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Let also U = up +Uo (14) represent a potential elastic energy with U -ipTKpp p-2 U, = 2CdTK,Zd, (16) where Kp = KT and KO = KT are positive definite matrices. In particular, Up expresses the elastic energy of a three-dimensional linear spring of constant stiffness Kp acting on the position displacement p. As for the contribution U,, even though this is expressed in terms of the vector part of a quaternion, it can be seen that it has a clear geometrical meaning. In fact, the following equality can be proven 2ZdTK,Zd = -Tr(RzAoR,) +%(Ao) (17) with KO = A, - Tr(Ao)I, (18) where the term on the right-hand side of (17) is the rotational elastic energy defined in [ 101 associated to a torsional spring of stiffness KO acting as to align frame Rd with R,; such stiffness is assumed to be constant in the desired frame. Force and moment contributions can be derived by consid- ering the associated powers. Taking the time derivative of (1 1) and accounting for (12) and (1 3) yields (19) where are respectively the inertial force and moment. Also, takmg the time derivative of (14) and accounting for (15) and (16) yields ir= fgp+pE dT-d w where fE = Kpp pi = 2ET(fj, Zd)KaZd are respectively the elastic force and moment. Suitable dissipative contributions can be added as fD = Dpp d pD -DOG - , where D, = DF and Do = DT are positive definite con- stant matrices characterizing a translational and rotational damping at the end effector, respectively. Therefore, a mechanical impedance at the end effector can be defined in terms of its translational part and its rotational part: MPp + Dpp + Kpp = f (27) (28) .d Ma& + DLGd + KLZd = pd, where DL = Do - MaS(w,d) KL = 2ET(fj, Zd)Ka. (30) (29) are the resulting time-varying rotational damping and stiff- ness matrices. The above energy-based formulation guarantees that the mapping between the vector of end-effector linear (angular) velocity and the vector of contact force (moment) is strictly passive. In fact, let u,=I,+u, (31) ua=I,+ua (32) respectively represent the Hamiltonian contributions asso- ciated to the translational and rotational motion, which are positive definite functions. Taking the time derivative of (3 l), (32) and accounting for (12), (13) and (15), (16) along with (27)-(30) yields -T '&!, = -p Dpp + f 'p '&!, = -WdTDac;rd + pdTGd (33) (34) which implies strict passivity of the mappings f c-) and pd I+ Gd [14]. It can be shown that the impedance in (27) and (28) guaran- tees asymptotic convergence to zero of end-effector position and orientation errors in the case of free motion. If f = 0, from (33) it is $ = 0 and thus it is immediate to see in (27) that p asymptotically tends to 0 from La Salle's theorem. On the other hand, if p = 0, from (34) it is Gd = 0 and thus it can be seen in (28) that Ed asymptotically tends to the invariant set described by (35) It can be proven that the only stable equilibrium is Zd = 0, implying Rd = I [6]. 3. Inverse dynamics control The dynamic model of an n-dof rigid robot manipulator can be written in the well-known form ,U$ = 2BT(fi, Zd)KaZd = 0. B(q)ii+C(q, 4)4+4q, 4)+9(4) = u-JT(q)k (36) where q is the (n x 1) vector of joint variables, B is the (n x n) symmetric positive definite inertia matrix, C4 is the (n x 1) vector of Coriolis and centrifugal torques, d is the (n x 1) vector of friction torques, g is the (n x 1) vector of gravitational torques, U is the (n x 1) vector of driving torques, h = [ f pT IT is the (6 x 1) vector of contact forces exerted by the end effector on the environment, and J is the (6 x n) Jacobian matrix relating joint velocities 4. to the (6 x 1) vector of end-effector velocities v = [ +T wT IT, i.e. which is assumed to be nonsingular. According to &e well-known concept of inverse dynamics, the driving torques are chosen as 'U = J(q)O, (37) U = B(s)Jt(n)(. - 4q, $4) + C(q, 414 + &, 4) + S(Q) + JT(q)h, (38) where Jt is the right pseudoinverse of J, ^d denotes the available estimate of the friction torques, and h is the mea- sured contact force. Notice that it is reasonable to as- sume accurate compensation of the terms in the dynamic model (36), e.g. as obtained by a parameter identification technique, except for the friction torques. Substituting the control law (38) in (36) and accounting for the time derivative of (37) gives ir=a-?J (39) that is a resolved end-effector acceleration for which the term (40) can be regarded as a disturbance. In the case of mismatching on other terms in the dynamic model (36), such a distur- bance would include additional contributions. In order to match the desired impedance behavior es- tablished by (27) and (28), t