Triangle-Decomposable Graphs for Isoperimetric Robots

TL;DR

This work advances isoperimetric robots built from inflatable triangle modules by proving how to partition an edge set into disjoint $K_3$ units and enumerating all minimally rigid triangle-decomposable graphs up to 9 nodes (7 triangles). It introduces three partitioning approaches, including a scalable pre-enumeration ILP, and provides a constructive method to combine partitioned graphs into larger rigid systems. Embedding strategies and workspace characterization reveal how graph topology and triangle partitioning influence reachable motion, with significant variability across partitions even for identical graphs. The results enable a scalable pathway to diverse, safe, shape-changing robots suitable for applications such as space exploration, and point to future hardware enhancements that expand deformable capabilities.

Abstract

Isoperimetric robots are large scale, untethered inflatable robots that can undergo large shape changes, but have only been demonstrated in one 3D shape -- an octahedron. These robots consist of independent triangles that can change shape while maintaining their perimeter by moving the relative position of their joints. We introduce an optimization routine that determines if an arbitrary graph can be partitioned into unique triangles, and thus be constructed as an isoperimetric robotic system. We enumerate all minimally rigid graphs that can be constructed with unique triangles up to 9 nodes (7 triangles), and characterize the workspace of one node of each these robots. We also present a method for constructing larger graphs that can be partitioned by assembling subgraphs that are already partitioned into triangles. This enables a wide variety of isoperimetric robot configurations.

Figures (6)

• Figure 1: (A) An isoperimetric robot that is constructed from 7 triangles (6 inflated triangles, and a rigid triangle on top that serves as an interaction surface. (B) An isoperimetric robot in the shape of a hexagonal bipyramid, formed from 6 triangles. (C) Illustration of the operating principle of the isoperimetric robot. Each triangle changes shape as the rollers move along the inflated tube, changing the position of the vertex, but maintaining a constant perimeter.
• Figure 2: Two different Hennenberg steps utilized to construct graphs. In an H1 step, a new node is added with 3 connecting edges. For an H2 step, a new node with four new edges is added, and an edge between the connecting nodes is deleted.
• Figure 3: All partitions of all minimally rigid graphs with 9 or fewer nodes that can be partitioned into triangles. Note that some graphs have multiple possible partitions. Graphs are grouped according to the number of component triangles. Graphs that are identical but have different partitions are grouped in blue boxes. For each graph, the normalized workspace volume NV of the top node is given.
• Figure 4: Shapes formed by combining octahedral units into chains and branching shapes
• Figure 5: Plot comparing runtimes for exhaustive search, pre-enumeration ILP, and end-to-end IQCQP. Center dots show the median runtime, while upper and lower whiskers show min / max runtimes respectively. All experiments ran with a time limit of $100$ seconds. Only two entries for exhaustive search are shown since the algorithm encountered out-of-memory errors after that point. End-to-end IQCQP runtimes beyond 27 nodes are omitted due to runs hitting the timeout limit.

Theorems & Definitions (7)

• Definition 1
• Definition 2
• Lemma 1: connelly1993rigidity
• Lemma 2
• proof
• Lemma 3
• proof