Experimental Architectural Design Study on Internally Tiled Pneumatic Surfaces for Adaptive Moldability

Adaptive moldable surfaces can repeatedly conform to different shapes, hold them rigidly, and reset to their original form. This adaptability has broad potential across various fields, including (1) medical applications like reusable orthopedic casts [1] and emergency braces [2], wheelchair positioning devices for stroke patients [3], and wearable rehabilitation aids [4] and upper arm support systems [5]; (2) human–computer interaction and consumer products, such as flexible displays with adjustable stiffness [6] and reconfigurable furniture or footwear [7]; (3) the entertainment industry, where wearables with haptic feedback enhance immersive gaming [8]; and (4) the transportation industry for securing cargo during transit [9]. For these types of applications, adaptive moldable surfaces provide three key subcapabilities: drapability, shapability, and rigidizability. Drapability requires a surface that can passively arrange itself over the shape of a target object under its weight in a fabric-like manner. Shapability requires a surface that can be manually conformed to the curvature and shape of a desired configuration. Rigidizability requires a surface that can change its stiffness on demand, holding its shape securely against external forces after being shaped. In most applications, transitioning between these subcapabilities must be fast and reversible. Designing adaptive moldable surfaces with all three subcapabilities involves tradeoffs, as the same design features affect each subcapability. In user-interacting products, the priority of these subcapabilities varies by application, further complicated by safety and packaging constraints like compactness and weight. Therefore, understanding the coupled relationship of the design features and their impact on all three subcapabilities is important for driving the design of effective products.

Extensive research has explored the use of traditional materials, smart materials, and pneumatically activated systems for adaptive moldable surfaces. Traditional materials like gallium–silicone composite [10], woven polyester with integrated polylactic acid (PLA) [11], and 3D-printed PLA [12] with embedded heating elements change stiffness through glass transition temperatures. These have enabled applications such as a transformable table clock/wearable biometric sensor [10], an active finger brace [11], and a custom-fitting musculoskeletal support system [12]. However, the activation temperature (30–48 °C) and operation time limit their use in user-interacting products. Smart materials such as nickel–titanium shape memory alloys (SMA) [13], polyurethane-based shape memory polymers (SMP) [14], and SMA wires with buckling springs [15] offer on-demand stiffness control. These materials have been applied to self-fitting spaceflight wearables [13], morphing airfoils for stiffness and strain control [14], and multilayered beams switching between compliant and stiff states [15]. Yet, the operating temperatures (∼20–120 °C for alloys, ∼20–55 °C for polymers) constrain design freedom in safe and user-friendly products.

A promising approach for user-facing applications involves pneumatically activated composites, where an airtight bladder contains a granular, fibrous, or layered material array. When air is vacuumed from the bladder, friction between the components increases, enabling a transition from soft to rigid states. Granular jamming has been extensively studied to determine how granule shape, size [16], and material type [17,18] affect stiffness in different states. Applications include universal robot grippers [17,19,20], transformable spaces [21], malleable user interfaces [22,23], and reusable molds for free-form concrete [24]. However, these systems lack compactness in both soft and rigid states due to the architecture of their systems. Research on fiber jamming has focused on how different fiber materials and surface finishes impact stiffness change, enabling applications such as soft robots [25,26] and variable stiffness surgical manipulators [27,28]. Yet, these implementations are limited to tubular form and involve complex fabrication and drive mechanisms. Layer jamming, using layers of sheets, offers a more compact cross section. Researchers have explored woven, interleaved, creased, and fish-scale-like layers [7,29] with affordable sheet materials like copy paper, sandpaper, fabric, polyethylene terephthalate (PET) foil [29,30], mylar [31], Kapton, polyester, polytetrafluoroethylene (PTFE), and PTFE-coated fiberglass [32]. Applications include medical uses like reusable casts [1], wheelchair posture devices [3], wearable rehabilitation structures [4,33], as well as human–computer interaction and consumer products like flexible displays [6] and reconfigurable furniture and footwear [7]. Layer jamming can be applied in the transportation industry, such as for seat belt dampers that switch quickly between soft and rigid states for comfort and safety [32]. However, traditional layer jamming structures using inextensible sheets lack the ability to bend in both transverse and longitudinal axes without creasing, limiting their conformability. Researchers have addressed this by using extensible woven fabric sheets [34], though the maximum stiffness is constrained by the fabric's inherent properties [35]. Adding more fabric layers in an airtight bladder could increase stiffness but would result in a bulky structure, similar to granular or fiber jamming systems. Hybrid uses of granular, fiber, and layer jamming approaches have also been studied [36], demonstrating the importance of interrelationships among different performance aspects such as conformability and stiffening.

A more recent approach, tile-based rigidization [4,37–39], addresses the drawbacks of granular, fiber, and traditional layer jamming by using opposing arrays of rigid tiles within an airtight bladder, allowing continuous stiffness control and design of structures with tunable energy dissipation properties [40]. With adjacent tiles able to move relative to each other, this approach enables a thin-profile surface capable of bending in two directions simultaneously, improving drapability and shapability. Thus, this architecture holds promise for the design of adaptive moldable surfaces. However, the key performances of the three subcapabilities inherently trade off against each other, as the same design features affect the mechanisms governing all three. These tradeoffs, combined with the broad design space where tile architecture configurations vary significantly, add complexity to designing with tile-based rigidization for novel products. Research to date has primarily focused on improving rigidization performance activated by vacuum pressure in isolation.

This article more broadly explores the coupled relationship of the architectural design features and the resulting tradeoffs in drapability, shapability, and rigidizability technology subcapability performances to drive the design of effective user-interactive adaptive moldable products. The surface architecture is introduced through three key example tile architectural classes defining the approach by which the tiles are held in a regular array: bladder-attached, internal sheet-attached, and mutually interlocking. The mechanisms by which the operational state (vacuum applied or released) enables each subcapability (drapability, shapability, and rigidizability) as a function of tile architectural class are discussed. To rigorously study the impact of tile architecture on the coupled subcapability performances, measurable engineering performance metrics are defined and experimentally characterized: draping angle for drapability, conformability and setability for shapability, and flexural rigidity and post-yield elasticity for rigidizability. A representative subset of the architectural design space of internally tiled pneumatic surfaces is systematically explored by conducting three studies, each exploring a different aspect of architecture: (1) a tile architectural class study, analyzing the effects of using different tile arrangement approaches, (2) a design coupling study, analyzing tradeoffs among performance metric pairs within and across tile architectural subclasses, and (3) an architectural feature variations study, analyzing the effects of shifting the tile array layers and including an intermediate friction layer. These studies develop an understanding of the coupled impacts of the architectural class and feature variations on the performance of internally tiled pneumatic surfaces, which encompasses a wide range of interrelated subcapabilities catering to the design of user-interacting adaptive moldability applications.

An internally tiled pneumatic surface transitions from providing one technology subcapability to another (drapability, shapability, and rigidizability) as vacuum pressure is introduced to the system as shown in Fig. 1. It exhibits fabric-like physical behavior in terms of being drapable over arbitrary forms at atmospheric pressure. However, the same pneumatic surface can be shaped manually by external forces and be posed into intended forms by applying a vacuum. While the vacuum pressure is held, a pneumatic surface holds its shape with increased stiffness. In the enhanced stiffness state, the pneumatic surface performs as a load-bearing rigid structure, resisting external forces that otherwise disturb the intended target form. An in-depth understanding of the trading-off relation between the technology subcapabilities obtained by applying or releasing the vacuum and the design features of the pneumatic surface is crucial for the implementation of these layered structures in user-interacting products. To achieve such an understanding, a generic architecture captures the basic components of internally tiled pneumatic surfaces. This provides a basis to categorize multiple types of tile architectural classes. The operation states (draping, shaping, rigidizing), which are induced by applying or releasing the vacuum pressure inside an airtight pneumatic surface, relate to the types of these architectural classes, enabling technology subcapabilities (drapability, shapability, rigidizability), which are combined to provide the overall capability of adaptive moldability.

The architecture of internally tiled pneumatic surfaces is composed of one or multiple layups that are based on layers of rigid tiles arrayed in regular patterns within an airtight bladder. Each one of these layups contains an exterior fabric layer and an inner airtight layer, which form the bladder skin when combined, and a layer (or layers) composed of an array of tiles. A vacuum port attached to one of these layups allows control over pressure inside the layered structure in relation to atmospheric pressure. By applying a vacuum, an internally tiled pneumatic surface can transition from soft to rigid states as tiles are pushed against each other due to the force generated by the external atmospheric pressure. The various adaptive moldability subcapabilities are functions of the applied vacuum. In Fig. 2, the architectural layup of a specific class of internally tiled pneumatic surface is shown as an example on the left, with a detail of the tile array on the right. An individual tile is designed with its shape (extruded triangle, hexagon, or complicated 3d form), size, height, or thickness, and features such as protrusions on the tile surface. The array pattern and layup are designed in the way these individual tiles are brought together, the number of tile array patterns encapsulated in the airtight bladder, and their relative position to each other.

The tile, array, and layup design affect the resulting performance of a pneumatic surface. Tile architectural classes can be categorized based on the approach by which the tiles are constrained into their array pattern. Each tile architectural class may be composed of one or more subclasses for which particular feature variations such as tile array pattern overlap and layup layers exist. Gaining a comprehensive understanding of the inherent characteristics of these architectural classes is important for designing the overall moldability performance of user-interacting adaptive moldable products. It is essential to consider that variables regarding the design of individual tiles, the selection of material, and the fabrication techniques vary based on the architectural class to which a pneumatic surface belongs.

Internally tiled pneumatic surfaces can be categorized into architectural classes based on the approaches that provide a defined spatial order among the adjacent tiles inside a bladder, hence facilitating the formation of the tile array patterns. Three primary examples of architectural classes are bladder-attached tiles, internal sheet-attached tiles, and mutually interlocking tiles. Each architectural class shows differences regarding the formation of the tile array layers, the strategies for the layup of these layers, and the relationship between the tile array layers and the bladder skin. Based on these differences, pneumatic surfaces provide varying levels of technology subcapability performance.

The bladder-attached tiles class contains tiles that are permanently attached to the bladder's interior skin, which ensures a regular array pattern among adjacent tiles. One subclass of this basic architecture, bladder-attached non-connected tiles, where the only connection between adjacent tiles is the bladder skin itself, is shown in Fig. 3 on the left. In this case, the bladder skin provides flexibility between individual tiles in a hinge-like manner. A variation of this architecture that accommodates tile design features to connect the adjacent tiles to each other is the bladder-attached connected tiles subclass, illustrated in Fig. 3 on the right. These additional tile design features enhance the overall stiffness of such a layered structure; however, it reduces the hinge-like ability of the non-connected tiles, negatively affecting the draping performance.

The internal sheet-attached tiles architectural class, unlike the bladder-attached tiles class, has no dependency on the bladder to maintain the tile array pattern. Instead of using permanent bonding to the bladder skin to create a defined tile array pattern, it makes use of an extra layer in a sheet form to which the individual tiles are attached to maintain their spatial locations within the airtight bladder. In Fig. 4, an example configuration of the internal sheet-attached tiles is shown, which consists of two opposing layups of hexagonal tiles facing one another. Each tile contains two parts: a bottom part with grooved pegs and a top part with through holes. The bottom part with pegs snaps into the top part through the sheet layer, which is precisely perforated, defining the gap between adjacent tiles, keeping the tile array pattern together, yet enabling individual tiles to have the capability of hinge-like motion.

The mutually interlocking tiles architectural class, similar to the internal sheet-attached class, has no bladder dependency for tile arrangement. However, it makes use of complementary tile features to connect adjacent tiles together to maintain a regular tile array pattern instead of using an extra sheet layer. A pneumatic surface made of the mutually interlocking tiles can function by containing only one tile array layer unlike the pneumatic surfaces made of tiles categorized under the other architectural classes, which necessitate at least two layers of tile arrays. Each tile that forms the array pattern has an equal number of complementary features; one instance of this is shown in Fig. 5, a tile making use of loops and legs. These complimentary design features enable any tile to be interlocked with its neighboring tiles. This provides flexibility since the legs can slide and rotate within their corresponding loops. Such flexibility allows these layered structures to be drapable in an unvacuumed mode and shapable during a transitioning mode. The size of the gap between adjacent tiles becomes minimal as the legs slide all the way back in the loops when the air is fully vacuumed out of the bladder, resulting in an enhanced overall stiffness of the pneumatic surface. Concurrently, the asymmetrical nature of the individual tiles induces a cinching motion because of the moment generated by the shrinkage of the bladder skin around the legs.

Internally tiled pneumatic surfaces operate between three states as a result of the application of vacuum inside an airtight bladder: (1) the draping state, where the pneumatic surface behaves like a fabric in the absence of a vacuum, (2) the shaping state, which is induced by manual external forces that enable the conformation of the pneumatic surface into an intended form while the air is being vacuumed out from the bladder, and (3) the rigidizing state, where the pneumatic surface becomes stiff while the vacuum pressure is held inside the bladder. Although the operation states are shared among all the tile architectural classes, the resulting physical behaviors and the performance quality of a pneumatic surface differ from one tile architectural class to another.

Internally tiled pneumatic surfaces exhibit fabric-like behavior in the unvacuumed operation mode. Gaps between adjacent tiles enable the individual tiles to shift, allowing them to have a hinge-like motion relative to each other. As shown in Fig. 6, this makes the pneumatic surface drapable, which facilitates the accommodation of any target object topography to a degree by bending to the target form under its weight. The tile architectural class affects the nature of the draping behavior of a pneumatic surface. Bladder-attached tiles have fixated locations on their corresponding bladder skins; thus, they are not free to move in translation and rotation relative to these skins. However, each tile array layer can move independently of one another, and each tile can have a hinge-like motion relative to its adjacent tiles. For internal sheet-attached tiles, the bladder skin does not restrict the draping behavior, but the perforated internal layer keeps individual tiles together as an array pattern does. Mutually interlocking tiles drape independent of the bladder skin restriction as well. However, the tile features such as legs and loops that interlock neighboring tiles to each other affect the draping behavior by limiting the maximum obtainable gap and angle between tiles. In addition, the asymmetrical design of these tiles produces hard-stops that constrain the draping behavior differently for convex and concave bending directions.

Manually conforming a pneumatic surface into the shape of a target object initializes the shaping state. The position of the individual tiles inside the bladder can be arranged relative to their adjacent tiles by applying an external force, for example, by using hands. In this way, the pneumatic surface can be conformed into varying shapes. However, without the presence of an external force, the individual tiles shift back from their conformed positions to their neutral draped positions. For this reason, the shaping state necessitates the transitional vacuum mode that generates permanent and distributed external forces over the pneumatic surface. During the application of vacuum inside the bladder, the opposing tiles and the adjacent tiles begin jamming against each other, restricting the relative motion of the tile array layers and setting the pneumatic surface into the conformed shape. The shaping behavior, composed of conforming and setting behaviors, shows differences among the tile architectural classes. The asymmetrical nature of the mutually interlocking tiles biases the structure to cinch toward a particular direction when the vacuum pressure is applied. Thus, it is relatively more difficult to conform and set the pneumatic surface made of mutually interlocking tiles to an arbitrary target shape. For pneumatic surfaces belonging to other tile architectural classes, such bias does not exist since their layup is symmetrical with the accommodation of two opposing tile array layers inside the bladder. Depending on the tile design, as the relative vacuum pressure level increases, the pneumatic surface may spring back as shown in Fig. 7. The springing back motion occurs due to the rotational force generated by the tiles as they are compressed on each other and by the bladder skin as it is being shrunk against the gap between each tile, thus reducing the shaping performance of the pneumatic surface.

The rigidizing operation state is accomplished by vacuuming most of the air out from the airtight bladder. In this state, the opposing tiles jam against each other, restricting the tile array layers from shifting relative to each other due to friction and/or interference and minimizing folding or bending along the gap lines as shown in Fig. 8. A rigidized pneumatic surface, by nature of its flexural rigidity, can resist deformation under external loads to a degree. Permanent deformation of a rigidized shape occurs when the external forces are so large that the individual tiles can no longer maintain their relatively fixed positions inside the bladder. This results in yielding, where opposing tiles begin sliding over each other and the tile array layers potentially get disengaged, although some of this post-yield deformation is recovered through residual elasticity. The level of deformation is related to the flexural rigidity of the pneumatic surface, which is influenced by the bending stiffness properties of the tile material and the bladder skin composite. For pneumatic surfaces that are made of either internal sheet-attached tiles or mutually interlocking tiles, the contact force between the tile array layers and the bladder skin increases when the vacuum is applied, increasing friction. As a result, it is relatively more difficult for any tile to shift its relative position inside the bladder as opposed to the case in the draping state. In addition, the bladder skin forces adjacent tiles laterally together by pulling into the gap lines. This improves the overall stiffness of the final configuration of a pneumatic surface. However, for pneumatic surfaces that are made of bladder-attached tiles, improved rigidity is achieved only by jamming opposing tile array layers and pulling into gap lines since each tile is already permanently bonded to the bladder skin.

The internally tiled pneumatic surface technology affords a plethora of architectures. The selection of a tile architectural class/subclass, the architectural feature variations corresponding to each class/subclass, and the tile shape all simultaneously affect the performances of the three key technology subcapabilities provided by these pneumatic surfaces. This creates a useful and broad design space in which the performances of the key subcapabilities are inherently interrelated. There is a need for a rigorous study enabling the quantified understanding of the relative contribution of using different architectural approaches to render this emerging technology useful in the design of various adaptive moldable user-interacting applications for different contexts. For example, while shapability and rigidizability performances would be of higher priority for a moldable medical cast application that needs to be operated by a medical expert, drapability performance would be of a higher priority for the design of a moldable active cargo blanket that needs to be draped over arbitrary objects by an end-user. Since the overall moldability performance of any internally tiled pneumatic surface is a set of combined performances obtained at the different operation states, the effects of the tile architectural classes/subclasses and architectural feature variations on the pneumatic surface's performance at each operation state need to be quantified. In Fig. 9, the overall performance of an internally tiled pneumatic surface is decomposed into a combination of the interrelated performances of technology subcapabilities and their corresponding quantifiable engineering performance metrics, which are simultaneously affected by the architectural design.

Without an applied vacuum, a pneumatic surface exhibits drapability. Shapability occurs as the vacuum pressure is applied. While vacuum pressure is held, a pneumatic surface exhibits rigidizability. The characterization of a pneumatic surface's performance over all the technology subcapabilities requires the definition of sets of quantifiable engineering performance metrics that are identified in relation to these subcapabilities. Each subcapability has one or more corresponding metrics, with the draping angle metric characterizing the drapability performance, the conformability and setability metrics characterizing the shapability performance, and the flexural rigidity and post-yield elasticity metrics characterizing the rigidizability performance. A set of experimental procedures is devised to collect data for quantifying each engineering performance metric to enable the comparison between the performances of the internally tiled pneumatic surface prototypes that vary in terms of the tile architectural class/subclass and architectural feature variations. This guides the architectural design of applications using the pneumatic surface technology to meet the targeted performance requirements driven by the design context.

To provide a quantified understanding of the pneumatic surface performance across different architectural approaches, experimental characterization of representative prototypes is required. Fabricating these representative prototypes for experimental characterization requires two main steps: the fabrication of tiles and the fabrication of bladders. While the fabrication steps for tiles show variations depending on the type of the tile architectural class to which particular prototype belongs, the fabrication steps for bladders are global and applicable to any prototype that is categorized under any architectural class. The fabrication processes employed for all three architectural classes are available in the Supplemental Materials A on the ASME Digital Collection. A set of pneumatic surface prototypes representing each tile architectural class is shown in Fig. 10, which demonstrates their unvacuumed and vacuumed states. While the prototypes seem almost identical in the absence of vacuum, the differences across their tile array layer details become visible when vacuum is applied.

An internally tiled pneumatic surface's moldability performance is a combination of performances of its technology subcapabilities: drapability, shapability, and rigidizability. The performance of each subcapability is characterized by a set of quantifiable engineering performance metrics: the drapability performance is quantified by one metric, draping angle, and the shapability, and rigidizability performances are quantified by sets of two metrics each. The shapability performance can be divided into conformability and setability metrics, and the rigidizability performance can be broken down into flexural rigidity and post-yield elasticity metrics. For the quantification of each one of these engineering performance metrics, an experimental procedure that enables testing and data collection was developed. The units and ranges of the underlying measurements vary from one performance metric to another, making the direct understanding of the significance of a score on any particular metric difficult, as well as making the direct comparison among metrics complicated. To convey a direct understanding of the merit of a score and enable direct comparison, the raw measurements were transformed into a natural functional scale in which the score naturally ranges from zero to one and units become dimensionless. In this scale, zero represents the theoretical worst case where no functionality toward the performance metric is observed (i.e., not draping at all, not rigid at all), and one represents the theoretical best case where the functionality toward the performance metric is ideal (i.e., unrestricted draping, solidly rigid). This enables the direct comparison of the engineering performance metric scores of pneumatic surfaces representing different architectural approaches. Supplemental Materials B details how the performance metrics are defined and experimentally characterized for each of the technology subcapabilities.

To gain an understanding of how tile architectural classes, subclasses, and feature variations affect the performance of internally tiled pneumatic surfaces, a series of controlled experiments is required. The design space is broad and open ended such that conducting a full design of experiments is not feasible. Therefore, the following series of three consecutive studies aim to demonstrate a design approach for obtaining insights into the moldability performance of pneumatic surfaces with different tile architectural classes, subclasses, and feature variations, rather than establishing quantitative performance distinctions between classes or uncovering a parametric relationship focused on a specific class or variation. These studies include the following: (1) a comparative performance study across tile architectural classes, generating representative prototypes from each class and measuring quantifiable metrics for drapability, shapability, and rigidizability performances to analyze overall moldability performance by comparing these metrics across classes; (2) a design coupling study within architectural subclasses, using correlation matrices to show dependencies between performance metric pairs for representatives with different tile shapes in two subclasses; and (3) a comparative performance study evaluating the effects of architectural feature variations (opposing tile layup and intermediate friction layer) on the overall performance of one representative architecture.

Tile architectural class defines the method by which the tiles are held in regular arrangements and has a major influence on the form of a pneumatic surface, hence on its drapability, shapability, and rigidizability performances. Understanding the effects of the different tile arrangement approach on the pneumatic surface's overall performance provides essential insights for the initial stages of the design process. Eight prototypes that contain various tile array layers representing three different tile architectural classes were designed and fabricated. Table 1 summarizes the specifications of the prototypes used throughout the performance comparison study of the tile architectural classes. Six prototypes represent the bladder-attached tiles class, half of which are from the non-connected tiles subclass, while the other half are from the connected tiles subclass. One prototype each represents the internal sheet-attached tiles and the mutually interlocking tiles classes. Each of these six prototypes employs different tile shapes. In Fig. 11, the shapes of the individual tiles, their size relative to each other, and how each tile is brought together to form a patch of a tile array layer are illustrated. The dimensions of the resulting tile array layers vary somewhat since the shape and size of each individual tile forming their respective tile array layers are not identical, but a discrete number of tiles must be used.

The performance in all five metrics was tested for each prototype, and the relevant engineering performance metric scores were quantified by processing and transforming the raw data into the corresponding natural functional scale, between zero and one. In Fig. 12, the performance metric scores of eight different prototypes representing three different tile architectural classes are compared on the natural functional scale, where the dashed black line shows the corresponding group average over all eight samples for each metric.

The performance range of the metrics varied across prototypes. The mean scores for conformability, setability, and post-yield elasticity were high on the natural functional scale, indicating strong overall performance. Most bladder-attached tile prototypes showed perfect conformability, while those from other classes scored lower. All prototypes performed well in setability, with no significant differences between classes. In post-yield elasticity, scores showed minimal variation, with bladder-attached tiles recovering deformation more successfully than others. Figure 13 illustrates the average performance of the bladder-attached tiles class alongside representatives from the other two classes, comparing their relative merits rather than individual sample performances.

The average of the bladder-attached tiles class (BAavr) performed best in conformability and post-yield elasticity but ranked below average in draping angle and setability, and significantly worse in flexural rigidity. The hinge-like flexibility, enhanced by the straight folding lines of the square and triangular array patterns, contributes to high conformability. However, the relatively thin and lightweight nature of the tiles and the elasticity of the connections between them hampers the drapability. In a vacuumed state, the interlocking opposing layers of tiles tend to return to their initial positions when disturbed, providing strong post-yield elasticity, though the discrete nature of the interlocking positions tends to induce some spring back, which hampers the setability. The small tile size and rounded shapes significantly reduce the flexural rigidity as they can easily move relative to their interlocked positions under vacuum. The prototype (IS1) representing the internal sheet-attached tile class achieved the highest flexural rigidity score across all classes, though its other performance metric scores were below average. The tiles in this class are made of stiffer material, have a larger base area, and feature the thickest tile array layer combinations with surface protrusions, increasing friction between opposing layers. These factors contribute to superior flexural rigidity. However, the honeycomb pattern of the tile array creates nonlinear folding lines, relatively reducing the ability to drape and conform. The prototype (MI1) representing the mutually interlocking tile class performed best in the draping angle. Its setability and flexural rigidity scores were slightly above average, but it had the lowest conformability and post-yield elasticity scores. The design allows flexibility as the legs can slide and rotate within loops when unvacuumed, enhancing the draping angle. However, the same features limit conformability due to hard-stops where tiles collide. The asymmetrical tile array induces a cinching effect under vacuum, improving setability by reducing spring back during the setability test.

Each architectural class has strengths and weaknesses, with no single class excelling across all performance metrics. As a result, performance tradeoffs exist between metrics. For example, the internal sheet-attached tiles offer outstanding flexural rigidity due to larger and taller tiles but perform poorly in draping angle and conformability. Similarly, the mutually interlocking tiles provide the best draping angle but score the lowest in conformability and post-yield elasticity. While the bladder-attached tiles perform best in conformability and post-yield elasticity, they trade off with flexural rigidity due to the use of smaller and shorter tiles. This suggests the potential of developing better-performing pneumatic surfaces in the bladder-attached tiles class by changing the tile geometry design variables.

It is important to recognize that performance differences among the sampled prototypes may arise from architectural class/subclass variations, differences in individual tile geometry, or a combination of both. Additionally, improving the overall moldability performance of any prototype, regardless of architectural class, could be achieved through parametric studies focused on optimizing tile geometry. However, as demonstrated by the performance of the sample prototypes in this study, inherent tradeoffs between drapability, shapability, and rigidizability performances are highly likely to persist. Depending on the design context, choosing a particular tile architectural class can be more favorable than choosing another one. For example, a certain design application might require an excellent rigidizability performance for which the drapability and shapability performances are of secondary importance. In such a case, choosing an internal sheet-attached tiles class and further exploring the architectural feature variations in that class would likely yield better performance results matching the desired moldability criteria. If above-average level drapability, shapability, and rigidizability performances are required for another design application, choosing a bladder-attached tiles class would provide a better ground for further design space exploration.

To understand effects within an architectural class and across subclasses, the bladder-attached tiles class is selected for the next study. However, the comparative analysis between the tile architectural subclasses in this tile class is not straightforward due to the absence of a single prototype that simultaneously performs well across all performance metrics. Another useful technique that can provide insights for understanding the effects of the tile architectural classes/subclasses on the resulting performances is to analyze the relationships between the pairs of performance metric scores of the prototypes to evaluate important design couplings.

Design tradeoffs typically exist between competing factors. While it is possible to improve certain performance metrics of a pneumatic surface by adjusting its architectural features and tile geometry, negative impacts on other metrics are inevitable. Assessing the direction and the strength of correlations between performance metric pairs helps identify which bladder-attached tile subclass exhibits the least negative dependency. The absence of negative dependency (and more so the presence of positive dependency) suggests that with proper design, metric performance can be improved independently with minimal tradeoffs.

Design freedom in architectural features and tile geometry increases by selecting the tile architectural subclass with the least negative correlation between its performance metrics. It is necessary to assess how the direction and strength of correlations between metric pairs change as the prototype grouping is narrowed. Correlation matrices are generated first for the bladder-attached tiles class (n = 6), then for the attached (n = 3) and non-attached (n = 3) subclasses. Tables 2–4 present Pearson product–moment correlation coefficients for each metric pair, with correlation strengths categorized based on Evans' guide [41], highlighting negative and positive associations, respectively. A value of one or negative one indicates the strongest linear association, while a value of zero indicates no association.

The correlation matrix for the bladder-attached tiles class (Table 2) shows mostly moderate or weak correlations, with negative directions dominating. Draping angle has weak negative correlations with conformability and post-yield elasticity and a moderate negative correlation with setability. Conformability is moderately positively correlated with setability but negatively correlated with post-yield elasticity. There is also a moderate positive association between flexural rigidity and post-yield elasticity. While there are no strong negative dependencies, moderate negative correlations are more frequent than positive ones. However, analyzing correlations for both subclasses taken together obfuscates differences between the two subclasses and does not provide information to aid understanding the effects of each subclass on the resulting performances. Separating prototypes by subclass allows for a clearer comparison of the quantity and strength of dependencies across the groups.

In the bladder-attached connected tiles subclass, all prototypes share the same conformability metric score, preventing correlation coefficient calculations for metrics involving conformability. Additionally, the range of the flexural rigidity scores for these three prototypes was extremely narrow (0.008 on a 0–1 scale), making the resulting correlation values highly sensitive to be easily influenced by other metrics with wider ranges. Therefore, both conformability and flexural rigidity metrics are excluded from this analysis. Some of the very strong negative correlations between the draping angle and other metrics can be attributed to the elastic connections between tiles in this subclass. Increasing the draping angle requires more flexible connections, which negatively impacts post-yield elasticity, as it relies on more rigid connections. Setability shows a strong positive correlation with post-yield elasticity, which is consistent from a physical standpoint since better setability would help a prototype maintain its configuration under external loads, as shown in Table 3.

In the bladder-attached non-connected tiles subclass, both the quantity and strength of negative correlations between performance metric pairs are relatively lower, while the positive correlations are stronger and more frequent, as shown in Table 4. However, similar to the previous subclass's narrow flexural rigidity range, the range of post-yield elasticity scores in this subclass was also considerably narrow (0.046 on a 0–1 scale) and was therefore excluded from further analysis. A moderate negative correlation is observed between the draping angle and conformability. Conformability shows strong positive correlations with both setability and flexural rigidity, while setability and flexural rigidity have a one-to-one positive linear association. A potential explanation for these strong positive associations is the absence of physical connections between adjacent tiles. The gaps between tiles similarly impact the tile array's ability to conform to target shapes and rigidly maintain set configurations under external loads when the air is fully vacuumed from the bladder.

In Table 5, quantities of the correlation coefficients that exist in each grouping of prototypes are shown (note: not including the excluded ones). While the combined prototype grouping mostly consists of weak to moderate correlations, the two separate subclass groupings contain moderate to very strong correlations. The ratio of the number of positive to negative correlations becomes greater by separating the subclasses, and the non-connected tiles subclass shows the largest ratio of positive to negative correlations. Based on the comparison between the sum of correlations regarding quantity and strength, the pneumatic surface prototypes representing the bladder-attached non-connected tiles subclass are found to be more flexible in terms of providing design freedom. The quantity and strength of the negative dependencies between its performance metric pairs in the non-connected tiles subclass are relatively minimized, indicating minimal design tradeoffs, while the quantity and the strength of the positive correlations are relatively maximized, indicating that multiple metrics can be improved simultaneously.

In this subclass, the prototype made of equilateral triangle tiles is selected to demonstrate the next study corresponding to the architectural feature variations. It supports more architectural features regarding the tile layer and architectural layup and lends itself to varying tile geometry design to improve and tailor pneumatic surface performance.

Two main variations of the architectural features are explored in this experimental study by using the equilateral triangle tiles representing the bladder-attached non-connected tiles subclass. The effects of alternating the way the tile array layers overlap with each other and introducing a layer of thin (0.3 mm) soft thermoplastic polyurethane (TPU) sheet between the tile array layers on the pneumatic surface performance were measured from which the most favorable combination of architectural feature variations is identified.

The selected pneumatic surface prototype (BN3) is made of two tile array layers, where all the individual non-connected tiles perfectly overlap with their opposing ones. Such perfect alignment, in combination with the effect of the gap between adjacent tiles, provides a folding pattern on the pneumatic surface prototype, enhancing the hinge-like ability of the individual tiles and enabling relative rotational motion. This is expected to have an observable negative effect on the resulting flexural rigidity performance metric score.

To improve the flexural rigidity performance of the downselected architectural variation (Fig. 14(a)) that contains perfectly overlapping tile array layers, an alternative version (Fig. 14(b)) containing a shifted formation of the tile array layers, where the center of mass of every other overlapping tile pair is in alignment, was fabricated without changing any other design variable.

To assess the effect of shifting the tile array layer overlap, the set of measured performance metric scores for the two prototypes is shown relative to the average scores over all three classes (highlighted with a black dashed line) in the plot in Fig. 15. Accordingly, shifting the tile array layers had a major positive influence on the flexural rigidity metric score, which was increased by 43%. However, it negatively affected the draping angle and setability metrics by reducing the scores by 40% and 20%, respectively. Conformability and post-yield elasticity metric scores showed no variation. The overall performance decreased by only 3% on average, showing the rebalancing of tradeoffs by shifting the layers.

The increase in the resulting flexural rigidity performance metric score and the decrease in the draping angle and setability scores were expected because the hinge-like ability of the individual tiles is reduced by shifting the way the tile array layers overlap with each other. While there is no significant negative effect on the overall pneumatic surface performance, accommodating shifted opposing tile array layers as an architectural feature is selected for further analysis since it improves upon the flexural rigidity, which was the primary weakness of this architectural class relative to the other classes.

To improve the level of drapability and setability lost by introducing shifted opposing tile layers, an additional architectural feature variation is introduced: an intermediate layer layup. The selected pneumatic surface prototype's operation is affected by the gap between the neighboring tiles forming the tile array layers. In the course of draping or shaping the pneumatic surface over arbitrary objects, it is likely for an individual tile to be blocked by the edges of tiles that form the opposing tile layer, resulting in a reduced range of translational and rotational motion. One way of addressing this drawback is to introduce a continuous soft surface between the tile array layers to lubricate this sliding for better drapability (Fig. 16) as well as to provide a positive grip during the setting process.

The effects of inserting a thin sheet of soft TPU layer with a thickness of 0.3 mm in between the tile array layers were measured. A pneumatic surface prototype (BN3.2) identical to the shifted tile prototype but containing a thin soft TPU layer was fabricated and tested. The difference between performance metrics for the two different versions, having one sheet of TPU layer in-between the tile array layers and having none, is shown in Fig. 17.

As expected, inserting a TPU layer in between the tile array layers as a part of the pneumatic surface layup had a large positive effect on the draping angle by providing a lubricating surface between the opposing tiles, thus increasing the normalized score by 81%. In addition, the soft TPU layer becomes compressed between the tile array layers when the vacuum is applied. This increases the friction between overlapping tiles, resulting in improved setability and flexural rigidity performances. As shown in Fig. 17, the setability and flexural rigidity performance metric scores were increased by 27%, and 17%, respectively, while conformability and post-yield elasticity were left almost unchanged.

The overall pneumatic surface performance was improved by 24% on average when compared to the performance of the pneumatic surface that does not contain any TPU layer in between its tile array layers. Introducing an intermediate layer layup as an architectural variation overcame the losses introduced by shifting the tiles. As a result of this study, having a TPU layer in between the tile array layers and shifting the layers relative to each other are identified as a useful combination of architectural feature variations.

In this article, a series of architectural design studies is presented, focusing on analyzing the coupled relationship of the architectural design (classes/features) of internally tiled pneumatic surfaces and the resulting tradeoffs in drapability, shapability, and rigidizability performances to drive the design of effective user-interactive adaptive moldable products. Internally tiled pneumatic surfaces are built on the tile-based rigidization technique, which enables the technology capability of moldability into complex geometries with a thin, flat, and lightweight structure, addressing the drawbacks of pneumatic surfaces built on granular, fibrous, and layer jamming approaches. The architecture of internally tiled pneumatic surfaces consists of layers of rigid tiles arrayed in regular patterns within an airtight bladder, where the stiffness of the pneumatic surface is controlled as a function of the vacuum pressure applied. The overall pneumatic surface architecture is introduced with three key example tile architectural classes (and the associated subclasses), based on the approach by which the tiles are held in a regular array in the airtight bladder system: bladder-attached tiles, internal sheet-attached tiles, and mutually interlocking tiles. The overall moldability performance of an internally tiled pneumatic surface is broken down as a combination of performances of three key technology subcapabilities: drapability, shapability, and rigidizability. To provide an understanding of how the architectural form influences the performances of these key technology subcapabilities, the mechanisms by which these subcapabilities are achieved based on the operational state (application or release of vacuum) are discussed, specifically focusing on how they vary depending on the tile architectural class and features. A set of engineering performance metrics is defined, one for each technology subcapability performance to quantitatively capture all aspects of the overall pneumatic surface performance. Experimental procedures to collect data enable quantification of the sets of engineering performance metrics, where the draping angle quantifies drapability, conformability and setability quantify shapability, and flexural rigidity and post-yield elasticity quantify rigidizability performance. Since the units of the underlying measurements differ from one performance metric to another, the direct comparison among performance metrics is enabled by transforming the raw data into a dimensionless natural functional scale. In this scale, the scores range from the theoretical worst case (zero), where no functionality toward the performance metric is observed, to the theoretical best case (one), where the ideal functionality toward the performance metric is observed. A representative subset of the architectural design space of these surfaces is systematically explored by conducting three consecutive studies, each exploring a different aspect of architecture by performing comparative analyses: (1) a tile architectural class study, analyzing the effects of using different tile arrangement approaches, (2) a design coupling study, analyzing tradeoffs among performance metric pairs within and across tile architectural subclasses, and (3) an architectural feature variations study, analyzing the effects of shifting the tile array layers and including an intermediate friction layer.

As a part of the first study, the assessment of the relative strengths and weaknesses of each tile architectural class, bladder-attached, internal sheet-attached, and mutually interlocking, on moldability performance, is achieved by analyzing the plots of normalized performance metric scores of the prototypes, aided by the natural functional scale rendering all the metrics directly comparable. While none of the three tile architectural classes studied clearly outperforms the others in all metrics simultaneously, each one presents distinct advantages that can be leveraged depending on the design context. For example, the internally sheet-attached tile class exhibits relatively high rigidizability performance (239% above the group average), whereas the bladder-attached tile class exhibits relatively balanced performance across all metrics. As a part of the second study, the bladder-attached tile class, which contains two subclasses, connected and non-connected tiles, with three tile geometry designs in each, is used to demonstrate a correlation-based tile architectural subclass comparison technique. This technique enables a complete view of the strengths of the pairwise design tradeoffs and synergies among a particular tile architectural class across all the metrics. Pairwise correlation matrix analyses show that the bladder-attached non-connected tiles subclass provided the minimized quantity and the strength of negative dependencies between its performance metric pairs, reducing design tradeoffs. In the bladder-attached non-connected subclass, which contains three pneumatic surface designs with varying tile geometries, the pneumatic surface that is made of equilateral triangles was used to study the effects of architectural feature variations due to its simple tile geometry, which provides effective packaging, ease of manufacturability, and affordance for more architectural feature variations. The third and the last study uses a direct comparison approach to measure the effects of alternating the way the tile array layers overlap with each other and inserting a layer of thin and soft elastomeric sheet between the tile array layers on the performance of this particular pneumatic surface in the bladder-attached non-connected tile subclass. Shifting the opposing tile array layers relative to each other provides a positive impact on the flexural rigidity metric score (43% relative increase), while negatively affecting the draping angle and setability metric scores (40% and 20% relative decrease, respectively). Introducing a layer of thin and soft elastomeric sheet overcomes not only these negative effects on draping angle and setability metric scores (81% and 27% relative increase, respectively) but also further improves the flexural rigidity metric score (17% relative increase) without causing any decline in the performance of the remaining metrics.

The experimental systematic architectural design analysis approach presented in this article relates the impacts of the variations in architectural class and features to relevant performance metrics. This enables the understanding of the inherent tradeoffs among the drapability, shapability, and rigidizability performances as the design decisions regarding the architectural class and feature variations contribute to and impact the performances of all three subcapabilities. In the high-payoff application space of user-interacting adaptive moldable products, the priority of the performances of these subcapabilities can significantly vary depending on the application. This article provides a good understanding of the coupled relationship of the architectural design and all three subcapabilities simultaneously to drive the design of effective adaptive moldable products.

The authors would like to thank the General Motors/University of Michigan Collaborative Research Lab on Multifunctional Vehicle Systems.

There are no conflicts of interest. This article does not include research in which human participants were involved. Informed consent not applicable. This article does not include any research in which animal participants were involved.

The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request.