# Input-to-State Stability of Variable Impedance Control for Robotic Manipulator
## 2 Variable Impedance Control in Joint Space  
Impedance control is usually derived and described in the task (or operational) space. However, this paper deals with it in the joint space because the task space dynamics can be easily obtained from the joint space dynamics only if we know the kinematic structure of the robotic manipulator. Consider the well-known dynamics of the robotic manipulator in [9]:  
\[H(q)\ddot{q}+C(q,\dot{q})\dot{q}+g(q)=\tau+J^{T}(q)f_{ext}, \tag{1}\]  
where \(H(q)\in\Re^{n\times n}\) is a symmetric and positive definite inertia (or mass) matrix, \(C(q,\dot{q})\in\Re^{n\times n}\) the Coriolis and centrifugal matrix, \(g(q)\in\Re^{n}\) the gravity force/torque vector, and \(J(q)\in\Re^{m\times n}\) the Jacobian matrix obtained from the kinematic structure of robotic manipulator, \(q,\dot{q},\dot{q}\in\Re^{n}\) the joint position, velocity and acceleration, respectively, \(\tau\in\Re^{n}\) the joint force/torque, \(f_{ext}\in\Re^{m}\) the externally applied force/torque, \(n\) the dimension of joint space, \(m\) the dimension of task space, and finally, the friction and unknown disturbances were ignored in the dynamics in order to avoid the difficulties of theoretical derivation, although the frictions affect the control performance badly, especially, in case of the impedance control. Also, it is obvious in [10] that the inertia matrix \(H(q)\) is uniformly bounded and the following property always holds:  
\[\lambda_{m}(H(q))I\leq H(q)\leq\lambda_{M}(H(q))I, \tag{2}\]  
where \(I\in\Re^{n\times n}\) is an identity matrix, \(\lambda_{m}(\bullet)\) and \(\lambda_{M}(\bullet)\) denote the minimum and maximum eigenvalues of matrix (\(\bullet\)), respectively.  
The impedance control makes the closed-loop control system act like a sort of linear mass-spring-damper system, and it is referred to as the generalized dynamic impedance in case of robotic manipulators in [9]. For simplicity, the stiffness parameter is chosen as a time-varying one. Now, let us assume that the generalized dynamic impedance is given as follow:  
\[H(q)(\ddot{q}-\ddot{q}_{d})+D(\dot{q}-\dot{q}_{d})+K(t)(q-q_{d})=J^{T}(q)f_{ext}, \tag{3}\]  
where \(D=D^{T}\in\Re^{n\times n}\) denotes constant positive definite damping matrix and \(K(t)=K^{T}(t)\in\Re^{n\times n}\) implies time-varying positive definite stiffness matrix. In order to implement the given generalized dynamic impedance, the inverse dynamics control method in [9] is designed as the following form:  
\[\tau_{c}=H(q)u_{0}+C(q,\dot{q})\dot{q}+g(q)-J^{T}(q)f_{ext}, \tag{4}\]  
where \(\tau_{c}\) denotes a control input to be applied to the robotic manipulator described by Equation (1), here \(\tau_{c}=\tau\), and \(u_{0}\) is an auxiliary control input to be designed from the generalized dynamic impedance of Equation (3) and it has the following form:  
\[u_{0}=\ddot{q}_{d}+H^{-1}(q)[D(\dot{q}_{d}-\dot{q})+K(t)(q_{d}-q)+J^{T}(q)f_{ ext}], \tag{5}\]  
where \(q_{d},\dot{q}_{d},\ddot{q}_{d}\in\Re^{n}\) represent the desired joint position, velocity and acceleration, respectively. If Equation (5) is applied to \(u_{0}\) term of Equation (4), then we have the variable impedance control in the joint space as follow:  
\[\tau_{c}=H(q)\ddot{q}_{d}+D(\dot{q}_{d}-\dot{q})+K(t)(q_{d}-q)+C(q,\dot{q}) \dot{q}+g(q). \tag{6}\]  
It is noted that the externally applied force/torque term disappears in the control algorithm as shown in Equation (6), in other words, Equation (6) does not require any explicit force/torque measurement feedback. As a result, when the control input of Equation (6) is applied to the robotic manipulator of Equation (1), we can obtain the generalized dynamic impedance of Equation (3) as the behavior of closed-loop control system. From now on, we will use Equation (3) as variable impedance model in the next sections.