONSE$)$

The robot shown in Figure $2$ has the differential equations of motion given. Symbols $m_1,m_2,$

$I_1,I_2,l_1,$ and $g$ are constant parameters, representing the characteristics of the rigid body links.

Quantities ${}_1$ and $d_2$ are the coordinate variables and are functions of time. The inputs are ${}_1$

and ${}_2.$ Consider the operating point ${}_1={}_1^{}={}_1^{}=0,d_2=3,$ and $d_2^{}=d_2^{}=0.$ What are

the constant values of ${}_1$ and ${}_2$ so that the given operating point is an equilibrium? Linearize

about this equilibrium and express the resultant linear equations in state variable form. What

are the physical meanings of the state variables and of the input that appear in your linear state

equations?

Figure $2$: Problem $3$ robot.

$(m_1{l}_1^2+I_1+I_2+m_2{d}_2^2){}_1^{}+2m_2{d}_2{}_1^{}{d}_2^{}+(m_1{l}_1+m_2{d}_2)gcos{}_1={}_1$

$m_2{d}_2-m_2{d}_2{}_1^{}{?}^2+m_2$gsin${}_1={}_2$