Hardware Reference
In-Depth Information
x
10
x
20
x
30
x
40
(a)
L
L
0
x
1−min
x
2−min
x
3−min
x
4−min
x
1
x
2
x
3
x
4
F
(b)
L
L
d
L
max
L
0
Figure 9.28
Springs with minimum length constraints (a); during compression, spring
S
2
reaches its minimum length and cannot be compressed any further (b).
=
S
i
∈
Γ
f
L
f
x
i
min
(9.36)
1
S
i
∈
Γ
v
K
v
=
.
(9.37)
1
k
i
Whenever there exists some spring for which Equation (9.34) gives
x
i
<x
i
min
, the
length of that spring has to be fixed at its minimum value, sets Γ
f
and Γ
v
must be
updated, and Equations (9.34), (9.35), (9.36) and (9.37) recomputed for the new set
Γ
v
. If there is a feasible solution, that is, if
L
d
≥
L
min
=
i
=1
x
i
min
, the iterative
process ends when each value computed by Equations (9.34) is greater than or equal
to its corresponding minimum
x
i
min
.
COMPRESSING TASKS' UTILIZATIONS
When dealing with a set of elastic tasks, Equations (9.34), (9.35), (9.36) and (9.37)
can be rewritten by substituting all length parameters with the corresponding utiliza-
tion factors, and the rigidity coefficients
k
i
and
K
v
with the corresponding elastic
coefficients
E
i
and
E
v
. Similarly, at each instant, the set Γ can be divided into two
subsets: a set Γ
f
of fixed tasks having minimum utilization, and a set Γ
v
of variable
tasks that can still be compressed. Let
U
i
0
=
C
i
/T
i
0
be the nominal utilization of task
=
i
=1
U
i
0
τ
i
,
U
0
be the sum of the
nominal utilizations of tasks in Γ
v
, and
U
f
be the total utilization factor of tasks in Γ
f
.
Then, to achieve a desired utilization
U
d
<U
0
be the nominal utilization of the task set,
U
v
0
each task has to be compressed up to
Search WWH ::
Custom Search