Example $5\mathrm{.}3($Elbow Manipulator $-$ Complete Solution$).$ To summarize the geometric approach for solving the inverse kinematics equations, we give here one solution to the inverse kinematics of the six degree$-$of$-$freedom elbow manipulator shown in Figure $5\mathrm{.}4,$ which has no joint offsets and a spherical wrist.

Given

$o=\left[\begin{array}{c}{o}_x\\ o_y\\ o_z\end{array}\right],R=\left[\begin{array}{ccc}{r}_{11}& r_{12}& r_{13}\\ r_{21}& r_{22}& r_{23}\\ r_{31}& r_{32}& r_{33}\end{array}\right]$

then with

$x_c=o_x-d_6{r}_{13}$

$y_c=o_y-d_6{r}_{23}$

$z_c=o_z-d_6{r}_{33}$

a set of DH joint variables is given by

The other possible solutions are left as an exercise $($Problem $5-10).$

$5-10$ Find all other solutions to the inverse kinematics of the elbow manipulator of Section $5\mathrm{.}3\mathrm{.}2.$

$3.$ Solve Problem $5-10$ on page $162$ to find all other analytical solutions of Example $5\mathrm{.}3$ of page $154.$ In doing so$,$ assume $d=0,$ i$.$e$.$ no offset. $(10$ pts $)$

$4.$ Use MATLAB's inverseKinematics to solve for the same elbow manipulator of Problem $5-10.$ First, build the manipulator using rigidBodyTree for numerical DH values and joint values of your choice. Second, use any desired end effector pose of your choice to find its inverse kinematic solution. Finally, verify that the numerical inverse kinematic solution returned by inverseKinematics corresponds to one of the analytical solutions found in Problem $5-10$ or shown in Example $5\mathrm{.}3.(10$ pts$)$