## 2 Resolution of the Kinematic Problem by Traditional Methods
### Inverse Kinematics Using Traditional Methods  
#### 2.2.1 Inverse Kinematics of the PUMA 560 by the Geometric Method  
Based on the link coordinate systems presented in Figure 3, and inspired in the geometry of a human arm, various arm configurations can be identified for a PUMA robot with the assistance \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Dimension [mm]** \\ \hline \(L_{1}\) & 28.0 \\ \(L_{2}\) & 58.0 \\ \(L_{3}\) & 110.0 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Parameter dimensions of the hexapod’s leg. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Link** & **\(\theta\)** & **\(d\)** & **\(a\)** & **\(\alpha\)** \\ \hline
1 & \(q_{1}\) & 0 & \(L_{1}\) & \(\pi/2\) \\
2 & \(q_{2}\) & 0 & \(L_{2}\) & \(\pi\) \\
3 & \(q_{3}+(\pi/2)\) & 0 & \(L_{3}\) & 0 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Denavit-Hatenberg Parameters of the hexapod’s leg. of three configuration indicators: ARM, ELBOW, and WRIST [3]. The first two, ARM and ELBOW, are associated with the solution of the first three joints, while WRIST is related to the solution of the last three. For a six-axis manipulator, like the PUMA 560, there are four possible solutions for the first three joints (\(q_{1},q_{2}\) and \(q_{3}\)), commonly referred as configurations, and for each one of these configurations, there are two possible solutions for the last three joints (\(q_{4}\), \(q_{5}\) and \(q_{6}\)) [3]. As was stated in Section 2.1.1, we will focus exclusively in the first three joints of the PUMA robotic arm. Thus, we only need to find the four possible solutions for \(q_{1}\) to \(q_{3}\), which are related to the first two configuration indicators: ARM and ELBOW. The ARM indicator can have two possible values, LEFT and RIGHT. LEFT indicates that positive values of \(q_{2}\) will move the robot's wrist in the negative \(\mathbf{Z_{0}}\) direction when the third joint, \(q_{3}\), is not activated, while RIGHT will move it in the positive \(\mathbf{Z_{0}}\) direction. ELBOW also has two possible values, ABOVE and BELOW, which indicate whether the third joint of the robot is above the imaginary line described between the top part of the robot's body and its wrist, or below this line, respectively. The combination of the two possible values of the ARM indicator with the two of ELBOW gives the four possible configurations that constitute the four solutions of the Inverse Kinematics Problem of the PUMA 560. If the position vector \(\mathbf{\vec{p}}(\mathbf{p_{x}},\mathbf{p_{y}},\mathbf{p_{z}})\) is projected onto the \(\mathbf{X_{0}}-\mathbf{Y_{0}}\) plane, as shown in Figure 4, \(q_{1}\) could be obtained solving the equations shown in Equation (3). The \"\(L\)\" and \"\(R\)\" superscripts added to \(q_{1}\) differentiate between the LEFT and RIGHT arm configurations. \[ \begin{split}\vec{OA}=&\ d_{2}\\ \vec{OE}=&\ R_{2}=\sqrt{(p_{x})^{2}+(p_{y})^{2}}\\ \vec{AE}=&\ r_{a2}=\sqrt{(p_{x})^{2}+(p_{y})^{2}-(d_ {2})^{2}}\\ \alpha=&\ \text{atan2}\Big{(}\frac{d_{2}}{R_{2}}, \frac{r_{a2}}{R_{2}}\Big{)}\\ \varphi=&\ \text{atan2}\Big{(}\frac{p_{y}}{R_{2}}, \frac{p_{x}}{R_{2}}\Big{)}\\ q_{1}^{L}=&\ \varphi-\alpha;\ \ \ \ \ q_{1}^{R}= \varphi+\alpha+\pi;\end{split}  \tag{3}\]  
To calculate \(q_{3}\), the position vector \(\mathbf{\vec{p}}\) is projected onto the plane \(\mathbf{X_{2}}-\mathbf{Y_{2}}\), as shown in Figure 5, obtaining the geometric equations presented in Equation (4):  
Figure 4: Geometric solution for joint 1 (\(q_{1}\)), showing both possible configurations: “LEFT Arm” (**left**) and “RIGHT Arm” (**right**) [3].