Building multi-stable structures based on Baranov truss part I: theoretical foundations

Abstract Multi-stable structures can adopt multiple stable configurations without continuous actuation, offering opportunities for deployable, adaptive and energy-efficient systems. Traditional approaches typically locate these states through iterative procedures that couple nonlinear equilibrium searches with stability checks—a process that is computationally costly and prone to missing solutions. We here present a geometry-first framework that decouples compatibility from elasticity. Using bilateration formulations, we enumerate all admissible rigid embeddings in closed form and then embed elastic energy terms on selected links to transform these discrete configurations into a continuous energy landscape. Part I establishes the theoretical foundations of the workflow and demonstrates the approach on the 5/B1 Baranov truss with a single non-rigid link. By formulating the closure condition in closed form through bilateration matrices, we derive the compatible deformation paths and show how a simple elastic embedding may produce up to six compatible strain-free configurations. The present work therefore establishes the methodological basis for a broader geometry-energy approach to programmable multi-stability, with larger structural and mechanistic extensions to be developed in parts II and III.