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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.