Cartesian Impedance Control of Flexible Joint Robots: A Decoupling Approach

Whenever a robotic manipulator is supposed to get in contact with its environment, the achievement of a compliant behavior is relevant. This is a classical control problem for rigid body robots, which led to control approaches like impedance control, admittance control, or stiffness control (Hogan, 1985; Sciavicco & Siciliano,1996). In contrast to the approach introduced in this contribution most works on the Cartesian impedance control problem consider a robot model which does not include joint flexibility. In this work a decoupling based control approach for flexible joint robots is described. The considered control objective is the achievement of a desired compliant behavior between external generalized forces and the Cartesian end-effector motion of the robot. The design method will be based on some results from control theory for cascaded systems. The proposed controller will be designed in two steps. First, an inner feedback loop is used to bring the flexible joint robot model into cascaded form. Then, an additional outer control loop implements the desired compliant behaviour (Ott et al., 2003). The stability theory for cascaded control systems (Seibert & Suarez, 1990; Loria, 2001) can be applied to analyze the closed-loop system. When dealing with a robot model with flexible joints, the maybe most obvious approach for the design of an impedance controller is the singular perturbation approach (Spong, 1987; De Luca, 1996; Ott et al., 2002). Therein the fast subsystem, which is in our case the torque dynamics, is considered as a perturbation of the rigid body model. One can then use any controller designed for the rigid body robot dynamics and apply it in combination with an inner torque control loop to the flexible joint robot model. The main disadvantage of this approach is that it does not allow for a formal stability proof without referring to the approximate singular perturbation consideration. The controller structure proposed herein is somewhat related to the singular perturbation based controller. Also herein an inner torque control loop is combined with an outer impedance control loop. But these control loops will be designed in such a way that one can give a proof of asymptotic stability, based on the stability theory for cascaded systems. The proposed controller structure is also related to the controllers presented in (Lin & Goldenberg, 1995; 1996). But the following analysis focuses on the design of an impedance controller, while Lin and Goldenberg considered a position controller respectively a hybrid position/force-controller. The procedure for the stability analysis from these works cannot be applied to the impedance control problem considered herein in a straightforward way. The chapter is organized as follows. In Section 2 some relevant results of the stability theory for cascaded systems are reviewed. The considered dynamical model of a flexible joint robot is described in Section 3. In Section 4 the design