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In-Depth Information
Desired
Formation
Tendon
Formation
Control
System
[
r
ij
]
1x4
Cross-
Tensegrity
Kinematic
Model
Vehicle
UV
1
Vehicle
UV
2
Vehicle
UV
3
[
l
b_d
]
1x2
Cross
Tensegrity-
based
Model
[
l
tensegrity
]
1x4
Centralised
Formation
Control
[
ș
BS
]
1x4
Formation's
Orientation Angles
[
ȕ
]
1x2,
[
Į
]
1x2
P
v
Vehicle
UV
4
[
l
b_cur
]
1x2
Fig. 3.
Block Diagram of the cross-tensegrity based centralised formation control
In the demonstration, both of the bars' length were set at 30m in the beginning. Due
to the inherent geometry properties of the cross-tensegrity structure in the formation
control, the bar-string angle,
θ
BS
1
and the vehicles' interspacing distance,
r
21
were ini-
tially at 45 degree and 21.2m, respectively. The
l
b
1
was then dilated from 30m to 55m
at time 120s for a period of 200s. The reference input (
l
b
1
) is taken to be a ramp signal
with 0.3 (
m
/
s
) slope in order to avoid any sudden changes to the formation. Note that
the change of
l
b
1
causes the ratio to reduce from 1 to 0.5. The reduction of the ratio also
reduces the bar-string angle (45 to 29 degree) but increases the distance between the
vehicles from 21.2m to 31.3m.
The
l
b
1
was then contracted from 30m to 10m at time 520s for the same period. This
time the ratio value increased to 3, which also increased the angle of
θ
BS
1
to 71.6 degree
but reduced the interspacing of the
UV
1
and
UV
2
to 15.8m. Hence, a wide variation of
formation shape can be converged by regulating the ratio of bars' length,
l
b
2
/
l
b
1
. Figure
3 depicts the complete setup of the formation control system.
The matrix parameters are defined as [
l
b d
]
1
×
2
=[
l
b
1
d
,
l
b
2
d
]; [
l
tensegrity
]
1
×
4
=[
l
s
1
d
,
l
s
2
d
,
l
s
3
d
,
l
s
4
d
]; [
θ
BS
]
1
×
4
=[
θ
BS
1
,
θ
BS
2
,
θ
BS
3
,
θ
BS
4
]; [
r
ij
]
1
×
4
=[
s
1
,
s
2
,
s
3
,
s
4
]; [
α
]
1
×
2
=
[
β
2
] and [
P
v
]=[
P
v
x
,
P
v
y
,
P
v
z
].Where
l
b
1
d
is desired length of the
bar,
b
1
;
l
s
1
d
is desired length of the string,
s
1
and
l
b cur
is the current length of the bar
during formation changing. Note that the ratio is used to perform the formation's shape
changing while formation's rotation can be achieved by regulating the formation's bear-
ings,
α
1
,
α
2
]; [
β
]
1
×
2
=[
β
1
,
. The formation rotation task will be demonstrate in the Section V. The
extension to obstacle avoidance using the rotation formation technique is possible but
is not cover here.
β
and
α
4.2 Tendon Controller
As mentioned earlier, the tendon force,
f
ij
was applied on vehicle (
i
th
) with respect to
its neighbour vehicle (
j
th
). This force was designed to have a much larger elastic limit
compare to its proportional limit as defined mathematically in Equation 4 in [LN12]:
⎧
⎨
r
ij
l
tensegrity
K
ln
if 0
<
r
ij
≤
l
ultimate
r
ij
−
l
tensegrity
l
break
f
ij
=
(4)
K
exp(
−
) if
l
ultimate
<
r
ij
≤
l
break
⎩
0if
l
LF
>
l
break
Where
K
=
K
1
α
ij
ω
ij
.And
K
1
is a gain parameter that is proportional to the distur-
bance force,
d
f
and adapt the tendon controller to external disturbances,
K
1
∝
d
f
.
ω
ij
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