Programmable Deployment of Tensegrity Structures by Stimulus-Responsive Polymers

Tensegrity structures with detached struts are naturally suitable for deployable applications, both in terrestrial and outer-space structures, as well as morphing devices. Composed of discontinuous struts and continuous cables, such systems are only structurally stable when self-stress is induced; otherwise, they lose the original geometrical configuration (while keeping the topology) and thus can be tightly packed. We exploit this feature by using stimulus responsive polymers to introduce a paradigm for creating actively deployable 3D structures with complex shapes. The shape-change of 3D printed smart materials adds an active dimension to the configurational space of some structural components. Then we achieve dramatic global volume expansion by amplifying component-wise deformations to global configurational change via the inherent deployability of tensegrity. Through modular design, we can generate active tensegrities that are relatively stiff yet resilient with various complexities. Such unique properties enable structural systems that can achieve gigantic shape change, making them ideal as a platform for super light-weight structures, shape-changing soft robots, morphing antenna and RF devices, and biomedical devices.

The stress-strain behavior of the cable material was tested in controlled force mode on the DMA machine at room temperature (~25°C). A complete load cycle was performed at a very low speed (quasi-static) on a sample with dimensions 10mm×0.9mm×0.25mm. The printed sample was stretched to 3.1 MPa at loading rate of 0.5 MPa/min and then unloaded. The initial tangent elastic modulus was determined to be 16.43MPa from the stress strain curves (see Figure S5).

S1. Analysis of self-stressed tensegrity structures
Based on the design, linear analysis of self-stressed tensegrity helps us to correlate the initial tangent stiffness of the tensegrity structure to its self-stress level. Considering the small strain due to self-stress, we treat the strut material as linear elastic and take the initial modulus for the calculation to simplify the design process. The governing equation takes the form of a linear equation = Ku F , where K is the stiffness matrix, u contains the nodal displacements, and F contains the applied forces. Due to the self-stress, the stiffness matrix for a tensegrity structure is different from a normal truss structure 1,2 . The major difference is the additional contribution of the geometrical stiffness matrix G K . Thus the tangent stiffness matrix takes the form: E G = + K K K , (S1) where, E K is the linear stiffness matrix. For completeness, we summarize the derivation of G K here. Assume that for a member (either a cable or strut) i , its two nodes, length and self-stress induced force are a , b , i L and i T , respectively. We define the components of a connectivity matrix C as: 1, if member is connected to node , and 1, if member is connected to node , and 0, otherwise (S2) 3 We also define a diagonal matrix Q such that: The ratio i i T L is known as the force density 2 . Let T be the normalized self-stress induced force vector with maximum compression in struts equal to 1. Denoting γ as a scaling factor (which equals to the maximum compressive force in struts), we can rewrite Q as: Because T is an intrinsic property of a tensegrity design, it is a constant vector.
Thus, we can write the so-called force density matrix 2 as: Since the strains of the members caused by the self-stress are small, the geometric stiffness matrix can be expressed as: (S6) Finally, with the contribution of the linear stiffness matrix, the stiffness matrix of the tensegrity structure can be approximated for small deformation as: From the above derivation, we can see that the stiffness matrix of a tensegrity is a function of its self-stress level γ. We find that the higher the self-stress, the stiffer the tensegrity. Using this equation, we can find the relationship between the initial tangent modulus of a tensegrity under global uniaxial compression and the self-stress level γ. The initial tangent modulus is the ratio of the applied force over the compression magnitude (in terms of displacement). Figure S1 plots the curve of initial tangent modulus versus self-stress level, based on the material properties for 3 tensegrity designs. The tangent modulus shown here is calculated using the non-dimensional displacements, which is the downward compression displacements normalized by the heights of the tensegrity designs, and thus, the unit of the tangent modulus is in Newton (N). The two 3layer tensegrities (with different materials) yield almost identical curves, so only one is plotted for clarity. This curve does not start from (0,0) because the 3-layer tensegrity is kinematically determinate, thus its stiffness matrix is not singular when there is no prestress ( (0) According to the experimental compression tests of the active tensegrities, we can approximate the initial tangent modulus of a tensegrity. Then, based on the curves shown in Figure S1, we can inversely estimate how much self-stress we have successfully applied to the active tensegrity.

S2. The two critical loads for the slitted tubular struts
From the previous section, we can see that the (initial) stiffness of a tensegrity structure depends on the self-stress level. However, the achievable self-stress level of an active tensegrity is not arbitrary, as it is determined by two critical factors. The first factor is that the compression on struts should not prevent their full recovery. In the final stage of deployment, some SMP struts in the active tensegrity will be subject to compression before full recovery, with their tubular cross-section still open. The second critical factor is that, after deployment, the struts should not buckle under the self-stress compression. If the struts buckle, then the tensegrity will lose some self-stress and cannot completely reach the designated shape. In the following, we will derive analytical estimations of these two critical strut loads. We first compute the critical force during the recovery, when the tubular cross section of a strut is open, as shown in Figure S2a.
We make the following assumptions: (1) a tube can be analyzed using shell theory because the thickness is relatively small; (2) the mid-surface is subject to isometric deformation; (3) the static behavior of the SMP can be regarded as elastic when the temperature is fixed and the strain is relatively small. The meaning of the symbols used in the derivation is illustrated in Figure S2. Therefore, supposing that the changes in the curvatures along the two principle directions are ( 1 r -, 1 R ), we can write the total strain energy at the bending region as 3 : The symbol D denotes the flexural rigidity, defined as: where E is Young's modulus and ν is Poisson's ratio. The value of r is determined when U B is minimized 3 . Therefore, r=R. Then the bending moment is calculated as: (S10) At the final stage of the strut's recovery, a single kink about a quarter from the end of a strut is usually observed (see Movie S3). Thus, we can draw the shape schematically as shown in Figure   S2c. The regions that are not opened are much stiffer than the bending region. Therefore, we may treat those regions as rigid. Notice that, If the two applied forces are aligned along the same line, then equilibrium is obtained as: Thus the critical force can be calculated by: In our case, the typical value for l is around 0.25. The angle 1 q can be computed from the deformed length of the strut ( ). The equality holds when 1 q is small. The later expression is used because it is simple and conservative. The derivation requires a portion with fully opened cross section along the strut (which forms a "kink"), thus it is not accurate when the strut is almost straight (i.e. 1 q becomes very small), because in reality the opened cross section starts to enclose before the strut recovers to straight, so the deformation mode no longer has a "kink".
The critical load before the buckling of the struts after deployment is given by the Euler buckling formula 4 , (S12) The effective length L eff depends on the boundary conditions of the strut. In the compression tests, the fixture of the sample constrains the free rotation at the two ends, resulting in an effective length around 0.75L. However, in the tensegrity, the two ends are assumed to be pinned, and thus L eff = L. The minimum static moment of inertia I min is determined to be the static moment of inertia of the X-X axis at the geometric centroid GC, which is denoted as I GC-XX , (S13) We note that, in the experiment, the struts are not loaded at the geometric center (GC) of the cross section. Instead, the compressive forces are loaded at point O (at the center of the midsurface circle). As a consequence, the actual critical buckling force will be lower than the estimation, since the buckling mode involves a combination of bending and twisting.

S3. Design of cables
As explained in our paper, the self-stress in the tensegrity is induced by prescribed length differences between cables and struts. We assume that after successful deployment, the struts become straight and their deformation under compression is negligible (recall that the struts are much stiffer than the cables). Therefore, we control the level of self-stress magnitude by manipulating the initial length of cables. We did this for two reasons. First, we do not want the initial length of cables to be too long so that the deployed tensegrity cannot gain enough selfstress to become stable and stiff. Second, the initial lengths of cables should not be so short that the struts cannot recover during deployment or stay straight after deployment, due to the excessive self-stress magnitude.
Suppose that the desired self-stress level is γ and member i is a cable. Given the normalized force vector T (as defined in Section S1), we can determine the initial length of a cable as: (S14) In this equation, 0 denotes the initial length, d is the design length of the cables which is pulled from the geometry of the tensegrity design, i T g is the desired tension in the cable, A is the cross-sectional area of the cable, and d is the ineffective length at each end of a cable which changes very little. Considering the contact angles of cables and struts, d is generally 1.4~3 times the distance d shown in Figure S3. Typically, the force is small, and we can assume linear behavior for the cables. Hence the initial elastic modulus c E is used.
Such calculation provides an approximate guide for determining the initial lengths of cables based on the value of γ, which needs to be greater than 0, but less than the minimum critical load of the strut. In reality, the control of the self-stress level and final geometry will not be precise due to many practical factors, for example: the twisting of cables, the plasticity of the cable material, the printing accuracy, and the entanglement of the cables near the joints. In some cases, adjustment based on the experimental results is needed, especially for tensegrity designs with complex geometries.

S4. Detailed experimental analysis
The shape recovery behavior of the strut comes from the viscoelastic properties of the SMPs. The DMA tests are performed to investigate the viscoelastic properties of the printed strut materials. The storage modulus and loss tangent tan δ vs. temperature plots of the printed three strut materials are shown in Figure S4.
The Filaflex material exhibits rubber-like viscoelastic properties at room temperature.
Uniaxial tension tests are performed to investigate the mechanical properties of the printed Filaflex material. The stress vs. strain curve is shown in Figure S5. The specimen occupies the same cross section as the cables, which is a rectangle of 250µm-thick and 920µm-wide. The uniaxial stretch is up to ~40% of the initial length.

S5. Constitutive model for the SMP
The multi-branch model is used to describe the viscoelastic properties of the printed SMP materials. In this model, one elastic equilibrium branch and several thermo-viscoelastic nonequilibrium branches are arranged in parallel. The non-equilibrium branch is described by the Maxwell element, represented by a viscous damper and an elastic spring connected in series. The total stress of the material can be expressed as: where Eq E is the Young's modulus of the equilibrium branch and both (WLF) equation 6 and the Arrhenius-type equation 7 . If the temperature is higher than the reference temperature, the shifting factor can be expressed using the WLF equation: (S17) The (S19c) The symbol ω denotes the test frequency. By employing a nonlinear regression software 8,9 , the parameters non m E , R m t , 1 C , 2 C and AF c /k can be determined by fitting the tand and storage modulus from experimental DMA tests. The material parameters used in this paper are provided in Table   S1.
To show the capability of this model, the comparison of the DMA curves between the experiment and the simulation are shown in Figure S6. We can see that the multi-branch model explains the thermomechanical behavior of the printed strut materials in the temperature range used for programming and actuation processes.

S6. Finite element analysis
The recovery process and mechanical properties of struts are modeled using the FEA software ABAQUS (Simulia, Providence, RI, USA). The hybrid C3D8RHT element is used. We implement the multi-branch model based on Prony's series, which is defined as: where G is the total shear modulus, G Eq and G m are the shear modulus of the equilibrium branch and m-th non-equilibrium branches. Applying the incompressible condition, the shear modulus G is calculated as G m =E m /3, where E m is the elastic modulus from the multi-branch model. The material parameters for the multi branch model are elaborated upon in Section S5. To apply the temperature effects, the shift factors are calculated using the WLF equation and Arrhenius-type equation 6,7 . The UTRS subroutine is used to implement the WLF equation and Arrhenius-type equation.
Considering the symmetry of the strut and boundary conditions, only 1/4 of a strut is used for simulation of free recovery. The slit of the strut is first opened into a nearly flat configuration in the middle part of the strut at 65°C, which is above the T g of the BM (Verowhite). The pressure used to open the slit is applied on the inner surface of the slit near the opening. After the slit is opened, we fix the middle section of the strut (one end in the 1/4 model) and add a pressure load on the end of the strut and in the transverse direction of the strut to bend it into a "U"-shape.
To further deform the strut into the "W"-shape, we fix the 1/4 section of the strut and apply pressure at the end in the opposite direction of the previous step. After the deformation process is finished, we cool the temperature to 25°C, at which the material is in a glassy state. Then all the external loading and constraints are removed, and the deformed shape of the strut is "frozen" due to viscoelasticity. To simulate the recovery process, the temperature is increased to 65°C. The recovery process by the simulation is compared with the experimental results, as shown in Figure   2.
The strut under compression is also modeled to determine the after-recovery critical force (F buckling ). In this simulation, the whole strut is modeled to consider asymmetric deformation modes. We impose an ambient temperature of 65°C. One end of the strut is pinned in directions x, y, z within the central zone (radius of 1mm), creating a partially fixed end. At the other end, the center zone is pinned in x, y directions (partially fixed), and a displacement load of rate 0.25mm/s is applied in the -z direction. This boundary condition is similar to the case of the strut compression experiment, but more restrictive than the actual boundary condition as embedded in the tensegrity structures.
A similar procedure can be applied to predict the mechanical performance of struts made with various SMPs. Figure S1. Initial tangent modulus vs. maximum self-stress forces in struts.

Figure S5
The stress-strain curve of Filaflex material at room temperature (~25°C).

Figure S6
Comparison of the DMA curves between experimental data and numerical models for three SMP materials used in this paper.

Figure S7
Failed deployment of a 6-strut spherical tensegrity, due to physical contact between struts, as highlighted by the red circle. As discussed in the main text, when the cables are loose, the folded struts are almost free to move in space. In this example, a strut blocks the recovery of another strut.

Figure S8
The compression tests on the two-layer tensegrity and the three-layer tensegrity, whose struts are made with two SMPs. (A) Compression test of the resultant deployed 2-layer tensegrity using 2 different SMPs for struts (BM and DM-1). The red line indicates the loading process while the green line indicates the unloading process. Estimated maximal compression equals 0.12N. (B) Compression test of the deployed 3-layer tensegrity made with 2 different SMPs (BM and DM-1). Maximal compression in the struts is estimated to be around 0.14N. The three-layer tensegrity shows two dips in both the loading and unloading process. This is due to the inherent multi-stability behavior of such structures. That is, the structure has a multiple local minima of stored energy at different configurations. For example, when one layer of the tower is fully flattened, the structure is at an alternative stable state (other than the fully deployed configuration). Due to the contact of struts, the other stable configurations cannot be reached. However, it still leads to a reduction in stiffness of the structure (snap through). The 3-layer tensegrity in B illustrates this effect more clearly than the one in Figure 4D of the main content because the structure in A has more DM-1 struts, which are less stiff than DM-2 and BM struts in room temperature (~25˚C). Thus, when a contact between struts happens, the DM-1 struts will bend, leading the structure slightly closer to the ideal alternative stable configuration, although this state cannot be fully reached. Movie S1 Deployment of a 3-strut tensegrity.
Movie S2 Numerical simulation of the free recovery of a single strut with the slitted tubular cross section.
Movie S3 Deployment of a 6-strut spherical tensegrity, using the partial folding strategy.
Movie S4 Deployment of a 6-strut spherical tensegrity with 3 discrete attaching pieces of surface.
Movie S5 Deployment of a 6-strut spherical tensegrity with one continuous attaching piece of surface.
Movie S6 Programmed sequential deployment of a 2-layer tower tensegrity, whose struts are made with 2 SMPs.

Movie S7
Programmed sequential deployment of a 3-layer tower tensegrity, whose struts are made with 2 SMPs.

Movie S8
Programmed sequential deployment of a 3-layer tower tensegrity, whose struts are made with 3 SMPs.