Design a feedback control law to linearize the system. B$)$linearize the given system using backstepping integrator.$x_1^{}=x_2$

$x_2^{}=D^{-1}[S()-{}_e-G(x_1)-C(x_1,x_2)x_2]$

$y=x_1$

where as $x_1=[q_1,q_2{]}^T$ and $x_2=[q_1^{},q_2^{}{]}^T.$ Considering the

$2-$DOF robotic manipulator in a vertical plane, simulations are

conducted to verify the effectiveness of the proposed control.

Let $m_i$ and $l_i$ be the mass and length of link $i,l_{ci}$ be the

distance from joint $i-1$ to the center of mass of link $i,$ as

indicated in the figure, and $I_i$ be the moment of inertia of link

i about an axis coming out of the page passing through the

center of mass of link $i.$

The inertia matrix $D(x_1),$ centripetal and Coriolis torques

$C(x_1,x_2),$ and gravitational force $G(x_1)$ are defined as

$D(x_1)=\left[\begin{array}{ccc}{p}_1+p_2+2p_{3}cos{q}_2& p_2+p_{3}cos{q}_2& p_2\end{array}\right]$

$p_2+p_{3}cos{q}_2$

$C(x_1,x_2)=\left[\begin{array}{ccc}-p_3{q}_2^{}sin{q}_2& -p_3(q_1^{}+q_2^{})sin{q}_2& 0\end{array}\right]$

$p_3{q}_1^{}sin{q}_2$

$G(x_1)=\left[\begin{array}{c}{p}_{4}gcos{q}_1+p_{5}gcos(q_1+q_2)\end{array}\right]$

$p_5$gcos$(q_1+q_2)$

where

$p_1=m_1{l}_{c1}^2+m_2{l}_1^2+I_1$

$p_2=m_2{l}_{c2}^2+I_2$

$p_3=m_2{l}_1{l}_{c2}$

$p_4=m_1{l}_{c2}+m_2{l}_1$

$p_5=m_2{l}_{c2}.$