## 5 Simulation and Experimental Results
### Real-World Experimental Results  
Real-world experiments were conducted to demonstrate the applicability of the proposed approach. The optimal joint configuration computed by this method can serve as a reference for point-to-point motion planning. For the experiments, a cubic polynomial trajectory was used to compute the joint motion from an initial joint position \(\mathbf{q}_{i}\) to the joint reference \(\mathbf{q}_{r}\) at a desired run time \(T\). The experimentation consists of two phases. First, the proposed approach solves the inverse kinematics for a desired end-effector pose to estimate a joint reference \(\mathbf{q}_{r}\). Then, the manipulator robot follows a cubic polynomial trajectory based on the initial joint configuration \(\mathbf{q}_{i}\) and a run time \(T\). A 5 DOF KUKA YouBot was used for experimentation, as shown in Figure 10. The DH parameters are provided in Table 11 and the coordinate frame assignment is presented in Figure 11. The proposed algorithm was implemented using Matlab and the ROS environment. ROS components on the KUKA YouBot provide an internal PID controller in the joint space, which is convenient for the considered trajectory planning. A point-to-point trajectory planning is performed on an external computer with Matlab and the ROS toolbox. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Joint** & **a [m]** & **a [rad]** & **d [m]** & \(\Theta\) [rad] \\ \hline
1 & 0.033 & \(\pi/2\) & 0.147 & \(q_{1}\) \\
2 & 0.155 & 0 & 0 & \(q_{2}\) \\
3 & 0.135 & 0 & 0 & \(q_{3}\) \\
4 & 0 & \(\pi/2\) & 0 & \(q_{4}\) \\
5 & 0 & 0 & 0.2174 & \(q_{5}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 11: DH parameters of KUKA YouBot manipulator. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Random Pose** & **Position Error** & **Orientation Error** \\ \hline
1 & 0.47979 & 3.1913 \(\times 10^{-5}\) \\
2 & 0.73254 & 9.575 \(\times 10^{-5}\) \\
3 & 0.71156 & 3.0564 \(\times 10^{-4}\) \\
4 & 0.94969 & 3.2291 \(\times 10^{-5}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 10: Position and orientation error results of unreachable poses in [PERSON] robot. Figure 9: Unreachable pose experiment results. Coordinate frames in red represents the unreachable poses and coordinate frames in blue the achieved ones. The lower and upper joint limits are defined as  
\[\mathbf{q}_{l} = \left[-169\quad-65\quad-150\quad-102.5\quad-167.5\right]^{T}\] \[\mathbf{q}_{u} = \left[169\quad 90\quad 146\quad 102.5\quad 167.5\right]^{T}\]  
In Experiment 1, a run time of \(T=8\) seconds is set, and the desired pose of the end-effector is specified as follows:  
\[\mathbf{T}_{d}= \begin{bmatrix}\mathbf{R}_{d}&\mathbf{t}_{d}\\ \mathbf{0}&1\end{bmatrix} = \begin{bmatrix}1&0&0&0.2\\ 0&-1&0&-0.2\\ 0&0&1&0\\ 0&0&0&1\end{bmatrix}  \tag{18}\]  
The inverse kinematics solution obtained through the proposed approach is  
\[\mathbf{q}_{r}= \begin{bmatrix}44.15&39.31&-54.6&13.57&44.43\end{bmatrix} ^{T}\]  
The initial joint configuration is selected as  
\[\mathbf{q}_{i}= \begin{bmatrix}0&90&0&90&0\end{bmatrix} ^{T}\]  
The joint position and velocities for the motion planning in Experiment 1 are presented in Figure 12. Additionally, Figure 13 illustrates the final joint configuration along with the joint motion results as measured by the encoders. Figure 11: Coordinate frames’ assignment for the KUKA YouBot manipulator. Figure 10: KUKA YouBot manipulator with 5 DOF. For Experiment 2, a run time of \(T=4\) seconds is set, and the desired pose of the end-effector is defined as  
\[\mathbf{T}_{d}= \begin{bmatrix}\mathbf{R}_{d}&\mathbf{t}_{d}\\ \mathbf{0}&1\end{bmatrix} = \begin{bmatrix}1&0&0&0.2\\ 0&-1&0&0.2\\ 0&0&1&0\\ 0&0&0&1\end{bmatrix} \]  
The inverse kinematics solution for Experiment 2, obtained through the proposed approach, is  
\[\mathbf{q}_{r}= \begin{bmatrix}-44.10&40.46&-50.10&13.97&-45.37\end{bmatrix} ^{T}\]  
and the considered initial joint configuration for this test is  
\[\mathbf{q}_{i}= \begin{bmatrix}0&90&-45&45&0\end{bmatrix}