Deployable structures are physical mechanisms that can easily transition between two or more geometric configurations; such structures enable industrial, scientific, and consumer applications at a wide variety of scales. This paper develops novel deployable structures that can approximate a large class of doubly-curved surfaces and are easily actuated from a flat initial state via inflation or gravitational loading. The structures are based on two-dimensional rigid mechanical linkages that implicitly encode the curvature of the target shape via a user-programmable pattern that permits locally isotropic scaling under load. We explicitly characterize the shapes that can be realized by such structures---in particular, we show that they can approximate target surfaces of positive mean curvature and bounded scale distortion relative to a given reference domain. Based on this observation, we develop efficient computational design algorithms for approximating a given input geometry. The resulting designs can be rapidly manufactured via digital fabrication technologies such as laser cutting, CNC milling, or 3D printing. We validate our approach through a series of physical prototypes and present several application case studies, ranging from surgical implants to large-scale deployable architecture.

We present X-shells , a new class of deployable structures formed by an ensemble of elastically deforming beams coupled through rotational joints. An X-shell can be assembled conveniently in a flat configuration from standard elastic beam elements and then deployed through force actuation into the desired 3D target state. During deployment, the coupling imposed by the joints will force the beams to twist and buckle out of plane to maintain a state of static equilibrium. This complex interaction of discrete joints and continuously deforming beams allows interesting 3D forms to emerge. Simulating X-shells is challenging, however, due to unstable equilibria at the onset of beam buckling. We propose an optimization-based simulation framework building on a discrete rod model that robustly handles such difficult scenarios by analyzing and appropriately modifying the elastic energy Hessian. This real-time simulation method forms the basis of a computational design tool for X-shells that enables interactive design space exploration by varying and optimizing design parameters to achieve a specific design intent. We jointly optimize the assembly state and the deployed configuration to ensure the geometric and structural integrity of the deployable X-shell. Once a design is finalized, we also optimize for a sparse distribution of actuation forces to efficiently deploy it from its flat assembly state to its 3D target state. We demonstrate the effectiveness of our design approach with a number of design studies that highlight the richness of the X-shell design space, enabling new forms not possible with existing approaches. We validate our computational model with several physical prototypes that show excellent agreement with the optimized digital models.

We present RoboGrammar , a fully automated approach for generating optimized robot structures to traverse given terrains. In this framework, we represent each robot design as a graph, and use a graph grammar to express possible arrangements of physical robot assemblies. Each robot design can then be expressed as a sequence of grammar rules. Using only a small set of rules our grammar can describe hundreds of thousands of possible robot designs. The construction of the grammar limits the design space to designs that can be fabricated. For a given input terrain, the design space is searched to find the top performing robots and their corresponding controllers. We introduce Graph Heuristic Search - a novel method for efficient search of combinatorial design spaces. In Graph Heuristic Search, we explore the design space while simultaneously learning a function that maps incomplete designs (e.g., nodes in the combinatorial search tree) to the best performance values that can be achieved by expanding these incomplete designs. Graph Heuristic Search prioritizes exploration of the most promising branches of the design space. To test our method we optimize robots for a number of challenging and varied terrains. We demonstrate that RoboGrammar can successfully generate nontrivial robots that are optimized for a single terrain or a combination of terrains.