**Design and Stability Analysis of a Robust Impedance Control System for a Robot Manipulator**

**Mohammad Reza SOLTANPOUR**

Shahid Sattari Air University

Shamshiri Street, Tehran, 1599964511, Iran

**Seyed Ehsan SHAFIE**

Shahrood University of Technology

Daneshgah Street, Shahrood, 3619995161, Iran

**Abstract**: This paper presents a novel impedance control approach for a robot manipulator and analyzes its stability. In order to achieve desired impedance for the robot manipulator subject to applied force on the end-effector, a hybrid position/force control in the task space is developed. For this purpose, the both cases of known and unknown bounds of uncertainties are considered to design the nonlinear robust controller. It is proven that the closed loop control system shows global exponential stability under known bounds of uncertainties. In the second case, an adaptive controller is used to estimate the bounds of uncertainties. It is then proven that the closed loop system has a global asymptotically stability. The case study is a two-link elbow manipulator which is simulated. The simulation results confirm good performances of proposed control approaches.

**Keywords**: Robot Manipulator, Uncertainties, Robust Impedance Control, Adaptive Control.

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**CITE THIS PAPER AS**:

Mohammad Reza SOLTANPOUR, Seyed Ehsan SHAFIE, **Design and Stability Analysis of a Robust Impedance Control System for a Robot Manipulator**, *Studies in Informatics and Control*, ISSN 1220-1766, vol. 19 (1), pp. 5-16, 2010.

**1. Introduction**

In addition of moving through free space, industrial robots involve their environments while operating special tasks such as assembling, polishing, deburring, pushing and power-assisting [1-5]. So, the force and position control must put on simultaneously. In spite of that, in the most researches in position control of robots it is supposed that the robot does not involve to operation environment and significantly, small position error, fast response and practical implementation are concerned in the controller design. In such control systems, a simple contact with working surface may lead notable problems; because the system dynamics change and in consequence, the closed loop system stability is not guaranteed anymore.

Hybrid position/force control and impedance control are the most noticeable methods that used to control of robots which involve their environments [6-7]. In the hybrid control, the task space is partitioned into two distinct position and force subspaces by selection matrix , such that the position control is accomplished in the position space, and force control is accomplished in the force space. Despite of considering distinguish between position and force control in the hybrid method, the desired impedance of robot manipulator and dynamical behavior of environmental reaction force are not taken into account.

In the impedance control one can control the environment reaction through the end-effector path as well as position control at the same time and further, there is no need to selection matrix . The main drawback of this method is that it employs identical design parameters for both position and force control. As a result, the controller performance in dealing with dynamical behavior of environment reaction force and end-effector position is the same [7].

After presenting these approaches, researchers focused their studies on concurrent position/force control. Hybrid impedance control was proposed based on the concepts of the internal and external control loops where the position and force control achieved simultaneously by using of the exact model of system dynamics [8].

Although, the access of exact dynamical model assumption is not fulfilled in the presence of structured and unstructured uncertainties as load variation, friction, disturbance and un-modeled dynamics and on the other hand, these uncertainties produce restrictions in measurement techniques. The uncertainties affect on controller performances and closed loop system stability and, in much cases cause the closed loop system becomes unstable.

Adaptive control has been used to overcome parametric uncertainties in the dynamical model of robot manipulator and to achieve desired impedance [9-10]. By noting to the result of these studies it can be seen that adaptive control has effective performance against parametric uncertainties but these are only parts of all uncertainties in a real system. Another method which is proposed to deal with structured and unstructured uncertainties in impedance control of robot manipulator is sliding mode control [11]. The control law is designed such that the position is controlled when the robot is in free space and the force is controlled when the robot involves. This control law is very simple and in order to avoid discontinuity of control input, the system could have only uniformly bounded stability. In [12] the desired impedance attained based on relation between position tracking error and force tracking error by employing sliding mode control. Stability analysis shows that the closed loop system has uniformly bounded stability. Additionally, there are other effective works in the field of robust impedance control [13-16]. But these controllers are designed based on the dynamical model of robot manipulator in task space and therefore the computation magnitudes are high and they need fast processors in implementation phase. It is worth mentioning that the fuzzy method is also used to model-free impedance control of robot for quick tasks [17].

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