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 a_1 a_2 C_2+a_2^2\\ $

$ q_2=\pm \arccos \frac{w_1^2+w_2^2-a_1^2-a_2^2}{2 a_1 a_2}\\ $

$ \begin{aligned}\\ \left(a_1+a_2 C_2\right) C_1+\left(a_2 S_2\right) S_1 &=w_1 \\\\ \left(-a_2 S_2\right) C_1+\left(a_1+a_2 C_2\right) S_1 &=w_2\\ \end{aligned}\\ $

$ \begin{array}{r}\\ S_1=\frac{a_2 S_2 w_1+\left(a_1+a_2 C_2\right) w_2}{\left(a_2 S_2\right)^2+\left(a_1+a_2 C_2\right)^2} \\\\ C_1=\frac{\left(a_1+a_2 C_2\right) w_1-a_2 S_2 w_2}{\left(a_2 S_2\right)^2+\left(a_1+a_2 C_2\right)^2}\\ \end{array}\\ $

$ q_1=\operatorname{atan} 2\left[a_2 S_2 w_1+\left(a_1+a_2 \mathrm{C}_2\right) w_2,\left(a_1+a_2 \mathrm{C}_2\right) w_1-a_2 \mathrm{~S}_2 w_2\right]\\ $

$ q_3=d_1=d_4-w_3\\ $

$ q_4=\pi \ln \left|w_6\right|\\ $

$ q_4=q_1-q_2-q_{1-2-4}\\ $

Complete Solution:

Following is the complete algorithm for a four-axis SCARA robot.