# Design, Simulation, and Comparison of Advanced Control Strategies for a 3-Degree-of-Freedom Robot
## 3 Description of the System Under Study
### Kinematic Model of the Scorbot Robot  
The kinematics of the robot manipulator are obtained using the Denavit-Hartenberg (DH) algorithm, described in Table 2[27]. Using the parameters from the previous table, the transformation matrix of each element of the kinematic chain is obtained according to (6) to (14). For simplicity, the following definitions are used:  
\[\mathrm{C_{i}}=\cos\!\theta_{i} \tag{6}\] \begin{table}
\begin{tabular}{c c c} \hline \hline
**SCORBOT-ER V Plus Specifications** \\ \hline Mechanical structure & Vertical articulated \\ Number of axes & Five axes plus servo gripper (three axes are considered) \\ \hline Axis Movement & \\ Axis 1: base rotation (first DoF) & 310\({}^{\circ}\) \\ Axis 2: shoulder rotation (second DoF) & +130\({}^{\circ}\)/\(-\)35\({}^{\circ}\) \\ Axis 3: elbow rotation (third DoF) & \(\pm\)130\({}^{\circ}\) \\ Axis 4: wrist pitch (end effector) & \(\pm\)130\({}^{\circ}\) \\ Axis 5: wrist roll (end effector) & Unlimited (mechanically); \(\pm\)570\({}^{\circ}\) (electrically) \\ \hline Maximum operating radius & 610 mm (24.4\({}^{\prime\prime}\)) \\ \hline End effector & DC servo gripper with optical encoder, \\  & parallel finger motion; \\  & Measurement of object’s size/gripping force \\  & by means of gripper sensors and software. \\ \hline Maximum gripper opening & 75 mm (3\({}^{\prime\prime}\)) without rubber pads \\  & 65 mm (2.6\({}^{\prime\prime}\)) with rubber pads \\ \hline Hard home & Fixed position on each axis, \\  & found by means of microswitches \\ \hline Feedback & Optical encoder on each axis \\ \hline Actuators & 12V DC servo motors \\ \hline Motor capacity (axes 1–6) & 15 oz. in Peak Torque (stall) \\  & 70 W Power for Peak Torque \\ \hline  & Motors 1, 2, 3: 127.1:1 \\  & Gener ratios & Motors 4, 5: 65.5:1 \\  & Motor 6 (gripper) 19.5:1 \\ \hline Transmission & Gears, timing belts, and lead screw \\ \hline Maximum payload & 1 kg (2.2 lb.), including gripper \\ \hline Position repeatability & \(\pm\)0.5 mm (0.02\({}^{\prime\prime}\)) at TCP (tip of gripper) \\ \hline Weight & 11.5 kg (25 lb) \\ \hline Maximum path velocity & 600 mm/s (23.6\({}^{\prime\prime}\)/s) \\ \hline Ambient operating temperature & 2\({}^{\circ}\)–40 \({}^{\circ}\)C (36\({}^{\circ}\)–104\({}^{\circ}\)F) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Scorbot robot specifications [26]. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Link** & \(\theta_{i}\) & \(\mathbf{d}_{i}\) & \(\mathbf{a}_{i}\) & \(\mathbf{a}_{i}\) \\ \hline
1 & \(\theta_{1}\) & \(l_{1}\) & 0 & 90\({}^{\circ}\) \\
2 & \(\theta_{2}\) & 0 & \(l_{2}\) & 0 \\
3 & \(\theta_{3}\) & 0 & \(l_{3}\) & 0 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: DH parameters of the Scorbot robot. \[S_{i}=\text{sin}\theta_{i} \tag{7}\]  
\[{}^{0}A_{1}=\left[ \begin{array}{cccc}C_{1}&0&S_{1}&0\\ S_{1}&0&-C_{1}&0\\ 0&1&0&d_{1}\\ 0&0&0&1\end{array} \right] \tag{8}\]  
\[{}^{1}A_{2}=\left[ \begin{array}{cccc}C_{2}&-S_{2}&0&l_{2}C_{2}\\ S_{2}&C_{2}&0&l_{2}C_{2}\\ 0&0&1&0\\ 0&0&0&1\end{array} \right] \tag{9}\]  
\[{}^{2}A_{3}=\left[ \begin{array}{cccc}C_{3}&-S_{3}&0&l_{3}C_{3}\\ S_{3}&C_{3}&0&l_{3}S_{3}\\ 0&0&1&0\\ 0&0&0&1\end{array} \right] \tag{10}\]  
Using the previous matrices, the homogeneous transformation matrix of the complete kinematic chain can be obtained. \[T\left[ \begin{array}{cccc}C_{1}C_{2}C_{3}-C_{1}S_{2}S_{3}&-C_{1}C_{2}S_{3}- C_{1}S_{2}C_{3}&S_{1}&l_{3}C_{1}C_{2}C_{3}-l_{3}C_{1}S_{2}S_{3}+l_{2}C_{1}\\ C_{2}C_{3}-S_{1}S_{2}S_{3}&-S_{1}C_{2}S_{3}-S_{1}S_{2}C_{3}&-C_{1}&l_{3}S_{3}C_{ 2}C_{3}-l_{3}S_{1}S_{2}S_{3}+l_{3}S_{1}C_{2}\\ S_{2}C_{3}+C_{2}S_{3}&-S_{2}S_{3}+C_{2}C_{3}&0&l_{3}S_{2}C_{3}+l_{3}C_{2}S_{3}+ l_{2}S_{2}+l_{1}\\ 0&0&0&1\end{array} \right] \tag{11}\]  
The last column of matrix \(T\) in Equation (11) corresponds to the position equations of the end-effector.