Building multi-stable structures based on Baranov truss part III: limit point (dead-centre) detection in linkage mechanism and non-rigid motion paths computation

Abstract

Part I established a bilateration-based framework for programmable multi-stability in Baranov trusses, showing how discrete geometric configurations can be transformed into stable equilibria through elasticity. Part II extended this foundation into a general design toolkit, enabling scalable multi-stable assemblies through multiple non-rigid links, tessellation and higher-order trusses. In this final part, we extend the framework from trusses to planar linkage mechanisms. Limit-point configurations, classically regarded as dead-centre or singularity states, are here reinterpreted as the gateways between rigid compatibility paths and non-rigid deformation paths. Classical approaches to locating limit points in linkages often rely on Jacobian matrix analysis, which, while rigorous, can be computationally intensive and opaque in interpretation. Here, we present an alternative based on Assur group decomposition and bilateration geometry. Within this framework, a limit point is the rank loss of the active, driver-conditioned bilateration closure Jacobian of the receiving Assur group; this criterion reduces to the collapse of an oriented area for a dyad block and to rank loss of the differentiated closure Jacobian for a tetrad block. This provides a direct, scalable criterion for detecting limit points in one-degree-of-freedom linkages. Two detailed case studies are provided to illustrate the merit of our approach.