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)
The force vector ( Eq. 7.81 ) and mass matrix ( Eq. 7.82 ) are defined so that f ¼ Ma holds in the
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
of that axis with the displacement vector from the axis to the center of the frame ( Eq. 7.83 ) .
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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