18
The Dynamics of the
Class 1 Shell
Tensegrity Structure
18.1 Introduction
18.2 Tensegrity Definitions
A Typical Element • Rules of Closure for the Shell Class
18.3 Dynamics of a Two-Rod Element
18.4 Choice of Independent Variables and
Coordinate Transformations
18.5 Tendon Forces
18.6 Conclusion
Appendix 18.A Proof of Theorem 18.1
Appendix 18.B Algebraic Inversion of the Q Matrix
Appendix 18.C General Case for (n, m) = (i, 1)
Appendix 18.D Example Case (n,m) = (3,1)
Appendix 18.E Nodal Forces
Abstract
A tensegrity structure is a special truss structure in a stable equilibrium with selected members
designated for only tension loading, and the members in tension forming a continuous network of
cables separated by a set of compressive members. This chapter develops an explicit analytical
University of California, Los Angeles
8596Ch18Frame Page 389 Wednesday, November 7, 2001 12:18 AM
© 2002 by CRC Press LLCcontrol discipline followed the classical structure design, where the structure and control disciplines
were ingredients in a multidisciplinary system design, but no interdisciplinary tools were developed
to integrate the design of the structure and the control. Hence, in this modern era, the dynamics of
the structure and control were not cooperating to the fullest extent possible. The post-modern era
of structural systems is identified by attempts to unify the structure and control design for a common
objective.
The ultimate performance capability of many new products and systems cannot be achieved until
mathematical tools exist that can extract the full measure of cooperation possible between the
dynamics of all components (structural components, controls, sensors, actuators, etc.). This requires
new research. Control theory describes how the design of one component (the controller) should
be influenced by the (given) dynamics of all other components. However, in systems design, where
more than one component remains to be designed, there is inadequate theory to suggest how the
dynamics of two or more components should influence each other at the design stage. In the future,
controlled structures will not be conceived merely as multidisciplinary design steps, where a plate,
beam, or shell is first designed, followed by the addition of control actuation. Rather, controlled
structures will be conceived as an interdisciplinary process in which both material architecture and
feedback information architecture will be jointly determined. New paradigms for material and
structure design might be found to help unify the disciplines. Such a search motivates this work.
Preliminary work on the integration of structure and control design appears in Skelton
1,2
and
8,9
This class of structure, with a continuous network of tension members and a discontinuous
network of compression members, will be called a Class 1 tensegrity structure. The important
lessons learned from the tensegrity structure of the spider fiber are that
1. Structural members never reverse their role. The compressive members never take tension
and, of course, tension members never take compression.
2. Compressive members do not touch (there are no joints in the structure).
3. Tensile strength is largely determined by the local topology of tension and compressive
members.
Another example from nature, with important lessons for our new paradigms is the carbon
nanotube often called the Fullerene (or Buckytube), which is a derivative of the Buckyball. Imagine
8596Ch18Frame Page 390 Wednesday, November 7, 2001 12:18 AM
© 2002 by CRC Press LLCa 1-atom thick sheet of a graphene, which has hexagonal holes due to the arrangements of material
at the atomic level (see Figure 18.2). Now imagine that the flat sheet is closed into a tube by
choosing an axis about which the sheet is closed to form a tube. A specific set of rules must define
this closure which takes the sheet to a tube, and the electrical and mechanical properties of the
resulting tube depend on the rules of closure (axis of wrap, relative to the local hexagonal topol-
ogy).
10
Smalley won the Nobel Prize in 1996 for these insights into the Fullerenes. The spider fiber
and the Fullerene provide the motivation to construct manmade materials whose overall mechanical,
thermal, and electrical properties can be predetermined by choosing the local topology and the
FIGURE 18.2
Buckytubes.
amorphous
chain
β-pleated sheet
entanglement
hydrogen bond
y
z
x
6nm
8596Ch18Frame Page 391 Wednesday, November 7, 2001 12:18 AM
© 2002 by CRC Press LLC18.2 Tensegrity Definitions
Kenneth Snelson built the first tensegrity structure in 1948 (Figure 18.3) and Buckminster Fuller
coined the word “tensegrity.” For 50 years tensegrity has existed as an art form with some archi-
tectural appeal, but engineering use has been hampered by the lack of models for the dynamics.
In fact, engineering use of tensegrity was doubted by the inventor himself. Kenneth Snelson in a
letter to R. Motro said, “As I see it, this type of structure, at least in its purest form, is not likely
to prove highly efficient or utilitarian.” This statement might partially explain why no one bothered
to develop math models to convert the art form into engineering practice. We seek to use science
to prove the artist wrong, that his invention is indeed more valuable than the artistic scope that he
ascribed to it. Mathematical models are essential design tools to make engineered products. This
chapter provides a dynamical model of a class of tensegrity structures that is appropriate for space
structures.
(From Connelly, R. and Beck, A.,
American Scientist
, 86(2), 143, 1998. With permissions.)
8596Ch18Frame Page 392 Wednesday, November 7, 2001 12:18 AM
© 2002 by CRC Press LLCDefinition 18.4
A stable structure is said to be a “Class 2” tense grity structure if the members
in tension form a continuous set of members, and there are at most tw o members in compression
connected to each node.
Figure 18.4 illustrates Class 1 and Class 2 tensegrity structures.
Consider the topology of structural members given in Figure 18.5, where thick lines indicate
rigid rods which tak e compressi ve loads and the thin lines represent tendons. This is a Class 1
tense grity structure.
Definition 18.5
Let the topology of Figure 18.5 describe a three-dimensional structure by con-
necting points A to A, B to B, C to C,…, I to I. This constitutes a “Class 1 tense grity shell” if there
exists a set of tensions in all tendons (
α
= 1
t
αβγ
,
8596Ch18Frame Page 393 Wednesday, November 7, 2001 12:18 AM
© 2002 by CRC Press LLC18.2.1 A Typical Element
The axial members in Figure 18.5 illustrate only the pattern of member connections and not the
actual loaded configuration. The purpose of this section is two-fold: (i) to define a typical “element”
which can be repeated to generate all elements, and (ii) to define rules of closure that will generate
a “shell” type of structure.
Consider the members that make the typical
ij
element where
i
= 1, 2, …, n indexes the element
to the left, and
j
= 1, 2, …, m indexes the element up the page in Figure 18.5. We describe the
axial elements by vectors. That is, the vectors describing the
1
ij
,
r
2
ij
, where, within the
ij
element,
t
α
ij
is a vector whose tail is fixed at the specified end of tendon
number
α
, and the head of the vector is fixed at the other end of tendon number
and the vector
r
2
ij
lies along the rod
r
2
ij
. The first goal of this chapter is to derive the equations of motion for the
dynamics of the two rods in the
ij
element. The second goal is to write the dynamics for the entire
system composed of
nm
elements. Figures 18.5 and 18.7 illustrate these closure rules for the case
(
ij
element.
8596Ch18Frame Page 394 Wednesday, November 7, 2001 12:18 AM
© 2002 by CRC Press LLC (18.1)
This closes nodes n
2ij
and n
1(i+1)(j+1)
to a single node, and closes nodes n
3(i–1)j
and n
4i(j–1)
to a single
node (with ball joints). The nodes are closed outside the rod, so that all tension elements are on
the exterior of the tensegrity structure and the rods are in the interior.
m).
Definition 18.6
Consider the shell of Figures 18.4. and 18.5, which may be Class 1 or Class 2
depending on whether constraints (18.1) are applied. In the absence of dotted tendons (labeled t
9
and t
10
), this is called a primal tensegrity shell. When all tendons t
9
, t
10
are present in Figure 18.5,
it is called simply a Class 1 or Class 2 tensegrity shell.
The remainder of this chapter focuses on the general Class 1 shell of Figures 18.5 and 18.6.
18.2.2 Rules of Closure for the Shell Class
Each tendon exerts a positive force away from a node and
α
ij
is the external applied force at node
n
α
ij
,
α
= 1, 2, 3, 4. At
the base, the rules of closure, from Figures 18.5 and 18.6, are
t
9
i
1
= –
(18.3)
t
600
= –
t
2n1
(18.4)
t
901
= t
9n1
= –t
1n1
(18.5)
0 = t
10(i–1)0
= t
5i0
= t
7i0
= t
7(i–1)0,
i = 1, 2, …, n. (18.6)
FIGURE 18.7 Class 1 shell: (n,m) = (3,1).
−+ =
+=
t
10im
= –t
7im
(18.7)
t
100m
= –t
70m
= –t
7nm
(18.8)
t
2i(m+1)
= 0 (18.9)
0 = t
1i(m+1)
= t
9i(m+1)
= t
3(i+1)(m+1)
= t
1(i+1)(m+1)
= t
2(i+1)(m+1)
. (18.10)
At the closure of the circumference (where i = 1):
t
90j
= t
= f
10(i–1)(j–1)
, (18.13)
and for j = m where,
0 = f
1i(m+1)
= f
9i(m+1)
= f
3(i+1)(m+1)
= f
1(i+1)(m+1)
. (18.14)
Nodes n
11j
, n
3nj
, n
41j
for j = 1, 2, …, m are involved in the longitudinal “zipper” that closes the
structure in circumference. The forces at these nodes are written explicitly to illustrate the closure
rules.
In 18.4, rod dynamics will be expressed in terms of sums and differences of the nodal forces,
so the forces acting on each node are presented in the following form, convenient for later use.
The definitions of the matrices B
i
are found in Appendix 18.E.
The forces acting on the nodes can be written in vector form:
f = B
d
w
=
=
=
m
d
o
o
m
o
m
,,, ,
L
L
B
BB
BB
B
BB
BB
B
BB
B
B
B
do
=
6
5
6
4
5
8
12
7
2
7
00
00
00
00
0
0
00
L
OO M
OO
MOO
K
L
OO M
MOOO
MOO
LL
,
8596Ch18Frame Page 396 Wednesday, November 7, 2001 12:18 AM
© 2002 by CRC Press LLC
, e
2
, and e
3
. Similarly, we may represent a force vector as
Matrix notation will be used in most of the development to follow.
f
f
f
f
f
f
f
f
f
f
f
f
w
w
w
w
w
ij
o
ij
ij
d
ij
=
5
1
2
3
4
6
7
8
9
10
1
i
ˆ
f
i
ˆˆ
.fEf
i
i
=
8596Ch18Frame Page 397 Wednesday, November 7, 2001 12:18 AM
© 2002 by CRC Press LLC
We now consider a single rod as shown in Figure 18.8 with nodal forces and applied to
the ends of the rod.
The following theorem will be fundamental to our development.
Theorem 18.1 Given a rigid rod of constant mass m and constant length L, the governing
equations may be described as:
(18.17)
where
The notation denotes the skew symmetric matrix formed from the elements of r:
and the square of this matrix is
The matrix elements r
1
, r
2
, r
3
, q
1,
q
2,
=
+
−
1
2
12
21
f
ff
ff
H =
I
q
~
~
=
+
−
3
3
2
2
2
223
2
2
m
L
L
T
0
0
K
00
0qqI
r
~
r r =
~
=
−
−
−
,
r
~2
=
−−
−−
−−
r r rr rr
rr r r rr
rr rr r r
2
2
3
2
12 13
21 1
2
3
2
23
ij
ij
ij ij
ij ij
ij
ij
ij ij ij
ij
m
L
1
2
1
12
1
2
2
2
21
22
2
2
22 1
2
6
qff
qq q ff
q q q q
q q
..
m
ij ij ij ij ij
ij ij ij ij
ij ij ij
ij
ij
L
2
2
2
6
2
2
˙˙
ˆˆ
(
˙˙
)(
ˆˆ
)
˙
.
˙
.
˙˙
.
,
qff
qq q ff
q q q q
3
31
32
33
4
41
42
43
ij
ij
ij
ij
ij
ij
ij
ij
ij
ij
ij
ij
ij
ij
ij
ij
q
q
q
q
q
q
,,,,
qq q q q
11 2 3 4ij ij
T
ij
T
ij
T
ij
T
T
∆
,,
ij
L
ij
mm
ij ij
=
=
0
0
0
0
˜
,
˜
1
2
12
12
34
34
0
0
,
ˆˆ
ˆˆ
ˆˆ
, t
1ij
, for i = 1, 2, …, n, j = 1, 2, …, m, and i < n when j = 1}.
This section discusses the relationship between the q variables and the string and rod vectors
t
αβγ
and r
βij
. From Figures 18.5 and 18.6, the position vectors from the origin of the reference frame,
E, to the nodal points, p
1ij
, p
2ij
, p
3ij
, and p
4ij
, can be described as follows:
˙˙
,qqHf
ij ij ij ij ij
+=ΩΩ
ΩΩΩΩ
1
1
2
223
2
2
2
,
˙˙
,
qqI
qqI
ΩΩ
ΩΩ
ΩΩ
ij
ij
ij
=
1
2
0
0
,
qq qq q q q=
[]
11 1 12 2 1
T
n
TT
nm
T
T
rnnmnm
nnmnm
BlockDiag
BlockDiag
,..., , ,..., ,..., ,..., ,
,..., , ,..., ,..., ,..., ,
,..., , ,..., ,..., ,...,
.
ΩΩΩΩΩΩΩΩΩΩΩΩ
ˆ
ij
ρρρρ
ij
k
k
i
k
k
i
ik
ik
k
j
ij
k
−
8596Ch18Frame Page 400 Wednesday, November 7, 2001 12:18 AM
© 2002 by CRC Press LLC
(18.23)
We define
(18.24)
Then,
(18.25)
FIGURE 18.9 Choice of independent variables.
p
pr
p
pr
1
21
3
42
ij ij
ij ij ij
ij ij
ij ij ij
=
=+
=
=+
∆∆
∆∆
∆∆
∆∆
ρρ
ρρ
r
r
r
r
ˆ
q
q
q
q
q
II
II
II
=
∆∆
ij
ij
ρρ
ρρ
r
r
1
2
ˆ
8596Ch18Frame Page 401 Wednesday, November 7, 2001 12:18 AM
© 2002 by CRC Press LLC
In shape control, we will later be interested in the p vector to describe all nodal points of the
structure. This relation is
p = Pq P = BlockDiag […,P
1
, …,P
33
33
33
33
1
2
=
−
−
=
−
−
21
311
11
1
11 1
2
5
11
1
2
2
ij
k
k
i
kik
k
j
ik
k
j
k
i
ij
ij ij
ij
k
k
i
kik
ρρ
−
=
r
qr
2
42
ij
ij ij
l
r
r
t
t
ij
ij
ij
ij
ij
jm=
111
211
511
1
111
11
21
51
2=
=
I
IIII
I
B
II
I
IIII
I
=
−
=
−
−−
where the 12nm × 12nm matrix Q is composed of the 12 × 12 matrices A–H as follows:
(18.29)
n × n blocks of 12 × 12 matrices,
12n × 12n matrix,
Q
22
= BlockDiag […,C, …,C],
Q
32
= BlockDiag […,J, …,J],
C
II
I
III
I
D
II
II
=
−
−
00
0000
00
0000
,
E
II
II
F
II I
II I
=
−
−
=
II I
II I
=
=
−
00
11
21 22
21 32 22
21 32 32 22
21 32 32 32 22
00
0
LL
OM
OM
MMMMOO
,
Q
A
DB
DEB
DEEB
00LLL
OM
OM
OM
MMMOO
L
Q
F
DG
DEG
DEE
DEEEEG
21
=
Equation (18.28) provides the relationship between the selected generalized coordinates and an
independent set of the tendon and rod vectors forming l. All remaining tendon vectors may be
written as a linear combination of l. This relation will now be established. The following equations
are written by inspection of Figures 18.5, 18.6, and 18.7 where
(18.30)
and for i = 1, 2, …, n, j = 1, 2, …,m we have
(18.31)
For j = 1 we replace t
2ij
with
For j = m we replace t
6ij
and t
7ij
with
where and i + n = i. Equation (18.31) has the matrix form,
tr
11 1 11 11nnn
=+−ρρρρ
t = + r
t
tt + r = r
t r
2121
3
1
431 1 1
6
11 2
),(
() ()
()
()
()( )
mm
jm
ij ij i j
ij ij ij ij ij
ij
ij
ij i j ij ij
ij ij ij ij
)
ˆ
,( )
ˆ
()
ˆˆ
.
()( )
()
() ()
t
tr rtr
tr
tr
711
81
5
=+−+
=− +
++
++
ˆ
(
ˆ
)
ˆ
(
ˆ
).
() ()
() ()
ρρρρ
ρρρρ
ρρρρρρρρ
0 jnjojnj
∆∆∆∆,
ˆˆ
,
t
II
r
ij
d
ij
=
∆∆
t
t
t
t
t
t
1
3
3
1
2
1
+
−
0000
00 0
00 0
0000
0000
0000
0000
0000
ρρ
ρρ
8596Ch18Frame Page 404 Wednesday, November 7, 2001 12:18 AM
© 2002 by CRC Press LLC
(18.32)
+
−
−−
−
−−
−
II
II
I
II I
II
I
r
r
3
3
33
33
3
33 3
33
3
1
2
000
000
00
00
00 0
0
00
00 0
000 0
000 0
000 0
000 0
3
++
ρρ
ρρ
r
r
1
2
1
3
3
1
∆∆
00 0
00 0
00 0 0
00 0 0
00 0 0
00 0 0
00 0 0
ρρ
ρρ
ˆ
22
11
3
3
3
33
3
33 3
33
3
1
2
−()
ˆ
i
I
I
II
II
I
II I
II
I
r
r
+
−
+
i
i
1
3
3
00 000
0 000
0 000
=
−−
t
t
3
00 0 0
00 0 0
00 0 0
00 0 0
00 0 0
00 0 0
00 0 0
0000
00 0
00 0
0000
0000
0000
0000
00 0
()im
00
000
000
00
00
00 0
0
00
00 0
−
ρρ
ρρ
r
r
I
I
II
II
I
II I
II
I
1
2
1
3
3
3
33
3
33
33
3
ˆ
+
−−
00
00 0 0
00 0
00
+()
.
im1
ρρ
ρρ
r
r
II
I
II
I
q
1
2
1
2
3
1
2
3
ij
ij
00
000
00
000
t
t
t
t
t
t
t
t
t
q
ij
d
ij
ij
=
∆∆
2
3
4
6
7
8
9
10
1
11
1
2
)()
–
–
00 0 0
00 I I
00 I I
00 0 0
00 0 0
00 0 0
00 0 0
00 0 0
33
33
q
ij
∆∆
8596Ch18Frame Page 406 Wednesday, November 7, 2001 12:18 AM
© 2002 by CRC Press LLC
0000
0000
0000
II00
00II
–
––
–
–
q
ij
1
2
+
t
t
t
t
t
t
q
i
d
i
1
2
3
4
6
7
8
9
10
1
1
2
∆∆
00 0 0
00 I I
00 I I
00 0 0
00 0 0
00 0 0
00 0 0
00 0 0
33
33
(()ii−
+
+
−
11 1
1
2
1
2
II00
II 0 0
II 00
00 I I
00I I
II II
II00
00 II
II 00
0000
0000
0
33
33
33
33
33
33 33
33
33
+
−
−
, E
2
, E
3
, E
4
, Ê
4
, , E
5
, equations in
(18.34) are written in the form, where q
01
= q
n1
, q
(n+1)j
= q
ij
,
(18.36)
t
t
t
t
t
t
t
t
t
im
d
im
2
3
4
6
7
8
9
10
1
2
∆∆
00 I I
00 0 0
00 0 0
00 0 0
+
−
−
−−
qq
im i m() ()11
1
2
1
2
00 0 0
00 I I
00 I I
00 0 0
33
33
33
33
−−
−
−
−−
−
+
q
()
.
im1
tI
r
r
II
r
r
tIIqIIq
Eq
11 3
1
2
11
33
1
2
1
+
[]
=−
[]
+
[]
=
,,,
,
,,
ˆ
,
ˆ
,, ,,,,
000 00
00 00
7
11
21
1
6
312
7
312
11 0 1 0 0
312
n
n
n
n
,
,, ,, , , ,
,.
,
E0 0E
q
q
q
EE
0
L
M
E
4
tEq EqEq Eq
tEqEqEqEqEq
341im i m
+
+()
.
8596Ch18Frame Page 408 Wednesday, November 7, 2001 12:18 AM
© 2002 by CRC Press LLC
Now from (18.34) and (18.35), define
to get
(18.37)
and have the same structure as R
11
except E
4
is replaced by , and , respectively.
Equation (18.37) will be needed to express the tendon forces in terms of q. Equations (18.28) and
(18.37) yield the dependent vectors (t
1n1
, t
2
, t
3
, t
4
, t
6
, t
7
, t
9
, t
n
dT
T
=
[]
=
[]
1 1 11 21 1 12 2
11 1 2
,,,, ,, ,
,,,, ,
KKK
K
ll
dnmnmnmdnm
=∈∈∈
+×
Rq R R q R R
(+)
,,,,
()24 3 12 12 24 3
R
R
RR
RRR
RRR
RR
R
RR
0
11 12
21 11 12
21 11 12
21 11
12
21 11
24 12
0
312
11
34 2
234
00
0
0
00
0
LLL
OM
OM
OM
MO O
M OOO
LL
LL
O
ˆ
,,,
ij
R
234
234
4
423
12
5
5
5
5
i
kk
BlockDiag i
+
+
×
=> =>
=
[]
∈=→
=
[]
=−
[]
=
[]
00
00 00
00
if if
,
11
15
1
2
1
2
21 1 1
24 12
0
α
α
ij ij
ij
ij
F=
8596Ch18Frame Page 409 Wednesday, November 7, 2001 12:18 AM
© 2002 by CRC Press LLC
For tensegrity structures with some slack strings, the magnitude of the force F
αij
can be zero, for
taut strings F
αij
> 0. Because tendons cannnot compress, F
αij
cannot be negative. Hence, the
magnitude of the force is
(18.40)
where
(18.41)
where is the rest length of tendon t
αij
before any control is applied, and the control is u
αij
,
the change in the rest length. The control shortens or lengthens the tendon, so u
αij
can be positive
or negative, but . So u
αij
αij
= – K
αij
(q)q + b
αij
(q)u
αij
where
(18.45)
(18.46)
Hence,
Fk L
ij ij ij ijααα α
=−
()
t
k
if L
kifL
ij
ij ij
ij ij ij
αα
αααα
αααααα
∆∆
0
0
,
,
tq
ααααααij ij ij
nm
=∈
×
RR,
312
tq q
ααααααij
T
ij
T
ij
2
= RR
Kq q q K
ααααααααααααααij ij ij
oT
ij
T
ij ij ij
nm
kL() ,=
()
−
f
f
f
f
f
f
K
K
K
K
K
K
K
K
q
ij
d
ij
ij
ij
ij
ij
ij
ij
ij
ij
ij
ij
ij
ij
=−
2
3
4
7
8
9
10
2
3
4
6
7
8
9
10
ij
ij
ij
ij
ij
ij
ij
ij
ij
ij
ij
ij
ij
ij
ij
ij
u
u
ij
ij
ij
ij
u
u
=
=−
+
K
K
K
q
P
P
P
P
u
u
u
1
11
11
21
1
11
11
21
1
11
11
21
1
11
11
21
1
d
n
=−
+
u
11
1
ddd
f
f
f
f
K
K
K
q
P
P
P
u
u
u
Kq P
2
12
22
2
12
22
2
12
22
2
12
=−
+
j
d
j
d
=− +
K
K
K
K
P
P
P
P
d
d
d
m
d
d
d
d
m
d
=
8596Ch18Frame Page 411 Wednesday, November 7, 2001 12:18 AM
© 2002 by CRC Press LLC
f
d
= –K
d
q + P
d
u
d
.
Likewise, for forces (18.48),
(18.50)
f
o
= – K
o
q + P
o
u
o
.
Substituting (18.49) and (18.50) into (18.E.21) yields
f = –(B
d
K
d
+ B
o
K
q
P
P
P
u
u
u
1
11
21
1
11
21
1
11
21
1
11
21
1
o
o
o
n
o
o
o
n
o
+
nj
o
j
o
j
o
nj
o
j
o
j
o
nj
o
j
o
j
o
nj
o
=
1
2
1
2
1
2
1
2
MM O M
fKqPu
j
d
d
d
dd
dd
d
m
d
m
d
m
d
=
,
8596Ch18Frame Page 412 Wednesday, November 7, 2001 12:18 AM
© 2002 by CRC Press LLC
(18.53)
(18.54)
In vector in (18.54), u
1n1
appears twice (for notational convenience u
1n1
appears in and in
. From the rules of closure, t
9i1
= – t
1i1
and t
7im
= – t
10im
, i = 1, 2, …, n, but t
1i1
, t
7im
, t
9i1
, t
10im
all
appear in (18.54). Hence, the rules of closure leave only n(10m – 2) tendons in the structure, but
(18.54) contains 10nm + 1 tendons. To eliminate the redundant variables in (18.54) define =
Tu, where u is the independent set u , and is given by (18.54). We choose
m
o
m
o
=
11 2 2
72 23
73
2
7
00
0
0
00
5
3
5
44
5
74
++
++++
++
d
m
o
M
˜
,
ˆ
ˆ
u
u
u
u
u
u
u
u
u
u
u
u
u
u
u
u
u
u
u
u
u
=
1
2
3
4
1
2
3
4
1
2
3
4
1
2
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