Four-axis robot:

Following is the link coordinate diagram based on the Denvait Hartenberg algorithm. Using the steps from 0 to 7 we obtain the link coordinate diagram for the four-axis SCARA robot.

Next, we need to apply the D-H algorithm steps 8 to 13 in order to obtain the kinematic parameters as shown below:

$ \begin{array}{clcccc} \hline \text { Axis } & \boldsymbol{\theta} & d & a & \alpha & \text { Home } \\ \hline 1 & q_1 & d_1 & a_1 & \pi & 0 \\ 2 & q_2 & 0 & a_2 & 0 & 0 \\ 3 & 0 & q_3 & 0 & 0 & 100 \\ 4 & q_4 & d_4 & 0 & 0 & \pi / 2 \\ \hline \end{array} $

$\mathrm{d} 1-877 \mathrm{~mm}, \mathrm{~d} 4=200 \mathrm{~mm}$

$\mathrm{d} 3=\mathrm{d} 3$ because it's a joint variable where value can range from 0 to $195 \mathrm{~mm}$ $\mathrm{a} 1=425 \mathrm{~mm}, \mathrm{a} 2=375 \mathrm{~mm}$

The vector of joint yariables $q=\{\theta 1, \theta 2, \mathrm{~d} 3, \theta 4\}$

$\theta 1, \theta 2=$ revolute variables for the tool position p

D3= prismatic variable

$\Theta 4=$ revolute variable controls the tool orientation R

ARM Matrix equation $\mathrm{T}_{\text {base }}^{\text {tool }}$ for the SCARA robot is as given below: There is no need for portioning and we calculate the arm matrix equation as follows:

$ \begin{aligned}\\ &T_{\text {bate }}^{\text {tol }}=T_0^1 T_1^2 T_2^3 T_3^4\\ &=\left[\begin{array}{cccc} \mathrm{C}_1 & \mathrm{~S}_1 & 0 & a_1 \mathrm{C}_1 \\ \mathrm{~S}_1 & -\mathrm{C}_1 & 0 & a_1 \mathrm{~S}_1 \\ 0 & 0 & -1 & d_1 \\ 0 & 0 & 0 & 1 \end{array}\right]\left[\begin{array}{cccc} \mathrm{C}_2 & -\mathrm{S}_2 & 0 & a_2 \mathrm{C}_2 \\ \mathrm{~S}_2 & \mathrm{C}_2 & 0 & a_2 \mathrm{~S}_2 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right]\left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & q_3 \\ 0 & 0 & 0 & 1 \end{array}\right]\left[\begin{array}{cccc} \mathrm{C}_4 & -\mathrm{S}_4 & 0 & 0 \\ \mathrm{~S}_4 & \mathrm{C}_4 & 0 & 0 \\ 0 & 0 & 1 & d_4 \\ 0 & 0 & 0 & 1 \end{array}\right] \text {. }\\ &=\left[\begin{array}{cccc} \mathrm{C}_{1-2} & \mathrm{~S}_{1-2} & 0 & a_1 \mathrm{C}_1+a_2 \mathrm{C}_{1-2} \\ \mathrm{~S}_{1-2} & -\mathrm{C}_{1-2} & 0 & a_1 \mathrm{~S}_1+a_2 \mathrm{~S}_{1-2} \\ 0 & 0 & -1 & d_1 \\ 0 & 0 & 0 & 1 \end{array}\right]\left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & q_3 \\ 0 & 0 & 0 & 1 \end{array}\right]\left[\begin{array}{cccc} \mathrm{C}_4 & -\mathrm{S}_4 & 0 & 0 \\ \mathrm{~S}_4 & \mathrm{C}_4 & 0 & 0 \\ 0 & 0 & 1 & d_4 \\ 0 & 0 & 0 & 1 \end{array}\right]\\ &=\left[\begin{array}{cccc} \mathrm{C}_{1-2} & \mathrm{~S}_{1-2} & 0 & a_1 \mathrm{C}_1+a_2 \mathrm{C}_{1-2} \\ \mathrm{~S}_{1-2} & -\mathrm{C}_{1-2} & 0 & a_1 \mathrm{~S}_1+a_2 \mathrm{~S}_{1-2} \\ 0 & 0 & -1 & d_1-q_3 \\ 0 & 0 & 0 & 1 \end{array}\right]\left[\begin{array}{cccc} \mathrm{C}_4 & -\mathrm{S}_4 & 0 & 0 \\ \mathrm{~S}_4 & \mathrm{C}_4 & 0 & 0 \\ 0 & 0 & 1 & d_4 \\ 0 & 0 & 0 & 1 \end{array}\right] \end{aligned}\\ $

$ T_{\text {base }}^{\text {tool }}=\left[\begin{array}{lcc:c} \mathrm{C}_{1-2-4} & \mathrm{~S}_{1-2-4} & 0 & a_1 \mathrm{C}_1+a_2 \mathrm{C}_{1-2} \\ \mathrm{~S}_{1-2-4} & -\mathrm{C}_{1-2-4} & 0 & a_1 \mathrm{~S}_1+a_2 \mathrm{~S}_{1-2} \\ 0 & 0 & -1 & d_1-q_3-d_4 \\ \hdashline 0 & 0 & 0 & 1 \end{array}\right]\\ $

Such robots are mostly used in applications of assembly operations where components are required to insert in the circuit boards