An algebraic method to check the singularity-free paths for parallel robots

TL;DR

This paper introduces an algebraic approach using Gröbner basis methods to verify the singularity-free paths of parallel robots, enabling precise trajectory feasibility analysis in complex workspaces.

Contribution

It presents a novel algebraic method that avoids discretization, allowing continuous trajectory analysis and handling more complex workspace shapes for parallel robots.

Findings

Method successfully checks singularity-free paths for Orthoglide robot.

Algebraic approach handles complex workspace geometries.

Provides continuous trajectory feasibility analysis without discretization.

Abstract

Trajectory planning is a critical step while programming the parallel manipulators in a robotic cell. The main problem arises when there exists a singular configuration between the two poses of the end-effectors while discretizing the path with a classical approach. This paper presents an algebraic method to check the feasibility of any given trajectories in the workspace. The solutions of the polynomial equations associated with the tra-jectories are projected in the joint space using Gr{\"o}bner based elimination methods and the remaining equations are expressed in a parametric form where the articular variables are functions of time t unlike any numerical or discretization method. These formal computations allow to write the Jacobian of the manip-ulator as a function of time and to check if its determinant can vanish between two poses. Another benefit of this approach is to use a largest workspace with a more complex shape than a cube, cylinder or sphere. For the Orthoglide, a three degrees of freedom parallel robot, three different trajectories are used to illustrate this method.