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Singularity-Free Guiding Vector Field over Bézier's Curves Applied to Rovers Path Planning and Path Following

Alfredo González-Calvin, Lía García-Pérez, Juan Jiménez

TL;DR

This work addresses robust path following for land-based wheeled mobile robots by combining a Singularity-Free Guiding Vector Field (SF-GVF) with parametric Bezier curves. By lifting the path into an augmented space with a path parameter $w$, the SF-GVF removes field singularities and provides global convergence to the desired parametric path, which is projected onto the plane for the rover controller. A curvature-dependent speed setpoint complements the heading guidance, improving convergence under turning constraints, and Bézier splines (degree 5) with $C^2$ continuity ensure smooth, easily editable trajectories within a convex hull. The approach is implemented and validated in Paparazzi on real Rovers and in simulations, demonstrating accurate path tracking and adaptive speed control; the results suggest potential extensions to obstacle avoidance and applications to other mobile platforms such as autonomous surface vehicles.

Abstract

This paper presents a guidance algorithm for solving the problem of following parametric paths, as well as a curvature-varying speed setpoint for land-based car-type wheeled mobile robots (WMRs). The guidance algorithm relies on Singularity-Free Guiding Vector Fields SF-GVF. This novel GVF approach expands the desired robot path and the Guiding vector field to a higher dimensional space, in which an angular control function can be found to ensure global asymptotic convergence to the desired parametric path while avoiding field singularities. In SF-GVF, paths should follow a parametric definition. This feature makes using Bezier's curves attractive to define the robot's desired patch. The curvature-varying speed setpoint, combined with the guidance algorithm, eases the convergence to the path when physical restrictions exist, such as minimal turning radius or maximal lateral acceleration. We provide theoretical results, simulations, and outdoor experiments using a WMR platform assembled with off-the-shelf components.

Singularity-Free Guiding Vector Field over Bézier's Curves Applied to Rovers Path Planning and Path Following

TL;DR

This work addresses robust path following for land-based wheeled mobile robots by combining a Singularity-Free Guiding Vector Field (SF-GVF) with parametric Bezier curves. By lifting the path into an augmented space with a path parameter , the SF-GVF removes field singularities and provides global convergence to the desired parametric path, which is projected onto the plane for the rover controller. A curvature-dependent speed setpoint complements the heading guidance, improving convergence under turning constraints, and Bézier splines (degree 5) with continuity ensure smooth, easily editable trajectories within a convex hull. The approach is implemented and validated in Paparazzi on real Rovers and in simulations, demonstrating accurate path tracking and adaptive speed control; the results suggest potential extensions to obstacle avoidance and applications to other mobile platforms such as autonomous surface vehicles.

Abstract

This paper presents a guidance algorithm for solving the problem of following parametric paths, as well as a curvature-varying speed setpoint for land-based car-type wheeled mobile robots (WMRs). The guidance algorithm relies on Singularity-Free Guiding Vector Fields SF-GVF. This novel GVF approach expands the desired robot path and the Guiding vector field to a higher dimensional space, in which an angular control function can be found to ensure global asymptotic convergence to the desired parametric path while avoiding field singularities. In SF-GVF, paths should follow a parametric definition. This feature makes using Bezier's curves attractive to define the robot's desired patch. The curvature-varying speed setpoint, combined with the guidance algorithm, eases the convergence to the path when physical restrictions exist, such as minimal turning radius or maximal lateral acceleration. We provide theoretical results, simulations, and outdoor experiments using a WMR platform assembled with off-the-shelf components.

Paper Structure

This paper contains 19 sections, 1 theorem, 40 equations, 15 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Rover model
  3. Basic model assumptions
  4. Rover kinematic equations
  5. Parametric GVF for Rover guidance and path following
  6. Guidance problem
  7. Path following
  8. A note on field perturbation.
  9. Bézier's curves for path planning
  10. Speed Control Problem
  11. Speed controller
  12. Changing speed setpoint
  13. Results
  14. Experimental Platform
  15. Rover's Hardware
  16. ...and 4 more sections

Key Result

Proposition 1

Let $\xi(t)$ be the solution to $\dot{\xi}(t) = \mkern 1mu ^{aug}\chi(\xi(t))$ with $^{aug}\chi(\xi(t))$ as defined in equation (eqchiaug2), then $\xi(t)$ will converge to the augmented path $^{aug}\mathcal{P}$, defined in equation (eqaugpath) as $t\rightarrow \infty$.

Figures (15)

  • Figure 1: Representation of a simple four-wheeled car with wheelbase $l$. The car is shown in light blue, while the virtual front wheel is shown in red. The direction of motion is shown with a blue line, and the direction of the virtual wheel is shown in red.
  • Figure 2: Illustration of the relationship among $\mathcal{P}$, $^{aug}\mathcal{P}$, $\phi_1(\xi)$ and $\phi_2(\xi)$. $\mathcal{P}$ is a Bézier's curve of degree 4, with control points $\mathcal{C} =\{(1,1),(3,-4),(1,0),(3,4),(1,-1)\}$
  • Figure 3: Illustration of the relationship between $^{aug}\chi$ and $\chi^p$. The vector field $\chi^p$ (shown as black arrows) is the projection of $^{aug}\chi$ (shown as red arrows) onto the plane $(p_x,p_y)$ of constant $w$ (shown as a black dot in the red plane). The solid orange line represents the augmented path, while the dashed blue line represents the augmented trajectory $\xi$.
  • Figure 4: Rovers
  • Figure 5: Rover's electronics hardware setup in an outdoors experiment.
  • ...and 10 more figures

Theorems & Definitions (2)

  • Proposition 1
  • proof