# Numerical Quantification of Controllability in the Null Space for Redundant Manipulators
## 4 Conclusions  
In this paper, we have proposed the index that quantifies the manipulating ability of a robot in the null space. In addition, a novel control method that is more efficient in the null space, especially under low DoF condition, has been proposed. In order to accomplish these contributions, we analyzed the joint constraints caused by the null space projection and the joint and task space basis with respect to the SVD of the null projected Jacobian.  
Figure 5: Experimental results for obstacle avoidance. The schematic of experiment is shown in the left. The primary task is the stationary task by utilizing 6 DoF at the end-effector and the secondary task is obstacle avoidance at link 4 (yellow point) while the obstacle approaches to link 4 from the right side of the manipulator at a speed of 5 cm/s. (**a**) shows the distance between the obstacle and the control point. (**b**) shows the input force when the obstacle approaches. \(\rho_{0}\) is the distance of the obstacle that the manipulator begins to avoid. \(t_{0}\) is the time when the obstacle infiltrates the radius \(\rho_{0}\). The proposed method avoids the obstacle faster with the less force input.  
Based on these findings, one can find a controllable direction with these bases and control the coupled DoF efficiently by projecting the force to the bases.  
We conducted several simulations and experiment to validate the singularity and improved control performances. At the first simulation, the two types of singularity that appear in the null space was verified, one of them which cannot be explained solely with manipulability. We have also conducted simulation to validate the proposed method with respect to the step and sinusoidal input. As a result, the proposed method improved the overall control performance (\(-51.79\%\) and \(73.48\%\) for the rise time, \(-54.82\%\) and \(-72.35\%\) for the settling time, \(-33.94\%\) and \(-38.97\%\) for RMS error). Furthermore, we applied the proposed method to obstacle avoidance and observed that the task can be performed with a smaller force input with shorter duration of response time for safety.  
The null space quality proposed in this work can be a useful substitute for manipulability when we control the null projected system, because it considers the properties of null space that manipulability cannot consider. The proposed control method can be a basic task space control strategy for the null projected system with local optimization. Furthermore, The null space quality can be applied to wide range of applications with regards to the consistency of the projector. In the future, we plan to further enhance the control strategy, so that it can be integrated to redundant robots with more than twp excess DoF.  
Conceptualization, methodology, simulation setup, experimental setup drafted the paper and revised the paper, S.K.; experimental setup, reviewed the paper for submission, S.Y.; supervised overall work the paper, D.S. All authors have read and agreed to the published version of the manuscript. This work was supported by the National Research Foundation of Korea(NRF) grant funded by the Korea government. (MSIT) (NRF-2021R1A2C2008379). Furthermore, this study was supported by the Chung-Ang University Graduate Research Scholarship in 2019. Not applicable for studies not involving humans or animals. Not applicable for studies not involving humans. Not applicable for studies not involving humans. Not applicable for studies not involving humans. Not applicable to this article. The authors thank to people in the Human-Centered Robotics Laboratory at Chung-Ang University for their valuable comments and feedback. The authors declare no conflict of interest.