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Reachability-Aware Time Scaling for Path Tracking

Hossein Gholampour, Logan E. Beaver

Abstract

This paper studies tracking of collision-free waypoint paths produced by an offline planner for a planar double-integrator system with bounded speed and acceleration. Because sampling-based planners must route around obstacles, the resulting waypoint paths can contain sharp turns and high-curvature regions, so one-step reachability under acceleration limits becomes critical even when the path geometry is collision-free. We build on a pure-pursuit-style, reachability-guided quadratic-program (QP) tracker with a one-step acceleration margin. Offline, we evaluate this margin along a spline fitted to the waypoint path and update a scalar speed-scaling profile so that the required one-step acceleration remains below the available bound. Online, the same look-ahead tracking structure is used to track the scaled reference.

Reachability-Aware Time Scaling for Path Tracking

Abstract

This paper studies tracking of collision-free waypoint paths produced by an offline planner for a planar double-integrator system with bounded speed and acceleration. Because sampling-based planners must route around obstacles, the resulting waypoint paths can contain sharp turns and high-curvature regions, so one-step reachability under acceleration limits becomes critical even when the path geometry is collision-free. We build on a pure-pursuit-style, reachability-guided quadratic-program (QP) tracker with a one-step acceleration margin. Offline, we evaluate this margin along a spline fitted to the waypoint path and update a scalar speed-scaling profile so that the required one-step acceleration remains below the available bound. Online, the same look-ahead tracking structure is used to track the scaled reference.

Paper Structure

This paper contains 9 sections, 1 theorem, 48 equations, 5 figures, 3 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Controller and Time-Scaling Design
  4. Look-Ahead and One-Step Reachability Margin
  5. Offline Reachability-Aware Time Scaling
  6. Simulation Setup and Results
  7. Simulation Setup
  8. Results
  9. Conclusion

Key Result

Lemma 1

Consider the disturbance-free sampled model eq:discrete_v--eq:discrete_p, and suppose the unclipped control law eq:u_star_k is applied. Then the next-step velocity is Hence exact one-step position matching is compatible with the desired reference velocity $\mathbf{v}_{\mathrm{ref},k}$ if and only if $\blacktriangleleft$$\blacktriangleleft$

Figures (5)

  • Figure 1: Look-ahead geometry on the reference path. The closest point $p_c=p_{\mathrm{ref}}(\tau_c)$ is obtained by projecting $p_k$ onto the path, and the look-ahead target $p_{\mathrm{LA},k}=p_{\mathrm{ref}}(\tau_{\mathrm{LA},k})$ is selected by advancing arc length $s_{\mathrm{LA},k}$ from $\tau_c$.
  • Figure 2: Representative trial in the workspace with active time-scaled trajectory segments highlighted in red.
  • Figure 3: Representative offline time-scaling profile $\alpha(\tau)$.
  • Figure 4: Representative trial: $\delta_k$ versus time with the freezing period indicated (QP vs. QP+TS).
  • Figure 5: Histogram of per-trial mean $\delta$ over moving samples (50 trials).

Theorems & Definitions (2)

  • Lemma 1
  • proof