# Input-to-State Stability of Variable Impedance Control for Robotic Manipulator
## 1 Introduction  
External forces and torques appear in contact with environments or with humans. Dealing with external forces and torques is important in achieving a stable and safe cooperating task with wide robotic applications. Impedance control is one of the methods of handling expected or unexpected external forces and torques [1]. When the impedance control is applied to the robot, it behaves like a linear second-order system, such as a mass-spring-damper model that can be very intuitive to predict how the system works. Furthermore, [PERSON] and [PERSON] [2] proposed an interaction controller to adjust the force contacting with environments. In the paper, necessary and sufficient conditions are proposed to ensure the stability of the coupled linear impedance model under a single interaction port for passive environments. Recent research on robotic manipulation and cooperation has focused on the fact that the robotic manipulator brings accurate and fast results, as well as safe collaborations with the human. [PERSON] et al. [3] proposed a learning by demonstration approach for a robotic manipulator operating in the unstructured environment while interacting with a human operator. As a common method, impedance control uses constant parameters, but the method might bring performance limitations in conducting cooperative works between robot and human or between robot and objects having uneven surfaces and various elasticities. As part of that, impedance control using time-varying parameters, called variable impedance control, was proposed; for example, [PERSON] and [PERSON] [4] analyzed the characteristics of impedance parameters of the human arm itself in order to execute the human-robot cooperative tasks. The impedance of the human arm was approximated using the variable impedance model that the resultant motions were planned and implemented. Also, [PERSON] et al. [5] suggested the modulation scheme of Cartesian impedance parameters forredundant manipulators in order to improve the control performance during the physical interaction between humans and robots. The aforementioned methods dealt with how to make appropriate varying profiles of impedance parameters, ultimately, for achieving the desired interaction task motion and its planning. [PERSON] et al. [6] proposed the variable impedance control with an update rule on the stiffness parameter. The proposed method guarantees that the system is passive with any variable stiffness matrix, but the specific method to design the remaining impedance parameters was not suggested. [PERSON] and [PERSON] [7] proposed the concept of state-independent stability for the variable impedance control, and it broadens the possibility to design update rules on the stiffness and damping matrices. Nevertheless, choosing the desired inertia parameter correctly was a remaining issue. As a matter of fact, impedance control is closely related to joint torques applied externally, but the aforementioned references did not deal with the external joint torques for designing the update rule on the stiffness and damping matrices. This paper aims at finding the sufficient conditions of variable impedance control for guaranteeing the input-to-state stability (ISS) as well as deriving the stiffness update rule from the asymptotic stability of the closed-loop control system. The ISS in [8] can be interpreted in two viewpoints: one is that the system behavior should remain bounded if its inputs are bounded, and the other is that the system states should tend to equilibrium if the inputs tend to zero. These concepts can be usefully applied to the variable impedance control because the impedance control operates in both motion spaces such as the constrained motion space and free (or unconstrained) motion space. For example, if the robotic manipulator is in the free motion without any contacts, then the variable impedance control acts like the conventional position control without any variations of impedance parameters.