# Rational Parametrization of Linear Pentapod's Singularity Variety and the Distance to it

## TL;DR

This paper provides a rational parametrization of the singularity variety for linear pentapods and calculates the shortest distance to this variety, aiding in understanding and avoiding singular configurations in robotic manipulators.

## Contribution

It introduces a novel rational parametrization of the linear pentapod's singularity variety and computes the minimal distance to it, enhancing singularity analysis.

## Key findings

- Rational parametrization of the singularity variety achieved
- Shortest distance to the singularity variety computed
- Maximal singularity-free sphere radius determined

## Abstract

A linear pentapod is a parallel manipulator with five collinear anchor points on the motion platform (end-effector), which are connected via extendible legs to the base. This manipulator has five controllable degrees-of-freedom and the remaining one is a free rotation around the motion platform axis (which in fact is an axial spindle). In this paper we present a rational parametrization of the singularity variety of the linear pentapod. Moreover we compute the shortest distance to this rational variety with respect to a suitable metric. Kinematically this distance can be interpreted as the radius of the maximal singularity free-sphere.