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The rigid contact task and the joint elasticity, however, are never explicity considered. To fully comply with the requirements stated above, the hybrid position/force control algorithin described in (Ferretti, et al., 1995b), and sketched in Fig.4, is adopted. Vectors k e ~m and X e ~m are contact force set points and measures expressed in the contact frame, respectively, m being the number of degrees of freedom constrained by the contact (m = 3 for the RUB action). Force control is implemented as an outer loop cascaded to the independent motor position loops. It adds a bias ~F ~ 9tn (n being tile number of robot axes) to the nominal motor position --M set points qm and thus it integrates easily in a conunon functional architecture of an industrial controller, consisting of trajectory planning, kinematics inversion and joint position control. The bias is computed from the force error by means of three blocks: the force regulator, the computation of motor torques, and the torque conversion into motor displacement. The motor torques -F ~n required to actuate a %m force vector u ~ ~n are computed as: -F = pTjc(q) TK(q)T u T N (1) Equation (1) is derived from the constrained dynamic model of the manipulator (McClamroch and Wang, 1988) by imposing that Xm F controls the interaction forces without disturbing the programmed motion in the unconstrained directions (Ferretti, et al., 1995b). Matrix P[nxnl in (1) accounts for the kinematic relations among motor coordinates qm and joint coordinates q (in general, P is not diagonal, and Pii = Ni, Ni being the i-th gear ratio). Matrix Jc(q)[6×n] is the manipulator Jacobian, relating joint velocities to the linear and angular velocities of the compliant frame, expressed in the compliant frame itself. In practice, Jc(q) is computed from the Jacobian J(q), relating joint and compliant frame velocities expressed in the base frame, and from the robot direct kinematics (rotation matrix). Matrix K(q)lm×61 is a mapping between vector 7~ and the vector of contact forces and moments F c ~ 9t 6 exerted in the compliant frame, and can be directly derived by analysing the directions of constrained motion (Ferretti, et al., 1995b). For the RUB action, matrix K(q) is constant, and is given by: K(10=K=[013×21 I[3×31 013xl]]. The transfer function diagonal matrix C(s)~n×n! is introduced to transform motor torques i~n into corresponding motor rotations ~F. It accounts for position regulator transfer functions and for joint torsional flexibility. It is shown in (Ferretti, et al., 1993), that under reasonable assumptions, C(s) is a diagonal matrix whose elements Cii(s) are second order transfer functions whose singularities depend on Kpi, Kli, KDi, the proportional, integral and derivative gain, respectively, of the i-th PID regulator, and on h), the stiffness constant of the i-th joint. KI can be experimentally identified (Ferretti, et al., 1994ab). In particular, it results that C~(0) = 1/ K,. The force regulator implements decoupled integral control laws for each force component: matrix K F is thus constant and diagonal. This choice is motivated by the fact that the integral regulator, if properly designed, guarantees a crossover frequency of the force-control loop which is compatible with the phase shifts due to delays in the loop and to unmodelled dynamics.
This point is confirmed by recent results presented by Ferretti, et al. (1995a), which have shown the increasing stability of integral control with increasing contact stiffness. Finally, the vector of contact force measurements, can be obtained from the outputs of a wrist-mounted force sensor through a fixed transformation, where the compliant frame is rigidly connected to the end- effector. 4.3 Application architecture for hybrid control The AA for the algorithm previously discussed is reported as a data flow diagram (DFD) (Lawrence, 1988) in Fig. 5. Actually the DFD has a hierarchical representation, and each function (circle) of Fig. 5 may be represented by another DFD, and so on until a decomposition in sufficiently simple functions is obtained. Functions which cannot be further decomposed are called "leaves" of the overall DFD, and are described by the process specifications (PS). Functions marked with "," in Fig. 5 should be further decomposed, while those marked with "p" should be described by the PS. Non-filled circles Control functions joined to the right-hand colunm of circles (1, 2, 3, 4, 5) are those which are relevant to the forward path of the current C3G (see Fig. 4). They include the action command interpretation (circle 1), trajectory computation in joint or Cartesian space as time function polynomials (2), trajectory knot-point computation at each sampling time (3), inverse kinematics (4), microinterpolation and motor position servos (5). It is assumed here that the path is introduced directly from the work program; optionally, it might be stored in external files. Functions of the circles in the left leg (12, 13, 14, 15) group together the processing of sensory signals. The output of the motor position sensors is filtered (circle 13) and then transformed into joint positions (15); likewise circle 12 is in charge of the low-level processing of force sensor outputs, and circle 14 of yielding force measures in the sensor reference frame (SRF).