Solving the double-banana rigidity problem: a loop-based approach

Abstract. Rigidity detection is an important tool for structural synthesis of mechanisms, as it helps to unveil possible sources of inconsistency in Grübler's count of degrees of freedom (DOFs) and thus to generate consistent kinematical models of complex mechanisms. One case that has puzzled researchers over many decades is the famous "double-banana" problem, which is a representative counter-example of Laman's rigidity condition formula for which existing standard DOF counting formulas fail. The reason for this is the body-by-body and joint-by-joint decomposition of the interconnection structure in classical algorithms, which does not unveil structural isotropy groups for example when whole substructures rotate about an "implied hinge" according to Streinu. In this paper, a completely new approach for rigidity detection for cases as the "double-banana" counterexample in which bars are connected by spherical joints is presented. The novelty of the approach consists in regarding the structure not as a set of joint-connected bodies but as a set of interconnected loops. By tracking isolated DOFs such as those arising between pairs of spherical joints, rigidity/mobility subspaces can be easily identified and thus the "double-banana" paradox can be resolved. Although the paper focuses on the solution of the double-banana mechanism as a special case of paradox bar-and-joint frameworks, the procedure is valid for general body-and-joint mechanisms, as is shown by the decomposition of spherical joints into a series of revolute joints and their rigid-link interconnections.