# Numerical Quantification of Controllability in the Null Space for Redundant Manipulators
## 3 Simulation and Experiment
### Step and Sinusoidal Response  
When controlling the excessive DoF within the null space, there are situations where the desired direction and controllable direction may coincide or differ while disregarding the singular state of the manipulator. In order to validate the proposed method, simulations were conducted to evaluate the response of a robot based on both step and sinusoidal input as shown in Figure 4.  
For the first case, both the z-direction input and the proposed input can be controlled with ease (Figure 4a-d). However, the proposed input tends to converge faster, because the proposed input considered the controllable direction in the null space. As shown in Figure 4a, the proposed input converges faster than the z-axis direction input by the reason above. As shown in Figure 4b, the singular vector representing the controllable directions, \(V_{x}\), \(V_{y}\), and \(V_{z}\), changes simultaneously in real-time with respect to the joint configuration of the robot. The resulting motion of a robot with respect to these vectors is shown in Figure 4a. Furthermore, it can be noted that the singular vectors \(V_{y}\) and \(V_{z}\) are relatively larger than the \(V_{x}\), which implies that the robot can be controlled with ease when moving in the y and z directions, but with more difficulty in the x direction. The rise time and the settling time was decreased from 0.6684 [s] to 0.3222 [s] (\(-\)51.79%) and from 1.3425 [s] to 0.6061 [s] (\(-\)54.82%), respectively. The sine wave input with a cycle per 2 s was simulated as \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Joint** & **a (m)** & **d (m)** & **\(\boldsymbol{\alpha}\) (rad)** & **\(\boldsymbol{\theta}\) (rad)** \\ \hline Joint 1 & 0 & 0.333 & 0 & \(\theta_{1}\) \\ \hline Joint 2 & 0 & 0 & \(-\frac{\pi}{2}\) & \(\theta_{2}\) \\ \hline Joint 3 & 0 & 0.316 & \(\frac{\pi}{2}\) & \(\theta_{3}\) \\ \hline Joint 4 & 0.0825 & 0 & \(\frac{\pi}{2}\) & \(\theta_{4}\) \\ \hline Joint 5 & \(-\)0.0825 & 0.384 & \(-\frac{\pi}{2}\) & \(\theta_{5}\) \\ \hline Joint 6 & 0 & 0 & \(\frac{\pi}{2}\) & \(\theta_{6}\) \\ \hline Joint 7 & 0.088 & 0 & \(\frac{\pi}{2}\) & \(\theta_{7}\) \\ \hline Flange & 0 & 0.107 & 0 & 0 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The table of the Denavit–Hartenberg parameters for PANDA derived following [PERSON]’s convention, further information on the PANDA can be accessed from the cited link [24]. shown in Figure 4c. The RMS error was decreased from 0.0109 [m] to 0.0072 [m] (\(-\)33.94%) by using the proposed method. For comparison, the second case shows the response of a robot moving in the y-direction, even with \(V_{y}\) is low (Figure 4e-h). In Figure 4e, the y position initially cannot follow the desired input, but later follows the desired input as the \(V_{y}\) increases as shown in Figure 4f. Unlike the first case, the control performance can be significantly increased in this case with the proposed force input. The difference here is the force direction, where the proposed input requires greater z-direction than the y-direction at the beginning, as shown in Figure 4f. The rise time and the settling time is decreased from 1.3187 [s] to 0.3497 [s] (\(-\)73.48%) and from 2.2736 [s] to 0.6286 [s] (\(-\)72.35%), respectively. The sine wave input was simulated as shown in Figure 4g. The results show that the RMS error decreased from 0.0136 [m] to 0.0083 [m] (\(-\)38.97%) by using the proposed method. Figure 3: The two types of singular state configurations are shown. (**a**) shows the null space quality vs. the manipulability of a manipulator when the manipulator moves to the type 1 singular state by the step input. Type 1 singularity is caused by the Jacobian, which reflects the kinematics of the controlled body. (**b**) shows the dot product between the joint basis of the null space projector and the Jacobian of the secondary task. The nonzero dot product subset of the joint bases of Jacobian forms the bases after the projection. The dot product is the dimensionless value.