Graphics Reference
In-Depth Information
r
i
Frame
i - 1
d
i
Frame
i
u
i
FIGURE 7.29
Vectors relating one coordinate frame to the next:
u
i
is the axis of revolution associated with
Frame
i
,
r
i
is the
displacement vector from the center of Frame
i1
to the center of Frame
i
; d
i
is the displacement vector from the
axis of revolution to the center of
Frame
i
.
a
a
^
¼
(7.80)
spatial notation.
f
t
f
¼
(7.81)
0
I
0
M ¼
(7.82)
The spatial axis is given by combining the axis of rotation of a revolute joint and the cross-product
u
i
u
i
d
i
S
i
¼
(7.83)
Using
Equation 7.84
to represent the matrix cross-product operator, if
r
is the translation vector and
R
is the rotation matrix between coordinate frames, then
Equation 7.85
transforms a value from frame
F
to frame
G
.
2
3
0
r
z
r
y
4
5
r
¼
r
z
0
r
x
(7.84)
r
y
r
x
0
R
0
r
RR
G
^
F ¼
(7.85)
Armed with this notation, a procedure that implements the Featherstone equations can be written
succinctly [
5
]
[
15
]. The interested reader is encouraged to refer to Mirtich's Ph.D. dissertation [
15
] for a
more complete discussion.
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