LIBRARY ı6ıul 0) - Cranfield University

In the off-line programming module, the referred transformation matrix is
obtained by defining four points, three relative to the robot world co-ordinates frame,
p0, p1, p2 , and one relative to the table co-ordinates frame, po (see Figure 3.13). p0
is used to define the origin of the table co-ordinates frame in robot world co-
ordinates. p1 is used to determine the direction of the X-axis of the table co-ordinates
frame, Xt , in robot world co-ordinates and p2 is used to define the direction of the Y-
axis of the table co-ordinates frame, Yt , also in robot world co-ordinates. po defines
the origin of the workpiece (CAD) co-ordinates frame relative to the table co-
ordinates frame. Equations (3.17) to (3.34) show how the transformation matrix is
obtained.
(p0), = [pORobot pOyobot poRobot]
(p1)T = [PiRobot
r
(3.17)
1 (3.18)
Plyob« p1Robor
(p2)T = [p2zRobot
p2yRobot
p2Robot
(3.19)
(po)t = [porbu potable po ýable (3.20)
where the subscripts x, y and z indicate vector components in the X, Y and Z
directions. The superscript Robot indicates that the co-ordinates are referred to the
robot world co-ordinates frame. The superscript table indicates that the co-ordinates
are referred to the table co-ordinates frame. The superscript T on the vector
representation means transposed.
M table =
Robot -
ýz
m13 POxý«
m21 m22 m23 PO y
'
0001
Robot
x1132 m33 POzabot
(3.21)
The matrix components m
, m21 and m, 3, form a unit vector which gives the
direction of the X-axis of the table co-ordinates frame relative to the robot world coordinates
frame. Their values are obtained as follows:
= P1ROba - PORoba
11p1- poll
plyRobot - p0y°b°t
m21 (Ipl
- poll
79
(3.22)
(3.23)