## 5 Simulation Results
### Multi-Point Control  
In this experiment, multiple control points were added to the continuum robot. In the first arrangement, only a single control point was added that was situated at the end effector. In the second arrangement, 2 control points were added, one at the end effector and one at the end of link 5. Finally, in the third arrangement, the 3 control points were added that were evenly distributed throughout the robot's kinematic chain. The placements of the control point for each arrangement are summarized in Table 3.  
These arrangements are depicted in Figure 4. Three reaching tasks were preformed with each arrangement. All desired poses were within the robot's configuration space. See Figure 5. The reaching tasks increased in complexity in terms of dexterity required. We note that arrangement 1 corresponds to kinematic-model-free control of the end-effector pose. Thus, this experiment also serves as a comparison with the kinematic-model-free orientation controller in Reference [5]. The results are shown in Figure 6 and tabulated in Table 4.  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Degrees of Freedom** & **Steps to Reach** \\ \hline
2 & 21 steps \\
3 & 40 steps \\
4 & 42 steps \\
5 & 35 steps \\
6 & 41 steps \\
7 & 76 steps \\
8 & 127 steps \\
9 & 150 steps \\ \hline \hline \end{tabular}
\end{table}
Table 2: Reaching steps for different degrees of freedom enabled.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Arrangement** & **Control Point 1** & **Control Point 2** & **Control Point 3** \\ \hline
1 & end effector & - & - \\
2 & end of link 5 & end effector & - \\
3 & end of link 3 & end of link 6 & end effector \\ \hline \hline \end{tabular}
\end{table}
Table 3: Control points placement.  
Figure 4: The three arrangements of the control points (CP’s) along the robot’s kinematic chain. In the first arrangement, only one control point was used which was placed at the end effector. In the second and third arrangements, two and three equally spaced control points were used, respectively.  
Figure 5: The three reaching tasks to be performed. The translucent pose is the initial configuration from which the robot is to actuate towards the opaque pose.  
Figure 6: The controller performed three reaching tasks. In all 3 tasks, the controller converged faster when using more control points (CPs).  
Only the pose distance of the control point situated at the end effector is shown. It can be seen that the controller converges faster to the desired pose when using more control points situated along the robot's kinematic chain. In task 1, we note that a 66% improvement in convergence speed when using 3 control points. In task 2 and 3, there was a 38% and 50% improvement was measured, respectively. These results indicate that adding more control points will improve convergence and resolve redundancies in hyper-redundant systems. As for accuracy, the termination criteria was set to DQdist < 0.03, achieving a close enough reaching accuracy.  
Furthermore, to demonstrate the joints' position during an execution, the joint position are shown in Figure 7. The larger jitters in joint positions correspond to exploratory behavior, which is triggered whenever there is deficiency in local kinodynamic information.