Derivation of Analytical, Closed‐form Formulas for the Calculations of Instantaneous Screw Axes of Arbitrary Rigid 3D Multi‐Body Systems

Jonas Boungard,Jens Wackerfuß

doi:10.1002/pamm.202000103

Abstract

AbstractA method to calculate the parameters for each instantaneous and relative instantaneous screw axis in an arbitrary rigid multi‐body system is presented. At first a kinematic analysis is performed which calculates the displacements of all nodes and the rotation of each body. In the next step the displacements of the nodes and the rotation of the bodies are used to calculate the instantaneous screw axis of each body. With the information on the instantaneous screw axes the instantaneous relative screw axes can be computed too. An example of the calculation and the visualization of the instantaneous screw axes is given.

Highlights

Any spatial rigid body motion of a point p on a body bi can be expressed as a combination of a rotation ψbi and a translation ubtrians. along a screw axis with a distance to the point rbpi
The screw axis is defined by the direction vector nbi, one position vector XSbi and the pitch hbi
With the displacements at three distinct points ub1i,2,α of each body bi the rotation ψbi of each body and the direction of the screw axis nbi can be computed using equation (3)1. This is done with the help of geometric helper matrices Tbi and Rbi which have already been constructed in [4] to generate the constraint matrix C. This approach has the advantage to other formulas found in the literature, e.g. [2], that the three displacements vectors can be linearly dependent

Summary

Introduction

Any spatial rigid body motion of a point p on a body bi can be expressed as a combination of a rotation ψbi and a translation ubtrians. along a screw axis with a distance to the point rbpi. Any spatial rigid body motion of a point p on a body bi can be expressed as a combination of a rotation ψbi and a translation ubtrians. Along a screw axis with a distance to the point rbpi. The screw axis is defined by the direction vector nbi , one position vector XSbi and the pitch hbi. The relative motion of two bodies can be expressed with the relative screw axis.

Kinematic Method

Calculation of the Instantaneous Screw Axis

Calculation of the Relative Instantaneous Screw Axis

Section 1: Multi-body dynamics

Visualization