# Input-to-State Stability of Variable Impedance Control for Robotic Manipulator
## 3 ISS of Variable Impedance Model
### Update Rule of Stiffness Matrix  
The variable impedance control was designed to improve the control performance by handling the disturbance such as the external joint torque. However, it was difficult to define the desired performance of the variable impedance control. [PERSON] et al. [13] proposed the strategy that chooses the impedance parameters by using force and velocity. [PERSON] et al. [14] suggested the neural network method to solve the interaction problem. [PERSON] et al. [15] proposed the selection strategy of the impedance parameters using a passivity index and an extreme value of force applied by the human operator. In this section, the stiffness update rule is proposed. For this purpose, we assume a situation that the manipulator interacts with an environment that has an uneven surface and various stiffnesses. For example, consider the scene of a disaster: several interactive tasks such as handling the tools and cutting the objects would be required. Even in this situation, the robotic manipulator should achieve a target position by updating its stiffness property, in fact, it is designed to be adjusted according to the external joint torque, i.e., \(d\), in the following Theorem. **Theorem 3**: _Assume that Theorem 2 holds. If the following update rule:_  
\[\dot{K}(t)=\left\{ \begin{array}{ll}\frac{2}{\alpha\|\varepsilon\|}\dot{e} \dot{d}^{T}+\frac{1}{\alpha}\dot{H}(q)&\quad\text{if}\quad\|\dot{e}\|\geq \epsilon\\ 0&\quad\text{otherwise}\end{array} \right. \tag{26}\]  
_is utilized for the variable stiffness adjustment, then the closed-loop control system is either asymptotically stable or ISS, where \(\epsilon>0\) is some positive constant._  
**Proof. ** First, consider the condition of \(\|\dot{e}\|\geq\epsilon\), since \(x_{2}^{T}d\) of Equation (19) is a scalar function, we can get  
\[x_{2}^{T}d=d^{T}x_{2}=\frac{x_{2}^{T}x_{2}d^{T}x_{2}}{\|x_{2}\|^{2}}=x_{2}^{T} \left[\frac{x_{2}d^{T}}{\|x_{2}\|^{2}}\right]x_{2}. \tag{27}\]  
By using above property, we have the following inequality from Equation (19),  
\[\dot{V}\leq -x_{1}^{T}\left[\frac{(1-\alpha)}{2}K(t)-\frac{\alpha}{2}\dot{K}( t)\right]x_{1} \tag{28}\] \[-x_{2}^{T}\left[\frac{(1-\alpha)}{2}K(t)+D-\frac{1}{2}\dot{H}(q) -\frac{x_{2}d^{T}}{\|x_{2}\|^{2}}\right]x_{2}.\]Let us define the positive definite matrices \(A_{1}\in\Re^{n\times n}\) and \(A_{2}\in\Re^{n\times n}\),  
\[ \begin{split}\frac{(1-\alpha)}{2}K(t)-\frac{\alpha}{2}\dot{K}(t)& =A_{1}>0\\ \frac{(1-\alpha)}{2}K(t)+D-\frac{1}{2}\dot{H}(q)-\frac{x_{2}d^{T} }{\left\|x_{2}\right\|^{2}}&=A_{2}>0.\end{split}  \tag{29}\]  
By subtracting the above equation from the below, we have:  
\[\frac{\alpha}{2}\dot{K}(t)=\frac{x_{2}d^{T}}{\left\|x_{2}\right\|^{2}}+\frac{1 }{2}\dot{H}(q)+A_{2}-A_{1}-D. \tag{30}\]  
Since we can choose \(A_{2}=A_{1}+D>0\) with \(A_{1}>0\) and \(D>0\), we assume that \(A_{2}-A_{1}-D=0\). Now, we can define the update rule as follow,  
\[\frac{\alpha}{2}\dot{K}(t)=\frac{x_{2}d^{T}}{\left\|x_{2}\right\|^{2}}+\frac{ 1}{2}\dot{H}(q). \tag{31}\]  
Furthermore, the time derivative of Lyapunov function holds:  
\[\dot{V}\leq\ -x_{1}^{T}A_{1}x_{1}-x_{2}^{T}A_{2}x_{2}. \tag{32}\]  
Thus we can conclude the closed loop control system is asymptotically stable under the condition of \(\left\|\dot{e}\right\|\geq\epsilon\). Second, in case of \(\left\|\dot{e}\right\|<\epsilon\), the ISS can be easily proved from the previous Theorem 2. From the stiffness update rule of Equation (26), we can know that the time derivative of the inertia matrix of the robotic manipulator should influence the variation of the joint stiffness matrix and its size. In addition, since the time derivative of manipulator inertia is closely related to the Coriolis and centrifugal matrix, the stiffness update is also affected by the size of the Coriolis and centrifugal matrix related to the joint velocity. Also, the outer product of the joint external torque and the joint velocity error influences the stiffness matrix update.