A Minimum-Jerk Approach to Handle Singularities in Virtual Fixtures

TL;DR

A geometric interpretation of discontinuities of the Gauss-Newton algorithm is given, based on a linear quadratic tracking problem with minimum jerk command, to compare and validate the performances of the proposed framework in two different human-robot interaction scenarios.

Abstract

Implementing virtual fixtures in guiding tasks constrains the movement of the robot's end effector to specific curves within its workspace. However, incorporating guiding frameworks may encounter discontinuities when optimizing the reference target position to the nearest point relative to the current robot position. This article aims to give a geometric interpretation of such discontinuities, with specific reference to the commonly adopted Gauss-Newton algorithm. The effect of such discontinuities, defined as Euclidean Distance Singularities, is experimentally proved. We then propose a solution that is based on a Linear Quadratic Tracking problem with minimum jerk command, then compare and validate the performances of the proposed framework in two different human-robot interaction scenarios.

Figures (6)

• Figure 1: 2D visualization of an Euclidean distance singularity. Continuous gray lines represent the iso-lines with respect to the constraint path $\hbox{$\boldsymbol{\mu}$}(s)$ depicted in black. While a distance-based method could correctly update the reference position $\hbox{$\boldsymbol{\mu}$}(s_t)$ for the green trajectory, it fails to find a stable solution for the red trajectory, as ${\boldsymbol y}_2$ has the same distance $\Delta$ from 2 points in $\hbox{$\boldsymbol{\mu}$}(s)$.
• Figure 2: Trend of the cost in \ref{['eq:optimal_problem']} for different path deviations.
• Figure 3: Behaviour of the curvilinear parameter velocity $\dot s$ with varying velocity weight ${\boldsymbol c}_2$.
• Figure 4: Controller framework. Here VF denotes the virtual fixture constraint.
• Figure 5: Experimental results of the center-reaching task.