114
FI2 = X2 + [-Y/2 + (13/2)Z + h]2 ri22 > 0 (3.29)
FI3 = X2 + [-Y/2 (.3/2)Z + h]2 rl32 0 (3.30)
where subscripts E and I denote the external and internal
boundary, respectively.
The workspace of the platform of the manipulator with
infinitesimal platform is the common reachable region of the
three subworkspaces. Therefore the workspace of the
manipulator is the region where Eqs. (3.25) (3.30) are
satisfied simultaneously.
The workspace can be expressed graphically by plotting
the cross sections of the workspace on the planes normal to
the axes of the system OXYZ. For the cross sections, Eqs.
(3.25) (3.27) are rewritten respectively as follows:
On plane OYZ, X = constant
Y = (al + bl)2 X2 h (3.31)
Y = (,3)Z + 2h 2 J(a2 + b2)2 -X2 (3.32)
Y = -(f3)Z + 2h 2 (a3 + b32 X2 (3.33)
On plane OZX, Y = constant
X = (al + b1)2 (Y + h)2 (3.34)
X = (a2 + b2 [-Y/2 + (J3/2)Z + h]2 (3.35)
X = J(a3 + b3)2 [-Y/2 (J3/2)Z + h]2 (3.36)
On plane OXY, Z = constant
X = 1 J(a + bl)2 (Y + h)2 (3.37)