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orientation) of the hand to a global coordinate system by
homogeneous transformation matrices. The determination of
joint velocities using screws, presented by Mohamed and
Duffy [46] and Sugimoto [47], is adapted in this chapter to
describe the velocities and small displacements.
With consideration of input errors as differential
motions, we will obtain the corresponding error movement of
the platform, in which the hand is embedded, due to the
inaccuracies of the actuators.
5.2 Position Analysis
A subchain of a six-degree-of-freedom parallel
manipulator, where all the actuators are ground-mounted, is
shown schematically in Fig. 5.1. Let H denote the position
vector of the hand which is embedded on the platform, and
1, m and n denote the unit vectors describing the hand
orientation of the moving system. By using homogeneous
transformation, we can describe the relative position and
orientation between any two coordinate frames. Therefore,
we can have the coordinate transformation matrix Tp as
follow:
1, mx nx Hx
lTp y my ny Hy
Tp = (5.1)
1i mz n, Hz
0 0 0 1