Mobility and Constraint Analysis of Interconnected Hybrid Flexure Systems Via Screw Algebra and Graph Theory

Abstract

This paper introduces a general method for analyzing flexure systems of any configuration, including those that cannot be broken into parallel and serial subsystems. Such flexure systems are called interconnected hybrid flexure systems because they possess limbs with intermediate bodies that are connected by flexure systems or elements. Specifically, the method introduced utilizes screw algebra and graph theory to help designers determine the freedom spaces (i.e., the geometric shapes that represent all the ways a body is permitted to move) for all the bodies joined together by compliant flexure elements within interconnected hybrid flexure systems (i.e., perform mobility analysis of general flexure systems). This method also allows designers to determine (i) whether such systems are under-constrained or not and (ii) whether such systems are over-constrained or exactly constrained (i.e., perform constraint analysis of general flexure systems). Although many flexure-based precision motion stages, compliant mechanisms, and microarchitectured materials possess topologies that are highly interconnected, the theory for performing the mobility and constraint analysis of such interconnected flexure systems using traditional screw theory does not currently exist. The theory introduced here lays the foundation for an automated tool that can rapidly generate the freedom spaces of every rigid body within a general flexure system without having to perform traditional computationally expensive finite element analysis. Case studies are provided to demonstrate the utility of the proposed theory.

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“…The theory of interconnected-hybrid-flexure-system mobility analysis 36 was adapted to enable designers to calculate the number of DOFs achieved by general CRAM tessellations of any boundary configuration. To demonstrate the approach, consider the simple CRAM example shown in Supplementary Fig.…”

“…The directions of the graph’s arrows are arbitrary, but they establish an important convention that must be maintained for future steps. The second step is to create the graph’s incidence matrix, [Inc], according to previously published instructions 36 . For the example case study, the incidence matrix is The third step is to find the transpose of the matrix [ Q ] that satisfies [Inc] T [ Q ] = [0].…”

“…For the example case study, [ Q ] T = [−1 −1 1 1]. The fourth step is to create the system’s freedom-topology matrix, [FT], by populating [ Q ] T with the appropriately ordered twist vectors, T j , that mathematically model the rotational DOFs at their corresponding joints 36 . For the example case study, .…”