Reachability-Aware Time Scaling for Path Tracking

Hossein Gholampour; Logan E. Beaver

cs.ROApr 1, 2026

Reachability-Aware Time Scaling for Path Tracking

Authors: Hossein Gholampour, Logan E. Beaver

Organizations: Department of Mechanical and Aerospace Engineering and the Institute of Autonomous and Connected Systems, Old Dominion University, Norfolk, VA 23529, USA

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.

Figures & tables

Fig. 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 $\tau_{c}$ .

Fig. 2 : Representative trial in the workspace with active time-scaled trajectory segments highlighted in red.

Parameter	Value
$t_{s}$ (control period)	$0.0125$ s $(80Hz)$
Nominal path horizon	$2$ s
$v_{\max}$	$1.0$ m/s
$a_{\max}$	$2.5$ m/s 2
$L_{x},L_{y}$	$0.5$ m, $0.5$ m
$\mathbf{p}_{\text{start}}$	$[0.05,\;0.05]^{T}$ m

TABLE I : Simulation and planning parameters.

Fig. 3 : Representative offline time-scaling profile

\alpha(\tau)

Metric	Value
Minimum $\alpha$	$0.163\pm 0.053$
Mean $\alpha$	$0.846\pm 0.036$
Modified path points (%)	$26.98\pm 9.66$

TABLE II : Offline time-scaling profile statistics over 50 trials.

Metric	QP	QP + Offline TS
Mean $\delta$ (m/s 2 )	$9.924\pm 9.916$	$\mathbf{0.382\pm 3.366}$
5th percentile of $\delta$ (m/s 2 )	$-1.849\pm 0.363$	$\mathbf{-1.890\pm 0.272}$
Samples with $\delta_{k}>0$ (%)	$21.03\pm 10.18$	$\mathbf{8.96\pm 4.27}$
Max $\delta$ (m/s 2 )	$130.24\pm 89.42$	$\mathbf{42.59\pm 46.47}$
Mean speed (m/s)	$0.288\pm 0.054$	$\mathbf{0.157\pm 0.052}$
Max speed (m/s)	$0.367\pm 0.072$	$\mathbf{0.344\pm 0.050}$

TABLE III : Summary over 50 randomized trials over the moving interval. Values are mean

\pm

standard deviation across trials.

Fig. 4 : Representative trial: $\delta_{k}$ versus time with the freezing period indicated (QP vs. QP+TS).

Fig. 5 : Histogram of per-trial mean $\delta$ over moving samples (50 trials).

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Track A*: Fast Visibility-Aware Trajectory Planning for Active Target Tracking

Hanxuan Chen, Kangli Wang, Ji Pei

Autel Robotics

Trajectory Generation Motion Planning Robot Motion Planning

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Offline reference trajectories for active target tracking are needed both for building multi-modal tracking datasets and for benchmarking online tracking planners under repeatable conditions. We present Track A star (TA star), an offline search-based trajectory planner that targets the visibility-aware target tracking objective on a discretized four-dimensional spatio-temporal grid (x, y, z, t). TA star combines a layered Directed Acyclic Graph (DAG) search with three engineering optimizations: cross-time obstacle distance caching against a Bounding Volume Hierarchy (BVH), per-layer beam pruning, and a configurable multi-ray visibility evaluator. TA star employs a beam-pruned heuristic search on this discrete graph to efficiently find high-quality tracking trajectories. While it trades strict theoretical optimality for practical scalability, our empirical results demonstrate robust, near-baseline visibility performance at a fraction of the computational cost. On a 1000-scenario stress test across eight CARLA Optimized maps, TA star converges on all scenarios and completes in 45 s using 32 workers; on a 248-scenario controlled comparison against an unoptimized priority-queue A star baseline (BinaryHeap implementation) under identical scenario inputs and a 5 x 10^6 expansion cap, TA star reduces mean planning time by 23.0x and worst-case planning time by 11.8x, while raising convergence from 56.9% to 100%. On the n=141 baseline-converged subset, TA star changes average visibility by only -0.15 percentage points (pp), with no scenario exceeding a 5 pp drop. We position TA star as a practical offline reference planner under these specific conditions, with limitations and failure cases discussed for environments such as Town07 dense vegetation.

Abstract

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