157
Cylindric joint Ai 11i = li(~li, Ai x Sli)
12i = v2i(0, li)
Revolute joint Bi !3i = W3i(S3i, Oi x 3i)
Spherical joint Ci 14i = L4i(-, OCi x j)
l5i = "Si(i, OCi x i)
16i = 66i(k, oCi x k)
where Sli = ai and 33i = Obi-
Since we deal with instantaneous kinematics, the
spherical joint Ci may be considered as a combination of
three revolute joints whose concurrent axes are parallel to
the axes of the OXYZ system.
Mohamed and Duffy [46] used orthogonal screws to study
the instantaneous kinematics of the end-effector platform of
fully parallel robot type devices. They demonstrated that
for any parallel device the twist representing the
instantaneous motion of the end-effector platform is equal
to the sum of its partial twists. A partial twist is
defined as the twist representing the instantaneous motion
of the end-effector when all actuators other than the input
actuator are locked.
We can find the reciprocal screws, r$1i and r'2i,
reciprocal to all other screws in subchain i but 1li and
12i, respectively. As shown in Fig. 5.4, the reciprocal
screw r$1i passes through the center of spherical joint Ci,
intersects the axis of revolute joint Bi and also lies on
the plane P which is perpendicular to the axis of cylindric
joint Ai. In Fig. 5.5, the reciprocal screw r$2i passes