IOP PUBLISHING
SMART MATERIALS AND STRUCTURES
Smart Mater. Struct. 18 (2009) 065012 (8pp)
doi:10.1088/0964-1726/18/6/065012
Shape morphing hinged truss structures
A Y N Sofla1 , D M Elzey2 and H N G Wadley2
1
Department of Mechanical and Industrial Engineering, University of Toronto, Toronto,
Canada
2
Materials Science and Engineering Department, University of Virginia, Charlottesville,
VA, USA
E-mail: [email protected]
Received 25 November 2008, in final form 26 February 2009
Published 6 May 2009
Online at stacks.iop.org/SMS/18/065012
Abstract
Truss structures are widely used for the support of structural loads in applications where
minimum mass solutions are required. Their nodes are normally constructed to resist rotation to
maximize their stiffness under load. A multi-link node concept has recently been proposed that
permits independent rotation of tetrahedral trusses linked by such a joint. High authority shape
morphing truss structures can therefore be designed by the installation of linear displacement
actuators within the truss mechanisms. Examples of actuated structures with either linear or
planar shapes are presented and their ability to bend, twist and undulate is demonstrated. An
experimental device has been constructed using one-way shape memory wire actuators in
antagonistic configurations that permit reversible actuated structures. It is shown that the
actuated structure displacement response is significantly amplified by use of a mechanically
magnified design.
(Some figures in this article are in colour only in the electronic version)
deployable actuated structure [8]. Although VGT structures
were originally conceived as the longitudinal repetition of
an octahedral truss module [9], tetrahedral truss units were
also used in later designs [10]. The application of VGTs in
robotic manipulator arm is arguably the first shape morphing
truss structure [11]. Such a shape morphing truss structure is
different from a one time deployable structure that is unfolded
from its initial (packed) form to a final end state configuration.
Shape morphing structures are required to reversibly change
shape on demand upon application of a suitable stimulus.
Several groups have explored the development of shape
morphing truss structures. Haftka and Adelman [12] and
Matunaga and Onada [13] showed that high precision control
of parabolic tetrahedral truss structures could be achieved by
replacing some of the truss members with actuators. These
actuated trusses were able to precisely compensate structural
changes resulting from launch loads or during operations in
a space environment. They identified optimal placements
for the actuators. Salama et al used the combination of
lead screws and piezo-actuators to experimentally demonstrate
shape control of an erectable, doubly curved tetrahedral truss
structure [14]. More recently, Gullapalli et al proposed an
alternate approach to achieve optical quality space mirrors by
using piezoelectric inchworm micro-actuators [15].
1. Introduction
An ideal space truss is defined as a three-dimensional system
of bars connected at their nodes by frictionless hinges or
joints which is subjected to forces applied only at the joint
centers [1]. The conventional fixed shape space trusses
consisted of tetrahedral truss units, which provide high
stiffness and strength to weight. They can be designed as
doubly curved structural systems such as the roof of the Eden
Project’s structure [2] and Buckminster Fuller’s geodesic dome
in Montreal [3]. The high specific stiffness of space trusses
also makes them well suitable for large space structures [4],
where the high cost of orbital insertion drives the design of
mass efficient concepts [5].
Shape morphing structures can be fabricated from truss
structures by replacing some of the trusses with linear
displacement actuators. Such actuated trusses, also known
as adaptable trusses, have been proposed for deployable
truss structures which are transported in a tightly packed
form and then deployed [6]. Onoda et al proposed a twodimensionally deployable ‘hexapod’ truss structure, which
could be replicated in two dimensions to create a parabolic
truss structure [7]. Variable geometry truss (VGT) structures
have also been proposed by Miura and Furuya as a type of
0964-1726/09/065012+08$30.00
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Smart Mater. Struct. 18 (2009) 065012
A Y N Sofla et al
The doubly curved truss structures proposed for antenna
or mirror support are statically indeterminate. As a result any
length change of a truss member results in the development
of states of self-stress within the structure. Actuators in such
structures therefore do significant work against the structure
leading to sometimes significant inefficiencies. Recently
Hutchinson et al proposed and analyzed a class of planar,
pin-jointed trusses based on a combination of tetrahedral and
planar Kagome (basket weave) truss structures in which some
truss members were replaced with linear actuators [16, 17].
These statically determinate actuated structures promised
higher authority actuation compared with previous designs.
Experimental efforts by dos Santos et al led to Kagome based
truss systems capable of significant bending and twisting of
the structure using linear actuators [18]. In an alternate
approach to create shape changing truss structure for shape
morphing of aircraft wing, Deepak et al deigned a tendon
actuated compliant structure in which actuation by pulling on
one set of tendons while controlling the release of another
creates the desired deformations. Six nodded octahedral unit
cell with diagonal tendon actuation were used for bending the
structure [19].
The shape morphing truss structures described above have
utilized conventional actuators and rigidly connected trusses.
They result in often heavy and sometimes bulky structures
that are difficult to miniaturize. The use of rigid nodes
leads can lead to premature failure (by fatigue) in heavily
loaded systems. Shape memory alloy (SMA) actuators can
be used as an alternative to conventional electromechanical
actuation. They exhibit a large recovery stress (several MPa)
and relatively large strain recovery (up to 7%) [20] when
heated above a critical transformation temperature. Dunlop
and Garcia used SMA actuators to deform Stewart platform
type structures [21]. The platform links were replaced by bowlike components with SMA wires as the bowstring. Heating
the SMA caused contraction of the wire and the elastic curved
components to retract. Several of these Stewart platforms could
then be stack to create a shape changing truss arm [21].
Here we describe an approach to convert statically
determinate trusses to shape morphing truss structures by the
replacement of the struts with linear displacement actuators.
The shape morphing truss structure is capable of bending,
twisting and undulating deformations. The structure consists
of tetrahedral truss unit cells, which are connected using a
spherical freely rotating joint [22]. The joint provides a
means for connecting several struts at a node while ensuring
sufficient rotational freedom. Both linear (beam) and planar
(plate) designs are illustrated. A design for infinitely large
planar mechanisms is developed for potential use as shape
morphing surface. At the end, the application of one-way
shape memory alloys as linear actuators is discussed. To
create reversible actuation the SMA actuators are arranged
in an antagonistic manner. A test structure which combines
mechanical amplification with reversible antagonistic actuation
has been constructed to validate the approach. The rotation
angles of a NiTi actuated truss unit cell are predicted
and compared with the measured responses of the test
structure. The shape morphing trusses can be replaced as
Figure 1. A 1-DOF hinged tetrahedral truss (HTT) is a mechanism
consisting of two tetrahedral trusses with one shared truss
member (a). A high authority morphing hinged truss (MHT) can be
formed by the connection of the top of pyramids in a HTT with a
linear displacement actuator (b).
the reconfigurable backbone structures of existing conventional
structures to convert them to adaptive structures. Examples
include, but are not limited to, buildings and infrastructure,
vehicles and aerospace structures.
2. Design and analysis
2.1. Basic concepts
A simple one degree of freedom mechanism, referred to here
as a hinged tetrahedral truss (HTT) structure can be created
by a combination of two pin-jointed tetrahedral truss units,
figure 1(a). It can be seen that the triangular bases of the two
tetrahedral truss pyramids share a truss member about which
they are free to rotate by means of rotational joints (nodes).
The HTT allows a pair of tetrahedral trusses to freely rotate
about their shared truss.
Maxwell’s stability criterion [23] enables determination of
the number of inextensional mechanisms, M , of such a pinjointed structure in terms of the number of non-foundation
joints ( j ) and non-foundation truss members (b ). The threedimensional form of the criterion applicable to the tetrahedral
system shown in figure 1(a) can be written:
M = 3 j − b.
(1)
For the HTT structure shown in figure 1, b = 8 and j = 3
(one of the base triangles consisting of three nodes and three
bars is the foundation). For the design shown in figure 1(a),
M = 1 which denotes a kinematically indeterminate structure
(or mechanism) with one degree of freedom.
Conventional space truss structures are usually connected
with rigid nodes, which restrict rotations of the truss members
at the nodes. They do not satisfy the pin-joint requirements
of a statically determinate structure. A hexa-pivotal joint
(HP joint) design was earlier developed as a solution to this
issue [22]. The use of HP joints as the revolute nodes of the
HTT, figure 1(a), would allow rotation of the tetrahedral trusses
about their common truss member, and allows the geometry to
act as a mechanism prior to the attachment of linear actuators.
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A Y N Sofla et al
Figure 3. A planar morphing hinged truss (PMHT) applies three
linear actuators and a closed chain hexa-pivotal joint.
same amount, results in pure bending of the truss upward,
figure 2(b), while the alternating contraction and extension of
the actuators results in the pure twisting of this linear morphing
hinged truss (LMHT) structure, figure 2(c). Such linear truss
actuators could be used as a space truss manipulator, a robotic
arm and other applications.
The basic HHT (figure 1) can also be repeated in two
dimensions to create a planar truss mechanism, figure 3. This
hexagonal structure has three degree of freedom (b = 27, j =
10, M = 3 from equation (1)). Any arbitrary three adjacent
pairs of pyramid tops (the top node of the tetrahedral) can
therefore be connected by linear actuators (dashed lines) to
create a planar morphing hinged truss (PMHT). However,
the use of more than three actuators results in the states of
self-stress upon the extension/contraction of the actuators and
significantly reduces the device authority. On the other hand,
using two or less actuators leads to kinematically indeterminate
structures (mechanisms). A closed chain hexa-pivotal joint
is used at the center of the device, which connects 12 truss
members and provides rotational freedom at central node. It
has been shown that the closed chain HP joint posses three
degree of freedom and is therefore suitable for converting a
3-DOF hexagonal structure to a PMHT [22].
Several hexagonal structures can be assembled in planar
patterns to create larger planar truss mechanisms, figure 4.
Each tetrahedral in figure 4 is represented by its pyramid base
(the out of plane truss members of the tetrahedral are not
shown for clarity). Two patterns, to create a plate-like structure
are proposed in figure 4. In the ‘a-pattern’, each internal
hexagon is surrounded by six other hexagons. The smallest
‘a-pattern’ structure is therefore a seven hexagon HHT. We
refer to such a mechanism as an ‘a-pattern’ mechanism with
k = 1 (because the central hexagon is surrounded by only
one ring of hexagons). Such truss mechanism posses 15
DOF (equation (1), b = 195, j = 70, M = 15). Larger
truss mechanisms can be created by the addition of extra
rings of hexagons. In general, the DOF of an ‘a-pattern’
truss mechanism can be determined from Maxwell’s stability
Figure 2. (a) A linear morphing hinged truss (LMHT), (b) LMHT
undergoes bending by the contraction of all the actuators, (c) LMHT
undergoes twisting by the alternate contraction and extension of the
actuators.
The hexa-pivotal joint consists of spherical shell elements
(links), which can rotate with respect to each other without
interference. A pivot pin passes through the common center
of curvature of each couple of neighboring links. To ensure
the links remain in sliding contact, they are fabricated from
spherical shells where the outer radius of the smaller sphere is
equal to the inner radius of the larger one, figure 1(a).
Connecting the pyramid tops of the single degree of
freedom HTT in figure 1(a) with a strut, results in a statically
determinate truss (b = 9, j = 3 therefore M = 0 from
equation (1)), which means if an external load is applied
to the structure, the force in every truss member can be
determined from the equations of mechanical equilibrium at
the nodes. Such a structure is also kinematically determinate
since the location of the joints can be uniquely determined.
If the pyramid tops are connected with a linear displacement
actuator, figure 1(b), the resulting statically and kinematically
determinate truss structure can exhibit high authority shape
morphing. Such actuated truss structures are called morphing
hinged trusses (MHT) here.
The basic HHT in figure 1(a) can be linearly replicated to
create linear truss mechanisms. In figure 2(a) eight tetrahedral
are hinged at their bases. For this case, b = 38, j = 15,
and therefore M = 7 (from equation (1)). Seven linear
actuators (denoted by dashed lines in the picture) sequentially
connect the pyramid tops to form an actuated truss beam.
The truss beam is capable of bending and twisting as well
as a combination of both those deformations. For instance,
simultaneous contraction of all the linear actuators by the
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Figure 5. A planar morphing hinged truss (PMHT) consisting of 18
actuators and 36 tetrahedra.
It is apparent that for all nonzero k , the average number of
DOF’s per tetrahedral for the b-pattern is larger than for the
a-pattern. The parabolic increase in the DOF for the b-pattern
also indicate that if an infinite plane is filled with the ‘b-pattern’
(k → ∞) then the average DOF per tetrahedral approaches
0.25 or equivalently 1.5 DOF per hexagon. Therefore an
infinitely large ‘b-pattern’ truss mechanism can be converted
to a statically determinate actuated truss by using an actuator
density of three actuators per two hexagons. Alternatively if an
infinitely large plane is filled with the ‘a-pattern’ (k → ∞),
then the average DOF per tetrahedral, approaches zero that
means the structure is too rigid.
The large total number of degrees of freedom for the
‘b-pattern’ suggests that the pattern can be used to design
precision shape controlled actuated truss structures such as
antennas. A first order (k = 1) ‘b-pattern’ PMHT, is shown
in figure 5. The actuators in the figure are represented with
the dash lines. There are a total 18 actuators in the PMHT
in figure 5, because the DOF for the corresponding truss
mechanism is 18, equation (3). The actuators are assembled
to connect the pyramid tops of 18 pairs of adjacent tetrahedra.
The 18 locations can be any arbitrary selection of the total
42 distinct available locations provided no more than three
actuators are placed in each hexagon. This restriction is
enforced because the closed chain hexa-pivotal joint at the
center of the hexagons has three degrees of freedom and
therefore using more than three actuators will result in the local
states of self-stress [22]. The location of the actuators can
be selected to fit the targeted shapes for the shape morphing
structure.
Figure 6(a) demonstrates the unfolding of the first order
PMHT to a flat shape and then its deformation to an undulating
shape, figure 6(b). The tetrahedral in the figure are represented
by colored triangles for visibility and the actuators are not
shown. The sequence can depict unfolding a deployable space
truss. Figure 7 shows a second order ‘b-pattern’, k = 2, with
60 DOF which deforms from a flat structure to a cup shape.
Application oriented arbitrary shapes can be achieved by the
control of the actuators displacement and location.
Figure 4. Large aspect ratio PMHT can be designed by the assembly
of several hexagons with different patterns. The b-pattern provides
greater total degrees of freedom.
criterion and is given by:
DOF = 12k + 3.
(2)
The DOF of the structure identify the total number of actuators,
which must be assembled to convert the mechanism to a
statically determinate planar morphing tetrahedral system of
trusses.
A second, ‘b-pattern’ arrangement, figure 4, can be created
by leaving hexagonal holes in the ‘a-pattern’ mechanism to
achieve a larger DOF per hexagon. The total DOF of the
‘b-pattern’ can again be determined from Maxwell’s stability
criterion;
DOF = 9k 2 + 15k − 6.
(3)
The truss order (k ) is shown for both the ‘a’- and ‘b’-patterns
in figure 4.
Comparison of the average degrees of freedom per
tetrahedral (equivalently the base triangles in figure 4) for the
two patterns, equations (4) and (5), reveals that the b-pattern
has a larger total number of degrees of freedom;
DOF per tetrahedral for pattern (a): =
12k + 3
(4)
6 (3 k 2 + 3 k + 1 )
DOF per tetrahedral for pattern (b): =
9k 2 + 15k − 6
. (5)
36k 2
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A Y N Sofla et al
Figure 8. One-way shape memory alloy actuators can connect the
pyramid tops of a hinged bi-pyramid to create a reversible truss. The
contraction of actuator 1 by heating rotates the cell about a revolute
joint by an angle θ and causes the extension of actuator 2, and
vice versa.
linear displacement [20]. However, a biasing force is needed
to restrain the contracted actuator and therefore complete
a reversible actuation cycle. We have previously used an
antagonistic approach [24] to create the biasing force, and
successfully designed and fabricated several actuated beams
and devices [25]. Here we exploit the implementation of the
method in the truss mechanisms.
The antagonistic approach requires that the basic HTT,
figure 1, be modified to accommodate a pair of countering
SMA actuators.
Hence, mirror tetrahedral trusses are
assembled on the opposite sides of the original pyramid base,
creating hinged bi-pyramids, figure 8. The strain actuation
Figure 6. The shape morphing of a ‘b-pattern’ PMHT with 18
actuators (not shown). (a) Unrolling to a planar shape.
(b) Undulating from a flat shape.
2.2. Antagonistic shape memory actuation
One-way shape memory alloy (SMA) wire and ribbon made
from equi-atomic NiTi alloys can be used as the linear
displacement actuator in the design of MHT’s. The resistive
heating can be accomplished by passing electric current
through a strained one-way actuator to create contractile
Figure 7. The undulating deformation of a second order b-pattern PMHT, k = 2 with 60 actuators (not shown).
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Figure 9. A shape memory actuated antagonistic hinged bi-pyramid.
shown in figure 9 were made. Near equi-atomic NiTi wires
were used in this study to connect the pyramid tops, figure 9.
Several samples, with different pre-strains were fabricated.
The NiTi wire diameter was 0.4 mm. The truss members
were selected to have high plastic buckling resistance and were
fabricated from stainless steel precision tubes with outside
diameters of 1.47 mm and thickness of 0.2 mm. The H–P joint
links were 1.0 mm thick with the dimension of 10 mm × 10 mm
and 12 mm × 10 mm (the outer links are wider), figure 1(b).
The H–P links were machined from stainless steel. The NiTi
wire was passed through the tubes and connected at the H–
P joint to mechanically magnify the recovered shape memory
strain of the NiTi wires. Parts of the NiTi wires which
pass through the tubes were electrically insulated by Teflon
sleeving. Mechanical fasteners were used to firmly connect the
NiTi wires to the truss tubes at the H–P joints. The rotation of
the bi-pyramids about the shared truss member, figure 8, was
measured for different shape memory pre-strain by actuated
(deformed) bi-pyramid’s picture. The camera was placed along
the shared strut of the bi-pyramid to accurately record the
rotation angle. The rotation angles were later measured from
the photos.
A first order ‘b-pattern’ planar morphing hinged truss
(PMHT) is also designed and fabricated. The prototype
unit cells shown in figure 8 are used in the design. The
PMHT includes 18 pairs of actuators and total 36 bi-pyramids.
Several different shapes are attained for the PMHT by actuating
different sets of actuators.
capability is then imparted to the antagonistic flexural unit
cell (AFC) [24] by mechanically stretching at least one of the
two opposing actuators connecting the bi-pyramid tops. The
initial pre-straining should take place at low temperature when
the SMA is in its martensitic or R-phase state [25] prior to
assembly to the cell. This pre-strain in actuator 1 in figure 8
is denoted ε1s . That in actuator 2 is ε2s . Upon heating the prestrained actuator, it will begin to contract and cause rotation of
the bi-pyramids about their shared pivoting truss. This rotation
requires extension (straining) of actuator 2 and results in a
remnant strain, ε , in actuator 1. Given a pre-strain value, and
cell geometry, the cell rotation angle, θ , can be calculated and
is given by [24]:
2(1 + ε − ε1s )2
−1
−1
θ = 2 tan (L/H ) − cos 1 −
(6)
1 + (H /L)2
where, L is the SMA actuator length in the initial configuration
(prior to the activation of the assembled cell), H is the distance
between the tops of each bi-pyramid, figure 8. The strain,
ε, refers to the strain in actuator 1 measured with respect to
the original length (prior to the pre-straining) of the actuator.
Note that, for relatively small rotation angles (smaller than ten
degrees), the strain developed in actuator 2, can be related to
the strain in actuator 1 [25];
Strain in actuator 2 = (ε1s + ε2s ) − ε.
(7)
If only one of the actuators, say actuator 1, is pre-strained then
equation (7) becomes;
Strain in actuator 2 = ε1s − ε.
4. Results and discussion
(8)
Heating the top NiTi wire, figures 8 and 9, above its martensite
finish temperature, Af , results in the upward rotation of the cell
to an active equilibrium position [25]. Upon cooling below the
martensite start temperature, Ms , the top NiTi wire undergoes
a martensitic transformation which results in the extension of
3. Experimental assessments
To determine the relationship between the rotation angle of the
bi-pyramids to the shape memory pre-strain, the test structures
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A Y N Sofla et al
also observed in this study that the shape recovery of NiTi
wires is similar to the ribbons and therefore the data can be
used to estimate the equilibrium rotation angles of the truss
unit cell. The mechanical amplification method requires the
SMA wires be attached to the HP joint rather than the pyramid
tops. The recovered strain, ε , must therefore be multiplied by
the amplification factor which is defined by the ratio of the
length of the wire divided by the distance between the pyramid
tops. For the prototype, figure 8, this amplification factor was
determined to be 3.2 (80/25 mm).
The active and inactive rotation angle as a function of prestrain have been determined [25] and are plotted in figure 10. In
addition, the responses of prototypes fabricated from actuators
with 2.5%, 4%, 5%, 6%, and 7% pre-strains are marked in
the figure for comparison. The figure shows that there is good
agreement between the measured and predicted deformations
for smaller pre-strains. However the measured ranges are
smaller than the predicted values at the larger pre-strains
probably due to the frictional effects, which have been ignored
in the analysis. The mechanical amplification method used in
this design has made large rotation angle of over 10◦ feasible.
The promising deformations achieved in this design from
relatively small strain recovery of restrained SMA actuators
(less than 4%) makes the method suitable for high authority
shape morphing applications.
The ‘b-pattern structure has also been investigated.
Figure 11 shows the symmetric deformations of a first order
‘b-pattern’ PMHT. The bending of the device is shown in
figure 11(a), the twisting in figure 11(b) and undulating in
figure 11(c). Local deformation resulted from the weight of
the truss is visible at the attaching point of the truss to the vice.
Figure 10. Rotation angle of the actuated bi-pyramids at mechanical
equilibrium.
the wire due to the de-twinning and causes the downward
rotating relaxation of the unit cell. At the mechanical
equilibrium (inactive equilibrium) further displacement ceases.
The new rotation angle of the cell remains unchanged without
requiring additional power and is therefore suitable for long
term applications. The heating of the bottom actuator above
Af later in the cycle results in the contraction of the heated
wire (shape recovery) to a second active equilibrium state
resulting in a further downward rotation. The difference of
the two active rotation angles is the active rotation range of
the cell and is the maximum deformation that the unit cell
can achieve. Cooling the bottom SMA wire in turn relaxes
the unit cell and the bi-pyramids slightly rotate upward to a
new inactive configuration. The inactive rotation range is the
maximum attainable deformation range for the unit cell at low
temperature. The rotation angle depends on the shape memory
pre-strain in the top wire, or actuator 1, ε1s , the cell geometry
and the recovered strain, ε , equation (6).
The recovered strain of the antagonistic actuators as a
function of shape memory pre-strain for ribbons of identical
NiTi alloy has been measured and analyzed [25], and it was
5. Conclusion
Hinged truss mechanisms can be designed by using a novel
spherical–pivotal joint (H–P joint). Several truss members
can be connected at the truss nodes and freely rotate about
revolute truss members of the hinged trusses. The assembly of
a number of actuators equivalent to the total degree of freedom
of the hinged truss mechanism can create high authority planar
morphing hinged trusses (PMHT) with possible applications
Figure 11. Deformations of a prototype PMHT. (a) Bending, (b) twisting and (c) undulating.
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A Y N Sofla et al
such as antenna supports or shape changing constructions.
One-way shape memory wires can also be used to actuate
the hinged trusses. An antagonistic hinged truss mechanism
is designed to allow the shape memory alloy actuation of the
PMHTs. Using SMA wires longer than the original intended
actuator length has resulted in the significant amplification
of the response of the PMHT. As a result of the mechanical
amplification, visible rotation range of over 10◦ can be
achieved by a single PMHT unit cell. It was shown that
large planar actuated truss structures, ‘b-pattern’, capable of
bending, twisting and undulating deformation can be created
by linking unlimited number of the unit cells.
[10] Subramaniam M and Kramer S N 1992 The inverse kinematic
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experimental validation Smart Mater. Struct. 2 240–8
[15] Gullapalli S, Flood R, Hyeok E and Lih S S 2003 New
technologies for the actuation and control of large aperture
lightweight optical quality mirrors IEEE Aerosp. Proc. 4
1717–28
[16] Hutchinson Wicks N, Evans A G, Fleck N A and
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[18] dos Santos e Lucato S L, Wang J, McMeeking R M and
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41 3521–43
[19] Ramrakhyani D, Lesieutre G, Bharti S and Frecker M 2005
Aircraft structural morphing using tendon actuated
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[20] Otsuka K and Wayman C M 1998 Shape Memory Materials
(New York: Cambridge University Press)
[21] Dunlop R and Garcia A C 2002 A nitinol wire actuated stewart
platform Proc. 2002 Australian Conf. on Robotics and
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[22] Sofla A Y N, Elzey D M and Wadley H N G 2007 A rotational
joint for shape morphing space trusses
Smart Mater. Struct. J. 16 1277–84
[23] Maxwell J C 1864 On the calculation of the equilibrium and
stiffness of frames Phil. Mag. 27 294
[24] Sofla A Y N, Elzey D M and Wadley H N G 2004 An
antagonistic flexural unit cell for design of shape morphing
structures Proc. IMECE2004 (Anaheim, CA: ASME)
[25] Sofla A Y N, Elzey D M and Wadley H N G 2008 Two-way
antagonistic shape actuation based on the one-way shape
memory effect J. Intell. Mater. Syst. Struct. 19 1017–27
Acknowledgments
The research was supported in part by the Defense Advanced
Research Projects Agency (Leo Christodoulou, program
manager) and the Office of Naval Research (Steve Fishman,
program manager) under grant number N00014-02-1-0614.
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