145
z= Zm GzL/,Gy2 + GZ2 (4.52)
The simultaneous solution of Eqs. (4.50) (4.52) and
(4.45) represents the equation of the external boundary of
the CRSW of the first subchain on the plane, correlating Ym
and Zm expressed in the new system Oxyz.
Finally we can use the inverse transformation A-1
x X
y Y
= A-1
z Z
1 1
and transform the coordinates of the point on the boundary
defined by Eq. (4.50) (4.52) combined with Eq. (4.45) to
be expressed with respect to the global system OXYZ.
That is the method of transforming the SWIP in the
plane to be expressed with respect to a new reference where
the given plane is perpendicular to one of the axes. Then
in the new reference the gradient of the boundary in the
given plane and the coordinates of the point on the CRSW or
NRSS are respectively calculated. Finally, the results are
transformed back to be expressed with respect to the global
system using inverse transformation. This procedure will be
taken for the three subworkspaces with complete rotatability
or nonrotatability of the platform, and the corresponding
workspace of the manipulator is obtained as the common
region of these three subworkspaces.