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Abstract-This work is focused on impedance control of robot manipulators performing six-degree-of-freedom interaction tasks. An energy-based formulation leads to formally deriving the dynamic equation character- izing a mechanical impedance at the end effector. An inverse dynamics strategy with contact force and mo- ment measurement is adopted to obtain a configuration- independent desired impedance. For given contact force and moment, an impedance control scheme is pro- posed acting on both translational displacements and rotational displacements where end-effector orientation is described using a singularity-free representation in terms of a unitary quaternion. Experimental results on an industrial robot with open control architecture are presented. 37006
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Keywords-Robot manipulators; impedance control; in- verse dynamics; quaternion.
1. Introduction Impedance control is a well-established framework for man- aging interaction of the end effector of a robot manipulator with a compliant environment [ 11. The goal is to keep the contact force limited both during transient and at steady state (as for stiffness control [2]) by acting on the end- effector displacement. Most interaction control schemes refer to three-degree-of- freedom (dof) tasks in that they handle end-effector position and contact linear force. On the other hand, those control schemes for six-dof tasks typically utilize a minimal de- scription of end-effector orientation, e.g. in terms of three Euler angles, which in turn requires the contact moment to be expressed in the operational space [3]. A drawback of the above description is the occurrence of representation singularities of the analytical Jacobian [4]. These can be avoided by resorting to a four-parameter singularity-free description of end-effector orientation, e.g. in terms of a unitary quaternion [5]. Such a description has already been successfully used for the attitude motion control problem of spacecrafts [6,7] and manipulators [8]. Another drawback is that the rotational parameters of the Following the basic formulation in [9], recent research ef- forts have been devoted to extending the concepts of stiff- ness and impedance to six-dof tasks by using a geometric rather than an analytical approach [ 10,111. The advantage of such technique lies in the possibility of referring to suit- able energy contributions of clear physical interpretation. The goal of this work is to present a six-dof impedance control scheme, where the impedance equation is charac- terized for both its translational part and its rotational part. A kinetic energy function is defined in terms of the end- effector linear and angular velocity error, and a potential energy function is defined in terms of the end-effector po- sition and orientation error (viz., quaternion). The single terms of the impedance are then obtained by considering power contributions and dissipative terms are introduced to guarantee a passive behaviour. In order to obtain a configuration-independent desired impedance, an inverse dynamics strategy with contact force and moment measurement is adopted leading to a resolved acceleration scheme [
121. An inner loop acting on the end- effector position and orientation error is used to ensure good disturbance rejection [ 131. The proposed control scheme is tested on an experimental set-up constituted by an industrial robot with open con- trol architecture and forcekorque wrist sensor. Numerical results of a case study are presented. 2. Six-degree-of-freedom impedance In robotics it is customary to describe manipulation tasks in terms of a (3 x 1) position vector pd and a (3 x 3) rotation matrix & that represent the origin and orientation of a desired frame with respect to a base frame. Let also p and R denote the origin and orientation of a frame attached to the manipulator’s end effector. When the manipulator moves in free space, the end-effector frame is required to match the desired frame. Instead, when the end effector interacts with the environment, it is worth introducing another frame specified by p, and R, so as to ensure a compliant behavior; in this case, the end-effector frame is required to match the compliant frame. impedance cannot be directly related to the task geometry. The desired velocity is given by the linear velocity vector being S( .) the skew-symmetric matrix operator performing vector product. As usual, the superscript denotes the frame to which a quantity (vector or matrix) is referred; the super- script is dropped whenever a quantity is referred to the base frame. Also, the desired linear and angular acceleration are respectively given by pd and wd. Similarly to above, the velocity and acceleration of the end- effector (compliant) frame are respectively given by p, w (p,, w,) and 9, GJ (p,, hc). The mutual position and orientation between the compliant and the desired frame can be characterized by the position error P = Pc - Pd (2) and the orientation error Rd = R~R,. (3) Differentiating (3) with respect to time gives where b=wc-wd (5) is the angular velocity error. By adopting an anglelaxis description, Rd can be generated by the rotation of an angle 8 about an axis of unit vector fd; note that this vector has the same representation when referred to the compliant frame, i.e. fd = P. Hence, the orientation error can be expressed in terms of a four- parameter description e ij = cos - 2 i.e. a unitary quaternion. This has the advantage over other (minimal) descriptions of orientation, e.g. Euler angles, to be singularity-free. The relationship between the time derivative of the quater- nion and the angular velocity error is established by the so-called quaternion propagation: fj=--c l-dT-d w 2 + (10) More specific material about quaternions can be found in [5-81. The goal is to derive a mechanical impedance at the end effector for both its translational part and its rotational part. For a given force, the former should act on the position er- ror (2) and its derivatives by a stiffness, damping and mass, whereas for a given moment, the latter should act on the orientation error (7) and the angular velocity and accelera- tion errors by a rotational stiffness, rotational damping and inertia. To this purpose, it is convenient to refer to an energy-based formulation. Let 7qp+z (1 1) represent a kinetic energy with 1- 7 - -wdTM0Gd, O-2 where Mp = MF and MO = MF are positive definite matrices. In particular, when Mp = mpI, lP expresses the translational kinetic energy of a rigid body with mass mp, whose center of mass has a linear velocity p; otherwise, in the general case, Mp can be regarded as a generalized mass which is assumed to be constant in the base frame. On the other hand, 7, expresses the rotational kinetic energy of a rigid body with inertia tensor MO and angular velocity 6; such tensor is assumed to be constant in the desired frame.