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In-Depth Information
X
n
j¼
1
ða
ij
f
j
Þ
d
i
ðtÞ¼b
i
þ
(7.74)
€
v
A
þ o
A
ðtÞ
p
A
ðtÞ¼_
r
A
ðtÞþo
A
ðtÞðo
A
ðtÞ
r
A
ðtÞÞ
(7.75)
v
A
(
t
) is the linear acceleration as a result of the total force acting on object
A
divided by its mass. A force
f
j
acting in direction
n
j
(
t
0
) produces
f
j
/
m
A
n
j
(
t
0
). Angular acceleration,
o
A
(
t
), is formed by differentiating
Equation 7.48
and produces
Equation 7.76
, in which the first term
contains torque and therefore relates the force
f
j
to
p
A
(
t
) while the second term is independent of the
force
f
j
and so is incorporated into a constant term,
b
i
.
o
A
ðtÞ¼I
1
In
Equation 7.75
,
A
ðtÞt
A
ðtÞþI
1
ðtÞðL
A
ðtÞo
A
ðtÞÞ
(7.76)
The torque from force
f
j
is (
p
j
x
A
(
t
0
))
f
j
n
j
(
t
0
). The total dependence of
p
A
(
t
)on
f
j
is shown in
p
B
(
t
)on
f
j
can be computed. The results are combined as
N
j
ðt
0
Þ
m
A
þðI
1
f
j
A
ðt
0
Þð
p
j
x
A
ðt
0
Þ
N
j
ðt
0
ÞÞ
r
A
(7.77)
Collecting the terms not dependent on an
f
j
and incorporating a term based on the net external
force,
F
A
(
t
0
), and net external torque,
t
A
(
t
0
) acting on
A
, the part of
p
A
(
t
) independent of the
f
j
is shown
in
Equation 7.78
. A similar expression is generated for
p
B
(
t
). To compute
b
i
in
Equation 7.74
,
the
constant parts of
p
A
(
t
)and
p
B
(
t
) are combined and dotted with
n
i
(
t
0
). To this, the term 2
n
i
(
t
0
)(
p
A
(
t
0
)
p
B
(
t
0
)) is added.
F
A
ðt
0
Þ
m
A
þ I
1
r
A
ÞþðI
1
A
ðtÞt
A
ðtÞþo
A
ðtÞ o
A
ðtÞ
A
ðtÞðL
A
ðtÞo
A
ðtÞÞÞ
r
A
(7.78)
Equation 7.74
must be solved subject to the constraints in
Equations 7.72 and 7.73
.
These systems
of equations are solved by
quadratic programming
. It is nontrivial to implement. Baraff [
1
]
uses a pack-
age from Stanford [
7
]
[
8
]
[
9
][
10
]. Baraff notes that the quadratic programming code can easily handle
d
i
(
t
)
0 instead of
d
i
(
t
)
0 in order to constrain two bodies to never separate. This enables the model-
ing of hinges and pin-joints.
¼
7.4.3
Dynamics of linked hierarchies
Applying forces to a linked figure, such as when falling under the force of gravity or taking a blow to the
chest from a heavy object, results in complex reactions as the various links pull and push on each other.
There are various ways to deal with the resulting motion depending on the quality of motion desired and
the amount of computation tolerated. We consider two approaches here: constrained dynamics and the
Featherstone equations.
Constrained dynamics
Geometric point constraints can be used as an alternative to rigid body dynamics [
18
]
. In a simple case,
particles are connected together with distance constraints. A mass particle (e.g., representing the torso
of a figure) reacts to an applied force resulting in a spatial displacement. The hierarchy is traversed
starting at the initially displaced particle. During traversal, each particle is repositioned in order to
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