Theory and Primary Computational Simulations of the Numerical Reuleaux Method (NRM)

Abstract

The Classical Reuleaux Method (CRM) is frequently used in Dynamics, Mechanics, and Bioengineering to determine the Instantaneous Rotation Centre (IRC) of a Rigid Body (RB) in arbitrary movement. The generic mathematical CRM only can be applied on RBs, whose shape remains constant during the movement. If the solid in movement is a Pseudo-Rigid Body (PRB), the CRM has to be modified numerically to conform the shape changes and adapt on the density distribution variations of the PRB (we denominate it, in this case , The Numerical Reuleaux Method, NRM). This Geometrical-Numerical Approximation Method is based on the division of the PRB into small volume parts (voxels, roughly speaking parallelepiped, but taking the calculation points in a tetrahedral way (each parallelepiped voxel can be divided into 2 tetrahedral volumes, with several points’ combinations)). Theoretical basis of the method are strictly shown, with the necessary Theorems and Propositions of the model. Nonlinear Optimization Techniques that support the initial Theory have been developed, and the Error Boundaries/Reduction Techniques are determined. Computational Simulations have been carried out to prove the NRM Theoretical Model feasibility, and veracity of the Propositions, Theorems, and Error Boundaries. Appropriate software was made to carry out these simulations conveniently. The initial results agree to the theoretical calculations, and the IRC computation for 2 voxels shows to be simple and easy. Finally, some guidelines for a theoretical development of this algorithm, applied on large PRBs, by using Monte-Carlo techniques, are explained.