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3.1
Tensegrity Structure Definition
In the tensegrity structure, let
n
be the number of nodes in a structure given by the
three-dimensional vectors,
n
i
,where
i
= 1
,...,
n
. The position vector of the
i
th
node
in the structure is defined as
n
i
=[
n
ix
,
n
iy
,
n
iz
]. Hence, the configuration of the entire
tensegrity structure can be described as the node vector,
n
=[
n
1
...
n
n
]
T
.Let
m
be the
vector that describes the member (edge) in the structure that connecting any two nodes.
In cross tensegrity structures such as one shown in Figure 1, members are represented
by strings/elastic springs (
s
1
,
s
2
,
s
3
,
s
4
) and bars (
b
1
,
b
2
), hence
m
can be written as:
m
=
m
s
m
b
T
=
s
1
s
2
s
3
s
4
b
1
b
2
T
3
m
∈
ℜ
(2)
The synchronisation of vehicles' positions in the formation requires the consideration
of Equation 3. This kinematic Equation defines the position of four vehicles (repre-
sented by nodes,
n
1
,
n
2
,
n
3
,
n
4
in the tensegrity structure) according to the virtual leader
(represented by the cross point of bars,
O
in the tensegrity structure) in the plane by
referring to the virtual leader's fixed body axes frame (
X
B
,
Y
B
,
Z
B
).
P
1
=
l
b
2
[
cos
(
1
)
,
sin
(
1
)
,
sin
(
β
1
+
ψ
v
)
cos
(
α
β
1
+
ψ
v
)
cos
(
α
α
1
)]+
P
v
P
2
=
l
b
2
[
sin
(
−
β
2
+
ψ
v
)
cos
(
α
2
)
,
cos
(
−
β
2
+
ψ
v
)
cos
(
α
2
)
,
sin
(
α
2
)]+
P
v
(3)
P
3
=
l
b
2
[
−
cos
(
β
1
+
ψ
v
)
cos
(
α
1
)
,
−
sin
(
β
1
+
ψ
v
)
cos
(
α
1
)
,
−
sin
(
α
1
)]+
P
v
P
4
=
l
b
2
[
−
sin
(
−
β
2
+
ψ
v
)
cos
(
α
2
)
,
−
cos
(
−
β
2
+
ψ
v
)
cos
(
α
2
)
,
−
sin
(
α
2
)]+
P
v
Where
v
is the heading angle of the virtual leader while
P
v
is the position of leader in
the plane.
ψ
1
is the angle between the bar,
b
1
to the horizontal body axis of the virtual
leader (
X
B
) while
β
β
2
is the angle between the bar,
b
2
and the virtual leader's vertical
body axis (
Y
B
).
α
2
is
the angle between
X
B
Y
B
-body axes frame to the bar,
b
2
.
l
b
represents the total length of
the bar while
α
1
is the angle between
X
B
Y
B
-body axes frame to bar,
b
1
while
θ
BSi
is the bar-string angle between any given bar and the corresponding
string (
s
i
) in Equation 3 and its use in formation control will be elaborated further in
Section 4 (A).
4
Formation Control Methodology
In this section, a centralised tensegrity-based formation control system is described
which makes use of the tendon forces in formation changing. The use of autopilot in
the formation's manoeuvring task is also explained.
4.1
Formation Controller Design
The formation control method should be flexible so that the shape changes can be effi-
ciently carried out to adopt the changes in the unknown environment. Here, the forma-
tion control is achieved by maintaining the distances (
l
b
/
2) between the nominated pair
of vehicle and the virtual leader by using the concept of cross-tensegrity as depicted in
Figure 1 and Equation 3.
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