# Input-to-State Stability of Variable Impedance Control for Robotic Manipulator
## 5 Conclusions  
The sufficient conditions for the ISS and the stiffness update rule were proposed in the paper. In summary, the time derivative of the stiffness matrix was upper-bounded by the size of the stiffness matrix itself and lower-bounded by the size of the time derivative of the inertia matrix, respectively. The sufficient conditions of Equations (12) and (13) could guarantee the ISS of the variable impedance control even when the external force exists. Besides, the variable impedance control expressed by the Equation (6) does not require measuring the external force, although the stiffness update rule requires the use of joint torque sensor. As an alternative, the momentum observer [16] and external force observer [18] to estimate the external joint torques will be necessary.  
Figure 5: Simulation data while performing the simulation suggested in Figure 4: (**a**) Lyapunov function value of Equation (14) with \(\alpha=0.75\), where it was designed using kinetic plus potential energies. (**b**) Time derivative of Lyapunov function of Equation (21), where we could confirm whether the closed-loop control system is being stable. (**c**) \(K(t)\) and \(\dot{K}(t)\) were expressed using the proposed update rule Equation (26). (**d**) Second sufficient condition, that is, \(\left\|\frac{(1-\alpha)}{2}K(t)+D\right\|_{l}-\left\|\frac{1}{2}\dot{H}(q) \right\|\geq 0\) of Equation (13), where we could confirm that Equation (13) was satisfied even though the impedance parameters of \(K(t)\) and \(H(q)\) (its time derivative \(H(q)\)) were changed according to the joint configuration. (**e**) External joint torques obtained using Equation (7). (**f**) Velocity error of the each joint. (**g**) Position error of each joint.  
The limitations of the proposed method are summarized as follows. The update rule for the damping matrix was not proposed. Modifying the Lyapunov function to suggest the damping update rule will be necessary for the whole system. An improved update rule for the stiffness and damping is our next goal. The selection of the impedance parameters such as stiffness and damping is still complicated because of the Equation (13) consists of the time derivative of the inertia matrix. Simplifying the sufficient conditions for the ISS will be one of our challenging works. Also, consideration of the model uncertainty of the dynamic parameters will be our future work. Further, experimental verification will be necessary for the near future.  
[PERSON] is a PhD student of Hanyang University and [PERSON] is a Professor of Hanyang University. For this research article with two authors, conceptualization, [PERSON]; methodology, [PERSON]; software, [PERSON]; validation, [PERSON]; formal analysis, [PERSON]; investigation, [PERSON]; writing--original draft preparation, [PERSON]; writing--review and editing, Y.C.; visualization, [PERSON]; supervision, Y.C.; project administration, Y.C.; funding acquisition, Y.C. All authors have read and agreed to the published version of the manuscript.  
This work was supported in part by the Convergence Technology Development Program for Bionic Arm through the National Research Foundation of Korea funded by the Ministry of Science, ICT & Future Planning (NRF-2015M3C1B2052811), in part by the Technology Innovation Program (or Industrial Strategic Technology Development Program) (20001856, Development of robotic work control technology capable of grasping and manipulating various objects in everyday life environment based on multimodal recognition and using tools) funded By the Ministry of Trade, Industry & Energy(MOTIE, Korea), Republic of Korea.  
The authors declare no conflict of interest.