On the line-symmetry of self-motions of linear pentapods

TL;DR

This paper demonstrates that all self-motions of certain linear pentapods are generated by line-symmetric motions, bridging historical and modern studies, and provides new insights into the Borel Bricard problem and design conditions.

Contribution

It establishes that all self-motions of Type 1 and Type 2 linear pentapods are line-symmetric, connecting classical and recent research and offering new solution sets.

Findings

Self-motions of Type 1 and Type 2 linear pentapods are line-symmetric.

A new solution set for the Borel Bricard problem is derived.

Conditions for designing self-motion free linear pentapods are provided.

Abstract

We show that all self-motions of pentapods with linear platform of Type 1 and Type 2 can be generated by line-symmetric motions. Thus this paper closes a gap between the more than 100 year old works of Duporcq and Borel and the extensive study of line-symmetric motions done by Krames in the 1930's. As a consequence we also get a new solution set for the Borel Bricard problem. Moreover we discuss the reality of self-motions and give a sufficient condition for the design of linear pentapods of Type 1 and Type 2, which have a self-motion free workspace.