Modeling of planar elastically coupled rigid bodies: Geometric algebra methods and applications

Abstract

This study presents two new, generic methods to modeling planar elastically coupled rigid body systems using Geometric Algebra. The two methods are twist-based potential energy function method and twistor-based potential energy function method. In this research, the rigid body motion in the plane is modeled as a twist or twistor motion in which the rotational motion and translational motion happen simultaneously. The twist is denoted as a bivector using Geometric Algebra which facilitates the notation and computation. A twistor is defined in an intermediate frame half way between two displacement frames. The twistor parameters intuitively represent the relative displacement between two frames. Both twist-based and twistor-based potential energy functions are shown to be frame-independent and body-independent. The kinematics is studied using twist and twistor parameters. The constitutive equations are derived in which the wrench exerted by a pair of elastic bodies is computable given twist or twistor displacements. To analyze large displacements, this study also provides two higher order polynomial potential energy functions of twist parameters and twistor parameters. The polynomial potential energy functions are also shown to be frame-independent and body-independent. They are generally applicable to analyze large displacements of elastically coupled rigid body systems. Several case studies are provided in this research to demonstrate the utility of the presented modeling methods. A micropositioning stage device is modeled as a flexural mechanism with 6 rigid bodies and 7 flexural joints. Simulation is performed using Scilab software. The simulation results show good agreement with actual experimental data. The methods are also applied to simulate the displacement of flexural four-bar linkages with various geometry and various flexural hinges. This case study shows that the presented methods in this research are generic and case-independent. In another case study, the higher order polynomial function method is applied to fit some randomly generated data which demonstrates the generality of the method and the applicability of the method in cases when only experimental data is available without knowing the geometry parameters of a mechanism. The case study of modeling electrostatic potential energy between liquid water molecules using polynomial function of twistor shows the potential utility of the method in the analysis of large displacement. The methods presented in this research have been shown to be generic, easily applicable, and easily computable.