he  new  control input  a = [U: 77  =  JB-'(d  -  2) can be chosen as [8] a,  =  Pc +  kvp(ljc  -  lj) +  kpP(pc  -  P) a, = w,  +  kvo(w,  -  w)  +  kpoz (41) (42) where C  is the vector part of  the quaternion associated to RTR, when referred  to  the base frame. Also, kv,, kpp in  (41),  and  kv0, kp,  in  (42) are suitable positive gains of  the position loop and the orientation loop, respectively; such inner loops provide robustness  to  the above distur- bance, which  otherwise could not be  effectively counter- acted through the impedance parameters. Finally, notice that  p,  and  R, and their associated derivatives can  be computed by  forward integration of  the differential equa- tions (27) and (28). 4. Experimental results The laboratory setup consists of an industrial robot Comau SMART-3  S. The  robot manipulator has a six-revolute-joint anthropomorphic geometry with nonnull shoulder and el- bow  offsets and non-spherical wrist. The joints are  ac- tuated by  brushless motors via gear trains; shaft absolute resolvers provide motor position measurements. The robot is controlled by  an open version of  the C3G 9000 control unit which has a VME-based architecture with  a bus-to- bus communication link to a PC Pentium 133. This is in charge of computing the control algorithm and passing the references  to the current servos through the communication link at 2 ms sampling rate. Joint velocities are reconstructed through numerical differentiation of  joint position readings. A 6-axis forcekorque sensor AT1  FT30- 100  with  force range of  f130 N and torque range of  f10  Nm is mounted at the wrist of  the robot manipulator. The sensor is connected to the PC by a parallel interface board which provides readings of six components of generalized  force at 2 ms. An end effector has been built as a steel stick with a thin plastic disk of  5.5 cm radius at the tip.  The end-effector frame has its origin at  the center of the disk and its approach axis normal to the disk surface and pointing outwards. The end-effector desired  task consists of a straight line mo- tion  with a vertical displacement of  -0.25  m along the z-axis of  the base frame. The trajectory along the path is generated according  to a 5th-order interpolating polynomial with null initial and final velocities and accelerations, and a duration of 5 s. The end effector is oriented so that the ap- proach axis of its frame forms an angle of  -5/6a  rad about the y-axis of the base frame; then the desired orientation is required to remain constant during the task. The environment is constituted by  a cardboard box, where the stiffness is of  the order of  lo4 N/m.  The  surface of the box is nearly flat and is placed (horizontally) in the sy- plane in such a way  as to obstruct the desired end-effector motion. The impedance paramctcrs  in  (27)-(30)  have been set to M, = 101, D, = 3001, K, = 500, and MO  = 0.251, Do = 31, KO  = 51.  Notice that K, and KO  have been chosen  so  as  to  ensure a compliant behavior at the end effector (limited  values of contact force and moment) during the constrained motion, while D, and Do  have been chosen so as to guarantee a well-damped behavior. The dynamic model of the robot manipulator has been iden- tified in terms of a minimum number of parameters, where the dynamics of the outer three joints has been simply cho- sen as purely inertial and decoupled. Only joint viscous friction has been included, since other types of friction (e.g. Coulomb and dry friction) are difficult  to model. The com- plete identified model is reported in [  151. The gains of  the control action in (41) and (42) have been set to kpp  =  kp,  = 2025 and kvp  = kvo = 56. Figure 1  shows the components of the end-effector position error pd  -  p  in the base frame. After the contact (occurring at t = 3.3  s) the component along the z-axis significantly deviates from zero, as expected, while an appreciable error cap be  seen also for the component along the y-axis.  In fact, contact is made at the edge of  the disk and then  the end  effector tends  to  anti-align the approach axis  of  its frame with the z-axis of  the base frame by  rotating about the x-axis; the presence of  friction at the contact causes a deviation of  the origin of  the end-effector frame off  the vertical direction (along the y-axis). 5. Conclusion The problem  of  impedance control for robot manipula- tors performing six-degree-of-freedom interaction tasks has been tackled in this work. A theoretical study has  been  developed to show how  the inertial and stiffness parameters of  the impedance can be clearly derived from kinetic and potential energy contribu- tions, respectively. Passivity between generalized contact force and end-effector velocity is then ensured by the intro- duction of  suitable damping parameters, as proven through a Hamiltonian argument. A  key  point of  the approach is the adoption of  a unitary quaternion to describe mutual orientation between the de- sired, the compliant and  the end-effector frame, with  the noticeable advantage of  avoiding representation singulari- ties. 六自由度电阻机器人英文文献和中文翻译(3):http://www.751com.cn/fanyi/lunwen_35660.html