A Cartesian Compliance Controller for a Manipulator Mounted on a Flexible Structure Christian Ott, Alin Albu-Sch¨ affer, and Gerd Hirzinger German Aerospace Center (DLR e.V.) Institute of Robotics and Mechatronics 82234 Wessling, Germany Email: christian.ott@dlr.de, alin.albu-schaeffer@dlr.de Abstract— In this paper the Cartesian compliance control of a manipulator mounted on a flexible base is considered. The proposed control law aims at achieving a desired stiffness and damping in Cartesian coordinates while taking account of the base flexibility. The controller does not use any measurement of the base motion, however a model of the base stiffness is required. For the closed loop system, asymptotic stability in case of free motion is proven. Furthermore, considering interaction tasks, it is shown that the controlled manipulator system has a useful passivity property. I. I NTRODUCTION It is well known that the flexibility of the base on which a robot manipulator is mounted can significantly influence the positioning accuracy [1]. Moreover, the base flexibility is also relevant from a stability point of view due to the presence of non-linear couplings between the robot dynamics and the base dynamics. In many applications it is not possible to measure the motion of the base reliably, but still the controller of the arm should take the base flexibility into account. Otherwise base vibrations and the gravity load of the arm on the flexible base will lead to degraded position accuracy of the end-effector. Typical examples for applications where this is relevant are for instance Micro/Macro-manipulator systems where the macro- manipulator represents the base, or mobile manipulation set- tings where a manipulator is mounted on a mobile base. If the mobile base is actuated by non-rigid wheels, this leads to a considerable elasticity. In order to achieve high position accuracy as required for fine manipulation tasks, one clearly should take this into account. The control of a robot mounted on a flexible base has been treated by several authors. Nenchev et al. proposed the co-called reaction null-space control [2]. In this method the dynamic redundancy of a kinematically redundant arm is exploited such that the robot can perform its tasks without exciting the vibrations of the base. Furthermore, a gravity- free environment was considered therein. In contrast to that our contribution focuses on a compensation of the static end- effector deviation due to a gravity induced base deflection. In particular it is also applicable both to redundant and to non- redundant manipulators. Therefore, the focus of our paper is somewhat complementary to [2]. Clearly, a combination of the results from [2] with our approach would be possible, and would particularly be useful when the control of a kinemati- cally redundant arm under the effect of gravity is considered. Another important work on the control of a manipulator mounted on a flexible structure was presented by Ueda and Yoshikawa in [3] where they analyze the robustness of a compliance controller with non-collocated position feedback in a gravity-free environment. Their analysis is based on a modal analysis of the linearized system and includes an additional feedback of the joint acceleration for improving robustness. In contrary to [3], the controller presented in this paper avoids the use of non-collocated feedback and a passivity analysis of the non-linear closed loop system is given. Several authors treated the vibration damping for a Micro/Macro-manipulator system. In [4] the reaction force of a short rigid manipulator mounted at the tip of a large flexible arm was considered as a control input, and the motion of the short manipulator was commanded such that the reaction force acting on the base produced a damping of the base vibration. Lew and Trudnowski [5] presented a control scheme in which the control torque from a joint level PD-controller is augmented by an additional feedback of the base motion in order to achieve enhanced vibration damping. In [6] this technique was combined with a special filtering of the command. A composite controller was proposed in [7] where the fast part of the control input deals with the joint angle dynamics while the slow part deals with the base motion. This composite controller was used together with an inner loop acceleration feedback. In this paper we propose a Cartesian compliance controller for a manipulator mounted on a flexible structure. Since the gravity load of the manipulator leads to a deflection of the base, the controller must compensate for this deflection. In particular, it is assumed that the motion of the base cannot be measured but the stiffness of the base is known. The presented controller is related to our recent results on the control of flexible joint robots [8], [9], [10], [11], [12], [13]. In [8], [9], [12] a passivity based approach for the com- pliance control of a flexible joint robot was presented. Therein a compensation term for the link side gravity components was computed based only on the motor side positions. This gravity compensation term was combined with a PD-like stiffness