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 structuresA Y N Sofla1, D M Elzey2 and H N G Wadley2

1 Department of Mechanical and Industrial Engineering, University of Toronto, Toronto,Canada2 Materials Science and Engineering Department, University of Virginia, Charlottesville,VA, USA

E-mail: sofla@mie.utoronto.ca

Received 25 November 2008, in final form 26 February 2009Published 6 May 2009Online at stacks.iop.org/SMS/18/065012

AbstractTruss structures are widely used for the support of structural loads in applications whereminimum mass solutions are required. Their nodes are normally constructed to resist rotation tomaximize their stiffness under load. A multi-link node concept has recently been proposed thatpermits independent rotation of tetrahedral trusses linked by such a joint. High authority shapemorphing truss structures can therefore be designed by the installation of linear displacementactuators within the truss mechanisms. Examples of actuated structures with either linear orplanar shapes are presented and their ability to bend, twist and undulate is demonstrated. Anexperimental device has been constructed using one-way shape memory wire actuators inantagonistic configurations that permit reversible actuated structures. It is shown that theactuated structure displacement response is significantly amplified by use of a mechanicallymagnified design.

(Some figures in this article are in colour only in the electronic version)

1. Introduction

An ideal space truss is defined as a three-dimensional systemof bars connected at their nodes by frictionless hinges orjoints which is subjected to forces applied only at the jointcenters [1]. The conventional fixed shape space trussesconsisted of tetrahedral truss units, which provide highstiffness and strength to weight. They can be designed asdoubly curved structural systems such as the roof of the EdenProject’s structure [2] and Buckminster Fuller’s geodesic domein Montreal [3]. The high specific stiffness of space trussesalso makes them well suitable for large space structures [4],where the high cost of orbital insertion drives the design ofmass efficient concepts [5].

Shape morphing structures can be fabricated from trussstructures by replacing some of the trusses with lineardisplacement actuators. Such actuated trusses, also knownas adaptable trusses, have been proposed for deployabletruss structures which are transported in a tightly packedform and then deployed [6]. Onoda et al proposed a two-dimensionally deployable ‘hexapod’ truss structure, whichcould be replicated in two dimensions to create a parabolictruss structure [7]. Variable geometry truss (VGT) structureshave also been proposed by Miura and Furuya as a type of

deployable actuated structure [8]. Although VGT structureswere originally conceived as the longitudinal repetition ofan octahedral truss module [9], tetrahedral truss units werealso used in later designs [10]. The application of VGTs inrobotic manipulator arm is arguably the first shape morphingtruss structure [11]. Such a shape morphing truss structure isdifferent from a one time deployable structure that is unfoldedfrom its initial (packed) form to a final end state configuration.Shape morphing structures are required to reversibly changeshape on demand upon application of a suitable stimulus.

Several groups have explored the development of shapemorphing truss structures. Haftka and Adelman [12] andMatunaga and Onada [13] showed that high precision controlof parabolic tetrahedral truss structures could be achieved byreplacing some of the truss members with actuators. Theseactuated trusses were able to precisely compensate structuralchanges resulting from launch loads or during operations ina space environment. They identified optimal placementsfor the actuators. Salama et al used the combination oflead screws and piezo-actuators to experimentally demonstrateshape control of an erectable, doubly curved tetrahedral trussstructure [14]. More recently, Gullapalli et al proposed analternate approach to achieve optical quality space mirrors byusing piezoelectric inchworm micro-actuators [15].

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Smart Mater. Struct. 18 (2009) 065012 A Y N Sofla et al

The doubly curved truss structures proposed for antennaor mirror support are statically indeterminate. As a result anylength change of a truss member results in the developmentof states of self-stress within the structure. Actuators in suchstructures therefore do significant work against the structureleading to sometimes significant inefficiencies. RecentlyHutchinson et al proposed and analyzed a class of planar,pin-jointed trusses based on a combination of tetrahedral andplanar Kagome (basket weave) truss structures in which sometruss members were replaced with linear actuators [16, 17].These statically determinate actuated structures promisedhigher authority actuation compared with previous designs.Experimental efforts by dos Santos et al led to Kagome basedtruss systems capable of significant bending and twisting ofthe structure using linear actuators [18]. In an alternateapproach to create shape changing truss structure for shapemorphing of aircraft wing, Deepak et al deigned a tendonactuated compliant structure in which actuation by pulling onone set of tendons while controlling the release of anothercreates the desired deformations. Six nodded octahedral unitcell with diagonal tendon actuation were used for bending thestructure [19].

The shape morphing truss structures described above haveutilized conventional actuators and rigidly connected trusses.They result in often heavy and sometimes bulky structuresthat are difficult to miniaturize. The use of rigid nodesleads can lead to premature failure (by fatigue) in heavilyloaded systems. Shape memory alloy (SMA) actuators canbe used as an alternative to conventional electromechanicalactuation. They exhibit a large recovery stress (several MPa)and relatively large strain recovery (up to 7%) [20] whenheated above a critical transformation temperature. Dunlopand Garcia used SMA actuators to deform Stewart platformtype structures [21]. The platform links were replaced by bow-like components with SMA wires as the bowstring. Heatingthe SMA caused contraction of the wire and the elastic curvedcomponents to retract. Several of these Stewart platforms couldthen be stack to create a shape changing truss arm [21].

Here we describe an approach to convert staticallydeterminate trusses to shape morphing truss structures by thereplacement of the struts with linear displacement actuators.The shape morphing truss structure is capable of bending,twisting and undulating deformations. The structure consistsof tetrahedral truss unit cells, which are connected using aspherical freely rotating joint [22]. The joint provides ameans for connecting several struts at a node while ensuringsufficient rotational freedom. Both linear (beam) and planar(plate) designs are illustrated. A design for infinitely largeplanar mechanisms is developed for potential use as shapemorphing surface. At the end, the application of one-wayshape memory alloys as linear actuators is discussed. Tocreate reversible actuation the SMA actuators are arrangedin an antagonistic manner. A test structure which combinesmechanical amplification with reversible antagonistic actuationhas been constructed to validate the approach. The rotationangles of a NiTi actuated truss unit cell are predictedand compared with the measured responses of the teststructure. The shape morphing trusses can be replaced as

Figure 1. A 1-DOF hinged tetrahedral truss (HTT) is a mechanismconsisting of two tetrahedral trusses with one shared trussmember (a). A high authority morphing hinged truss (MHT) can beformed by the connection of the top of pyramids in a HTT with alinear displacement actuator (b).

the reconfigurable backbone structures of existing conventionalstructures to convert them to adaptive structures. Examplesinclude, 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 hereas a hinged tetrahedral truss (HTT) structure can be createdby a combination of two pin-jointed tetrahedral truss units,figure 1(a). It can be seen that the triangular bases of the twotetrahedral truss pyramids share a truss member about whichthey are free to rotate by means of rotational joints (nodes).The HTT allows a pair of tetrahedral trusses to freely rotateabout their shared truss.

Maxwell’s stability criterion [23] enables determination ofthe number of inextensional mechanisms, M , of such a pin-jointed structure in terms of the number of non-foundationjoints ( j ) and non-foundation truss members (b). The three-dimensional form of the criterion applicable to the tetrahedralsystem 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 threebars 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 connectedwith rigid nodes, which restrict rotations of the truss membersat the nodes. They do not satisfy the pin-joint requirementsof a statically determinate structure. A hexa-pivotal joint(HP joint) design was earlier developed as a solution to thisissue [22]. The use of HP joints as the revolute nodes of theHTT, figure 1(a), would allow rotation of the tetrahedral trussesabout their common truss member, and allows the geometry toact as a mechanism prior to the attachment of linear actuators.

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Figure 2. (a) A linear morphing hinged truss (LMHT), (b) LMHTundergoes bending by the contraction of all the actuators, (c) LMHTundergoes twisting by the alternate contraction and extension of theactuators.

The hexa-pivotal joint consists of spherical shell elements(links), which can rotate with respect to each other withoutinterference. A pivot pin passes through the common centerof curvature of each couple of neighboring links. To ensurethe links remain in sliding contact, they are fabricated fromspherical shells where the outer radius of the smaller sphere isequal to the inner radius of the larger one, figure 1(a).

Connecting the pyramid tops of the single degree offreedom HTT in figure 1(a) with a strut, results in a staticallydeterminate truss (b = 9, j = 3 therefore M = 0 fromequation (1)), which means if an external load is appliedto the structure, the force in every truss member can bedetermined from the equations of mechanical equilibrium atthe nodes. Such a structure is also kinematically determinatesince the location of the joints can be uniquely determined.If the pyramid tops are connected with a linear displacementactuator, figure 1(b), the resulting statically and kinematicallydeterminate truss structure can exhibit high authority shapemorphing. Such actuated truss structures are called morphinghinged trusses (MHT) here.

The basic HHT in figure 1(a) can be linearly replicated tocreate linear truss mechanisms. In figure 2(a) eight tetrahedralare hinged at their bases. For this case, b = 38, j = 15,and therefore M = 7 (from equation (1)). Seven linearactuators (denoted by dashed lines in the picture) sequentiallyconnect the pyramid tops to form an actuated truss beam.The truss beam is capable of bending and twisting as wellas a combination of both those deformations. For instance,simultaneous contraction of all the linear actuators by the

Figure 3. A planar morphing hinged truss (PMHT) applies threelinear 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 ofthe actuators results in the pure twisting of this linear morphinghinged truss (LMHT) structure, figure 2(c). Such linear trussactuators could be used as a space truss manipulator, a roboticarm and other applications.

The basic HHT (figure 1) can also be repeated in twodimensions to create a planar truss mechanism, figure 3. Thishexagonal structure has three degree of freedom (b = 27, j =10, M = 3 from equation (1)). Any arbitrary three adjacentpairs of pyramid tops (the top node of the tetrahedral) cantherefore be connected by linear actuators (dashed lines) tocreate a planar morphing hinged truss (PMHT). However,the use of more than three actuators results in the states ofself-stress upon the extension/contraction of the actuators andsignificantly reduces the device authority. On the other hand,using two or less actuators leads to kinematically indeterminatestructures (mechanisms). A closed chain hexa-pivotal jointis used at the center of the device, which connects 12 trussmembers and provides rotational freedom at central node. Ithas been shown that the closed chain HP joint posses threedegree of freedom and is therefore suitable for converting a3-DOF hexagonal structure to a PMHT [22].

Several hexagonal structures can be assembled in planarpatterns 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 notshown for clarity). Two patterns, to create a plate-like structureare proposed in figure 4. In the ‘a-pattern’, each internalhexagon is surrounded by six other hexagons. The smallest‘a-pattern’ structure is therefore a seven hexagon HHT. Werefer to such a mechanism as an ‘a-pattern’ mechanism withk = 1 (because the central hexagon is surrounded by onlyone ring of hexagons). Such truss mechanism posses 15DOF (equation (1), b = 195, j = 70, M = 15). Largertruss mechanisms can be created by the addition of extrarings of hexagons. In general, the DOF of an ‘a-pattern’truss mechanism can be determined from Maxwell’s stability

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Figure 4. Large aspect ratio PMHT can be designed by the assemblyof several hexagons with different patterns. The b-pattern providesgreater 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 astatically determinate planar morphing tetrahedral system oftrusses.

A second, ‘b-pattern’ arrangement, figure 4, can be createdby leaving hexagonal holes in the ‘a-pattern’ mechanism toachieve a larger DOF per hexagon. The total DOF of the‘b-pattern’ can again be determined from Maxwell’s stabilitycriterion;

DOF = 9k2 + 15k − 6. (3)

The truss order (k) is shown for both the ‘a’- and ‘b’-patternsin figure 4.

Comparison of the average degrees of freedom pertetrahedral (equivalently the base triangles in figure 4) for thetwo patterns, equations (4) and (5), reveals that the b-patternhas a larger total number of degrees of freedom;

DOF per tetrahedral for pattern (a): = 12k + 3

6(3k2 + 3k + 1)(4)

DOF per tetrahedral for pattern (b): = 9k2 + 15k − 6

36k2. (5)

Figure 5. A planar morphing hinged truss (PMHT) consisting of 18actuators and 36 tetrahedra.

It is apparent that for all nonzero k, the average number ofDOF’s per tetrahedral for the b-pattern is larger than for thea-pattern. The parabolic increase in the DOF for the b-patternalso indicate that if an infinite plane is filled with the ‘b-pattern’(k → ∞) then the average DOF per tetrahedral approaches0.25 or equivalently 1.5 DOF per hexagon. Therefore aninfinitely large ‘b-pattern’ truss mechanism can be convertedto a statically determinate actuated truss by using an actuatordensity of three actuators per two hexagons. Alternatively if aninfinitely large plane is filled with the ‘a-pattern’ (k → ∞),then the average DOF per tetrahedral, approaches zero thatmeans 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 designprecision shape controlled actuated truss structures such asantennas. A first order (k = 1) ‘b-pattern’ PMHT, is shownin figure 5. The actuators in the figure are represented withthe dash lines. There are a total 18 actuators in the PMHTin figure 5, because the DOF for the corresponding trussmechanism is 18, equation (3). The actuators are assembledto connect the pyramid tops of 18 pairs of adjacent tetrahedra.The 18 locations can be any arbitrary selection of the total42 distinct available locations provided no more than threeactuators are placed in each hexagon. This restriction isenforced because the closed chain hexa-pivotal joint at thecenter of the hexagons has three degrees of freedom andtherefore using more than three actuators will result in the localstates of self-stress [22]. The location of the actuators canbe selected to fit the targeted shapes for the shape morphingstructure.

Figure 6(a) demonstrates the unfolding of the first orderPMHT to a flat shape and then its deformation to an undulatingshape, figure 6(b). The tetrahedral in the figure are representedby colored triangles for visibility and the actuators are notshown. The sequence can depict unfolding a deployable spacetruss. Figure 7 shows a second order ‘b-pattern’, k = 2, with60 DOF which deforms from a flat structure to a cup shape.Application oriented arbitrary shapes can be achieved by thecontrol of the actuators displacement and location.

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Figure 6. The shape morphing of a ‘b-pattern’ PMHT with 18actuators (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 madefrom equi-atomic NiTi alloys can be used as the lineardisplacement actuator in the design of MHT’s. The resistiveheating can be accomplished by passing electric currentthrough a strained one-way actuator to create contractile

Figure 8. One-way shape memory alloy actuators can connect thepyramid tops of a hinged bi-pyramid to create a reversible truss. Thecontraction of actuator 1 by heating rotates the cell about a revolutejoint by an angle θ and causes the extension of actuator 2, andvice versa.

linear displacement [20]. However, a biasing force is neededto restrain the contracted actuator and therefore completea reversible actuation cycle. We have previously used anantagonistic approach [24] to create the biasing force, andsuccessfully designed and fabricated several actuated beamsand devices [25]. Here we exploit the implementation of themethod in the truss mechanisms.

The antagonistic approach requires that the basic HTT,figure 1, be modified to accommodate a pair of counteringSMA actuators. Hence, mirror tetrahedral trusses areassembled on the opposite sides of the original pyramid base,creating hinged bi-pyramids, figure 8. The strain actuation

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.

capability is then imparted to the antagonistic flexural unitcell (AFC) [24] by mechanically stretching at least one of thetwo opposing actuators connecting the bi-pyramid tops. Theinitial pre-straining should take place at low temperature whenthe SMA is in its martensitic or R-phase state [25] prior toassembly to the cell. This pre-strain in actuator 1 in figure 8is denoted εs

1. That in actuator 2 is εs2. Upon heating the pre-

strained actuator, it will begin to contract and cause rotation ofthe bi-pyramids about their shared pivoting truss. This rotationrequires extension (straining) of actuator 2 and results in aremnant strain, ε, in actuator 1. Given a pre-strain value, andcell geometry, the cell rotation angle, θ , can be calculated andis given by [24]:

θ = 2 tan−1(L/H ) − cos−1

[1 − 2(1 + ε − εs

1)2

1 + (H/L)2

](6)

where, L is the SMA actuator length in the initial configuration(prior to the activation of the assembled cell), H is the distancebetween the tops of each bi-pyramid, figure 8. The strain,ε, refers to the strain in actuator 1 measured with respect tothe original length (prior to the pre-straining) of the actuator.Note that, for relatively small rotation angles (smaller than tendegrees), the strain developed in actuator 2, can be related tothe strain in actuator 1 [25];

Strain in actuator 2 = (εs1 + εs

2) − ε. (7)

If only one of the actuators, say actuator 1, is pre-strained thenequation (7) becomes;

Strain in actuator 2 = εs1 − ε. (8)

3. Experimental assessments

To determine the relationship between the rotation angle of thebi-pyramids to the shape memory pre-strain, the test structures

shown in figure 9 were made. Near equi-atomic NiTi wireswere 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 memberswere selected to have high plastic buckling resistance and werefabricated from stainless steel precision tubes with outsidediameters of 1.47 mm and thickness of 0.2 mm. The H–P jointlinks were 1.0 mm thick with the dimension of 10 mm×10 mmand 12 mm × 10 mm (the outer links are wider), figure 1(b).The H–P links were machined from stainless steel. The NiTiwire was passed through the tubes and connected at the H–P joint to mechanically magnify the recovered shape memorystrain of the NiTi wires. Parts of the NiTi wires whichpass through the tubes were electrically insulated by Teflonsleeving. Mechanical fasteners were used to firmly connect theNiTi wires to the truss tubes at the H–P joints. The rotation ofthe bi-pyramids about the shared truss member, figure 8, wasmeasured for different shape memory pre-strain by actuated(deformed) bi-pyramid’s picture. The camera was placed alongthe shared strut of the bi-pyramid to accurately record therotation angle. The rotation angles were later measured fromthe photos.

A first order ‘b-pattern’ planar morphing hinged truss(PMHT) is also designed and fabricated. The prototypeunit cells shown in figure 8 are used in the design. ThePMHT includes 18 pairs of actuators and total 36 bi-pyramids.Several different shapes are attained for the PMHT by actuatingdifferent sets of actuators.

4. Results and discussion

Heating the top NiTi wire, figures 8 and 9, above its martensitefinish temperature, Af, results in the upward rotation of the cellto an active equilibrium position [25]. Upon cooling below themartensite start temperature, Ms, the top NiTi wire undergoesa martensitic transformation which results in the extension of

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Figure 10. Rotation angle of the actuated bi-pyramids at mechanicalequilibrium.

the wire due to the de-twinning and causes the downwardrotating relaxation of the unit cell. At the mechanicalequilibrium (inactive equilibrium) further displacement ceases.The new rotation angle of the cell remains unchanged withoutrequiring additional power and is therefore suitable for longterm applications. The heating of the bottom actuator aboveAf later in the cycle results in the contraction of the heatedwire (shape recovery) to a second active equilibrium stateresulting in a further downward rotation. The difference ofthe two active rotation angles is the active rotation range ofthe cell and is the maximum deformation that the unit cellcan achieve. Cooling the bottom SMA wire in turn relaxesthe unit cell and the bi-pyramids slightly rotate upward to anew inactive configuration. The inactive rotation range is themaximum attainable deformation range for the unit cell at lowtemperature. The rotation angle depends on the shape memorypre-strain in the top wire, or actuator 1, εs

1, the cell geometryand the recovered strain, ε, equation (6).

The recovered strain of the antagonistic actuators as afunction of shape memory pre-strain for ribbons of identicalNiTi alloy has been measured and analyzed [25], and it was

also observed in this study that the shape recovery of NiTiwires is similar to the ribbons and therefore the data can beused to estimate the equilibrium rotation angles of the trussunit cell. The mechanical amplification method requires theSMA wires be attached to the HP joint rather than the pyramidtops. The recovered strain, ε, must therefore be multiplied bythe amplification factor which is defined by the ratio of thelength of the wire divided by the distance between the pyramidtops. For the prototype, figure 8, this amplification factor wasdetermined to be 3.2 (80/25 mm).

The active and inactive rotation angle as a function of pre-strain have been determined [25] and are plotted in figure 10. Inaddition, the responses of prototypes fabricated from actuatorswith 2.5%, 4%, 5%, 6%, and 7% pre-strains are marked inthe figure for comparison. The figure shows that there is goodagreement between the measured and predicted deformationsfor smaller pre-strains. However the measured ranges aresmaller than the predicted values at the larger pre-strainsprobably due to the frictional effects, which have been ignoredin the analysis. The mechanical amplification method used inthis design has made large rotation angle of over 10◦ feasible.The promising deformations achieved in this design fromrelatively small strain recovery of restrained SMA actuators(less than 4%) makes the method suitable for high authorityshape 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 infigure 11(a), the twisting in figure 11(b) and undulating infigure 11(c). Local deformation resulted from the weight ofthe truss is visible at the attaching point of the truss to the vice.

5. Conclusion

Hinged truss mechanisms can be designed by using a novelspherical–pivotal joint (H–P joint). Several truss memberscan be connected at the truss nodes and freely rotate aboutrevolute truss members of the hinged trusses. The assembly ofa number of actuators equivalent to the total degree of freedomof the hinged truss mechanism can create high authority planarmorphing hinged trusses (PMHT) with possible applications

Figure 11. Deformations of a prototype PMHT. (a) Bending, (b) twisting and (c) undulating.

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such as antenna supports or shape changing constructions.One-way shape memory wires can also be used to actuatethe hinged trusses. An antagonistic hinged truss mechanismis designed to allow the shape memory alloy actuation of thePMHTs. Using SMA wires longer than the original intendedactuator length has resulted in the significant amplificationof the response of the PMHT. As a result of the mechanicalamplification, visible rotation range of over 10◦ can beachieved by a single PMHT unit cell. It was shown thatlarge planar actuated truss structures, ‘b-pattern’, capable ofbending, twisting and undulating deformation can be createdby linking unlimited number of the unit cells.

Acknowledgments

The research was supported in part by the Defense AdvancedResearch Projects Agency (Leo Christodoulou, programmanager) and the Office of Naval Research (Steve Fishman,program manager) under grant number N00014-02-1-0614.

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