Linear Pentapods with a Simple Singularity Variety

TL;DR

This paper explores the geometric structure of linear pentapods' singularities, proposing simplified designs with reduced degree singularity varieties for easier computation and redundancy.

Contribution

It introduces mechanisms with simplified singularity varieties by restricting the cubic hypersurface, enabling easier singularity analysis and redundancy in linear pentapod design.

Findings

Reduced degree of singularity varieties for specific design restrictions

Simplified computation of singularity-free spheres in configuration space

Proposed kinematically redundant linear pentapod designs

Abstract

There exists a bijection between the configuration space of a linear pentapod and all points $(u,v,w,p_x,p_y,p_z)\in\mathbb{R}^{6}$ located on the singular quadric $\Gamma: u^2+v^2+w^2=1$, where $(u,v,w)$ determines the orientation of the linear platform and $(p_x,p_y,p_z)$ its position. Then the set of all singular robot configurations is obtained by intersecting $\Gamma$ with a cubic hypersurface $\Sigma$ in $\mathbb{R}^{6}$, which is only quadratic in the orientation variables and position variables, respectively. This article investigates the restrictions to be imposed on the design of this mechanism in order to obtain a reduction in degree. In detail we study the cases where $\Sigma$ is (1) linear in position variables, (2) linear in orientation variables and (3) quadratic in total. The resulting designs of linear pentapods have the advantage of considerably simplified computation of singularity-free spheres in the configuration space. Finally we propose three kinematically redundant designs of linear pentapods with a simple singularity surface.