# Input-to-State Stability of Variable Impedance Control for Robotic Manipulator
## 3 ISS of Variable Impedance Model  
Figure 1 illustrates a block diagram of the variable impedance control scheme for robotic manipulators. To begin with, we prove the ISS of the variable impedance model (closed-loop control system) of Equation (3) that is obtained by applying the impedance control of Equation (6) to the robotic manipulator of Equation (1). While proving the ISS, two sufficient conditions are derived from the stability criterion. For notational convenience, external joint torque vector that is distributed from the externally applied force/torque is newly introduced as follow:  
\[d\triangleq J^{T}(q)f_{ext}. \tag{7}\]where \(d\in\Re^{n}\) denotes \(n\)-dimensional external joint torque. Let us define a state vector using the position and velocity errors as follows:  
\[x\triangleq\begin{bmatrix}x_{1}\\ x_{2}\end{bmatrix}, \tag{8}\]  
where \(x\in\Re^{2n}\) is \(2n\)-dimensional state vector, and:  
\[x_{1} \triangleq e=q-q_{d}\] \[x_{2} \triangleq e=q-\dot{q}_{d},\]  
in which \(x_{1}=e\in\Re^{n}\) and \(x_{2}=\dot{e}\in\Re^{n}\) express \(n\)-dimensional position and velocity error, respectively. Now, the variable impedance model (VIM) can be simply expressed either by using the definitions of errors:  
\[H(q)\ddot{e}+D\dot{e}+K(t)e=d \tag{9}\]  
or by using the state vector notations:  
\[\dot{x}_{1} =x_{2} \tag{10}\] \[\dot{x}_{2} =-H^{-1}(q)[Dx_{2}+K(t)x_{1}-d].\]  
To help understanding the concept of ISS, it is shortly explained. For given system  
\[\dot{x}=f(t,x,d), \tag{11}\]  
it is assumed that the function \(f:[0,\infty)\times\Re^{2n}\times\Re^{n}\rightarrow\Re^{2n}\) is piecewise continuous in \(t\) and locally Lipschitz in \(x\) and \(d\), and furthermore, it is assumed that external joint torque \(d(t)\) is also piecewise continuous in \(t\) and bounded function for all \(t\geq 0\). Then the existence and uniqueness of the solution of first-order nonlinear differential equation of Equation (11) are guaranteed. Additionally, it is noted that \(\rho_{1}(a):[0,\infty)\rightarrow\Re^{+}\) is called a class \(K\) function if it is strictly increasing with \(\rho_{1}(0)=0\), and \(\rho_{2}(a):[0,\infty)\rightarrow\Re^{+}\) as a class \(K_{\infty}\) function if it belongs to the class \(K\) and \(\rho_{2}(a)\rightarrow\infty\) as \(a\rightarrow\infty\). Now, the well-known theorem regarding the ISS in [8, 11] is introduced in the following.  
**Theorem 1** (ISS Theorem, [8]).: _Let \(V(t,x):[0,\infty)\times\Re^{2n}\rightarrow\Re\) be a continuously differentiable function such that \(V(t,x)>0\) except the equilibrium \(x=0\). Here, if and only if there exist a class \(K_{\infty}\) function \(\gamma_{1}\) and a class \(K\) function \(\gamma_{2}\) such that the following inequality is satisfied:_  
\[\dot{V}\leq-\gamma_{1}(\|x\|)+\gamma_{2}(\|d\|),\]  
_then the system (11) is input-to-state stable(ISS)._  
Figure 1: Variable impedance control scheme; \(q,\dot{q},d\in\Re^{n}\) denote the joint position, velocity and the externally applied joint force/torque, respectively, and \(q_{d},\dot{q}_{d},\dot{q}_{d}\in\Re^{n}\) the desired joint position, velocity and acceleration, respectively, and \(\tau\in\Re^{n}\) the control input.  
The proof is omitted because we can easily find it from many references including [8]. On the other hand, suppose that there exists a function \(V(x,t)\) such that the following holds for all \(x\) and \(d\):  
\[\|x\|\geq\rho(\|d\|)\ \ \rightarrow\ \ \dot{V}\leq-\gamma_{3}(\|x\|)\]  
with class \(K_{\infty}\) functions \(\rho\) and \(\gamma_{3}\), then the system is ISS as well. In other words, if the size of the state vector is lower-bounded as a class \(K_{\infty}\) function \(\rho\), then the time derivative of the Lyapunov function is negative definite for all state vectors.