7.6 A three-revolute spherical wrist with an orthogonal architecture, i.e., with neighboring joint axes at right angles, is shown in Fig. 7.10. Assume that the moments of inertia of its three links with respect to O, the point of
(a) (b)
Figure 7.9: Three different pairs of coupled bodies
(c)
concurrency of the three axes, are given by constant diagonal matrices, in link-fixed coordinates, as
14 = diag(J1, J2, J3)
15 = diag(K1, K2, K3)
16 = diag(L1, L2, L3)
while the potential energy of the wrist is
Moreover, the motors produce torques T4 , T5 , and T6 , respectively, whereas the power losses can be accounted for via a dissipation function of the form
where bi and T( , for i= 4, 5, 6, are constants.
(a) Derive an expression for the matrix of generalized inertia of the wrist.
(b) Derive an expression for the term of Coriolis and centrifugal forces.
(c) Derive the dynamical model of the wrist. Hint: The kinetic energy T of a rigid body rotating about a fixed point O with angular velocity w
can be written as T = !wTlow, where Io is the moment-of-inertia
matrix of the body with respect to O.
