$5-13$a $($modified$)$ For the three$-$link planar manipulator illustrated below, where the manipulator

begins at $q_s$ and is attempting to reach $q_f,$ please:

analytically determine the Jacobian matrix at each relevant DH origin in terms of the

configuration, $J_{p,i}(q),$ where,

$q=\left[\begin{array}{c}{}_1\\ {}_2\\ {}_3\end{array}\right]$

calculate the Artificial Potential Forces $($APFs$)$ of attraction at each DH origin at $q_s,$

calculate the APFs of repulsion on all relevant DH origins at $q_s($analyze and specify

when origins are beyond the radius of influence$),$

sketch the configuration at $q_s$ in the workspace and draw $/$ label the APF vectors

to scale, and

calculate the resultant torque vector at $q_s,$

$(q_s)=\left[\begin{array}{c}{}_1\\ {}_2\\ {}_3\end{array}\right]$

You may assume all gains of attraction and repulsion, ${}_i={}_i=1$ and that the obstacle is a

single point with radius of influence ${}_0=1.$

$5\mathrm{.}13$a $($modified$)$

For the three$-$link planar manipulator illustrated below, where the manipulator

begins at