Solution:

Four-axis robot:

Following is the kink-coordinate diagram for a four-axis articulated robot:

To solve the inverse kinematics, we use the tool configuration vector as follows:

$\mathrm{W}(\mathrm{q})=\left[\begin{array}{c} \mathrm{a}_1 \mathrm{C}_1+\mathrm{a}_2 \mathrm{C}_{1-2} \\ \mathrm{a}_1 \mathrm{~S}_1+\mathrm{a}_2 \mathrm{~S}_{1-2} \\ \mathrm{~d}_1-\mathrm{q}_3-\mathrm{d}_4 \\ 0 \\ 0 \\ -\exp \left(\mathrm{q}_4 / \pi\right) \end{array}\right]$

$ w_1^2+w_2^2=a_1^2+2 …