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
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. 六自由度电阻机器人英文文献和中文翻译:http://www.751com.cn/fanyi/lunwen_35660.html