Graphics Reference
In-Depth Information
Let R
i
(
z
i
,q
i
) denote the rotation about the
z
i
axis of the frame F
i
through an angle q
i
.
Its homogeneous matrix is
cos
q
sin
q
00
00
Ê
ˆ
i
i
Á
Á
Á
˜
˜
˜
-
sin
q
cos
q
i
i
0
0
1
0
Ë
¯
0
0
0
1
Let T
i-1
(
x
i-1
,a
i-1
) denote the translation with translation vector a
i-1
x
i-1
. Its homoge-
neous matrix is
1 000
0100
0010
001
Ê
ˆ
Á
Á
Á
˜
˜
˜
Ë
¯
a
i-
1
Finally, let R
i-1
(
x
i-1
,a
i-1
) denote the rotation about the
x
i-1
axis of the frame F
i-1
through an angle a
i-1
. Its homogeneous matrix is
1
0
0
0
Ê
ˆ
Á
Á
Á
˜
˜
˜
0
cos
a
sin
a
0
i
-
1
i
-
1
0
-
sin
a
cos
a
0
i
-
1
i
-
1
Ë
¯
0
0
0
1
Then
(
)
(
)
(
)
(
)
dT
=
R
x
,
a
o
T
x
,
a
o
R
z
,
q
o
T
z
,
d
,
(4.22)
i
i
-
1
i
-
1
i
-
1
i
-
1
i
-
1
i
-
1
i
i
i
i
i
i
and multiplying the matrices for the corresponding maps together (but in reverse
order since matrices act on the right of a point), we get that the matrix dM
i
associ-
ated to the transformation dT
i
is defined by
cos
q
sin
q
cos
a
sin
q
sin
a
0
0
Ê
ˆ
i
i
i
-
1
i
i
-
1
Á
Á
Á
˜
˜
˜
-
sin
q
cos
q
cos
a
cos
q
sin
a
i
i
i
-
1
i
i
-
1
dM
=
(4.23)
i
0
-
sin
a
cos
a
0
1
i
-
1
i
-
1
Ë
¯
a
-
d
sin
a
d
cos
a
i
-
1
i
i
-
1
i
i
-
1
Equations (4.21-4.23) provide the solution to the forward kinematic problem. In the
two-dimensional robot case where a
i
and d
i
are zero, the matrices dM
i
specialize to
matrices dN
i
, where
cos
q
sin
q
00
00
Ê
ˆ
i
i
Á
Á
Á
˜
˜
˜
-
sin
q
cos
q
i
i
dN
=
(4.24)
i
0
0 1 0
001
Ë
¯
a
i
-
1
